Newton Matching for Generative Modeling: A Unified Framework for Fine-Tuning and Sampling

# Newton Matching for Generative Modeling:A Unified Framework for Fine-Tuning and Sampling
Source: [https://arxiv.org/html/2609.05727](https://arxiv.org/html/2609.05727)
Zeyang Li zeyang@mit\.eduAffiliation:Yunan Wang wangyn25@mit\.eduAffiliation:Paolo Giaretta pgiarett@mit\.eduAffiliation:Navid Azizan azizan@mit\.eduAffiliation:Massachusetts Institute of Technology

###### Abstract

Diffusion and flow models have achieved remarkable success, enabled in large part by conditional matching, which reduces distribution learning to a scalable supervised regression problem when samples from the target distribution are available\. However, many important tasks, such as fine\-tuning and sampling, fall into the opposite regime, in which the target probability density is specified only up to a normalizing constant and direct samples are unavailable\. Specifically, we consider target densities of the formπ⁡\(x\)∝μ⁡\(x\)​eτ​r​\(x\)\\pi\(x\)\\propto\\mu\(x\)e^\{\\tau r\(x\)\}, whereμ\\mudenotes the reference factor,rrdenotes the reward function, andτ\\taucontrols the emphasis on the reward\.μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}denotes the normalized terminal density from the pretrained model in the fine\-tuning task, whereasμ≡1\\mu\\equiv 1in the sampling task\.

Early approaches cast these tasks as reinforcement learning over the generative dynamics, lifting a terminal\-density problem to trajectory\-level optimization while leaving the conditional\-matching structure largely unexploited\. More recent work instead seeks to preserve the supervised\-regression form\. Reweighting\-based methods retain the standard matching objective but can become unstable when the proposal poorly covers the target density\. Another emerging class draws samples from the current model and uses reward\-based feedback to construct the next regression update\. These methods, however, arise from disparate principles and intertwine model representation, sampling construction, and feedback signal\. Consequently, their regression objectives rarely reveal the underlying population\-level update, obscuring which differences are fundamental, which reflect alternative exact realizations, and which arise from approximations that alter the update\. More fundamentally, how a single stage transforms the terminal density, whether it improves the objective, and whether the resulting iteration converges remain largely uncharacterized\.

In this paper, we develop*Newton Matching*, a unified framework for fine\-tuning and sampling\. Newton Matching rests on two paradigm shifts in perspective\. First, rather than treating a method as an isolated regression loss, we view learning as an iterative optimization process: the current model generates samples, each stage updates its terminal density based on these samples, and the resulting density determines the next model iterate\. Second, we exploit the conditional\-expectation structure of standard matching to define the optimization domain\. Although a terminal density can be realized by infinitely many generative dynamics, standard conditional matching selects a distinguished representation—the population minimizer that would be recovered if samples from that density were available—which we call its canonical model\. Canonical models retain the regression structure that makes diffusion and flow training scalable\. Under the compatible smooth\-realization assumptions, their velocity representations form a manifold in one\-to\-one correspondence with terminal densities\. On this shared domain, fine\-tuning and sampling become iterative optimization problems, and our framework disentangles three questions that existing methods often conflate: which distributional direction should govern the iteration, how a finite\-stepsize update along that direction transforms the terminal density, and how that transformation can be realized through matching\-style regression\. To specify a principled direction, Newton Matching chooses the reverse Kullback–Leibler \(KL\) objective, equips the terminal\-density manifold with the Fisher–Rao metric and the mixture connection, and transports both geometric structures to the canonical manifold\. Remarkably, the reverse\-KL Hessian under the mixture connection coincides with the Fisher–Rao metric, so the Newton direction is exactly the negative Fisher–Rao gradient\. This direction is generated by the regularized reward, which combines the task reward with the density\-ratio correction required by the prescribed target, and admits equivalent covariance and gradient representations, both learnable through sample\-wise matching without importance sampling or backpropagation through the full generation trajectory\. A Newton Matching stage first takes a damped or full tangential step and then canonicalizes this intermediate outcome, replacing it with the canonical model associated with the same terminal density\. This composition defines a canonical retraction and yields an exact finite\-stepsize characterization of the induced density update, from which we establish strict reverse\-KL descent throughout the full admissible stepsize range, global convergence under mild conditions, and local quadratic convergence of the full\-step iteration\. We then translate the ideal tangential update into scalable algorithms\. Covariance and gradient representations of the Newton direction can each be combined with forward or reverse regression\-pair constructions, yielding multiple sample\-wise regression objectives with the same population minimizer\. To broaden the computational design space, we further develop approximate Newton Matching, which allows the tangential update to be approximated in various principled ways\. We introduce critical\-point consistency to distinguish approximations whose tangential updates vanish exactly when the current terminal density equals the prescribed target\. Finally, this modular decomposition recovers representative existing methods as exact realizations, critical\-point\-consistent approximations, or variants that alter the underlying objective, thereby demystifying methods derived from seemingly disparate principles and furnishing a principled design space for new algorithms\.

††firstpage:1††footnotetext:Correspondence to Zeyang Li and Yunan Wang\.###### keywords

generative models, diffusion, flow, reinforcement learning, fine\-tuning, sampling

###### Contents

1. [1Introduction](https://arxiv.org/html/2609.05727#S1)
2. [2Preliminaries](https://arxiv.org/html/2609.05727#S2)1. [2\.1Diffusion and Flow](https://arxiv.org/html/2609.05727#S2.SS1) 2. [2\.2Differential Geometry on Banach Manifolds](https://arxiv.org/html/2609.05727#S2.SS2)
3. [3Problem Formulation](https://arxiv.org/html/2609.05727#S3)1. [3\.1Sampling and Fine\-Tuning](https://arxiv.org/html/2609.05727#S3.SS1) 2. [3\.2Optimization on Velocity Fields](https://arxiv.org/html/2609.05727#S3.SS2)
4. [4Canonical Geometry](https://arxiv.org/html/2609.05727#S4)1. [4\.1Global Geometry](https://arxiv.org/html/2609.05727#S4.SS1) 2. [4\.2First\-Order Geometry](https://arxiv.org/html/2609.05727#S4.SS2) 3. [4\.3Second\-Order Geometry](https://arxiv.org/html/2609.05727#S4.SS3)
5. [5Canonical Retraction and Value Ascent](https://arxiv.org/html/2609.05727#S5)1. [5\.1Canonical Retractions in Velocity and Density Coordinates](https://arxiv.org/html/2609.05727#S5.SS1) 2. [5\.2Value\-Ascent Certificate](https://arxiv.org/html/2609.05727#S5.SS2)
6. [6Newton Matching on the Canonical Manifold](https://arxiv.org/html/2609.05727#S6)1. [6\.1Fisher–Rao Gradient](https://arxiv.org/html/2609.05727#S6.SS1) 2. [6\.2Newton Direction](https://arxiv.org/html/2609.05727#S6.SS2) 3. [6\.3Newton’s Method with Canonical Retraction](https://arxiv.org/html/2609.05727#S6.SS3)
7. [7Convergence Guarantees of Newton Matching](https://arxiv.org/html/2609.05727#S7)1. [7\.1KL Descent and Global Convergence](https://arxiv.org/html/2609.05727#S7.SS1) 2. [7\.2Local Quadratic Convergence and Continuation Method](https://arxiv.org/html/2609.05727#S7.SS2)
8. [8Exact Newton Matching](https://arxiv.org/html/2609.05727#S8)1. [8\.1Tangent Representations](https://arxiv.org/html/2609.05727#S8.SS1) 2. [8\.2Forward and Reverse Constructions](https://arxiv.org/html/2609.05727#S8.SS2) 3. [8\.3Covariance Form](https://arxiv.org/html/2609.05727#S8.SS3) 4. [8\.4Gradient Form](https://arxiv.org/html/2609.05727#S8.SS4) 5. [8\.5Unification via Control Variates](https://arxiv.org/html/2609.05727#S8.SS5) 6. [8\.6Multi\-Time Supervision from Shared Samples](https://arxiv.org/html/2609.05727#S8.SS6) 7. [8\.7Canonicalization](https://arxiv.org/html/2609.05727#S8.SS7)
9. [9Approximate Newton Matching](https://arxiv.org/html/2609.05727#S9)1. [9\.1Approximate Covariance Forms with Critical\-Point Consistency](https://arxiv.org/html/2609.05727#S9.SS1) 2. [9\.2Approximate Gradient Forms with Critical\-Point Consistency](https://arxiv.org/html/2609.05727#S9.SS2) 3. [9\.3Regularization Trade\-Offs](https://arxiv.org/html/2609.05727#S9.SS3) 4. [9\.4Fixed\-Point Conditions via Zero Displacement](https://arxiv.org/html/2609.05727#S9.SS4)
10. [10Demystifying Existing Works](https://arxiv.org/html/2609.05727#S10)1. [10\.1Tilt Matching](https://arxiv.org/html/2609.05727#S10.SS1) 2. [10\.2DiffusionNFT](https://arxiv.org/html/2609.05727#S10.SS2) 3. [10\.3Reinforce Adjoint Matching](https://arxiv.org/html/2609.05727#S10.SS3) 4. [10\.4Adjoint Matching](https://arxiv.org/html/2609.05727#S10.SS4) 5. [10\.5Flow Sampling](https://arxiv.org/html/2609.05727#S10.SS5) 6. [10\.6Adjoint Sampling](https://arxiv.org/html/2609.05727#S10.SS6)
11. [11Conclusion](https://arxiv.org/html/2609.05727#S11)
12. [AGeometric Foundation](https://arxiv.org/html/2609.05727#A1)1. [A\.1Ambient Classes](https://arxiv.org/html/2609.05727#A1.SS1) 2. [A\.2Set\-Theoretic Definitions](https://arxiv.org/html/2609.05727#A1.SS2) 3. [A\.3Smooth Geometric Realization](https://arxiv.org/html/2609.05727#A1.SS3) 4. [A\.4Non\-Affineness of the Canonical Manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}](https://arxiv.org/html/2609.05727#A1.SS4)
13. [BConvergence Proofs for Newton Matching](https://arxiv.org/html/2609.05727#A2)1. [B\.1Proofs of Global Convergence](https://arxiv.org/html/2609.05727#A2.SS1) 2. [B\.2Proofs of Local Convergence](https://arxiv.org/html/2609.05727#A2.SS2)
14. [CNewton Matching for the Isotropic Gaussian Family](https://arxiv.org/html/2609.05727#A3)1. [C\.1Gaussian Closure](https://arxiv.org/html/2609.05727#A3.SS1) 2. [C\.2Convergence Guarantee](https://arxiv.org/html/2609.05727#A3.SS2)
15. [DStochastic Foundations and Computational Tools](https://arxiv.org/html/2609.05727#A4)1. [D\.1Common SDE Setup and Basic Identities](https://arxiv.org/html/2609.05727#A4.SS1) 2. [D\.2Backward Equation for Posterior Value](https://arxiv.org/html/2609.05727#A4.SS2) 3. [D\.3Posterior\-Preserving SDE](https://arxiv.org/html/2609.05727#A4.SS3) 4. [D\.4Universal Bridges](https://arxiv.org/html/2609.05727#A4.SS4) 5. [D\.5Log Density and Log\-Density Ratio](https://arxiv.org/html/2609.05727#A4.SS5) 6. [D\.6Initial\-State Sensitivity and Adjoint Calculus](https://arxiv.org/html/2609.05727#A4.SS6)
16. [EPosterior Stein Kernels](https://arxiv.org/html/2609.05727#A5)1. [E\.1Langevin Stein Identities](https://arxiv.org/html/2609.05727#A5.SS1) 2. [E\.2Stein Control Variates](https://arxiv.org/html/2609.05727#A5.SS2) 3. [E\.3Covariance–Gradient Identity of Posterior Stein Kernels](https://arxiv.org/html/2609.05727#A5.SS3) 4. [E\.4Posterior Sensitivity Kernel](https://arxiv.org/html/2609.05727#A5.SS4) 5. [E\.5Gaussian Stein Kernels and Posterior Moments](https://arxiv.org/html/2609.05727#A5.SS5)
17. [FProofs for Approximate Newton Matching](https://arxiv.org/html/2609.05727#A6)1. [F\.1Approximate Covariance Forms with Critical\-Point Consistency](https://arxiv.org/html/2609.05727#A6.SS1) 2. [F\.2Approximate Gradient Forms with Critical\-Point Consistency](https://arxiv.org/html/2609.05727#A6.SS2) 3. [F\.3Regularization Trade\-Offs](https://arxiv.org/html/2609.05727#A6.SS3)
18. [GCanonical Updates with Time\-Dependent Stepsizes](https://arxiv.org/html/2609.05727#A7)1. [G\.1Terminal\-Density Characterization](https://arxiv.org/html/2609.05727#A7.SS1) 2. [G\.2Monotonicity Guarantees](https://arxiv.org/html/2609.05727#A7.SS2)
19. [HExtensions to Alternative Coordinates and Interpolants](https://arxiv.org/html/2609.05727#A8)1. [H\.1Newton Matching under Score and Drift Coordinates](https://arxiv.org/html/2609.05727#A8.SS1) 2. [H\.2Newton Matching under One\-Sided Interpolants](https://arxiv.org/html/2609.05727#A8.SS2)
20. [IDetailed Derivations for Existing Works](https://arxiv.org/html/2609.05727#A9)1. [I\.1Tilt Matching](https://arxiv.org/html/2609.05727#A9.SS1) 2. [I\.2DiffusionNFT](https://arxiv.org/html/2609.05727#A9.SS2) 3. [I\.3Reinforce Adjoint Matching](https://arxiv.org/html/2609.05727#A9.SS3) 4. [I\.4Adjoint Matching](https://arxiv.org/html/2609.05727#A9.SS4) 5. [I\.5Flow Sampling](https://arxiv.org/html/2609.05727#A9.SS5) 6. [I\.6Adjoint Sampling](https://arxiv.org/html/2609.05727#A9.SS6)
21. [References](https://arxiv.org/html/2609.05727#bib)

## 1Introduction

Diffusion\[[42](https://arxiv.org/html/2609.05727#bib.bib2),[24](https://arxiv.org/html/2609.05727#bib.bib3)\]and flow models\[[33](https://arxiv.org/html/2609.05727#bib.bib1)\]have achieved remarkable success in image and video generation\[[41](https://arxiv.org/html/2609.05727#bib.bib4),[25](https://arxiv.org/html/2609.05727#bib.bib5),[16](https://arxiv.org/html/2609.05727#bib.bib6)\], and their applications are rapidly expanding to broader domains\[[9](https://arxiv.org/html/2609.05727#bib.bib7),[49](https://arxiv.org/html/2609.05727#bib.bib11),[54](https://arxiv.org/html/2609.05727#bib.bib13)\]\. At their core, these models learn a time\-dependent generative process that gradually transports a simple prior distribution, typically Gaussian noise, into a complex target distribution\. Their success, however, stems not only from the expressive power of these generative dynamics, but also from the remarkably simple and scalable way in which they can be learned\. When samples from the target distribution are available, both model families share a common conditional\-matching structure: intermediate states are constructed analytically along a prescribed noising process, and the model is trained by supervised regression against tractable sample\-wise targets\. At the population level, the optimal regressor is precisely the conditional expectation that drives the desired probability transport\. This conditional\-expectation structure reduces the difficult problem of learning a complex distribution to a scalable supervised regression problem\.

Many important tasks, however, violate precisely the assumption that makes this training recipe possible: samples from the desired target density are unavailable\. In reward\-based fine\-tuning, one starts from a pretrained generative model and seeks to shift probability mass toward high\-reward outputs\. In sampling from an unnormalized density, the target is specified by an energy function\. Although usually studied separately, both problems can be expressed through the shared target

π⁡\(x\)∝μ⁡\(x\)​eτ​r​\(x\),\\pi\(x\)\\propto\\mu\(x\)e^\{\\tau r\(x\)\},whereμ\\mudenotes the reference factor,rris the reward function, and the inverse temperatureτ\>0\\tau\>0controls the strength of the tilt\. For fine\-tuning,μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}is the terminal density induced by the pretrained model; for sampling,μ≡1\\mu\\equiv 1\. These two formulations encompass a broad range of applications\. For the fine\-tuning formulation, in which the terminal density of a pretrained model supplies the reference factor, representative examples are reward alignment of text\-to\-image models\[[18](https://arxiv.org/html/2609.05727#bib.bib14),[7](https://arxiv.org/html/2609.05727#bib.bib15)\]and offline reinforcement learning\[[48](https://arxiv.org/html/2609.05727#bib.bib9),[47](https://arxiv.org/html/2609.05727#bib.bib10)\]\. For the sampling formulation, in which the target is specified directly as a Boltzmann distribution, representative examples are equilibrium configuration sampling\[[37](https://arxiv.org/html/2609.05727#bib.bib8)\]and maximum\-entropy online reinforcement learning\[[20](https://arxiv.org/html/2609.05727#bib.bib12),[32](https://arxiv.org/html/2609.05727#bib.bib35)\]\. In this paper, we focus on theory and algorithms at the level of these two formulations rather than on any particular application; accordingly, our results apply whenever the desired target can be expressed in either form\.

In both cases, learning takes place in a*dataless*regime: the target densityπ⁡\(x\)∝μ⁡\(x\)​eτ​r​\(x\)\\pi\(x\)\\propto\\mu\(x\)e^\{\\tau r\(x\)\}is specified through its defining factors, rather than through samples drawn from it\. Therefore, the supervised construction that underpins the efficacy of diffusion and flow training can no longer be used directly: there are no target endpoints from which to form training pairs, and the generative dynamics must instead be learned from feedback evaluated on samples produced by the current or base model\. Turning this indirect signal into a stable and scalable learning procedure that recovers the prescribed target density is highly nontrivial\.

Across the literatures on fine\-tuning and sampling, a common early strategy is to recast the dataless learning problem as reinforcement learning \(RL\)\. At a high level, these approaches can be divided into discrete\-time and continuous\-time formulations\. In discrete time, the discretized generative trajectory is modeled as a Markov decision process \(MDP\): intermediate noisy samples serve as states, denoising transitions define the policy, and the terminal reward guides policy optimization\[[18](https://arxiv.org/html/2609.05727#bib.bib14),[7](https://arxiv.org/html/2609.05727#bib.bib15),[35](https://arxiv.org/html/2609.05727#bib.bib16),[51](https://arxiv.org/html/2609.05727#bib.bib17),[28](https://arxiv.org/html/2609.05727#bib.bib18)\]\. In continuous time, the analogous formulation is stochastic optimal control, where the drift or score is treated as the control and optimized against a terminal reward or energy, typically together with a pathwise control cost or a divergence from a reference process\[[44](https://arxiv.org/html/2609.05727#bib.bib19),[55](https://arxiv.org/html/2609.05727#bib.bib20),[46](https://arxiv.org/html/2609.05727#bib.bib21),[6](https://arxiv.org/html/2609.05727#bib.bib22),[40](https://arxiv.org/html/2609.05727#bib.bib23),[45](https://arxiv.org/html/2609.05727#bib.bib24),[57](https://arxiv.org/html/2609.05727#bib.bib25),[21](https://arxiv.org/html/2609.05727#bib.bib26)\]\. These formulations offer a straightforward way to propagate terminal feedback through the generative dynamics, but they treat diffusion and flow models as generic controlled processes\. The problem of learning the prescribed terminal density is thereby lifted to a trajectory\-level optimization problem, through transition\-wise objectives in discrete time or path\-space objectives in continuous time\. Since many dynamics can realize the same terminal law, this lift requires path\-level choices beyond the target specification, which can bias the endpoint distribution or tie the method to particular noise schedules and samplers\. Meanwhile, the specialized structure underlying scalable diffusion and flow training remains largely unused\.

More recently, a growing line of work has sought to preserve the native regression structure of diffusion and flow training in the dataless regime\. The most straightforward strategy retains the usual matching objective and incorporates the task signal through sample reweighting\. For fine\-tuning, this amounts to weighting matching losses by rewards or advantages evaluated on samples from a reference distribution\[[56](https://arxiv.org/html/2609.05727#bib.bib44),[17](https://arxiv.org/html/2609.05727#bib.bib43),[50](https://arxiv.org/html/2609.05727#bib.bib28)\]; for sampling, it corresponds to posterior\-mean estimation of the target score or velocity via self\-normalized importance sampling\[[1](https://arxiv.org/html/2609.05727#bib.bib27),[32](https://arxiv.org/html/2609.05727#bib.bib35)\]\. In both settings, insufficient coverage of the desired target can cause the weights to concentrate on a small fraction of samples, thereby reducing the effective sample size and leading to high\-variance, unstable updates\. To avoid the pitfalls of importance sampling, a more promising direction is to derive model\-dependent regression targets from samples generated by the current model\. This direction has emerged from various perspectives, including stochastic optimal control\[[14](https://arxiv.org/html/2609.05727#bib.bib30),[23](https://arxiv.org/html/2609.05727#bib.bib31),[34](https://arxiv.org/html/2609.05727#bib.bib42),[5](https://arxiv.org/html/2609.05727#bib.bib37)\], contrastive learning\[[58](https://arxiv.org/html/2609.05727#bib.bib36)\], fixed\-point iteration\[[22](https://arxiv.org/html/2609.05727#bib.bib34),[8](https://arxiv.org/html/2609.05727#bib.bib29)\], and velocity evolution under exponential tilting\[[39](https://arxiv.org/html/2609.05727#bib.bib32)\]\. Collectively, these works demonstrate the feasibility of retaining the simple and scalable regression form of standard generative model training, even in the absence of target samples\.

Despite this progress, the emerging landscape remains conceptually fragmented\. Even the basic design choices are tightly intertwined: the model may be represented by a score, velocity, or drift; constructing a regression pair involves choosing whether the endpoint is obtained from an ODE or SDE sampler and whether the corresponding intermediate state is taken from the same trajectory or sampled from a prescribed noising kernel; and the regression target may depend on the reward itself, its gradient, or an adjoint quantity\. These choices arise from distinct, often ad hoc and technically intricate derivations, and the distributional meaning of the resulting regression objectives is rarely transparent from their final form\. Consequently, the objectives alone do not reveal which differences are fundamental, which are merely alternative realizations of the same population\-level update, and which arise from approximations that alter its distributional meaning\. More fundamentally, these methods are often presented as generic regression losses, even though in practice they define an iterative model\-update procedure: the current model generates samples, regression produces an updated model, and the cycle repeats\. The iterative behavior of these methods remains largely uncharacterized, even at the level of a single stage: it remains unclear how the terminal density is updated, whether the procedure moves the model density towardπ⁡\(x\)∝μ⁡\(x\)​eτ​r​\(x\)\\pi\(x\)\\propto\\mu\(x\)e^\{\\tau r\(x\)\}, or whether it yields consistent improvement, let alone whether the full iterative procedure converges\.

Motivated by these gaps, we develop a unified framework for dataless fine\-tuning and sampling that reveals the shared structure underlying scalable matching methods that do not rely on importance sampling\. Our starting point is a simple but powerful observation: the conditional\-expectation structure is not merely a convenient way to train diffusion and flow models; it selects a distinguished model representation for every terminal density\. Indeed, the same terminal density can be generated by infinitely many dynamics\. Searching over all such representations introduces unnecessary redundancy and does not by itself ensure that intermediate models retain the structure needed for scalable regression\. We therefore restrict attention to the specific models, whether expressed as velocities, scores, or drifts, that standard diffusion or flow training would recover at the population level if samples from their terminal densities were available\. We call them*canonical models*and show that they form a structured geometric family, the*canonical manifold*, embedded in the generic model space and in one\-to\-one correspondence with terminal densities\. Fine\-tuning and sampling can then be viewed as iterative optimization over this manifold\. At each stage, the current model is updated toward the target density using a direction that admits a conditional\-expectation representation and can therefore be learned through sample\-wise regression in the same form as conditional flow matching\. The updated terminal density is then expressed again through its canonical model, providing the appropriate starting point for the next stage\. With this fundamental shift in perspective, the central question is no longer how to design a particular regression loss, but which principled optimization direction should govern the iteration, how a finite\-stepsize update along that direction changes the terminal density, and how the resulting update can be realized through scalable matching\.

![Refer to caption](https://arxiv.org/html/2609.05727v1/overview_a.png)𝒯\\mathcal\{T\}𝒞\\mathcal\{C\}∼\\simvelocity space𝒱\\mathscr\{V\}density manifold𝒫\\mathscr\{P\}vπ=𝒞⁡\(π\)v^\{\\pi\}=\\mathcal\{C\}\(\\pi\)π=𝒯⁡\(vπ\)\\pi=\\mathcal\{T\}\(v^\{\\pi\}\)vρk=𝒞⁡\(ρk\)v^\{\\rho\_\{k\}\}=\\mathcal\{C\}\(\\rho\_\{k\}\)vρk\+1=ℛ⁡\(v¯k\+1\)v^\{\\rho\_\{k\+1\}\}=\\mathcal\{R\}\(\\bar\{v\}^\{k\+1\}\)⋯\\cdotsρk=𝒯⁡\(vρk\)\\rho\_\{k\}=\\mathcal\{T\}\\\!\\left\(v^\{\\rho\_\{k\}\}\\right\)ρk\+1=𝒯⁡\(vρk\+1\)\\rho\_\{k\+1\}=\\mathcal\{T\}\\\!\\left\(v^\{\\rho\_\{k\+1\}\}\\right\)⋯\\cdotsξρk,r~ρk∈Tρk​𝒫\\xi^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\\in T\_\{\\rho\_\{k\}\}\\mathscr\{P\}v¯k\+1=vρk\+ηk​Γρk,r~ρk\\bar\{v\}^\{k\+1\}=v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}Γρk,r~ρk∈Tvρk​𝒱can\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\\in T\_\{v^\{\\rho\_\{k\}\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}ℛ\\mathcal\{R\}canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}↑∞\\uparrow\\inftyreverse KL0aGeometric illustration of Newton Matching\.𝒯:𝒱→𝒫\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{P\}is the terminal\-density map,𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is the canonical\-velocity map, andℛ:𝒱→𝒱can\\mathcal\{R\}:\\mathscr\{V\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}is the canonical projection\. The target densityπ∈𝒫\\pi\\in\\mathscr\{P\}is defined asπ⁡\(x\)∝μ⁡\(x\)​eτ​r​\(x\)\\pi\(x\)\\propto\\mu\(x\)e^\{\\tau r\(x\)\}, where the referenceμ≡1\\mu\\equiv 1andμ∈𝒫\\mu\\in\\mathscr\{P\}correspond to sampling and fine\-tuning, respectively\.rris the raw reward andr~ρ=r−1τ​log⁡ρμ\\tilde\{r\}^\{\\rho\}=r\-\\frac\{1\}\{\\tau\}\\log\\frac\{\\rho\}\{\\mu\}is the regularized reward\.Γρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}andξρ,r~ρ\\xi^\{\\rho,\\tilde\{r\}^\{\\rho\}\}define the Newton directions of reverse KL atvρ∈𝒱canv^\{\\rho\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}andρ∈𝒫\\rho\\in\\mathscr\{P\}, respectively\.![Refer to caption](https://arxiv.org/html/2609.05727v1/overview_b.png)t=0t=0t=1t=1p0p\_\{0\}ρk\\rho\_\{k\}π\\pip^t\\hat\{p\}\_\{t\}v¯k\+1​\(xt\)=𝔼X1∼p1\|tρk\(⋅\|xt\)\[vt\|1\(xt\|X1\)\+ηk\(vt\|1\(xt\|X1\)−vtρk\(xt\)\)\(r~ρk\(X1\)−Bt\(xt\)\)\]\(covariance form\)=𝔼X1∼p1\|tρk\(⋅\|xt\)\[vt\|1\(xt\|X1\)\+ηkαt​κtβt2Λtρk\(X1\|xt\)⊤∇r~ρk\(X1\)\]\(gradient form\)\\displaystyle\\begin\{aligned\} \\bar\{v\}^\{k\+1\}\(x\_\{t\}\)&=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|x\_\{t\}\)\}\\\!\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\+\\eta\_\{k\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\_\{k\}\}\(x\_\{t\}\)\\right\)\\left\(\\tilde\{r\}^\{\\rho\_\{k\}\}\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\\right\)\\right\]&\\hskip 16\.38895pt\\text\{\(covariance form\)\}&\\\\ &=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|x\_\{t\}\)\}\\\!\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\+\\eta\_\{k\}\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\Lambda\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla\\tilde\{r\}^\{\\rho\_\{k\}\}\(X\_\{1\}\)\\right\]&\\hskip 16\.38895pt\\text\{\(gradient form\)\}&\\end\{aligned\}Forward constructionReverse constructionODE forvρkv^\{\\rho\_\{k\}\}SDE forvρkv^\{\\rho\_\{k\}\}2\.Sample​Xt∼𝒩⁡\(αt​X1,βt2​I\)\\text\{Sample \}X\_\{t\}\\sim\\mathcal\{N\}\\\!\\left\(\\alpha\_\{t\}X\_\{1\},\\beta\_\{t\}^\{2\}I\\right\)\.1\.SampleX1X\_\{1\}withODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) or SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\)\.3\.Update network parameterswith reverse loss \([57](https://arxiv.org/html/2609.05727#S8.E57)\) or \([74](https://arxiv.org/html/2609.05727#S8.E74)\)\.3\.Update network parameterswith forward loss \([56](https://arxiv.org/html/2609.05727#S8.E56)\) or \([73](https://arxiv.org/html/2609.05727#S8.E73)\)\.1\.Sample​Xt∼p^t\\text\{Sample \}X\_\{t\}\\sim\\hat\{p\}\_\{t\}\.2\.SampleX1X\_\{1\}withposterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\)\.X1X\_\{1\}XtX\_\{t\}XtX\_\{t\}XtX\_\{t\}XtX\_\{t\}X1X\_\{1\}bTangential update\. At stagekk, givenvρkv^\{\\rho\_\{k\}\}, obtainv¯k\+1=vρk\+ηk​Γρk,r~ρk\\bar\{v\}^\{k\+1\}=v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.![Refer to caption](https://arxiv.org/html/2609.05727v1/overview_c.png)ODE for noncanonicalv¯k\+1\\bar\{v\}^\{k\+1\}t=0t=0t=1t=1X1X\_\{1\}XtX\_\{t\}1\.SampleX1∼ρk\+1X\_\{1\}\\sim\\rho\_\{k\+1\}with ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) forv¯k\+1\\bar\{v\}^\{k\+1\}\.2\.Sample​Xt∼𝒩⁡\(αt​X1,βt2​I\)\\text\{Sample \}X\_\{t\}\\sim\\mathcal\{N\}\\\!\\left\(\\alpha\_\{t\}X\_\{1\},\\beta\_\{t\}^\{2\}I\\right\)\.3\.Update network parameters with canonicalization loss \([106](https://arxiv.org/html/2609.05727#S8.E106)\)\.p0p\_\{0\}ρk\+1\\rho\_\{k\+1\}π\\picCanonicalization\. At stagekk, givenv¯k\+1\\bar\{v\}^\{k\+1\}, obtainvρk\+1=ℛ⁡\(v¯k\+1\)\{v\}^\{\\rho\_\{k\+1\}\}=\\mathcal\{R\}\(\\bar\{v\}^\{k\+1\}\)\.Figure 1:Overview of Newton Matching, which iteratively solves the optimization problemminv∈𝒱can⁡KL⁡\(𝒯⁡\(v\)∥π\)\\min\_\{v\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\\,\\operatorname\{KL\}\\left\(\\mathcal\{T\}\(v\)\\middle\\\|\\pi\\right\)by applying Newton’s method on the canonical manifold\.

We answer these questions with*Newton Matching*, whose overall architecture is illustrated in Figure[1](https://arxiv.org/html/2609.05727#S1.F1)\. Newton Matching casts fine\-tuning and sampling as iterative optimization on the canonical manifold, with the generative model itself as the optimization variable\. Although Figure[1](https://arxiv.org/html/2609.05727#S1.F1)and our main development use the velocity representation, the same canonical model admits equivalent representations in terms of velocity, score, drift, data prediction, and noise prediction\. Regardless of the chosen representation, restricting the search to the canonical manifold keeps the iterates within the class of models that retain the conditional\-expectation structure of standard diffusion and flow training\. The optimization objective is to minimize the reverse Kullback–Leibler \(KL\) divergence between the terminal density generated by the current model and the prescribed target density\. Unlike optimization in Euclidean space, where the standard geometric choices are usually left implicit, optimization on a manifold requires its geometry to be specified\. Given an objective, a first\-order method requires a Riemannian metric to define the gradient, while a second\-order method requires an affine connection to define the Hessian\. We equip the density manifold with the Fisher–Rao metric and the mixture connection; then, we transport both structures to the canonical manifold through a diffeomorphism\. For the reverse\-KL objective, we prove a remarkable identity: its Hessian under the mixture connection equals the Fisher–Rao metric as a bilinear form on the tangent space\. Consequently, the same intrinsic optimization direction is both the Newton direction and the negative Fisher–Rao gradient\. Crucially, this direction is computationally accessible rather than merely a geometric abstraction\. It is generated by the current*regularized reward*, which combines the task reward with the density\-ratio correction required by the prescribed target\. The direction admits two equivalent conditional\-expectation representations: a covariance form based on regularized\-reward values and a gradient form based on their gradients\. Either representation can be realized through matching\-style sample\-wise regression, without importance sampling or backpropagation through the entire generation trajectory\.

Each Newton Matching stage consists of the two operations depicted in Figure[1](https://arxiv.org/html/2609.05727#S1.F1)\. First, the*tangential update*takes a damped or full step along the Newton direction, realized through either the forward or the reverse construction\. Since the canonical manifold is generally non\-affine, this finite tangential step need not remain on the manifold\. The subsequent*canonicalization*preserves the updated terminal density while replacing its dynamical representation with the unique canonical model associated with that density, thereby providing a valid starting point for the next stage\. The composition of these two operations defines a*canonical retraction*that realizes one damped or full Newton update on the canonical manifold and can be implemented through scalable matching\-style regression losses\. At the same time, the induced terminal\-density update admits an exact finite\-stepsize characterization\. Thus, a single Newton Matching stage has three mutually consistent interpretations: intrinsically, it is a Newton step; computationally, it is realized through conditional\-matching regression; and distributionally, its finite\-stepsize effect is exactly analyzable\.

The*exact Newton Matching*framework admits multiple population\-exact realizations, all of which preserve the scalable regression structure of diffusion and flow training\. These realizations differ in the sampling constructions and auxiliary computations used to form their regression targets, yielding a range of computational options\. To broaden the design space and make the trade\-off between population exactness and computational efficiency explicit, we further develop*approximate Newton Matching*, which replaces the exact tangential update with computationally cheaper surrogates while preserving the canonical\-manifold iteration\. To distinguish approximations that preserve the prescribed target at the level of stationarity from those that do not, we introduce*critical\-point consistency*: an approximate tangential update is critical\-point consistent if its tangential displacement vanishes exactly when the current terminal density equals the target densityπ\\pi\. Together, exact and approximate Newton Matching organize the fragmented landscape of existing matching methods into a principled hierarchy, revealing the population\-level update implemented by each method and clarifying its trade\-offs among computational cost, stagewise exactness, and fidelity to the prescribed target\.

Finally, we summarize the contributions of this paper as follows\.

1. 1\.A paradigm shift to iterative optimization over canonical models\.We introduce two fundamental shifts in how fine\-tuning and sampling in generative modeling are conceptualized\. First, rather than viewing a method as an isolated regression loss, we view learning as an iterative optimization process: the current model generates samples, each stage updates its terminal density, and the resulting density determines the model used to initialize the next stage\. Second, we elevate the conditional\-expectation structure of standard diffusion and flow training from a regression identity to a principle for selecting the optimization domain\. Since every terminal density admits infinitely many dynamical representations, optimization over arbitrary models introduces redundant degrees of freedom and need not preserve the structure required for scalable matching\. We thereby restrict attention to canonical models, each defined as the population minimizer of the standard conditional\-matching objective for its terminal density\. This new perspective disentangles three questions that existing methods often conflate: which distributional direction should govern the iteration, how a finite\-stepsize update along that direction transforms the terminal density, and how that update can be realized through matching\-style regression\.
2. 2\.The canonical manifold and its differential geometry\.Under the compatible smooth\-realization assumption, we identify canonical velocity fields with a split embedded Banach submanifold of the velocity space\. The canonical\-velocity map is a diffeomorphism from the density manifold onto this canonical manifold, with the terminal\-density map restricted to canonical fields as its inverse\. This correspondence induces a smooth, idempotent canonical projection that maps any admissible velocity field to the unique canonical representative of the same terminal density\. We then characterize the tangent and cotangent spaces and derive canonical lifts of terminal\-density perturbations, with equivalent representations as posterior\-value gradients and posterior covariances\. Finally, we transport the Fisher–Rao metric and the mixture connection from the density manifold to the canonical manifold, furnishing the first\-order and second\-order geometric structures required for optimization\.
3. 3\.Canonical retractions and finite\-stepsize value ascent\.Before specifying a particular objective or search direction, we develop an objective\-agnostic mechanism for taking finite\-stepsize updates on the canonical manifold\. Given a canonical velocity model and a tangent direction, we first take a displacement in the velocity space and then canonicalize the result without changing its terminal density\. This two\-stage construction defines a canonical retraction, whose transport through the canonical diffeomorphism yields an equivalent retraction on the density manifold\. For tangent directions generated by admissible terminal observables, we derive an exact evolution identity along every characteristic of the updated flow, its density\-level counterpart, and an explicit formula for the updated terminal density\. These results yield a finite\-stepsize value\-ascent certificate: the expectation of every nonconstant terminal observable increases strictly, and the gain decomposes exactly into a stepsize\-scaled KL divergence from the updated density to the current density and a nonnegative expected path\-dissipation term\.
4. 4\.Newton Matching as Fisher–Rao gradient descent and Newton’s method\.For both fine\-tuning and sampling, we specialize the canonical optimization framework to reverse\-KL minimization\. The negative Fisher–Rao gradient is generated by the current regularized reward, and its canonical lift admits equivalent posterior\-value\-gradient and conditional\-covariance representations, making the direction directly learnable through matching\-style regression\. We then prove that the reverse\-KL Hessian under the mixture connection equals the Fisher–Rao metric as a bilinear form, so the negative Fisher–Rao gradient coincides exactly with the Newton direction under the mixture connection\. This Newton characterization is essential: it identifiesη=τ\\eta=\\tauas the full step solving the linearized stationarity equation and underlies the local quadratic convergence established later\. Combining this direction with the canonical retraction defines Newton Matching, identifies the prescribed target as the unique stationary terminal density, and yields an exact characterization of the induced terminal\-density update\.
5. 5\.Finite\-stepsize KL descent, global convergence, and local quadratic convergence\.For a single Newton Matching stage, we derive a dissipation\-corrected three\-point identity and an equivalent KL\-proximal characterization\. These results establish strict reverse\-KL descent for every stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\], including the full Newton step, unless the target has already been reached\. For the resulting iteration, we prove the global convergence of the terminal densities and canonical models to the target under mild assumptions\. Locally, full\-step Newton Matching converges quadratically for both terminal densities and canonical models\. As an optional complement, we further develop a continuation scheme along the inverse\-temperature path that initializes each successive stage within the quadratic\-convergence neighborhood of its target, thereby ensuring quadratic convergence throughout the continuation procedure\.
6. 6\.Exact realizations of Newton Matching\.We convert both operations in an ideal Newton Matching stage, the tangential update and canonicalization, into population\-exact sample\-wise regression objectives\. For the tangential update, the framework separates two independent design choices: how regression pairs are generated and how the Newton direction is represented\. The*forward construction*first samples an endpoint from the current model and then generates an intermediate state through the prescribed noising kernel\. The*reverse construction*first draws an intermediate state from a flexible proposal and then samples one or more endpoints from the corresponding canonical posterior\. Within the same\-marginal SDE family, we prove that this posterior requirement uniquely determines the*posterior\-preserving SDE*, whose terminal transition law is exactly the required canonical posterior\. We further establish*bridge universality*: conditioning on the states at any two times, the intervening path law of the posterior\-preserving SDE is independent of the terminal density and admits an explicit Gaussian representation\. This allows endpoint pairs produced by the forward construction to be augmented with exact posterior\-compatible paths whenever pathwise targets are required\. The Newton direction itself admits two exact representations\. The*covariance form*uses regularized\-reward values, for which we derive exact evaluations of the required terminal log densities and log\-density ratios using either ODE or SDE calculus\. To obtain the*gradient form*, we introduce posterior Stein kernels and establish a covariance–gradient identity that rewrites the same posterior covariance as a conditional expectation involving regularized\-reward gradients\. We then construct the*posterior sensitivity kernel*, a particular exact posterior Stein kernel obtained from the initial\-state sensitivity of the posterior\-preserving SDE\. Its action can be evaluated without forming sensitivity matrices through adjoint calculus, yielding exact Mayer and Bolza realizations of the gradient\-form target\. In contrast to methods based on stochastic optimal control, which require deriving the Hamilton–Jacobi–Bellman \(HJB\) equation and associated optimality conditions, our adjoint equations arise naturally as matrix\-free realizations of the initial\-state sensitivity underlying the posterior Stein kernel\. Stein control variates further unify the covariance and gradient forms into a broader family of population\-exact targets\. We further show that exactness is preserved under multi\-time supervision, allowing a single endpoint or posterior\-compatible path and its associated computations to be shared across multiple regression pairs\. We prove that all resulting tangential\-update losses share the same unique population minimizer and complete the stage with an explicit canonicalization loss based on conditional flow matching, together with an implicit alternative under suitable conditions\. A summary of losses for the tangential update in exact Newton Matching is provided in Table[1](https://arxiv.org/html/2609.05727#S8.T1)\.
7. 7\.Approximate realizations of Newton Matching and critical\-point consistency\.We broaden the computational design space by replacing the exact Newton tangential update with more efficient surrogates while retaining the canonical\-manifold iteration\. To formalize target correctness, we introduce*critical\-point consistency*: an approximate tangential update is critical\-point consistent if it vanishes exactly when the current terminal density equals the prescribed target\. As concrete examples, we derive four families satisfying this criterion: direct and split posterior\-ratio linearizations in covariance form, together with reference\-adjoint and Gaussian\-kernel approximations in gradient form\. These approximations generally differ from the exact Newton tangential update away from the target, while preserving the target as the unique density at which the update vanishes\. We also study regularization trade\-offs obtained by approximating the density\-ratio correction or removing it altogether\. Approximate regularization can reduce computation but may sacrifice critical\-point consistency, whereas complete removal turns the update into direct reward ascent\. Complementary to the stage\-by\-stage construction, we develop a zero\-displacement fixed\-point perspective that translates the additive update rules underlying both exact and approximate tangential updates into stop\-gradient objectives and precisely characterizes their population\-stationary points\.
8. 8\.A unification of existing methods and a modular design space for new algorithms\.We show that representative matching\-based methods for fine\-tuning and sampling can be recovered as particular configurations of exact or approximate Newton Matching, obtained through different choices of common building blocks, including the tangent representation, regression\-pair construction, density\-ratio regularization, posterior or sensitivity approximation, baselines, and stepsizes, among others\[[39](https://arxiv.org/html/2609.05727#bib.bib32),[58](https://arxiv.org/html/2609.05727#bib.bib36),[5](https://arxiv.org/html/2609.05727#bib.bib37),[14](https://arxiv.org/html/2609.05727#bib.bib30),[22](https://arxiv.org/html/2609.05727#bib.bib34),[23](https://arxiv.org/html/2609.05727#bib.bib31)\]\. As summarized in Table[2](https://arxiv.org/html/2609.05727#S9.T2), this correspondence places algorithms motivated by seemingly distinct principles within a unified design space and reveals the population\-level update implemented by each\. It distinguishes methods that realize the exact Newton Matching update from those that retain the prescribed target only at the level of critical\-point consistency or alter the underlying objective\. Conversely, the same modular perspective turns this taxonomy into a constructive design principle: new exact and approximate matching algorithms can be derived systematically by recombining these building blocks\.
9. 9\.Analytic validation and broader extensions\.We complement the general infinite\-dimensional theory with an analytically tractable specialization to the isotropic Gaussian family\. When the current and target densities are both Gaussian, one ideal Newton Matching stage produces another Gaussian; consequently, the entire iteration reduces exactly to a finite\-dimensional recursion for the mean and variance\. Analyzing this recursion directly, we prove that both forward and reverse KL divergences converge to zero and exhibit local quadratic convergence near the target\. We also extend the finite\-stepsize analysis by allowing time\-dependent stepsize profiles\. We derive their exact terminal\-density formula and show that nondecreasing stepsize profiles preserve finite\-stepsize value ascent and, for the regularized\-reward case, reverse\-KL descent\. Finally, we show that Newton Matching is not tied to velocity prediction: common parameterizations represent the same canonical model, and we explicitly demonstrate Newton Matching in score and drift coordinates\. We further extend the framework to one\-sided interpolants with deterministic initial states\.

## 2Preliminaries

### 2\.1Diffusion and Flow

In this subsection, we review the basic concepts and constructions underlying diffusion and flow models\.

#### 2\.1\.1Probability Paths and Interpolants

Letp0p\_\{0\}andp1p\_\{1\}be a source density and a terminal density onℝd\\mathbb\{R\}^\{d\}, respectively\. In diffusion and flow, the goal is to learn a deterministic or stochastic evolution that transports a simple source densityp0p\_\{0\}to a complex terminal densityp1p\_\{1\}\. A family of intermediate densities\(pt\)t∈\[0,1\]\\left\(p\_\{t\}\\right\)\_\{t\\in\[0,1\]\}is called the marginal probability path\.\(X0,X1\)∼ν0,1\(X\_\{0\},X\_\{1\}\)\\sim\\nu\_\{0,1\}is called a coupling ofp0p\_\{0\}andp1p\_\{1\}\. In this paper, we adopt the standard setting with Gaussian sourcep0=𝒩⁡\(0,I\)p\_\{0\}=\\mathcal\{N\}\(0,I\)and independent couplingν0,1​\(x0,x1\)=p0​\(x0\)​p1​\(x1\)\\nu\_\{0,1\}\(x\_\{0\},x\_\{1\}\)=p\_\{0\}\(x\_\{0\}\)p\_\{1\}\(x\_\{1\}\)\.

The central idea for making training tractable is to introduce a conditional probability path\(pt\|Z\)t∈\[0,1\]\\left\(p\_\{t\|Z\}\\right\)\_\{t\\in\[0,1\]\}, whose marginalization recovers the desired probability path fromp0p\_\{0\}top1p\_\{1\}\. HereZZdenotes the conditioning variable\. Common choices include the one\-sidedZ=X1Z=X\_\{1\}and the two\-sidedZ=\(X0,X1\)Z=\(X\_\{0\},X\_\{1\}\)\.

The standard approach for constructing a conditional probability path is to introduce the linear interpolantXt=αt​X1\+βt​X0X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}, where\(αt,βt\)\(\\alpha\_\{t\},\\beta\_\{t\}\)is referred to as the interpolant schedule\. For one\-sided conditioning andt∈\[0,1\)t\\in\[0,1\),

pt\|1​\(xt\|x1\)=1βtd​p0​\(xt−αt​x1βt\)=𝒩⁡\(xt,αt​x1,βt2​I\)\.p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\frac\{1\}\{\\beta\_\{t\}^\{d\}\}p\_\{0\}\\left\(\\frac\{x\_\{t\}\-\\alpha\_\{t\}x\_\{1\}\}\{\\beta\_\{t\}\}\\right\)=\\mathcal\{N\}\\left\(x\_\{t\};\\alpha\_\{t\}x\_\{1\},\\beta\_\{t\}^\{2\}I\\right\)\.For two\-sided conditioning,

pt\|0,1​\(xt\|x0,x1\)=δ⁡\(xt−\(αt​x1\+βt​x0\)\)p\_\{t\|0,1\}\(x\_\{t\}\|x\_\{0\},x\_\{1\}\)=\\delta\(x\_\{t\}\-\(\\alpha\_\{t\}x\_\{1\}\+\\beta\_\{t\}x\_\{0\}\)\)is a Dirac measure concentrated on the interpolant\.

For well\-posedness of the flow matching framework, we impose the following regularity conditions on the schedule\(αt,βt\)\(\\alpha\_\{t\},\\beta\_\{t\}\)\. We assume that

\(αt\)t∈\[0,1\],\(βt\)t∈\[0,1\]∈C1\(\[0,1\];\[0,\+∞\)\),\(\\alpha\_\{t\}\)\_\{t\\in\[0,1\]\},\\ \(\\beta\_\{t\}\)\_\{t\\in\[0,1\]\}\\in C^\{1\}\(\[0,1\];\[0,\+\\infty\)\),withαt\>0\\alpha\_\{t\}\>0andβt\>0\\beta\_\{t\}\>0for allt∈\(0,1\)t\\in\(0,1\), and that they satisfy the boundary conditions

α0=0,β0=1,α1=1,β1=0\.\\alpha\_\{0\}=0,\\qquad\\beta\_\{0\}=1,\\qquad\\alpha\_\{1\}=1,\\qquad\\beta\_\{1\}=0\.Fort∈\(0,1\)t\\in\(0,1\), define

κt:=βt2​dd​t​log⁡αtβt=βtαt​\(α˙t​βt−αt​β˙t\)\.\\kappa\_\{t\}:=\\beta\_\{t\}^\{2\}\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}\}=\\frac\{\\beta\_\{t\}\}\{\\alpha\_\{t\}\}\\left\(\\dot\{\\alpha\}\_\{t\}\\beta\_\{t\}\-\\alpha\_\{t\}\\dot\{\\beta\}\_\{t\}\\right\)\.We further assume that the signal\-to\-noise ratioαt2βt2\\frac\{\\alpha\_\{t\}^\{2\}\}\{\\beta\_\{t\}^\{2\}\}has strictly positive derivative on\(0,1\)\(0,1\), which impliesκt\>0\\kappa\_\{t\}\>0for allt∈\(0,1\)t\\in\(0,1\)\. The boundary conditions may be slightly relaxed in practice to avoid endpoint singularities\. We note that the above regularity conditions are satisfied by commonly used schedules in diffusion and flow models\.

We use the following notation convention throughout the paper\. The symbolsX0,Xt,X1X\_\{0\},X\_\{t\},X\_\{1\}are reserved for the variables of the linear interpolantXt=αt​X1\+βt​X0X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}, whereasYtY\_\{t\}denotes the state of an ordinary differential equation \(ODE\) or stochastic differential equation \(SDE\)\. The dynamics induce their own path\-space distribution for\(Yt\)t∈\[0,1\]\(Y\_\{t\}\)\_\{t\\in\[0,1\]\}and agree with the interpolant only at the level of one\-time marginals:

Xt∼pt,Yt∼pt\.X\_\{t\}\\sim p\_\{t\},\\qquad Y\_\{t\}\\sim p\_\{t\}\.However, this agreement does not imply that they have the same joint distribution across time\. In particular, the conditional distribution ofY1Y\_\{1\}givenYtY\_\{t\}need not coincide with that ofX1X\_\{1\}givenXtX\_\{t\}\. When a terminal state generated by a trajectory is subsequently used as an endpoint of the interpolant, we make this change of role explicit by writingX1:=Y1X\_\{1\}:=Y\_\{1\}\.

#### 2\.1\.2Deterministic Dynamics and Flow Maps

When sampling with deterministic dynamics, the stateYtY\_\{t\}evolves according to the ODE

d​Ytd​t=vt​\(Yt\),Y0∼p0,\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=v\_\{t\}\(Y\_\{t\}\),\\qquad Y\_\{0\}\\sim p\_\{0\},\(1\)wherevt:ℝd→ℝdv\_\{t\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}is the marginal velocity field\. The probability transport is governed by the continuity equation

∂tpt​\(xt\)\+∇⋅\(pt​\(xt\)​vt​\(xt\)\)=0,p0=𝒩⁡\(0,I\)\.\\partial\_\{t\}p\_\{t\}\(x\_\{t\}\)\+\\nabla\\cdot\(p\_\{t\}\(x\_\{t\}\)v\_\{t\}\(x\_\{t\}\)\)=0,\\qquad p\_\{0\}=\\mathcal\{N\}\(0,I\)\.\(2\)
For ODE sampling, given a velocity fieldvtv\_\{t\}, it is convenient to define flow maps over the time interval\[0,1\]\[0,1\]\. Fort1,t2∈\[0,1\]t\_\{1\},t\_\{2\}\\in\[0,1\], the flowΦt1→t2v:ℝd→ℝd\\Phi\_\{t\_\{1\}\\to t\_\{2\}\}^\{v\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}carries a state at timet1t\_\{1\}to the corresponding state at timet2t\_\{2\}along the ODE:

dd​s​Φt1→sv​\(x\)=vs​\(Φt1→sv​\(x\)\),\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\Phi\_\{t\_\{1\}\\to s\}^\{v\}\(x\)=v\_\{s\}\\left\(\\Phi\_\{t\_\{1\}\\to s\}^\{v\}\(x\)\\right\),Φt1→t1v​\(x\)=x\.\\displaystyle\\Phi\_\{t\_\{1\}\\to t\_\{1\}\}^\{v\}\(x\)=x\.Botht1≤t2t\_\{1\}\\leq t\_\{2\}andt1\>t2t\_\{1\}\>t\_\{2\}are allowed, corresponding to traversing the ODE solution forward and backward in time, respectively\. Under standard well\-posedness assumptions, the flow maps satisfyΦt1→t2v=\(Φt2→t1v\)−1\\Phi\_\{t\_\{1\}\\to t\_\{2\}\}^\{v\}=\\left\(\\Phi\_\{t\_\{2\}\\to t\_\{1\}\}^\{v\}\\right\)^\{\-1\}\.

#### 2\.1\.3Conditional Flow Matching

Although infinitely many velocity fields can transportp0p\_\{0\}top1p\_\{1\}, they are generally not available in closed form\. Flow matching provides a scalable way to learn one such field by making use of conditional velocity fieldsvt\|Z​\(x\|Z\)v\_\{t\|Z\}\(x\|Z\)\. For two\-sided conditioning,

vt\|0,1​\(xt\|x0,x1\)=α˙t​x1\+β˙t​x0\.v\_\{t\|0,1\}\(x\_\{t\}\|x\_\{0\},x\_\{1\}\)=\\dot\{\\alpha\}\_\{t\}x\_\{1\}\+\\dot\{\\beta\}\_\{t\}x\_\{0\}\.For one\-sided conditioning andt∈\[0,1\)t\\in\[0,1\),

vt\|1​\(xt\|x1\)=α˙t​x1\+β˙t​xt−αt​x1βt=β˙tβt​xt\+α˙t​βt−β˙t​αtβt​x1\.v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\dot\{\\alpha\}\_\{t\}x\_\{1\}\+\\dot\{\\beta\}\_\{t\}\\frac\{x\_\{t\}\-\\alpha\_\{t\}x\_\{1\}\}\{\\beta\_\{t\}\}=\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\+\\frac\{\\dot\{\\alpha\}\_\{t\}\\beta\_\{t\}\-\\dot\{\\beta\}\_\{t\}\\alpha\_\{t\}\}\{\\beta\_\{t\}\}x\_\{1\}\.They lead to the same results, though in different contexts one may be more convenient than the other for ease of presentation\.

Given a parameterized velocity fieldvtθ​\(xt\)v\_\{t\}^\{\\theta\}\(x\_\{t\}\), the conditional flow matching \(CFM\) loss is defined as

ℒCFM​\(θ\)=\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)=𝔼t∼U\(0,1\),X1∼p1,Xt∼pt\|1\(⋅\|X1\)​\[‖vtθ​\(Xt\)−vt\|1​\(Xt\|X1\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim p\_\{1\},\\\\ X\_\{t\}\\sim p\_\{t\|1\}\(\\cdot\|X\_\{1\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]=\\displaystyle=𝔼t∼U\(0,1\),X1∼p1,X0∼p0,Xt∼pt\|0,1\(⋅\|X0,X1\)​\[‖vtθ​\(Xt\)−vt\|0,1​\(Xt\|X0,X1\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim p\_\{1\},\\,X\_\{0\}\\sim p\_\{0\},\\\\ X\_\{t\}\\sim p\_\{t\|0,1\}\(\\cdot\|X\_\{0\},X\_\{1\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|0,1\}\(X\_\{t\}\|X\_\{0\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.\(3\)Its population minimizer is the conditional expectation

vt​\(xt\)=\\displaystyle v\_\{t\}\(x\_\{t\}\)=𝔼⁡\[vt\|0,1​\(Xt\|X0,X1\)\|Xt=xt\]=𝔼⁡\[α˙t​X1\+β˙t​X0\|Xt=xt\],\\displaystyle\\mathbb\{E\}\\left\[v\_\{t\|0,1\}\(X\_\{t\}\|X\_\{0\},X\_\{1\}\)\\middle\|X\_\{t\}=x\_\{t\}\\right\]=\\mathbb\{E\}\\left\[\\dot\{\\alpha\}\_\{t\}X\_\{1\}\+\\dot\{\\beta\}\_\{t\}X\_\{0\}\\middle\|X\_\{t\}=x\_\{t\}\\right\],\(4\)which is one specific velocity field that can transportp0p\_\{0\}top1p\_\{1\}\.

#### 2\.1\.4Stochastic Dynamics

When sampling with stochastic dynamics, we use the SDE

dYt=\(vt\(Yt\)\+σt22∇logpt\(Yt\)\)dt\+σtdWt,Y0∼p0,\\mathrm\{d\}Y\_\{t\}=\\left\(v\_\{t\}\(Y\_\{t\}\)\+\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\\nabla\\log p\_\{t\}\(Y\_\{t\}\)\\right\)\\mathrm\{d\}t\+\\sigma\_\{t\}\\mathrm\{d\}W\_\{t\},\\qquad Y\_\{0\}\\sim p\_\{0\},\(5\)whereWtW\_\{t\}denotes standard Brownian motion andσt\\sigma\_\{t\}is a scalar noise schedule\. The evolution of the density is governed by the Fokker–Planck equation

∂tpt​\(xt\)\+∇⋅\(pt​\(xt\)​bt​\(xt\)\)=σt22​Δ​pt​\(xt\),p0=𝒩⁡\(0,I\),\\partial\_\{t\}p\_\{t\}\(x\_\{t\}\)\+\\nabla\\cdot\\left\(p\_\{t\}\(x\_\{t\}\)b\_\{t\}\(x\_\{t\}\)\\right\)=\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\\Delta p\_\{t\}\(x\_\{t\}\),\\qquad p\_\{0\}=\\mathcal\{N\}\(0,I\),wherebt​\(xt\)=vt​\(xt\)\+σt22​∇xt​log⁡pt​\(xt\)b\_\{t\}\(x\_\{t\}\)=v\_\{t\}\(x\_\{t\}\)\+\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}\(x\_\{t\}\)denotes the drift\. The SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) and the ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) produce the same marginal probability path\{pt\}t∈\[0,1\]\\\{p\_\{t\}\\\}\_\{t\\in\[0,1\]\}for any choice ofσt≥0\\sigma\_\{t\}\\geq 0\.

#### 2\.1\.5Equivalent Predictions

Diffusion and flow models admit equivalent predictions in terms of velocity, score, drift, data, and noise under standard conditions\. The relation between velocity and score provides one example\. For the Gaussian source and linear interpolant, the score∇xt​log​pt​\(xt\)\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}\(x\_\{t\}\)satisfies

∇xt​log​pt​\(xt\)=−1βt​𝔼​\[X0\|Xt=xt\]\.\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}\(x\_\{t\}\)=\-\\frac\{1\}\{\\beta\_\{t\}\}\\mathbb\{E\}\\left\[X\_\{0\}\\middle\|X\_\{t\}=x\_\{t\}\\right\]\.Together with the interpolant identity

xt=αt​𝔼​\[X1\|Xt=xt\]\+βt​𝔼​\[X0\|Xt=xt\],x\_\{t\}=\\alpha\_\{t\}\\mathbb\{E\}\\left\[X\_\{1\}\\middle\|X\_\{t\}=x\_\{t\}\\right\]\+\\beta\_\{t\}\\mathbb\{E\}\\left\[X\_\{0\}\\middle\|X\_\{t\}=x\_\{t\}\\right\],it suffices to know either the velocity or the score, since one can be derived from the other\. This observation extends to other prediction forms as well\. In this paper, we focus on the standard flow models \(i\.e\., predicting the velocity\); extensions of our method to other prediction forms are provided in Appendix[H](https://arxiv.org/html/2609.05727#A8)\.

### 2\.2Differential Geometry on Banach Manifolds

This subsection briefly reviews the differential geometry of Banach manifolds\[[31](https://arxiv.org/html/2609.05727#bib.bib48)\]\. Most notations used in this review are generic and should be understood as local to this subsection\.

#### 2\.2\.1Banach Space

A real Banach space𝔹\\mathbb\{B\}is a normed vector space that is complete under its norm‖⋅‖𝔹\\left\\\|\\cdot\\right\\\|\_\{\\mathbb\{B\}\}\. The Banach space𝔹\\mathbb\{B\}may be of finite or infinite dimension\.

Consider an open subset𝒪𝔹⊂𝔹\\mathcal\{O\}\_\{\\mathbb\{B\}\}\\subset\\mathbb\{B\}and another Banach space𝔸\\mathbb\{A\}\. A mapF:𝒪𝔹→𝔸F:\\mathcal\{O\}\_\{\\mathbb\{B\}\}\\to\\mathbb\{A\}is Fréchet differentiable atx∈𝒪𝔹x\\in\\mathcal\{O\}\_\{\\mathbb\{B\}\}if there exists a continuous linear mapDx​F:𝔹→𝔸\\mathrm\{D\}\_\{x\}F:\\mathbb\{B\}\\to\\mathbb\{A\}such that

F⁡\(x\+h\)=F⁡\(x\)\+Dx​F​\[h\]\+o⁡\(‖h‖𝔹\)F\(x\+h\)=F\(x\)\+\\mathrm\{D\}\_\{x\}F\[h\]\+o\(\\left\\\|h\\right\\\|\_\{\\mathbb\{B\}\}\)ash→0h\\to 0\. The mapDx​F\\mathrm\{D\}\_\{x\}Fis called the Fréchet derivative ofFFatxx\.

Higher\-order Fréchet derivatives are defined recursively by differentiating the derivative mapx↦Dx​Fx\\mapsto\\mathrm\{D\}\_\{x\}F\. A map is called smooth if Fréchet derivatives of all orders exist and are continuous\.

#### 2\.2\.2Banach Manifold

A smooth Banach manifoldℳ\\mathscr\{M\}, modeled on a Banach space𝔹\\mathbb\{B\}, is a Hausdorff topological space equipped with a collection of charts whose domains coverℳ\\mathscr\{M\}\. A chart is a pair\(U,φ\)\(U,\\varphi\), whereU⊂ℳU\\subset\\mathscr\{M\}is open andφ:U→φ⁡\(U\)⊂𝔹\\varphi:U\\to\\varphi\(U\)\\subset\\mathbb\{B\}is a homeomorphism onto an open subsetφ⁡\(U\)⊂𝔹\\varphi\(U\)\\subset\\mathbb\{B\}\.φ⁡\(p\)\\varphi\(p\)is the coordinate representation ofpp\.

Whenever two charts\(U,φ\)\(U,\\varphi\)and\(V,ψ\)\(V,\\psi\)overlap, their coordinate\-change maps

ψ∘φ−1\|φ⁡\(U∩V\):φ⁡\(U∩V\)→ψ⁡\(U∩V\),φ∘ψ−1\|ψ⁡\(U∩V\):ψ⁡\(U∩V\)→φ⁡\(U∩V\)\\psi\\circ\\left\.\\varphi^\{\-1\}\\right\|\_\{\\varphi\(U\\cap V\)\}:\\varphi\(U\\cap V\)\\to\\psi\(U\\cap V\),\\qquad\\varphi\\circ\\left\.\\psi^\{\-1\}\\right\|\_\{\\psi\(U\\cap V\)\}:\\psi\(U\\cap V\)\\to\\varphi\(U\\cap V\)are smooth in the Fréchet sense\.

Letℳ\\mathscr\{M\}and𝒩\\mathscr\{N\}be smooth Banach manifolds\. A mapF:ℳ→𝒩F:\\mathscr\{M\}\\to\\mathscr\{N\}is called smooth if, for everyp∈ℳp\\in\\mathscr\{M\}, there exist a chart\(U,φ\)\(U,\\varphi\)ofℳ\\mathscr\{M\}aroundppand a chart\(V,ψ\)\(V,\\psi\)of𝒩\\mathscr\{N\}aroundF⁡\(p\)F\(p\), withF⁡\(U\)⊂VF\(U\)\\subset V, such that the coordinate representation

ψ∘F∘φ−1:φ⁡\(U\)→ψ⁡\(V\)\\psi\\circ F\\circ\\varphi^\{\-1\}:\\varphi\(U\)\\to\\psi\(V\)is smooth in the Fréchet sense\.

A smooth bijectionF:ℳ→𝒩F:\\mathscr\{M\}\\to\\mathscr\{N\}whose inverseF−1:𝒩→ℳF^\{\-1\}:\\mathscr\{N\}\\to\\mathscr\{M\}is also smooth is called a diffeomorphism, denoted byF:ℳ→∼𝒩F:\\mathscr\{M\}\\xrightarrow\{\\sim\}\\mathscr\{N\}orℳ≅𝒩\\mathscr\{M\}\\cong\\mathscr\{N\}\. Diffeomorphic manifolds have equivalent smooth structures\. In particular, their tangent vectors and the geometric structures introduced below can be transported through the differential ofFF\.

A subset𝒮⊂ℳ\\mathscr\{S\}\\subset\\mathscr\{M\}is called a split embedded Banach submanifold if, for everyp∈𝒮p\\in\\mathscr\{S\}, there exist a chart\(U,φ\)\(U,\\varphi\)ofℳ\\mathscr\{M\}aroundppand a closed complemented linear subspace𝔹0⊂𝔹\\mathbb\{B\}\_\{0\}\\subset\\mathbb\{B\}such that

φ⁡\(U∩𝒮\)=φ⁡\(U\)∩𝔹0\.\\varphi\(U\\cap\\mathscr\{S\}\)=\\varphi\(U\)\\cap\\mathbb\{B\}\_\{0\}\.

#### 2\.2\.3Tangent Space

A tangent vector atp∈ℳp\\in\\mathscr\{M\}represents a first\-order perturbation ofppthat remains in the manifold\. Consider a smooth curve

\(pε\)ε∈\(−δ,δ\)⊂ℳ,p0=p\(p^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{M\},\\qquad p^\{0\}=pfor someδ\>0\\delta\>0\. In a chart\(U,φ\)\(U,\\varphi\)aroundpp, two smooth curves\(pε\)ε∈\(−δ,δ\)\(p^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}and\(qε\)ε∈\(−δ,δ\)\(q^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}throughppare said to represent the same tangent vector if

dd​ε​φ​\(pε\)\|ε=0=dd​ε​φ​\(qε\)\|ε=0∈𝔹\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\varphi\(p^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\varphi\(q^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\}\\in\\mathbb\{B\}\.This equivalence relation is independent of the selected chart\.

A tangent vector is an equivalence class of such smooth curves\. The tangent vector represented by\(pε\)ε∈\(−δ,δ\)\(p^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}is denoted by

ξ=dd​ε​pε\|ε=0,\\xi=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}p^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\},where the derivative is understood through any local chart as above\.

The tangent space ofℳ\\mathscr\{M\}atppis defined as the collection of all tangent vectors atpp, denoted by

Tpℳ=\{dd​εpε\|ε=0:\(pε\)ε∈\(−δ,δ\)⊂ℳsmooth,p0=p,δ\>0\}\.T\_\{p\}\\mathscr\{M\}=\\left\\\{\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}p^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}:\(p^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{M\}\\text\{ smooth\},\\,p^\{0\}=p,\\,\\delta\>0\\right\\\}\.Herep∈ℳp\\in\\mathscr\{M\}is called the base point\. Every chart aroundppidentifiesTp​ℳT\_\{p\}\\mathscr\{M\}with the Banach model space𝔹\\mathbb\{B\}\. In particular,Tp​ℳT\_\{p\}\\mathscr\{M\}is itself a Banach space:

Tp​ℳ≅𝔹\.T\_\{p\}\\mathscr\{M\}\\cong\\mathbb\{B\}\.
LetF:ℳ→𝒩F:\\mathscr\{M\}\\to\\mathscr\{N\}be smooth\. The differential ofFFatp∈ℳp\\in\\mathscr\{M\}is the continuous linear map

dp​F:Tp​ℳ→TF⁡\(p\)​𝒩,dp​F​\[ξ\]:=dd​ε​F​\(pε\)\|ε=0,\\mathrm\{d\}\_\{p\}F:T\_\{p\}\\mathscr\{M\}\\to T\_\{F\(p\)\}\\mathscr\{N\},\\qquad\\mathrm\{d\}\_\{p\}F\[\\xi\]:=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}F\(p^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\},where\(pε\)ε∈\(−δ,δ\)\(p^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}is any smooth curve satisfyingp0=pp^\{0\}=panddd​ε​pε\|ε=0=ξ\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}p^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}=\\xi\. The valuedp​F​\[ξ\]\\mathrm\{d\}\_\{p\}F\[\\xi\]is independent of the selected path representingξ\\xi\. Thus, the differential maps a first\-order perturbation ofppto the corresponding first\-order perturbation ofF⁡\(p\)F\(p\)\.

Throughout this paper, lowercasedp​F\\mathrm\{d\}\_\{p\}Fdenotes the intrinsic differential of a smooth map between Banach manifolds, whereas uppercaseDx​F\\mathrm\{D\}\_\{x\}Fdenotes the Fréchet derivative of a map between Banach spaces, or of a coordinate representation after charts have been selected\. More precisely, for charts\(U,φ\)\(U,\\varphi\)aroundppand\(V,ψ\)\(V,\\psi\)aroundF⁡\(p\)F\(p\), the two notations are related by

Dφ⁡\(p\)​\(ψ∘F∘φ−1\)=dF⁡\(p\)​ψ∘dp​F∘\(dp​φ\)−1\.\\mathrm\{D\}\_\{\\varphi\(p\)\}\\left\(\\psi\\circ F\\circ\\varphi^\{\-1\}\\right\)=\\mathrm\{d\}\_\{F\(p\)\}\\psi\\circ\\mathrm\{d\}\_\{p\}F\\circ\\left\(\\mathrm\{d\}\_\{p\}\\varphi\\right\)^\{\-1\}\.Thus,dp​F\\mathrm\{d\}\_\{p\}Fis intrinsic and independent of the selected charts, while the Fréchet derivative on the left is taken in the corresponding Banach coordinates\. When the manifolds are open subsets of Banach spaces, the two are identified through the canonical identifications of their tangent spaces\.

A tangent vector fieldXXonℳ\\mathscr\{M\}is a smoothly varying assignment

p↦Xp∈Tp​ℳ\.p\\mapsto X\_\{p\}\\in T\_\{p\}\\mathscr\{M\}\.The space of smooth tangent vector fields onℳ\\mathscr\{M\}is denoted by𝔛⁡\(ℳ\)\\mathfrak\{X\}\(\\mathscr\{M\}\)\. In summary, a tangent vector describes one perturbation direction at one point, whereas a tangent vector field specifies a perturbation direction at every point of the manifold\.

#### 2\.2\.4Cotangent Space

The cotangent space ofℳ\\mathscr\{M\}atppis the continuous dual of the tangent space:

Tp∗ℳ:=\(Tpℳ\)∗=\{continuous linear functionalTpℳ→ℝ\}\.T\_\{p\}^\{\*\}\\mathscr\{M\}:=\(T\_\{p\}\\mathscr\{M\}\)^\{\*\}=\\left\\\{\\text\{continuous linear functional \}T\_\{p\}\\mathscr\{M\}\\to\\mathbb\{R\}\\right\\\}\.
An elementℓ∈Tp∗​ℳ\\ell\\in T\_\{p\}^\{\*\}\\mathscr\{M\}is called a cotangent vector\. Its action on a tangent vectorξ∈Tp​ℳ\\xi\\in T\_\{p\}\\mathscr\{M\}is denoted byℓ⁡\[ξ\]∈ℝ\\ell\[\\xi\]\\in\\mathbb\{R\}\.

LetJ:ℳ→ℝJ:\\mathscr\{M\}\\to\\mathbb\{R\}be a smooth objective\. Its differential atppis the cotangent vector

dp​J∈Tp∗​ℳ,dp​J​\[ξ\]:=dd​ε​J​\(pε\)\|ε=0,\\mathrm\{d\}\_\{p\}J\\in T\_\{p\}^\{\*\}\\mathscr\{M\},\\qquad\\mathrm\{d\}\_\{p\}J\[\\xi\]:=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}J\(p^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\},where\(pε\)ε∈\(−δ,δ\)\(p^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}is any smooth curve satisfyingp0=pp^\{0\}=panddd​ε​pε\|ε=0=ξ\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}p^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}=\\xi\. The valuedp​J​\[ξ\]\\mathrm\{d\}\_\{p\}J\[\\xi\]is independent of the selected path representingξ\\xi\. It describes the first\-order change of the objectiveJJalong the tangent directionξ\\xi\.

For a tangent vector fieldX∈𝔛⁡\(ℳ\)X\\in\\mathfrak\{X\}\(\\mathscr\{M\}\), the directional derivative ofJJalongXXis defined by

X⁡\[J\]:ℳ→ℝ,X⁡\[J\]​\(p\):=dp​J​\[Xp\]\.X\[J\]:\\mathscr\{M\}\\to\\mathbb\{R\},\\qquad X\[J\]\(p\):=\\mathrm\{d\}\_\{p\}J\[X\_\{p\}\]\.Consider a smooth curve\(pε\)ε∈\(−δ,δ\)⊂ℳ\(p^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{M\}which passes throughppatε=0\\varepsilon=0and satisfiesXp=dd​ε​pε\|ε=0X\_\{p\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}p^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}\. Then, we can also write

X⁡\[J\]​\(p\)=dp​J​\[Xp\]=dd​ε​J​\(pε\)\|ε=0\.X\[J\]\(p\)=\\mathrm\{d\}\_\{p\}J\[X\_\{p\}\]=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}J\(p^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\}\.

#### 2\.2\.5Riemannian Metric and Gradient

A Riemannian metric onℳ\\mathscr\{M\}is a smoothly varying family of continuous, symmetric, and positive\-definite bilinear forms

gp:Tp​ℳ×Tp​ℳ→ℝ,p∈ℳ\.g\_\{p\}:T\_\{p\}\\mathscr\{M\}\\times T\_\{p\}\\mathscr\{M\}\\to\\mathbb\{R\},\\qquad p\\in\\mathscr\{M\}\.The metric induces a continuous linear map

gp♭:Tpℳ→Tp∗ℳ,gp♭\(ξ\)\[ζ\]:=gp\(ξ,ζ\),ξ,ζ∈Tpℳ,g\_\{p\}^\{\\flat\}:T\_\{p\}\\mathscr\{M\}\\to T\_\{p\}^\{\*\}\\mathscr\{M\},\\qquad g\_\{p\}^\{\\flat\}\(\\xi\)\[\\zeta\]:=g\_\{p\}\(\\xi,\\zeta\),\\qquad\\xi,\\zeta\\in T\_\{p\}\\mathscr\{M\},which is called the flat map\. Positive definiteness implies thatgp♭g\_\{p\}^\{\\flat\}is injective\. Its inverse is defined as

\(gp♭\)−1:Im⁡\(gp♭\)→Tp​ℳ\.\(g\_\{p\}^\{\\flat\}\)^\{\-1\}:\\operatorname\{Im\}\(g\_\{p\}^\{\\flat\}\)\\to T\_\{p\}\\mathscr\{M\}\.For a cotangent vectorℓ∈Im⁡\(gp♭\)\\ell\\in\\mathrm\{Im\}\(g\_\{p\}^\{\\flat\}\), the unique tangent vector\(gp♭\)−1​\(ℓ\)∈Tp​ℳ\(g\_\{p\}^\{\\flat\}\)^\{\-1\}\(\\ell\)\\in T\_\{p\}\\mathscr\{M\}is called the metric dual ofℓ\\ell\. The metric is called strong ifgp♭:Tp​ℳ→Tp∗​ℳg\_\{p\}^\{\\flat\}:T\_\{p\}\\mathscr\{M\}\\to T\_\{p\}^\{\*\}\\mathscr\{M\}is a Banach\-space isomorphism for everyp∈ℳp\\in\\mathscr\{M\}; otherwise, it is called weak\. For a weak Riemannian metric, it may hold thatIm⁡\(gp♭\)⊊Tp∗​ℳ;\\mathrm\{Im\}\(g\_\{p\}^\{\\flat\}\)\\subsetneq T\_\{p\}^\{\*\}\\mathscr\{M\};in this case, not every cotangent vector has a metric\-dual tangent representation\.

LetJ:ℳ→ℝJ:\\mathscr\{M\}\\to\\mathbb\{R\}be a smooth objective\. Wheneverdp​J∈gp♭​\(Tp​ℳ\)\\mathrm\{d\}\_\{p\}J\\in g\_\{p\}^\{\\flat\}\(T\_\{p\}\\mathscr\{M\}\), the gradient ofJJatppis the unique tangent vectorgradg⁡J⁡\(p\)∈Tp​ℳ\\operatorname\{grad\}\_\{g\}J\(p\)\\in T\_\{p\}\\mathscr\{M\}satisfying

gp​\(gradg⁡J⁡\(p\),ξ\)=dp​J​\[ξ\]g\_\{p\}\\bigl\(\\operatorname\{grad\}\_\{g\}J\(p\),\\xi\\bigr\)=\\mathrm\{d\}\_\{p\}J\[\\xi\]for everyξ∈Tp​ℳ\\xi\\in T\_\{p\}\\mathscr\{M\}\.

Thus, the differentialdp​J\\mathrm\{d\}\_\{p\}Jcontains the intrinsic first\-order information of the objective, whereas the gradient is its tangent\-vector representation under the selected metric\. For a weak Riemannian metric, the gradient need not exist for every smooth objective, and its existence must be verified in the setting under consideration\.

#### 2\.2\.6Affine Connection

Tangent vectors at different points belong to different tangent spaces and therefore cannot be directly subtracted\. An affine connection provides a rule for differentiating one tangent vector field along another\.

An affine connection onℳ\\mathscr\{M\}is a local covariant derivative on the tangent bundle\. In global\-section notation, it is written as a map

∇:𝔛⁡\(ℳ\)×𝔛⁡\(ℳ\)→𝔛⁡\(ℳ\),\(X,Y\)↦∇XY,\\nabla:\\mathfrak\{X\}\(\\mathscr\{M\}\)\\times\\mathfrak\{X\}\(\\mathscr\{M\}\)\\to\\mathfrak\{X\}\(\\mathscr\{M\}\),\\qquad\(X,Y\)\\mapsto\\nabla\_\{X\}Y,satisfying

∇f​X\+h​YZ=f​∇XZ\+h​∇YZ,\\displaystyle\\nabla\_\{fX\+hY\}Z=f\\nabla\_\{X\}Z\+h\\nabla\_\{Y\}Z,\(6a\)∇X\(a​Y\+b​Z\)=a​∇XY\+b​∇XZ,\\displaystyle\\nabla\_\{X\}\(aY\+bZ\)=a\\nabla\_\{X\}Y\+b\\nabla\_\{X\}Z,\(6b\)∇X\(f​Y\)=X⁡\[f\]​Y\+f​∇XY,\\displaystyle\\nabla\_\{X\}\(fY\)=X\[f\]Y\+f\\nabla\_\{X\}Y,\(6c\)for all tangent vector fieldsX,Y,Z∈𝔛⁡\(ℳ\)X,Y,Z\\in\\mathfrak\{X\}\(\\mathscr\{M\}\), smooth scalar functionsf,h:ℳ→ℝf,h:\\mathscr\{M\}\\to\\mathbb\{R\}, and constantsa,b∈ℝa,b\\in\\mathbb\{R\}\. The last equation is the Leibniz rule in its second argument\. The same notation is used for vector fields defined on an open subsetU⊂ℳU\\subset\\mathscr\{M\}, compatibly with restriction to smaller open subsets\.

Note that, for consistency with the differential\-geometry literature, we use∇\\nablato denote an affine connection\. The same symbol denotes the usual gradient operator when applied to scalar functions onℝd\\mathbb\{R\}^\{d\}; its meaning is clear from context\.

#### 2\.2\.7Covariant Hessian

LetJ:ℳ→ℝJ:\\mathscr\{M\}\\to\\mathbb\{R\}be a smooth objective, and let∇\\nablabe an affine connection onℳ\\mathscr\{M\}\. The covariant Hessian ofJJatp∈ℳp\\in\\mathscr\{M\}is the bilinear form

Hess∇⁡Jp:Tp​ℳ×Tp​ℳ→ℝ,Hess∇⁡Jp​\(ξ,ζ\):=X⁡\[Y⁡\[J\]\]​\(p\)−dp​J​\[\(∇XY\)p\],\\operatorname\{Hess\}^\{\\nabla\}J\_\{p\}:T\_\{p\}\\mathscr\{M\}\\times T\_\{p\}\\mathscr\{M\}\\to\\mathbb\{R\},\\qquad\\operatorname\{Hess\}^\{\\nabla\}J\_\{p\}\(\\xi,\\zeta\):=X\[Y\[J\]\]\(p\)\-\\mathrm\{d\}\_\{p\}J\\bigl\[\(\\nabla\_\{X\}Y\)\_\{p\}\\bigr\],whereX,YX,Yare arbitrary smooth local tangent vector fields satisfyingXp=ξX\_\{p\}=\\xi,Yp=ζY\_\{p\}=\\zeta\. The valueHess∇⁡Jp​\(ξ,ζ\)\\operatorname\{Hess\}^\{\\nabla\}J\_\{p\}\(\\xi,\\zeta\)is independent of the selected extensionsXXandYY\. The covariant Hessian can equivalently be viewed as a continuous linear operator

Hess∇⁡Jp:Tp​ℳ→Tp∗​ℳ,\(Hess∇⁡Jp​\[ξ\]\)​\[ζ\]:=Hess∇⁡Jp​\(ξ,ζ\)\.\\operatorname\{Hess\}^\{\\nabla\}J\_\{p\}:T\_\{p\}\\mathscr\{M\}\\to T\_\{p\}^\{\*\}\\mathscr\{M\},\\qquad\\bigl\(\\operatorname\{Hess\}^\{\\nabla\}J\_\{p\}\[\\xi\]\\bigr\)\[\\zeta\]:=\\operatorname\{Hess\}^\{\\nabla\}J\_\{p\}\(\\xi,\\zeta\)\.For a fixed objectiveJJ, the covariant Hessian depends on the selected affine connection, does not require a Riemannian metric, and need not be symmetric in general\. For brevity, we refer toHess∇⁡J\\operatorname\{Hess\}^\{\\nabla\}Jsimply as the Hessian ofJJthroughout this paper\.

## 3Problem Formulation

### 3\.1Sampling and Fine\-Tuning

In this paper, we consider the following target density:

πμ,τ,r​\(x\):=μ⁡\(x\)​eτ​r​\(x\)∫ℝdμ⁡\(z\)​eτ​r​\(z\)​𝑑z∝μ⁡\(x\)​eτ​r​\(x\),\\pi\_\{\\mu,\\tau,r\}\(x\):=\\frac\{\\mu\(x\)e^\{\\tau r\(x\)\}\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\mu\(z\)e^\{\\tau r\(z\)\}\\mathrm\{d\}z\}\\propto\\mu\(x\)e^\{\\tau r\(x\)\},whereμ:ℝd→\(0,∞\)\\mu:\\mathbb\{R\}^\{d\}\\to\(0,\\infty\)is a reference factor,r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}is a reward function, andτ\>0\\tau\>0is an inverse temperature\. The normalizing constant satisfies

0<∫ℝdμ⁡\(z\)​eτ​r​\(z\)​𝑑z<∞\.0<\\int\_\{\\mathbb\{R\}^\{d\}\}\\mu\(z\)e^\{\\tau r\(z\)\}\\mathrm\{d\}z<\\infty\.The additional regularity assumptions required by the subsequent analysis are stated in Appendix[A](https://arxiv.org/html/2609.05727#A1)\. The reference factorμ\\muhas two cases of interest:

- •μ⁡\(x\)=1\\mu\(x\)=1, which corresponds to sampling from the Boltzmann target density π1,τ,r​\(x\)∝exp⁡\(τ​r​\(x\)\)\.\\pi\_\{1,\\tau,r\}\(x\)\\propto\\exp\(\\tau r\(x\)\)\.
- •μ​\(x\)=ρbase​\(x\)\\mu\(x\)=\\rho^\{\\mathrm\{base\}\}\(x\), whereρbase\\rho^\{\\mathrm\{base\}\}is the terminal density of a base modelvtbasev\_\{t\}^\{\\mathrm\{base\}\}trained with standard diffusion or flow losses\. This corresponds to fine\-tuning the base model to sample from the tilted density πρbase,τ,r​\(x\)∝ρbase​\(x\)​eτ​r​\(x\)\.\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\(x\)\\propto\\rho^\{\\mathrm\{base\}\}\(x\)e^\{\\tau r\(x\)\}\.Typically, we have access to the base model, but its terminal densityρbase\\rho^\{\\mathrm\{base\}\}is implicit and not directly available\. We can sample fromρbase\\rho^\{\\mathrm\{base\}\}by solving the ODE or SDE induced byvtbasev\_\{t\}^\{\\mathrm\{base\}\}\.

Our goal is to train a diffusion or flow model to sample from the target densityπμ,τ,r\\pi\_\{\\mu,\\tau,r\}, thereby covering both sampling from unnormalized densities and fine\-tuning pretrained models\. In both cases, the task is distinct from standard generative modeling, where samples from the target density are directly available, and the model can be trained via supervised learning\. In contrast, here we only have access to the reward functionr⁡\(x\)r\(x\), and possibly its gradient whenrris differentiable and the algorithm requires it\. The model must therefore be trained by leveraging the reward signal, which places the problem in the reinforcement learning \(RL\) setting\.

Existing works often treat the two casesμ=1\\mu=1andμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}separately, with different algorithms developed for each\. In this paper, we propose a unified approach that handles both cases simultaneously\. We present our method primarily in the standard velocity\-prediction formulation of flow models, but the theory and algorithms also apply to other commonly used prediction forms\. We illustrate extensions to score and drift forms in Appendix[H](https://arxiv.org/html/2609.05727#A8)\.

### 3\.2Optimization on Velocity Fields

Given a velocity fieldvv, the ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) transports the source densityp0p\_\{0\}to a terminal densityρv:=p1v\\rho^\{v\}:=p\_\{1\}^\{v\}\. Specifically,ρv\\rho^\{v\}is the pushforward ofp0p\_\{0\}under the flow mapΦ0→1v\\Phi\_\{0\\to 1\}^\{v\}:

ρv=\(Φ0→1v\)\#​p0\.\\rho^\{v\}=\\left\(\\Phi\_\{0\\to 1\}^\{v\}\\right\)\_\{\\\#\}p\_\{0\}\.
To align the terminal densityρv\\rho^\{v\}with the target densityπμ,τ,r\\pi\_\{\\mu,\\tau,r\}, we introduce an objective that measures the discrepancy between them\. Choosing the reverse KL divergence as the objective, we seek to solve

minv⁡KL⁡\(ρv∥πμ,τ,r\)\.\\min\_\{v\}\\,\\operatorname\{KL\}\\left\(\\rho^\{v\}\\middle\\\|\\pi\_\{\\mu,\\tau,r\}\\right\)\.\(7\)Thus, the task can be viewed as a*functional optimization*problem over the space of velocity fields\. Any velocity fieldv⋆v^\{\\star\}that transportsp0p\_\{0\}toπμ,τ,r\\pi\_\{\\mu,\\tau,r\}satisfiesρv⋆=πμ,τ,r\\rho^\{v^\{\\star\}\}=\\pi\_\{\\mu,\\tau,r\}and achieves the global minimumKL⁡\(ρv⋆∥πμ,τ,r\)=0\\operatorname\{KL\}\\left\(\\rho^\{v^\{\\star\}\}\\middle\\\|\\pi\_\{\\mu,\\tau,r\}\\right\)=0\.

Although optimization problem \([7](https://arxiv.org/html/2609.05727#S3.E7)\) provides a natural formulation of our task, it is not directly amenable to scalable optimization\. The dependence of the terminal densityρv\\rho^\{v\}on the velocity fieldvvis specified only implicitly through the associated continuity equation \([2](https://arxiv.org/html/2609.05727#S2.E2)\)\. Consequently, a local perturbation ofvvat an intermediate time can propagate through the entire flow and alterρv\\rho^\{v\}globally, making the mapv↦ρvv\\mapsto\\rho^\{v\}difficult to handle numerically\. More importantly, the terminal KL objective does not expose the conditional\-expectation structure that underlies the scalability of the flow\-matching framework\. Optimizing it naïvely would generally require lifting the terminal\-density objective to path space, constructing a suitable surrogate objective, and, most likely, backpropagating through the entire sampling trajectory\. This would forgo the central advantage of CFM, which transforms an intractable marginal objective into a tractable sample\-wise regression loss\. We therefore need to improve the formulation in \([7](https://arxiv.org/html/2609.05727#S3.E7)\) so that it admits CFM\-like optimization and leads to scalable algorithms\.

To develop such a formulation, we reinterpret the flow\-matching construction as a map from terminal densities to velocity fields\. We first observe that, once the source densityp0p\_\{0\}and the schedule\(αt,βt\)\(\\alpha\_\{t\},\\beta\_\{t\}\)are fixed, two central ingredients are determined independently of the terminal densityp1p\_\{1\}\. The first is the conditional probability path

pt\|1​\(xt\|x1\)=1βtd​p0​\(xt−αt​x1βt\),p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\frac\{1\}\{\\beta\_\{t\}^\{d\}\}p\_\{0\}\\left\(\\frac\{x\_\{t\}\-\\alpha\_\{t\}x\_\{1\}\}\{\\beta\_\{t\}\}\\right\),\(8\)and the second is the conditional velocity field

vt\|1​\(xt\|x1\)=β˙tβt​xt\+α˙t​βt−αt​β˙tβt​x1=β˙tβt​xt\+αt​κtβt2​x1\.v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\+\\frac\{\\dot\{\\alpha\}\_\{t\}\\beta\_\{t\}\-\\alpha\_\{t\}\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{1\}=\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\+\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}x\_\{1\}\.\(9\)Both depend only on the source densityp0p\_\{0\}and the selected schedule, not on the terminal densityp1p\_\{1\}\.

By contrast, several induced quantities depend on the terminal density\. Letρ\\rhobe a terminal density and setp1=ρp\_\{1\}=\\rhoin the flow\-matching construction\. To make this dependence explicit, we attachρ\\rhoas a superscript to the corresponding quantities\. The first is the marginal probability path

ptρ​\(xt\)=∫ℝdρ⁡\(x1\)​pt\|1​\(xt\|x1\)​d​x1\.p\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\,\\mathrm\{d\}x\_\{1\}\.The second is the posterior

p1\|tρ​\(x1\|xt\)=ρ⁡\(x1\)​pt\|1​\(xt\|x1\)ptρ​\(xt\)\.p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)=\\frac\{\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\.\(10\)The third is the marginal velocity field, which is the population minimizer of the CFM loss:

vtρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|X1\)\]\.v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]\.\(11\)By construction,vρv^\{\\rho\}transportsp0p\_\{0\}toρ\\rho\. Equivalently,vρv^\{\\rho\}andptρp\_\{t\}^\{\\rho\}satisfy the continuity equation

∂tptρ​\(xt\)\+∇xt⋅\(ptρ​\(xt\)​vtρ​\(xt\)\)=0,p0ρ=p0,p1ρ=ρ\.\\partial\_\{t\}p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\nabla\_\{x\_\{t\}\}\\cdot\\left\(p\_\{t\}^\{\\rho\}\(x\_\{t\}\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)=0,\\qquad p\_\{0\}^\{\\rho\}=p\_\{0\},\\qquad p\_\{1\}^\{\\rho\}=\\rho\.
Given a terminal densityρ\\rho, there are generally infinitely many velocity fields that transportp0p\_\{0\}toρ\\rho, as illustrated in Figure[2](https://arxiv.org/html/2609.05727#S3.F2)\. The flow\-matching construction selects the particular fieldvρv^\{\\rho\}through the conditional\-expectation representation \([11](https://arxiv.org/html/2609.05727#S3.E11)\)\. We refer to velocity fields obtained in this way as*canonical velocity fields*\. Their conditional\-expectation structure is especially valuable algorithmically: in the standard generative modeling setting, where samples fromρ\\rhoare available, it allowsvρv^\{\\rho\}to be learned through sample\-wise regression\.

In the tasks considered in this paper, we do not have direct access to samples from the target densityπμ,τ,r\\pi\_\{\\mu,\\tau,r\}\. Nevertheless, the target density determines a canonical velocity fieldvπμ,τ,rv^\{\\pi\_\{\\mu,\\tau,r\}\}\. Our goal is not merely to find any velocity field that transportsp0p\_\{0\}toπμ,τ,r\\pi\_\{\\mu,\\tau,r\}, but specifically to recover this canonical field\. Moreover, throughout the iterative optimization process, we require all intermediate velocity fields to remain canonical\. This leads to the constrained optimization problem

minv\\displaystyle\\min\_\{v\}KL⁡\(ρv∥πμ,τ,r\)\\displaystyle\\operatorname\{KL\}\\left\(\\rho^\{v\}\\middle\\\|\\pi\_\{\\mu,\\tau,r\}\\right\)\(12\)s\.t\.\\displaystyle\\text\{s\.t\.\}v​is a canonical velocity field\.\\displaystyle v\\text\{ is a canonical velocity field\}\.We restate this problem precisely in Section[6](https://arxiv.org/html/2609.05727#S6), after introducing the necessary ingredients\. In contrast to \([7](https://arxiv.org/html/2609.05727#S3.E7)\), the formulation \([12](https://arxiv.org/html/2609.05727#S3.E12)\) restricts the search to canonical velocity fields, each corresponding to a terminal densityρ\\rhothrough \([11](https://arxiv.org/html/2609.05727#S3.E11)\)\. This choice is both natural and essential: a canonical velocity field is accompanied by a well\-defined marginal probability path and posterior density, thereby preserving the conditional\-expectation structure required for CFM\-like optimization\.

t=0t=0t=1t=1p0=𝒩⁡\(0,I\)p\_\{0\}=\\mathcal\{N\}\(0,I\)ρ=p1vρ=p1v^\\rho=p\_\{1\}^\{v^\{\\rho\}\}=p\_\{1\}^\{\\hat\{v\}\}canonicalvρv^\{\\rho\}noncanonicalv^\\hat\{v\}Figure 2:Nonuniqueness of velocity representations for terminal densities\. The canonical velocity fieldvρv^\{\\rho\}and a distinct noncanonical velocity fieldv^\\hat\{v\}both transportp0=𝒩⁡\(0,I\)p\_\{0\}=\\mathcal\{N\}\(0,I\)to the same terminal densityρ\\rho\.This viewpoint motivates us to characterize this special class of velocity fields\. For fixedp0p\_\{0\}and\(αt,βt\)\(\\alpha\_\{t\},\\beta\_\{t\}\), each terminal densityρ\\rhoinduces a unique canonical velocity fieldvρv^\{\\rho\}, while each canonical velocity field determines its terminal density through its flow\. Thus, canonical velocity fields and terminal densities are in one\-to\-one correspondence\. Geometrically, the image of the mapρ↦vρ\\rho\\mapsto v^\{\\rho\}can be viewed as a*canonical manifold*embedded in the space of all velocity fields\. We next develop this geometric perspective, which forms the foundation of our Newton Matching framework\.

## 4Canonical Geometry

Section[3\.2](https://arxiv.org/html/2609.05727#S3.SS2)illustrates that a terminal density does not uniquely determine a velocity representation: the map from velocity fields to terminal densities is many\-to\-one\. The canonical construction resolves this ambiguity by selecting, for each terminal density, a distinguished velocity representative with the conditional\-expectation structure underlying scalable CFM training\.

This section develops the geometry of this canonical correspondence and equips the canonical manifold with the differential structures required for first\-order and second\-order optimization\. Section[4\.1](https://arxiv.org/html/2609.05727#S4.SS1)establishes the global geometry: the canonical velocity fields form an embedded submanifold of the velocity space, and the canonical\-velocity map is a diffeomorphism from the density manifold onto this submanifold\. The first\-order development in Section[4\.2](https://arxiv.org/html/2609.05727#S4.SS2)characterizes the tangent and cotangent spaces, constructs canonical lifts of density perturbations, and transports the Fisher–Rao metric to the canonical manifold\. These structures provide the admissible directions used by the canonical retraction in Section[5](https://arxiv.org/html/2609.05727#S5)and the metric used to derive the Fisher–Rao gradient in Section[6\.1](https://arxiv.org/html/2609.05727#S6.SS1)\. The second\-order development in Section[4\.3](https://arxiv.org/html/2609.05727#S4.SS3)characterizes the mixture connection through variations of Fisher–Rao pairings against fixed terminal observables and transports this connection to the canonical manifold\. It provides the covariant differentiation needed to define the Hessian and Newton direction in Section[6\.2](https://arxiv.org/html/2609.05727#S6.SS2)\. The Banach\-manifold realization and the regularity assumptions underlying these constructions are deferred to Appendix[A](https://arxiv.org/html/2609.05727#A1)\.

### 4\.1Global Geometry

Throughout this paper, the source densityp0p\_\{0\}and the interpolant schedule\(αt,βt\)t∈\[0,1\]\(\\alpha\_\{t\},\\beta\_\{t\}\)\_\{t\\in\[0,1\]\}are fixed\. Unless stated otherwise, Assumptions[67](https://arxiv.org/html/2609.05727#Thmtheorem67)and[69](https://arxiv.org/html/2609.05727#Thmtheorem69)are in force throughout the remainder of the paper\. Stronger regularity assumptions are invoked only for the particular first\-order or second\-order calculations that require them\.

This subsection develops the global geometry of the canonical construction\. We first formalize the relationship between velocity fields and terminal densities, and then use this relationship to define the canonical manifold and the canonical projection\. The resulting structure is summarized in Figure[3](https://arxiv.org/html/2609.05727#S4.F3)\.

As introduced in Section[3\.2](https://arxiv.org/html/2609.05727#S3.SS2), two objects are central to the present construction: the velocity field, which is the trainable object defining the sampling dynamics, and the terminal density induced by its flow\. The following definition specifies the spaces in which these objects live and the maps that relate them\.

###### Definition 1\.

The velocity space𝒱\\mathscr\{V\}is a Banach space satisfying

𝒱⊂\{v=\(vt\)t∈\[0,1\]:\[0,1\]×ℝd→ℝd\}\.\\mathscr\{V\}\\subset\\left\\\{v=\\left\(v\_\{t\}\\right\)\_\{t\\in\[0,1\]\}:\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}\\right\\\}\.
The density manifold𝒫\\mathscr\{P\}is a Banach manifold satisfying

𝒫⊂\{ρ∈C⁡\(ℝd,\(0,∞\)\):∫ℝdρ⁡\(x\)​𝑑x=1\}\.\\mathscr\{P\}\\subset\\left\\\{\\rho\\in C\\left\(\\mathbb\{R\}^\{d\};\(0,\\infty\)\\right\):\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(x\)\\,\\mathrm\{d\}x=1\\right\\\}\.
The terminal\-density map𝒯\\mathcal\{T\}transforms a velocity fieldv∈𝒱v\\in\\mathscr\{V\}to the terminal densityρv∈𝒫\\rho^\{v\}\\in\\mathscr\{P\}, i\.e\.,

𝒯:𝒱→𝒫,v↦ρv:=\(Φ0→1v\)\#​p0,\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{P\},\\qquad v\\mapsto\\rho^\{v\}:=\(\\Phi\_\{0\\to 1\}^\{v\}\)\_\{\\\#\}p\_\{0\},where\#\\\#denotes the pushforward operator\.

The canonical\-velocity map𝒞\\mathcal\{C\}is defined as

𝒞:𝒫→𝒱,ρ↦vρ,vtρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|X1\)\],\\mathcal\{C\}:\\mathscr\{P\}\\to\\mathscr\{V\},\\qquad\\rho\\mapsto v^\{\\rho\},\\qquad v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\],where the posterior densityp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)is given in \([10](https://arxiv.org/html/2609.05727#S3.E10)\)\.vρ∈𝒱v^\{\\rho\}\\in\\mathscr\{V\}is called the canonical velocity field of the terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}\.

Recall the nonuniqueness illustrated in Figure[2](https://arxiv.org/html/2609.05727#S3.F2): the terminal\-density map𝒯:𝒱→𝒫\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{P\}forgets the particular velocity representation and retains only its terminal density through its flow\. For a fixed densityρ∈𝒫\\rho\\in\\mathscr\{P\}, the preimage𝒯−1​\(ρ\)\\mathcal\{T\}^\{\-1\}\(\\rho\)generally consists of different velocity fields that transportp0p\_\{0\}toρ\\rho\. The canonical\-velocity map𝒞:𝒫→𝒱\\mathcal\{C\}:\\mathscr\{P\}\\to\\mathscr\{V\}selects the CFM representativevρ∈𝒱v^\{\\rho\}\\in\\mathscr\{V\}specified by the conditional\-expectation formula above\. The key compatibility property, as stated in Theorem[3](https://arxiv.org/html/2609.05727#Thmtheorem3), is that the canonical velocity fieldvρv^\{\\rho\}also realizes the terminal densityρ\\rho, i\.e\.,

𝒯⁡\(vρ\)=ρ\.\\mathcal\{T\}\(v^\{\\rho\}\)=\\rho\.
Having associated each terminal density with a distinguished canonical velocity field, we now regard the collection of all such fields as a geometric subset of the velocity space\. This subset serves as the intrinsic search space for Newton Matching\. We also introduce a map that replaces an arbitrary velocity field by the canonical representative associated with its terminal density\.

###### Definition 2\.

The canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is the collection of canonical velocity fields, i\.e\.,

𝒱can:=𝒞⁡\(𝒫\)⊂𝒱\.\\mathscr\{V\}\_\{\\mathrm\{can\}\}:=\\mathcal\{C\}\(\\mathscr\{P\}\)\\subset\\mathscr\{V\}\.
The canonical projectionℛ\\mathcal\{R\}is defined as

ℛ:=𝒞∘𝒯:𝒱→𝒱can,v↦vρv=𝒞⁡\(𝒯⁡\(v\)\)\.\\mathcal\{R\}:=\\mathcal\{C\}\\circ\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\},\\qquad v\\mapsto v^\{\\rho^\{v\}\}=\\mathcal\{C\}\(\\mathcal\{T\}\(v\)\)\.

By construction,𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}consists of all canonical velocity fields\. The compositionℛ=𝒞∘𝒯\\mathcal\{R\}=\\mathcal\{C\}\\circ\\mathcal\{T\}admits a simple two\-step interpretation\. Given an arbitrary velocity fieldv∈𝒱v\\in\\mathscr\{V\}, the terminal\-density map first extracts its terminal densityρv=𝒯⁡\(v\)\\rho^\{v\}=\\mathcal\{T\}\(v\); the canonical\-velocity map then reconstructs the corresponding canonical velocity fieldvρv=𝒞⁡\(ρv\)v^\{\\rho^\{v\}\}=\\mathcal\{C\}\(\\rho^\{v\}\)\. Thus, once the identity𝒯∘𝒞=Id𝒫\\mathcal\{T\}\\circ\\mathcal\{C\}=\\mathrm\{Id\}\_\{\\mathscr\{P\}\}is established, all velocity fields with the same terminal density are mapped byℛ\\mathcal\{R\}to the same canonical representative, without changing their terminal density\. These relationships are summarized in Figure[3](https://arxiv.org/html/2609.05727#S4.F3)\.

![Refer to caption](https://arxiv.org/html/2609.05727v1/canonical.png)velocity space𝒱\\mathscr\{V\}density manifold𝒫\\mathscr\{P\}canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}vρ=𝒞⁡\(ρ\)v^\{\\rho\}=\\mathcal\{C\}\(\\rho\)vvρ=𝒯⁡\(v\)\\rho=\\mathcal\{T\}\(v\)𝒯\\mathcal\{T\}𝒞\\mathcal\{C\}∼\\simℛ=𝒞∘𝒯\\mathcal\{R\}=\\mathcal\{C\}\\circ\\mathcal\{T\}Figure 3:Global geometry of the canonical construction\.The definitions above describe the canonical construction at the set\-theoretic level\. The following theorem establishes its smooth geometric structure\. In particular, it shows that𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}, which is diffeomorphic to𝒫\\mathscr\{P\}, is a split embedded Banach submanifold of𝒱\\mathscr\{V\}, whereasℛ\\mathcal\{R\}is a smooth idempotent projection that preserves terminal densities\.

###### Theorem 3\.

Under Assumption[67](https://arxiv.org/html/2609.05727#Thmtheorem67), the following conclusions hold\.

1. 1\.The terminal\-density map𝒯:𝒱→𝒫\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{P\}and the canonical\-velocity map𝒞:𝒫→𝒱\\mathcal\{C\}:\\mathscr\{P\}\\to\\mathscr\{V\}are both smooth\. Consequently, the canonical projectionℛ=𝒞∘𝒯:𝒱→𝒱can\\mathcal\{R\}=\\mathcal\{C\}\\circ\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}is smooth\.
2. 2\.The canonical\-velocity map𝒞:𝒫→𝒱\\mathcal\{C\}:\\mathscr\{P\}\\to\\mathscr\{V\}preserves the terminal density, i\.e\., 𝒯∘𝒞=Id𝒫\.\\mathcal\{T\}\\circ\\mathcal\{C\}=\\mathrm\{Id\}\_\{\\mathscr\{P\}\}\.Consequently,𝒯\\mathcal\{T\}is surjective and𝒞\\mathcal\{C\}is injective\.
3. 3\.The canonical manifold𝒱can=𝒞⁡\(𝒫\)\\mathscr\{V\}\_\{\\mathrm\{can\}\}=\\mathcal\{C\}\(\\mathscr\{P\}\)is a split embedded Banach submanifold of the velocity space𝒱\\mathscr\{V\}, and 𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a diffeomorphism with inverse 𝒞−1=𝒯\|𝒱can\.\\mathcal\{C\}^\{\-1\}=\\mathcal\{T\}\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\.
4. 4\.The canonical projectionℛ:𝒱→𝒱can\\mathcal\{R\}:\\mathscr\{V\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}is idempotent and preserves the terminal density, i\.e\., ℛ2=ℛ,ℛ\|𝒱can=Id𝒱can,𝒯∘ℛ=𝒯\.\\mathcal\{R\}^\{2\}=\\mathcal\{R\},\\qquad\\left\.\\mathcal\{R\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathrm\{Id\}\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\},\\qquad\\mathcal\{T\}\\circ\\mathcal\{R\}=\\mathcal\{T\}\.

###### Proof\.

The detailed proof of this theorem is deferred to Appendix[A](https://arxiv.org/html/2609.05727#A1)\. Under Assumption[67](https://arxiv.org/html/2609.05727#Thmtheorem67), Definition[66](https://arxiv.org/html/2609.05727#Thmtheorem66)states that𝒯:𝒱→𝒫\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{P\}and𝒞:𝒫→𝒱\\mathcal\{C\}:\\mathscr\{P\}\\to\\mathscr\{V\}are smooth\. In particular, Proposition[68](https://arxiv.org/html/2609.05727#Thmtheorem68)proves thatℛ=𝒞∘𝒯\\mathcal\{R\}=\\mathcal\{C\}\\circ\\mathcal\{T\}is smooth, that𝒱can=𝒞⁡\(𝒫\)\\mathscr\{V\}\_\{\\mathrm\{can\}\}=\\mathcal\{C\}\(\\mathscr\{P\}\)is a split embedded Banach submanifold of𝒱\\mathscr\{V\}, and that𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a diffeomorphism\. Corollary[63](https://arxiv.org/html/2609.05727#Thmtheorem63)proves that𝒯∘𝒞=Id𝒫\\mathcal\{T\}\\circ\\mathcal\{C\}=\\mathrm\{Id\}\_\{\\mathscr\{P\}\}, which further implies that𝒯\\mathcal\{T\}is surjective and𝒞\\mathcal\{C\}is injective\. Theorem[65](https://arxiv.org/html/2609.05727#Thmtheorem65)proves that𝒞−1=𝒯\|𝒱can\\mathcal\{C\}^\{\-1\}=\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\},ℛ2=ℛ\\mathcal\{R\}^\{2\}=\\mathcal\{R\},ℛ\|𝒱can=Id𝒱can\\left\.\\mathcal\{R\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathrm\{Id\}\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}, and𝒯∘ℛ=𝒯\\mathcal\{T\}\\circ\\mathcal\{R\}=\\mathcal\{T\}\. ∎

Theorem[3](https://arxiv.org/html/2609.05727#Thmtheorem3)provides two structural ingredients used in the remainder of the paper\. First, the identity

𝒯∘ℛ=𝒯\\mathcal\{T\}\\circ\\mathcal\{R\}=\\mathcal\{T\}allows an ambient velocity field to be canonicalized without changing its terminal density\. This property underlies the canonical retraction introduced in Section[5](https://arxiv.org/html/2609.05727#S5)\.

Second, the diffeomorphism between the density manifold𝒫\\mathscr\{P\}and the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}allows us to equip the canonical optimization domain with a geometry that has a direct distributional interpretation\. Since the optimization objective and the terminal effects of model updates are naturally expressed in terms of probability densities, Sections[4\.2](https://arxiv.org/html/2609.05727#S4.SS2)and[4\.3](https://arxiv.org/html/2609.05727#S4.SS3)first formulate the relevant first\-order and second\-order differential structures on𝒫\\mathscr\{P\}, and then transport them through the diffeomorphism

𝒞:𝒫→∼𝒱can\.\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.At the same time, the canonical representation is computationally natural, since training and model updates are carried out in velocity\-field coordinates\. Accordingly, Section[5](https://arxiv.org/html/2609.05727#S5)defines the retraction directly on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}, while the corresponding terminal\-density update is obtained through the inverse diffeomorphism

𝒞−1=𝒯\|𝒱can:𝒱can→∼𝒫\.\\mathcal\{C\}^\{\-1\}=\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}:\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\xrightarrow\{\\sim\}\\mathscr\{P\}\.Thus,𝒞\\mathcal\{C\}transports the selected density\-space geometry to the canonical velocity representation, whereas𝒞−1\\mathcal\{C\}^\{\-1\}identifies the distributional effect of an update performed in velocity\-field coordinates\. These complementary roles are summarized in Figure[4](https://arxiv.org/html/2609.05727#S4.F4)\.

canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}density manifold𝒫\\mathscr\{P\}tangentΓρ,f\\Gamma^\{\\rho,f\}tangentξρ,f\\xi^\{\\rho,f\}metricgcang^\{\\mathrm\{can\}\}metricgFRg^\{\\mathrm\{FR\}\}connection∇can\\nabla^\{\\mathrm\{can\}\}connection∇mix\\nabla^\{\\mathrm\{mix\}\}retractionℛ⁡\(vρ\+η​Γρ,f\)\\mathcal\{R\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\)retraction𝒯⁡\(vρ\+η​Γρ,f\)\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\)𝒞\\mathcal\{C\}𝒞−1\\mathcal\{C\}^\{\-1\}Figure 4:Geometric structures and retraction operations under the canonical diffeomorphism\. The tangent representation, Fisher–Rao metric, and mixture connection on𝒫\\mathscr\{P\}are transported to𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}through𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. Conversely, the canonical retraction on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}corresponds through𝒞−1=𝒯\|𝒱can:𝒱can→∼𝒫\\mathcal\{C\}^\{\-1\}=\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}:\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\xrightarrow\{\\sim\}\\mathscr\{P\}to the induced retraction on𝒫\\mathscr\{P\}\.
### 4\.2First\-Order Geometry

Having established the diffeomorphism𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}, we develop the first\-order structures needed to construct updates on the canonical manifold in this subsection\. The canonical retraction introduced in Section[5](https://arxiv.org/html/2609.05727#S5)takes a tangent direction at a canonical velocity field and converts it into a finite\-stepsize velocity\-field update\. We therefore begin by characterizing tangent vectors on𝒫\\mathscr\{P\}and their canonical lifts to𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}in Section[4\.2\.1](https://arxiv.org/html/2609.05727#S4.SS2.SSS1)\. An objective function, however, initially provides first\-order information through its differential, which is a cotangent vector rather than an update direction\. We identify the cotangent spaces of the two manifolds in Section[4\.2\.2](https://arxiv.org/html/2609.05727#S4.SS2.SSS2)\. Finally, we equip𝒫\\mathscr\{P\}with the Fisher–Rao metric, which provides a tangent\-vector representation of a differential whenever its metric dual exists, and transport this metric to𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. As summarized in Figure[4](https://arxiv.org/html/2609.05727#S4.F4), all first\-order geometric structures defined on the density manifold𝒫\\mathscr\{P\}are transported to the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

#### 4\.2\.1Tangent Spaces and Canonical Lifts

In the CFM framework, the trainable object is the velocity field, so an update should ultimately be represented as a velocity\-field perturbation\. The canonical\-velocity update introduced in Section[5](https://arxiv.org/html/2609.05727#S5)takes a tangent direction on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}as input\. Accordingly, we characterize the relevant tangent spaces below\.

Since the canonical\-velocity map𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a diffeomorphism, it is sufficient to first characterizeTρ​𝒫T\_\{\\rho\}\\mathscr\{P\}\. Every density tangent vectorξ∈Tρ​𝒫\\xi\\in T\_\{\\rho\}\\mathscr\{P\}has a unique canonical lift

dρ​𝒞​\[ξ\]∈Tvρ​𝒱can\.\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\xi\]\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.\(13\)Since𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is an embedded submanifold of the Banach space𝒱\\mathscr\{V\}, this canonical lift can be used directly as a velocity\-field perturbation in Section[5](https://arxiv.org/html/2609.05727#S5)\.

We first characterize a tangent vectorξ∈Tρ​𝒫\\xi\\in T\_\{\\rho\}\\mathscr\{P\}\. Consider any smooth curve\(ρε\)ε∈\(−δ,δ\)⊂𝒫\(\\rho^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{P\}which passes throughρ\\rhoatε=0\\varepsilon=0and satisfiesξ=dd​ε​ρε\|ε=0\\xi=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}\. Since eachρε\\rho^\{\\varepsilon\}is normalized, we have

∀ε∈\(−δ,δ\),∫ℝdρε​\(x\)​𝑑x=1\.\\forall\\varepsilon\\in\(\-\\delta,\\delta\),\\,\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho^\{\\varepsilon\}\(x\)\\,\\mathrm\{d\}x=1\.As formalized in Theorem[72](https://arxiv.org/html/2609.05727#Thmtheorem72), we identifyξ\\xiwith its density\-function representation onℝd\\mathbb\{R\}^\{d\}\. Therefore,

∫ℝdξ⁡\(x\)​𝑑x=dd​ε​∫ℝdρε​\(x\)​𝑑x\|ε=0=0\.\\int\_\{\\mathbb\{R\}^\{d\}\}\\xi\(x\)\\,\\mathrm\{d\}x=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho^\{\\varepsilon\}\(x\)\\,\\mathrm\{d\}x\\right\|\_\{\\varepsilon=0\}=0\.Writingf=ξρ\+constf=\\frac\{\\xi\}\{\\rho\}\+\\mathrm\{const\}, we obtain

ξ=ξρ,f:=ρ⋅\(f−𝔼ρ​\[f\]\)\.\\xi=\\xi^\{\\rho,f\}:=\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\)\.Here, the termf−𝔼ρ​\[f\]f\-\\mathbb\{E\}\_\{\\rho\}\[f\]centers the variation of the density\. More precisely, the functionffshould belong to a regular function class\. Evidently, for any constantc∈ℝc\\in\\mathbb\{R\},

ξρ,f\+c=ξρ,f\.\\xi^\{\\rho,f\+c\}=\\xi^\{\\rho,f\}\.
###### Proposition 4\.

Under Assumption[70](https://arxiv.org/html/2609.05727#Thmtheorem70), the tangent space of𝒫\\mathscr\{P\}at any densityρ∈𝒫\\rho\\in\\mathscr\{P\}is a vector space satisfying

Tρ𝒫⊂\{ρ⋅\(f−𝔼ρ\[f\]\):f∈C\(ℝd;ℝ\),𝔼ρ\[\|f\|\]<∞\}\.T\_\{\\rho\}\\mathscr\{P\}\\subset\\left\\\{\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\):f\\in C\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}\),\\mathbb\{E\}\_\{\\rho\}\[\\left\|f\\right\|\]<\\infty\\right\\\}\.

###### Proof\.

The proof is deferred to Theorem[72](https://arxiv.org/html/2609.05727#Thmtheorem72)in Appendix[A\.3\.2](https://arxiv.org/html/2609.05727#A1.SS3.SSS2)\. ∎

The representationξρ,f=ρ⋅\(f−𝔼ρ​\[f\]\)\\xi^\{\\rho,f\}=\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\)describes an infinitesimal perturbation of the terminal density, but it is not yet a velocity\-field update\. We therefore transport it throughdρ​𝒞:Tρ​𝒫→Tvρ​𝒱can\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}:T\_\{\\rho\}\\mathscr\{P\}\\to T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}and define its canonical lift by \([13](https://arxiv.org/html/2609.05727#S4.E13)\)\. To make this lifted directiondρ​𝒞​\[ξρ,f\]\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\xi^\{\\rho,f\}\]usable in the flow\-matching framework, we seek an explicit representation in terms of the posterior associated with the current canonical field\. For this purpose, define the posterior value operator

Vtρ\[f\]:ℝd→ℝ,Vtρ\[f\]\(xt\):=𝔼X1∼p1\|tρ\(⋅\|xt\)\[f\(X1\)\],V\_\{t\}^\{\\rho\}\[f\]:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\},\\qquad V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\):=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(X\_\{1\}\)\\right\],\(14\)wheref∈C⁡\(ℝd,ℝ\)f\\in C\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}\)is called a terminal observable\. Att=1t=1, setV1ρ​\[f\]​\(x1\):=f⁡\(x1\)V\_\{1\}^\{\\rho\}\[f\]\(x\_\{1\}\):=f\(x\_\{1\}\)\.

The following proposition shows that the canonical lift can be written equivalently as the gradient of a posterior value or as a posterior covariance\. Both representations make its conditional\-expectation structure explicit and yield a CFM\-style learnable update in Section[8\.1](https://arxiv.org/html/2609.05727#S8.SS1)\.

###### Proposition 5\.

Consider a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}and its canonical velocity fieldvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. For any tangent vectorξρ,f=ρ⋅\(f−𝔼ρ​\[f\]\)∈Tρ​𝒫\\xi^\{\\rho,f\}=\\rho\\cdot\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\)\\in T\_\{\\rho\}\\mathscr\{P\}, its canonical lift

Γρ,f:=dρ​𝒞​\[ξρ,f\]∈Tvρ​𝒱can\\Gamma^\{\\rho,f\}:=\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\xi^\{\\rho,f\}\]\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}can be represented, for everyt∈\(0,1\)t\\in\(0,1\), by

Γtρ,f\(xt\)=κt∇Vtρ\[f\]\(xt\)=CovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),f\(X1\)\)\.\\Gamma^\{\\rho,f\}\_\{t\}\(x\_\{t\}\)=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),f\(X\_\{1\}\)\\right\)\.

###### Proof\.

First, we proveΓtρ,f=κt∇Vtρ\[f\]\\Gamma^\{\\rho,f\}\_\{t\}=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\. Consider any smooth curve\(ρε\)ε∈\[−δ,δ\]⊂𝒫\(\\rho^\{\\varepsilon\}\)\_\{\\varepsilon\\in\[\-\\delta,\\delta\]\}\\subset\\mathscr\{P\}which passes throughρ\\rhoatε=0\\varepsilon=0, whereξρ,f=dd​ε​ρε\|ε=0=ρ⋅\(f−𝔼ρ​\[f\]\)\\xi^\{\\rho,f\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}=\\rho\\cdot\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\)\. Note that

Vtρ​\[f\]​\(xt\)=\\displaystyle V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=∫ℝd\(f⁡\(x1\)−𝔼ρ​\[f\]\)​p1\|tρ​\(x1\|xt\)​d​x1\+𝔼ρ​\[f\]\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\(f\(x\_\{1\}\)\-\\mathbb\{E\}\_\{\\rho\}\[f\]\)p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\,\\mathrm\{d\}x\_\{1\}\+\\mathbb\{E\}\_\{\\rho\}\[f\]=\\displaystyle=∫ℝd\(f⁡\(x1\)−𝔼ρ​\[f\]\)​ρ​\(x1\)​pt\|1​\(xt\|x1\)​d​x1ptρ​\(xt\)\+𝔼ρ​\[f\]\\displaystyle\\frac\{\\int\_\{\\mathbb\{R\}^\{d\}\}\(f\(x\_\{1\}\)\-\\mathbb\{E\}\_\{\\rho\}\[f\]\)\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\+\\mathbb\{E\}\_\{\\rho\}\[f\]=\\displaystyle=∫ℝddd​ε​ρε​\(x1\)\|ε=0​pt\|1​\(xt\|x1\)​d​x1ptρ​\(xt\)\+𝔼ρ​\[f\]\\displaystyle\\frac\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\(x\_\{1\}\)\\right\|\_\{\\varepsilon=0\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\+\\mathbb\{E\}\_\{\\rho\}\[f\]=\\displaystyle=dd​ε​ptρε​\(xt\)\|ε=0ptρ​\(xt\)\+𝔼ρ​\[f\]\\displaystyle\\frac\{\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}p\_\{t\}^\{\\rho^\{\\varepsilon\}\}\(x\_\{t\}\)\\right\|\_\{\\varepsilon=0\}\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\+\\mathbb\{E\}\_\{\\rho\}\[f\]=\\displaystyle=dd​ε​\(log⁡ptρε​\(xt\)\)\|ε=0\+𝔼ρ​\[f\]\.\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\left\(\\log p\_\{t\}^\{\\rho^\{\\varepsilon\}\}\(x\_\{t\}\)\\right\)\\right\|\_\{\\varepsilon=0\}\+\\mathbb\{E\}\_\{\\rho\}\[f\]\.We also have

dd​ε​vtρε​\(xt\)\|ε=0=dd​ε​\(α˙tαt​xt\+κt​∇xt​log⁡ptρε​\(xt\)\)\|ε=0=κt​∇xtdd​ε​\(log⁡ptρε​\(xt\)\)\|ε=0,\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}v\_\{t\}^\{\\rho^\{\\varepsilon\}\}\(x\_\{t\}\)\\right\|\_\{\\varepsilon=0\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\left\(\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\}\+\\kappa\_\{t\}\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}^\{\\rho^\{\\varepsilon\}\}\(x\_\{t\}\)\\right\)\\right\|\_\{\\varepsilon=0\}=\\kappa\_\{t\}\\nabla\_\{x\_\{t\}\}\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\left\(\\log p\_\{t\}^\{\\rho^\{\\varepsilon\}\}\(x\_\{t\}\)\\right\)\\right\|\_\{\\varepsilon=0\},which givesΓtρ,f=κt∇Vtρ\[f\]\\Gamma^\{\\rho,f\}\_\{t\}=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\.

Next, we compute∇Vtρ​\[f\]\\nabla V\_\{t\}^\{\\rho\}\[f\]\. By \([8](https://arxiv.org/html/2609.05727#S3.E8)\), we have∇xtpt\|1​\(xt\|x1\)=−xt−αt​x1βt2​pt\|1​\(xt\|x1\)\\nabla\_\{x\_\{t\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\-\\frac\{x\_\{t\}\-\\alpha\_\{t\}x\_\{1\}\}\{\\beta\_\{t\}^\{2\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\. In particular,

∇Vtρ​\[f\]​\(xt\)=\\displaystyle\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=∇xt\(∫ℝdf⁡\(x1\)​ρ​\(x1\)​pt\|1​\(xt\|x1\)​d​x1∫ℝdρ⁡\(x1\)​pt\|1​\(xt\|x1\)​d​x1\)\\displaystyle\\nabla\_\{x\_\{t\}\}\\left\(\\frac\{\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\_\{1\}\)\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\}\\right\)=\\displaystyle=∫ℝdf⁡\(x1\)​ρ​\(x1\)​∇xtpt\|1​\(xt\|x1\)​d​x1ptρ​\(xt\)−Vtρ​\[f\]​\(xt\)​∫ℝdρ⁡\(x1\)​∇xtpt\|1​\(xt\|x1\)​d​x1ptρ​\(xt\)\\displaystyle\\frac\{\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\_\{1\}\)\\rho\(x\_\{1\}\)\\nabla\_\{x\_\{t\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\-V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\\frac\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(x\_\{1\}\)\\nabla\_\{x\_\{t\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}=\\displaystyle=∫ℝd\(f⁡\(x1\)−Vtρ​\[f\]​\(xt\)\)​ρ⁡\(x1\)​∇xtpt\|1​\(xt\|x1\)ptρ​\(xt\)​d​x1\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\(f\(x\_\{1\}\)\-V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\\right\)\\frac\{\\rho\(x\_\{1\}\)\\nabla\_\{x\_\{t\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\\mathrm\{d\}x\_\{1\}=\\displaystyle=∫ℝd\(f⁡\(x1\)−Vtρ​\[f\]​\(xt\)\)​ρ⁡\(x1\)​pt\|1​\(xt\|x1\)ptρ​\(xt\)​αt​x1−xtβt2​d​x1\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\(f\(x\_\{1\}\)\-V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\\right\)\\frac\{\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\\frac\{\\alpha\_\{t\}x\_\{1\}\-x\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\mathrm\{d\}x\_\{1\}=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(f\(X1\)−Vtρ\[f\]\(xt\)\)αt​X1−xtβt2\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\left\(f\(X\_\{1\}\)\-V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\\right\)\\frac\{\\alpha\_\{t\}X\_\{1\}\-x\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\right\]=\\displaystyle=αtβt2CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)\\displaystyle\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)impliesκt∇Vtρ\[f\]\(xt\)=CovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),f\(X1\)\)\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),f\(X\_\{1\}\)\\right\)\. ∎

Based on Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5), we obtain the tangent space of𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}as follows\.

###### Corollary 6\.

The tangent space of𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}atvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}is

Tvρ​𝒱can=\{Γρ,f:ξρ,f∈Tρ​𝒫\}≅Tρ​𝒫\.T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}=\\left\\\{\\Gamma^\{\\rho,f\}:\\xi^\{\\rho,f\}\\in T\_\{\\rho\}\\mathscr\{P\}\\right\\\}\\cong T\_\{\\rho\}\\mathscr\{P\}\.Moreover, for any densityρ∈𝒫\\rho\\in\\mathscr\{P\}, regular terminal observablesf,g∈C⁡\(ℝd,ℝ\)f,g\\in C\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}\), and constantsa,b,c∈ℝa,b,c\\in\\mathbb\{R\}such thatξρ,f,ξρ,g∈Tρ​𝒫\\xi^\{\\rho,f\},\\xi^\{\\rho,g\}\\in T\_\{\\rho\}\\mathscr\{P\}, we have

Γρ,a​f\+b​g\+c=a​Γρ,f\+b​Γρ,g\.\\Gamma^\{\\rho,af\+bg\+c\}=a\\Gamma^\{\\rho,f\}\+b\\Gamma^\{\\rho,g\}\.

###### Proof\.

By Theorem[3](https://arxiv.org/html/2609.05727#Thmtheorem3), the canonical\-velocity map𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a diffeomorphism\. Therefore,dρ​𝒞:Tρ​𝒫→∼Tvρ​𝒱can\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}:T\_\{\\rho\}\\mathscr\{P\}\\xrightarrow\{\\sim\}T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a linear isomorphism; hence,

Tvρ​𝒱can=\{dρ​𝒞​\[ξ\]:ξ∈Tρ​𝒫\}=\{Γρ,f:ξρ,f∈Tρ​𝒫\}\.T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}=\\left\\\{\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\xi\]:\\xi\\in T\_\{\\rho\}\\mathscr\{P\}\\right\\\}=\\left\\\{\\Gamma^\{\\rho,f\}:\\xi^\{\\rho,f\}\\in T\_\{\\rho\}\\mathscr\{P\}\\right\\\}\.Moreover,

Γtρ,a​f\+b​g\+c=κt∇Vtρ\[af\+bg\+c\]=aκt∇Vtρ\[f\]\+bκt∇Vtρ\[g\]=aΓtρ,f\+bΓtρ,g,\\Gamma^\{\\rho,af\+bg\+c\}\_\{t\}=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[af\+bg\+c\]=a\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\+b\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[g\]=a\\Gamma^\{\\rho,f\}\_\{t\}\+b\\Gamma^\{\\rho,g\}\_\{t\},which finishes the proof\. ∎

We use theΓ\\Gamma\-representation as the basic learnable update direction throughout the remainder of the paper\. The next result establishes the interior\-time identifiability of this representation\. At any nondegenerate time slice, the lifted field determines the terminal observable up to an additive constant; equivalently, vanishing at a single interior time already forces the observable to be constant\.

###### Proposition 7\.

For a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}and a terminal observablef∈C⁡\(ℝd,ℝ\)f\\in C\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}\), consider the tangent vectorΓρ,f\\Gamma^\{\\rho,f\}\. The following three statements are equivalent:

f≡const⇔∀t∈\(0,1\),Vtρ\[f\]≡const⇔∃t0∈\(0,1\),Vt0ρ\[f\]≡const\.f\\equiv\\mathrm\{const\}\\quad\\Leftrightarrow\\quad\\forall t\\in\(0,1\),\\,V\_\{t\}^\{\\rho\}\[f\]\\equiv\\mathrm\{const\}\\quad\\Leftrightarrow\\quad\\exists t\_\{0\}\\in\(0,1\),\\,V\_\{t\_\{0\}\}^\{\\rho\}\[f\]\\equiv\\mathrm\{const\}\.In other words, we have

f≡const⇔∀t∈\(0,1\),Γtρ,f≡0⇔∃t0∈\(0,1\),Γt0ρ,f≡0\.f\\equiv\\mathrm\{const\}\\quad\\Leftrightarrow\\quad\\forall t\\in\(0,1\),\\,\\Gamma\_\{t\}^\{\\rho,f\}\\equiv 0\\quad\\Leftrightarrow\\quad\\exists t\_\{0\}\\in\(0,1\),\\,\\Gamma\_\{t\_\{0\}\}^\{\\rho,f\}\\equiv 0\.

###### Proof\.

Iff≡cf\\equiv cis constant, thenVtρ\[f\]\(xt\)=𝔼X∼p1\|tρ\(⋅\|xt\)\[c\]=cV\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[c\]=cis constant for allt∈\(0,1\)t\\in\(0,1\), which is equivalent toΓtρ,f≡0\\Gamma\_\{t\}^\{\\rho,f\}\\equiv 0for allt∈\(0,1\)t\\in\(0,1\)\.

IfVtρ​\[f\]≡cV\_\{t\}^\{\\rho\}\[f\]\\equiv cis constant for allt∈\(0,1\)t\\in\(0,1\), lett0∈\(0,1\)t\_\{0\}\\in\(0,1\)be arbitrary\. Then,Vt0ρ​\[f\]≡cV\_\{t\_\{0\}\}^\{\\rho\}\[f\]\\equiv cis constant, which is equivalent toΓt0ρ,f≡0\\Gamma\_\{t\_\{0\}\}^\{\\rho,f\}\\equiv 0\.

IfVt0ρ​\[f\]≡cV\_\{t\_\{0\}\}^\{\\rho\}\[f\]\\equiv cis constant for somet0∈\(0,1\)t\_\{0\}\\in\(0,1\), then we have

∫ℝd\(f⁡\(x1\)−c\)​ρ​\(x1\)​pt0\|1​\(xt0\|x1\)​d​x1=0\\int\_\{\\mathbb\{R\}^\{d\}\}\(f\(x\_\{1\}\)\-c\)\\rho\(x\_\{1\}\)p\_\{t\_\{0\}\|1\}\(x\_\{t\_\{0\}\}\|x\_\{1\}\)\\,\\mathrm\{d\}x\_\{1\}=0for everyxt0∈ℝdx\_\{t\_\{0\}\}\\in\\mathbb\{R\}^\{d\}\. Since the Gaussian convolution is injective andff,ρ\\rhoare continuous, we have\(f⁡\(x1\)−c\)​ρ​\(x1\)≡0\(f\(x\_\{1\}\)\-c\)\\rho\(x\_\{1\}\)\\equiv 0\. Therefore, the strict positivity ofρ\\rhoimpliesf≡constf\\equiv\\mathrm\{const\}\. ∎

Equivalently, for every fixedt0∈\(0,1\)t\_\{0\}\\in\(0,1\), the mapf↦Γt0ρ,ff\\mapsto\\Gamma\_\{t\_\{0\}\}^\{\\rho,f\}is injective modulo additive constants\. Thus, theΓ\\Gamma\-representation is nondegenerate: a nonconstant terminal observable satisfiesΓtρ,f≢0\\Gamma\_\{t\}^\{\\rho,f\}\\not\\equiv 0at every interior timet∈\(0,1\)t\\in\(0,1\)\.

#### 4\.2\.2Cotangent Spaces and Differentials

A tangent space describes the admissible infinitesimal perturbations of a density\. To describe the first\-order change of an objective along such perturbations, we need a continuous linear functional acting on the tangent space\. This motivates the cotangent space

Tρ∗𝒫=\{ℓ:Tρ𝒫→ℝ:ℓis continuous and linear\}\.T\_\{\\rho\}^\{\*\}\\mathscr\{P\}=\\left\\\{\\ell:T\_\{\\rho\}\\mathscr\{P\}\\to\\mathbb\{R\}:\\ell\\text\{ is continuous and linear\}\\right\\\}\.Its elements are called cotangent vectors\. Similarly, for anyv∈𝒱canv\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}, the cotangent space of𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}atvvis

Tv∗𝒱can=\{ℓcan:Tv𝒱can→ℝ:ℓcanis continuous and linear\}\.T\_\{v\}^\{\*\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}=\\left\\\{\\ell^\{\\mathrm\{can\}\}:T\_\{v\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to\\mathbb\{R\}:\\ell^\{\\mathrm\{can\}\}\\text\{ is continuous and linear\}\\right\\\}\.These two cotangent spaces are linearly isomorphic whenv=𝒞⁡\(ρ\)v=\\mathcal\{C\}\(\\rho\):

T𝒞⁡\(ρ\)∗​𝒱can=\{ℓ∘d𝒞⁡\(ρ\)​𝒞−1:ℓ∈Tρ∗​𝒫\}≅Tρ∗​𝒫\.T\_\{\\mathcal\{C\}\(\\rho\)\}^\{\*\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}=\\left\\\{\\ell\\circ\\mathrm\{d\}\_\{\\mathcal\{C\}\(\\rho\)\}\\mathcal\{C\}^\{\-1\}:\\ell\\in T\_\{\\rho\}^\{\*\}\\mathscr\{P\}\\right\\\}\\cong T\_\{\\rho\}^\{\*\}\\mathscr\{P\}\.
As recalled in Section[2\.2\.4](https://arxiv.org/html/2609.05727#S2.SS2.SSS4), the differential of a smooth objectiveJ:𝒫→ℝJ:\\mathscr\{P\}\\to\\mathbb\{R\}atρ\\rhois the cotangent vectordρ​J∈Tρ∗​𝒫\\mathrm\{d\}\_\{\\rho\}J\\in T\_\{\\rho\}^\{\*\}\\mathscr\{P\}defined as

dρ​J:Tρ​𝒫→ℝ,dρ​J​\[ξ\]:=dd​ε​J​\(ρε\)\|ε=0,\\mathrm\{d\}\_\{\\rho\}J:T\_\{\\rho\}\\mathscr\{P\}\\to\\mathbb\{R\},\\qquad\\mathrm\{d\}\_\{\\rho\}J\[\\xi\]:=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}J\(\\rho^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\},where\(ρε\)ε⊂𝒫\(\\rho^\{\\varepsilon\}\)\_\{\\varepsilon\}\\subset\\mathscr\{P\}is any smooth curve which passes throughρ\\rhoatε=0\\varepsilon=0and satisfiesdd​ε​ρε\|ε=0=ξ\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}=\\xi\. Thus,dρ​J\\mathrm\{d\}\_\{\\rho\}Jassigns to every admissible density perturbation its corresponding first\-order change in the objective\.

#### 4\.2\.3Fisher–Rao Metric

A cotangent vector records the first\-order variation of an objective, whereas the canonical retraction in Section[5](https://arxiv.org/html/2609.05727#S5)requires a tangent direction as the first\-order perturbation direction\. A Riemannian metric provides the link between these two objects\. Recall from Section[2\.2\.5](https://arxiv.org/html/2609.05727#S2.SS2.SSS5)that a Riemannian metric determines a unique tangent\-vector representation of a cotangent vector whenever such a representation exists\. The choice of metric is therefore part of the optimization method: different metrics generally produce different tangent\-vector representations of the same differential and hence different gradient directions\.

To identify objective differentials with tangent vectors, we equip𝒫\\mathscr\{P\}with theFisher–Rao metric\[[4](https://arxiv.org/html/2609.05727#bib.bib46)\]and transport this metric through the diffeomorphism𝒞\\mathcal\{C\}to the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. The Fisher–Rao metric is well suited to a density manifold since it measures density perturbations through their relative changesξρ\\frac\{\\xi\}\{\\rho\}:

gρFR:Tρ​𝒫×Tρ​𝒫→ℝ,gρFR​\(ξ1,ξ2\):=∫ℝdξ1​\(x\)​ξ2​\(x\)ρ⁡\(x\)​𝑑x\.g\_\{\\rho\}^\{\\mathrm\{FR\}\}:T\_\{\\rho\}\\mathscr\{P\}\\times T\_\{\\rho\}\\mathscr\{P\}\\to\\mathbb\{R\},\\qquad g\_\{\\rho\}^\{\\mathrm\{FR\}\}\(\\xi\_\{1\},\\xi\_\{2\}\):=\\int\_\{\\mathbb\{R\}^\{d\}\}\\frac\{\\xi\_\{1\}\(x\)\\xi\_\{2\}\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}x\.\(15\)In particular, for any two terminal observablesf1,f2f\_\{1\},f\_\{2\}, we have

gρFR​\(ξρ,f1,ξρ,f2\)=𝔼ρ​\[\(f1−𝔼ρ​\[f1\]\)​\(f2−𝔼ρ​\[f2\]\)\]\.g\_\{\\rho\}^\{\\mathrm\{FR\}\}\(\\xi^\{\\rho,f\_\{1\}\},\\xi^\{\\rho,f\_\{2\}\}\)=\\mathbb\{E\}\_\{\\rho\}\\left\[\(f\_\{1\}\-\\mathbb\{E\}\_\{\\rho\}\[f\_\{1\}\]\)\(f\_\{2\}\-\\mathbb\{E\}\_\{\\rho\}\[f\_\{2\}\]\)\\right\]\.Under Assumption[75](https://arxiv.org/html/2609.05727#Thmtheorem75), the mapgFR:\(ρ,ξ1,ξ2\)↦gρFR​\(ξ1,ξ2\)g^\{\\mathrm\{FR\}\}:\(\\rho,\\xi\_\{1\},\\xi\_\{2\}\)\\mapsto g\_\{\\rho\}^\{\\mathrm\{FR\}\}\(\\xi\_\{1\},\\xi\_\{2\}\)defines a smooth, possibly weak, Riemannian metric on the Banach manifold𝒫\\mathscr\{P\}\. In other words,𝒫\\mathscr\{P\}is a Riemannian manifold\.

Since the Fisher–Rao metric may be weak, a cotangent vectorℓ∈Tρ∗​𝒫\\ell\\in T\_\{\\rho\}^\{\*\}\\mathscr\{P\}need not admit a metric\-dual tangent representation\. Section[6\.1](https://arxiv.org/html/2609.05727#S6.SS1)verifies such a representation explicitly for the reverse\-KL differential\.

Consider anyv∈𝒱canv\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}and denoteρv=𝒯⁡\(v\)∈𝒫\\rho^\{v\}=\\mathcal\{T\}\(v\)\\in\\mathscr\{P\}\. For anyΓ1,Γ2∈Tv​𝒱can\\Gamma\_\{1\},\\Gamma\_\{2\}\\in T\_\{v\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}, the transported metric isgvcan:Tv​𝒱can×Tv​𝒱can→ℝg\_\{v\}^\{\\mathrm\{can\}\}:T\_\{v\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\times T\_\{v\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to\\mathbb\{R\},

gvcan​\(Γ1,Γ2\):=\\displaystyle g\_\{v\}^\{\\mathrm\{can\}\}\\left\(\\Gamma\_\{1\},\\Gamma\_\{2\}\\right\):=gρvFR​\(dv​𝒞−1​\[Γ1\],dv​𝒞−1​\[Γ2\]\)\.\\displaystyle g\_\{\\rho^\{v\}\}^\{\\mathrm\{FR\}\}\\left\(\\mathrm\{d\}\_\{v\}\\mathcal\{C\}^\{\-1\}\\left\[\\Gamma\_\{1\}\\right\],\\mathrm\{d\}\_\{v\}\\mathcal\{C\}^\{\-1\}\\left\[\\Gamma\_\{2\}\\right\]\\right\)\.\(16\)For any two terminal observablesf1f\_\{1\},f2f\_\{2\}, we have

gvcan​\(Γρv,f1,Γρv,f2\)=𝔼ρv​\[\(f1−𝔼ρv​\[f1\]\)​\(f2−𝔼ρv​\[f2\]\)\]\.g\_\{v\}^\{\\mathrm\{can\}\}\\left\(\\Gamma^\{\\rho^\{v\},f\_\{1\}\},\\Gamma^\{\\rho^\{v\},f\_\{2\}\}\\right\)=\\mathbb\{E\}\_\{\\rho^\{v\}\}\\left\[\(f\_\{1\}\-\\mathbb\{E\}\_\{\\rho^\{v\}\}\[f\_\{1\}\]\)\(f\_\{2\}\-\\mathbb\{E\}\_\{\\rho^\{v\}\}\[f\_\{2\}\]\)\\right\]\.By construction,𝒞\\mathcal\{C\}is an isometry between\(𝒫,gFR\)\(\\mathscr\{P\},g^\{\\mathrm\{FR\}\}\)and\(𝒱can,gcan\)\(\\mathscr\{V\}\_\{\\mathrm\{can\}\},g^\{\\mathrm\{can\}\}\)\. Thus,𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}inherits the same Riemannian, possibly weak, structure as𝒫\\mathscr\{P\}\.

### 4\.3Second\-Order Geometry

The first\-order structures in Section[4\.2](https://arxiv.org/html/2609.05727#S4.SS2)are defined separately at each base point\. To differentiate first\-order objects as the density varies, and thereby define the Hessian used in Section[6\.2](https://arxiv.org/html/2609.05727#S6.SS2), we require an affine connection\. The connection is part of the geometric specification of the optimization method: even for the same objective, different connections generally yield different Hessians and may therefore lead to different Newton directions\.

On the density manifold, we compare first\-order objects at neighboring densities by holding the terminal observable fixed as the base density varies\. An admissible terminal observable thereby induces a tangent vector field across densities\. Although the observable remains unchanged, the induced tangent vector and the Fisher–Rao metric both vary with the base density\. The mixture connection captures the resulting first\-order variation of the Fisher–Rao pairing\. We adopt this connection on𝒫\\mathscr\{P\}and transport it to𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}through the diffeomorphism𝒞\\mathcal\{C\}\.

To make this characterization precise, consider a terminal observablefffor which

ξ⋅,f∈𝔛⁡\(𝒫\),ξq,f:=q⋅\(f−𝔼q​\[f\]\),q∈𝒫,\\xi^\{\\cdot,f\}\\in\\mathfrak\{X\}\(\\mathscr\{P\}\),\\qquad\\xi^\{q,f\}:=q\\cdot\\left\(f\-\\mathbb\{E\}\_\{q\}\[f\]\\right\),\\qquad q\\in\\mathscr\{P\},defines a smooth tangent vector field on𝒫\\mathscr\{P\}\. We callξ⋅,f\\xi^\{\\cdot,f\}thefixed\-observable extensionofξρ,f∈Tρ​𝒫\\xi^\{\\rho,f\}\\in T\_\{\\rho\}\\mathscr\{P\}\. Along this extension, the terminal observableffis held fixed, whereas its density\-function representation varies with the base density through bothqqand𝔼q​\[f\]\\mathbb\{E\}\_\{q\}\[f\]\. Under the smooth\-realization assumptions of Appendix[A\.3](https://arxiv.org/html/2609.05727#A1.SS3), the values of admissible fixed\-observable extensions exhaust each tangent spaceTρ​𝒫T\_\{\\rho\}\\mathscr\{P\}\.

For anyQ∈𝔛⁡\(𝒫\)Q\\in\\mathfrak\{X\}\(\\mathscr\{P\}\), define the real\-valued function

gFR​\(Q,ξ⋅,f\):𝒫→ℝ,q↦gqFR​\(Qq,ξq,f\)\.g^\{\\mathrm\{FR\}\}\\left\(Q,\\xi^\{\\cdot,f\}\\right\):\\mathscr\{P\}\\to\\mathbb\{R\},\\qquad q\\mapsto g\_\{q\}^\{\\mathrm\{FR\}\}\\left\(Q\_\{q\},\\xi^\{q,f\}\\right\)\.The mixture connection records the first\-order change of this pairing when the base density varies\. Specifically, forP,Q∈𝔛⁡\(𝒫\)P,Q\\in\\mathfrak\{X\}\(\\mathscr\{P\}\), we define\(∇PmixQ\)ρ∈Tρ​𝒫\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\}\\in T\_\{\\rho\}\\mathscr\{P\}by the tangent vector satisfying

gρFR​\(\(∇PmixQ\)ρ,ξρ,f\)\\displaystyle g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)=P⁡\[gFR​\(Q,ξ⋅,f\)\]​\(ρ\)=dρ​\(gFR​\(Q,ξ⋅,f\)\)​\[Pρ\]\\displaystyle=P\\left\[g^\{\\mathrm\{FR\}\}\\left\(Q,\\xi^\{\\cdot,f\}\\right\)\\right\]\(\\rho\)=\\mathrm\{d\}\_\{\\rho\}\\left\(g^\{\\mathrm\{FR\}\}\\left\(Q,\\xi^\{\\cdot,f\}\\right\)\\right\)\[P\_\{\\rho\}\]\(17\)for every admissible terminal observableff\. Equivalently, for any smooth curve\(ρε\)ε∈\(−δ,δ\)⊂𝒫\(\\rho^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{P\}which passes throughρ\\rhoatε=0\\varepsilon=0and satisfiesdd​ε​ρε\|ε=0=Pρ\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}=P\_\{\\rho\},

gρFR​\(\(∇PmixQ\)ρ,ξρ,f\)=dd​ε​gρεFR​\(Qρε,ξρε,f\)\|ε=0\.g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\\rho^\{\\varepsilon\}\}^\{\\mathrm\{FR\}\}\\left\(Q\_\{\\rho^\{\\varepsilon\}\},\\xi^\{\\rho^\{\\varepsilon\},f\}\\right\)\\right\|\_\{\\varepsilon=0\}\.Here both the tangent vector fieldQρεQ\_\{\\rho^\{\\varepsilon\}\}and the base densityρε\\rho^\{\\varepsilon\}vary, whileffis fixed\. In particular, the variation ofρε\\rho^\{\\varepsilon\}changes both the Fisher–Rao pairing and the centered density representationξρε,f\\xi^\{\\rho^\{\\varepsilon\},f\}\. The proof of \([17](https://arxiv.org/html/2609.05727#S4.E17)\) is deferred to Theorem[80](https://arxiv.org/html/2609.05727#Thmtheorem80)\.

Under Assumption[78](https://arxiv.org/html/2609.05727#Thmtheorem78), the defining relation \([17](https://arxiv.org/html/2609.05727#S4.E17)\) uniquely determines\(∇PmixQ\)ρ∈Tρ​𝒫\\left\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Q\\right\)\_\{\\rho\}\\in T\_\{\\rho\}\\mathscr\{P\}for every\(P,Q,ρ\)\(P,Q,\\rho\), and the resulting assignment is smooth\. Therefore,

∇mix:𝔛⁡\(𝒫\)×𝔛⁡\(𝒫\)→𝔛⁡\(𝒫\),\(P,Q\)↦∇PmixQ,\\nabla^\{\\mathrm\{mix\}\}:\\mathfrak\{X\}\(\\mathscr\{P\}\)\\times\\mathfrak\{X\}\(\\mathscr\{P\}\)\\to\\mathfrak\{X\}\(\\mathscr\{P\}\),\\qquad\(P,Q\)\\mapsto\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q,is well\-defined\. Its construction, together with the proofs of uniqueness and smoothness, is deferred to Appendix[A\.3\.3](https://arxiv.org/html/2609.05727#A1.SS3.SSS3)\. The following proposition verifies the connection axioms \([6](https://arxiv.org/html/2609.05727#S2.E6)\)\.

###### Proposition 8\.

Under Assumption[78](https://arxiv.org/html/2609.05727#Thmtheorem78), the operator∇mix\\nabla^\{\\mathrm\{mix\}\}in \([17](https://arxiv.org/html/2609.05727#S4.E17)\) satisfies the definition of a connection \([6](https://arxiv.org/html/2609.05727#S2.E6)\)\. Consequently,∇mix\\nabla^\{\\mathrm\{mix\}\}is an affine connection on𝒫\\mathscr\{P\}\.

###### Proof\.

Under Assumption[78](https://arxiv.org/html/2609.05727#Thmtheorem78),∇mix\\nabla^\{\\mathrm\{mix\}\}is well\-defined and smooth\. Consider any tangent vector fieldsP,Q,Z∈𝔛⁡\(𝒫\)P,Q,Z\\in\\mathfrak\{X\}\(\\mathscr\{P\}\), smooth scalar functionsh1,h2:𝒫→ℝh\_\{1\},h\_\{2\}:\\mathscr\{P\}\\to\\mathbb\{R\}, and constantsa,b∈ℝa,b\\in\\mathbb\{R\}\. First, for everyξρ,f∈Tρ​𝒫\\xi^\{\\rho,f\}\\in T\_\{\\rho\}\\mathscr\{P\},

gρFR​\(\(∇h1​P\+h2​QmixZ\)ρ,ξρ,f\)=\\displaystyle g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\(\\nabla^\{\\mathrm\{mix\}\}\_\{h\_\{1\}P\+h\_\{2\}Q\}Z\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)=dρ​\(gFR​\(Z,ξ⋅,f\)\)​\[h1​\(ρ\)​Pρ\+h2​\(ρ\)​Qρ\]\\displaystyle\\mathrm\{d\}\_\{\\rho\}\\left\(g^\{\\mathrm\{FR\}\}\\left\(Z,\\xi^\{\\cdot,f\}\\right\)\\right\)\\left\[h\_\{1\}\(\\rho\)P\_\{\\rho\}\+h\_\{2\}\(\\rho\)Q\_\{\\rho\}\\right\]=\\displaystyle=h1​\(ρ\)​dρ​\(gFR​\(Z,ξ⋅,f\)\)​\[Pρ\]\+h2​\(ρ\)​dρ​\(gFR​\(Z,ξ⋅,f\)\)​\[Qρ\]\\displaystyle h\_\{1\}\(\\rho\)\\mathrm\{d\}\_\{\\rho\}\\left\(g^\{\\mathrm\{FR\}\}\\left\(Z,\\xi^\{\\cdot,f\}\\right\)\\right\)\\left\[P\_\{\\rho\}\\right\]\+h\_\{2\}\(\\rho\)\\mathrm\{d\}\_\{\\rho\}\\left\(g^\{\\mathrm\{FR\}\}\\left\(Z,\\xi^\{\\cdot,f\}\\right\)\\right\)\\left\[Q\_\{\\rho\}\\right\]=\\displaystyle=gρFR​\(\(h1​∇PmixZ\+h2​∇QmixZ\)ρ,ξρ,f\)\.\\displaystyle g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\(h\_\{1\}\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Z\+h\_\{2\}\\nabla^\{\\mathrm\{mix\}\}\_\{Q\}Z\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)\.The positive definiteness ofgρFRg\_\{\\rho\}^\{\\mathrm\{FR\}\}gives \([6a](https://arxiv.org/html/2609.05727#S2.E6.1)\)\. Then, for everyξρ,f∈Tρ​𝒫\\xi^\{\\rho,f\}\\in T\_\{\\rho\}\\mathscr\{P\},

gρFR​\(\(∇Pmix\(a​Q\+b​Z\)\)ρ,ξρ,f\)=\\displaystyle g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}\(aQ\+bZ\)\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)=dρ​\(gFR​\(a​Q\+b​Z,ξ⋅,f\)\)​\[Pρ\]\\displaystyle\\mathrm\{d\}\_\{\\rho\}\\left\(g^\{\\mathrm\{FR\}\}\\left\(aQ\+bZ,\\xi^\{\\cdot,f\}\\right\)\\right\)\\left\[P\_\{\\rho\}\\right\]=\\displaystyle=a​dρ​\(gFR​\(Q,ξ⋅,f\)\)​\[Pρ\]\+b​dρ​\(gFR​\(Z,ξ⋅,f\)\)​\[Pρ\]\\displaystyle a\\mathrm\{d\}\_\{\\rho\}\\left\(g^\{\\mathrm\{FR\}\}\\left\(Q,\\xi^\{\\cdot,f\}\\right\)\\right\)\\left\[P\_\{\\rho\}\\right\]\+b\\mathrm\{d\}\_\{\\rho\}\\left\(g^\{\\mathrm\{FR\}\}\\left\(Z,\\xi^\{\\cdot,f\}\\right\)\\right\)\\left\[P\_\{\\rho\}\\right\]=\\displaystyle=gρFR​\(\(a​∇PmixQ\+b​∇PmixZ\)ρ,ξρ,f\);\\displaystyle g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\(a\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Q\+b\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Z\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\);hence, we have \([6b](https://arxiv.org/html/2609.05727#S2.E6.2)\)\. Finally, \([6c](https://arxiv.org/html/2609.05727#S2.E6.3)\) is implied by

gρFR​\(\(∇Pmix\(h1​Q\)\)ρ,ξρ,f\)=\\displaystyle g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}\(h\_\{1\}Q\)\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)=dd​ε​gρεFR​\(h1​\(ρε\)​Qρε,ξρε,f\)\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g^\{\\mathrm\{FR\}\}\_\{\\rho^\{\\varepsilon\}\}\\left\(h\_\{1\}\(\\rho^\{\\varepsilon\}\)Q\_\{\\rho^\{\\varepsilon\}\},\\xi^\{\\rho^\{\\varepsilon\},f\}\\right\)\\right\|\_\{\\varepsilon=0\}=\\displaystyle=h1​\(ρ\)​dd​ε​gρεFR​\(Qρε,ξρε,f\)\|ε=0\+dd​ε​h1​\(ρε\)\|ε=0​gρFR​\(Qρ,ξρ,f\)\\displaystyle h\_\{1\}\(\\rho\)\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g^\{\\mathrm\{FR\}\}\_\{\\rho^\{\\varepsilon\}\}\\left\(Q\_\{\\rho^\{\\varepsilon\}\},\\xi^\{\\rho^\{\\varepsilon\},f\}\\right\)\\right\|\_\{\\varepsilon=0\}\+\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}h\_\{1\}\(\\rho^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\}g^\{\\mathrm\{FR\}\}\_\{\\rho\}\\left\(Q\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)=\\displaystyle=h1​\(ρ\)​gρFR​\(\(∇PmixQ\)ρ,ξρ,f\)\+P⁡\[h1\]​\(ρ\)​gρFR​\(Qρ,ξρ,f\)\\displaystyle h\_\{1\}\(\\rho\)g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Q\)\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)\+P\[h\_\{1\}\]\(\\rho\)g^\{\\mathrm\{FR\}\}\_\{\\rho\}\\left\(Q\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)=\\displaystyle=gρFR​\(h1​\(ρ\)​\(∇PmixQ\)ρ\+P⁡\[h1\]​\(ρ\)​Qρ,ξρ,f\)\.\\displaystyle g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(h\_\{1\}\(\\rho\)\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Q\)\_\{\\rho\}\+P\[h\_\{1\}\]\(\\rho\)Q\_\{\\rho\},\\xi^\{\\rho,f\}\\right\)\.Therefore,∇mix\\nabla^\{\\mathrm\{mix\}\}is an affine connection on𝒫\\mathscr\{P\}\. ∎

We also transport the mixture connection∇mix\\nabla^\{\\mathrm\{mix\}\}to𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. For everyP~,Q~∈𝔛⁡\(𝒱can\)\\widetilde\{P\},\\widetilde\{Q\}\\in\\mathfrak\{X\}\(\\mathscr\{V\}\_\{\\mathrm\{can\}\}\), define the tangent vector fieldsP,Q∈𝔛⁡\(𝒫\)P,Q\\in\\mathfrak\{X\}\(\\mathscr\{P\}\)by

Pρ=d𝒞⁡\(ρ\)​𝒞−1​\[P~𝒞⁡\(ρ\)\],Qρ=d𝒞⁡\(ρ\)​𝒞−1​\[Q~𝒞⁡\(ρ\)\]\.P\_\{\\rho\}=\\mathrm\{d\}\_\{\\mathcal\{C\}\(\\rho\)\}\\mathcal\{C\}^\{\-1\}\\left\[\\widetilde\{P\}\_\{\\mathcal\{C\}\(\\rho\)\}\\right\],\\qquad Q\_\{\\rho\}=\\mathrm\{d\}\_\{\\mathcal\{C\}\(\\rho\)\}\\mathcal\{C\}^\{\-1\}\\left\[\\widetilde\{Q\}\_\{\\mathcal\{C\}\(\\rho\)\}\\right\]\.Then, we define the transported mixture connection∇can\\nabla^\{\\mathrm\{can\}\}on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}by

∇P~canQ~∈𝔛⁡\(𝒱can\),\(∇P~canQ~\)𝒞⁡\(ρ\):=dρ​𝒞​\[\(∇PmixQ\)ρ\]\.\\nabla^\{\\mathrm\{can\}\}\_\{\\widetilde\{P\}\}\\widetilde\{Q\}\\in\\mathfrak\{X\}\(\\mathscr\{V\}\_\{\\mathrm\{can\}\}\),\\qquad\\left\(\\nabla^\{\\mathrm\{can\}\}\_\{\\widetilde\{P\}\}\\widetilde\{Q\}\\right\)\_\{\\mathcal\{C\}\(\\rho\)\}:=\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\\left\[\\left\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Q\\right\)\_\{\\rho\}\\right\]\.

## 5Canonical Retraction and Value Ascent

As discussed in Section[4](https://arxiv.org/html/2609.05727#S4), the diffeomorphism𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}allows the geometric structures defined on the density manifold𝒫\\mathscr\{P\}to be transported to the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. At the same time,𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is the natural computational domain, since training is carried out in velocity\-field coordinates\. Accordingly, this section constructs finite\-stepsize updates directly on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. As summarized in Figure[4](https://arxiv.org/html/2609.05727#S4.F4), through the inverse diffeomorphism𝒞−1:𝒱can→∼𝒫\\mathcal\{C\}^\{\-1\}:\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\xrightarrow\{\\sim\}\\mathscr\{P\}, the resulting canonical retraction admits an equivalent density\-coordinate representation on𝒫\\mathscr\{P\}\.

The purpose of this section is to construct an objective\-agnostic finite\-stepsize update rule on the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}that is compatible with the conditional\-expectation structure underlying standard diffusion and flow training\. At this stage, we do not yet specify whether the tangent direction is a gradient, a Newton direction, or another search direction\. Instead, given a canonical velocity fieldv∈𝒱canv\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}and a tangent vectorΓ∈Tv​𝒱can\\Gamma\\in T\_\{v\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}, we construct a finite\-stepsize canonical\-velocity update that returns another canonical velocity fieldv\+∈𝒱canv^\{\+\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}using tractable operations in the velocity space𝒱\\mathscr\{V\}\. Beyond establishing geometric feasibility, we also seek to understand how this update changes the entire induced probability path and its terminal density\. In general, this finite\-stepsize effect is nonlinear and nonlocal, admitting only an implicit characterization through the continuity equation\. However, by exploiting the posterior\-value\-gradient structure of canonical tangent directions, we obtain an exact characterization\. Section[6](https://arxiv.org/html/2609.05727#S6)instantiates this general objective\-agnostic update rule with the reverse\-KL Newton direction and thereby turns it into an optimization algorithm\.

Section[5\.1](https://arxiv.org/html/2609.05727#S5.SS1)constructs the canonical retraction through a tangential update followed by canonicalization, and proves that this update induces a retraction on the density manifold\. Together, these retractions provide velocity\-coordinate and density\-coordinate representations of the same finite\-stepsize update rule\. Section[5\.2](https://arxiv.org/html/2609.05727#S5.SS2)then establishes exact trajectory\-wise and density\-level identities that characterize the finite\-stepsize effect on the induced probability path\. These identities establish posterior\-value improvement at every intermediate time and, at the terminal time, yield an explicit formula for the updated density together with a finite\-stepsize value\-ascent certificate\.

### 5\.1Canonical Retractions in Velocity and Density Coordinates

Fix a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}, its canonical velocity fieldvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}, and a tangent vectorΓρ,f∈Tvρ​𝒱can\\Gamma^\{\\rho,f\}\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. A tangent vector represents an infinitesimal first\-order perturbation of the canonical velocity field, whereas a finite\-stepsize update is required in practice\. Since𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is an embedded submanifold of the Banach space𝒱\\mathscr\{V\}, the tangent vectorΓρ,f\\Gamma^\{\\rho,f\}can also be identified with a velocity\-field displacement in𝒱\\mathscr\{V\}\. As shown in Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5),Γρ,f:\[0,1\]×ℝd→ℝd\\Gamma^\{\\rho,f\}:\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}admits the representation

Γtρ,f\(xt\)=κt∇Vtρ\[f\]\(xt\)=CovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),f\(X1\)\)\.\\Gamma\_\{t\}^\{\\rho,f\}\(x\_\{t\}\)=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),f\(X\_\{1\}\)\\right\)\.We may therefore form the finite\-stepsize velocity\-space update

v¯:=vρ\+Γρ,f∈𝒱,\\bar\{v\}:=v^\{\\rho\}\+\\Gamma^\{\\rho,f\}\\in\\mathscr\{V\},\(18\)which is called thetangential update\.

The tangential update \([18](https://arxiv.org/html/2609.05727#S5.E18)\) is computationally useful since it preserves the conditional\-expectation structure required by CFM\. More precisely, the updated velocity field

v¯t\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|X1\)\+f\(X1\)\(vt\|1\(xt\|X1\)−vtρ\(xt\)\)\]\\bar\{v\}\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\+f\(X\_\{1\}\)\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\\right\]can be learned by shifting the sample\-wise conditional regression target\. Section[8](https://arxiv.org/html/2609.05727#S8)generalizes this target and develops population\-exact implementations of the resulting update\.

Although the updated velocity fieldv¯\\bar\{v\}is a valid element of the velocity space𝒱\\mathscr\{V\}, it does not necessarily lie on the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. The fact thatΓρ,f∈Tvρ​𝒱can\\Gamma^\{\\rho,f\}\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is tangent at the base pointvρ∈𝒱canv^\{\\rho\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}is only a first\-order property; it does not imply that the obtainedv¯\\bar\{v\}remains on the canonical manifold\. Since𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is generally non\-affine,

v¯=vρ\+Γρ,f∉𝒱can​in general\.\\bar\{v\}=v^\{\\rho\}\+\\Gamma^\{\\rho,f\}\\notin\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\text\{ in general\}\.Appendix[A\.4](https://arxiv.org/html/2609.05727#A1.SS4)gives an explicit Gaussian example for which this failure occurs\. Consequently, to continue the update loop, we need a new canonical velocity field\. Althoughv¯\\bar\{v\}is a valid velocity field with a well\-defined terminal density𝒯⁡\(v¯\)∈𝒫\\mathcal\{T\}\(\\bar\{v\}\)\\in\\mathscr\{P\}, it is generally noncanonical and therefore cannot serve directly as the next iterate\.

A retraction is the standard geometric device for converting such a tangent displacement into a feasible point on the manifold\. At a base pointvρ∈𝒱canv^\{\\rho\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}, a retraction is a smooth map

Retrvρ:Tvρ​𝒱can→𝒱can\\mathrm\{Retr\}\_\{v^\{\\rho\}\}:T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}satisfying

Retrvρ​\(0\)=vρ,d0​Retrvρ=IdTvρ​𝒱can\.\\mathrm\{Retr\}\_\{v^\{\\rho\}\}\(0\)=v^\{\\rho\},\\qquad\\mathrm\{d\}\_\{0\}\\mathrm\{Retr\}\_\{v^\{\\rho\}\}=\\mathrm\{Id\}\_\{T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\.Intuitively, the tangent vector specifies both an initial direction and a displacement magnitude, whereas the retraction specifies how that displacement is realized as a finite\-stepsize update on the manifold\. The first conditionRetrvρ​\(0\)=vρ\\mathrm\{Retr\}\_\{v^\{\\rho\}\}\(0\)=v^\{\\rho\}requires that a zero displacement leaves the current point unchanged\. The second conditiond0​Retrvρ=IdTvρ​𝒱can\\mathrm\{d\}\_\{0\}\\mathrm\{Retr\}\_\{v^\{\\rho\}\}=\\mathrm\{Id\}\_\{T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}indicates that the retraction preserves every prescribed tangent direction to first order\. Equivalently, for everyΓ∈Tvρ​𝒱can\\Gamma\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\},

dd​ε​Retrvρ​\(ε​Γ\)\|ε=0=Γ\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathrm\{Retr\}\_\{v^\{\\rho\}\}\(\\varepsilon\\Gamma\)\\right\|\_\{\\varepsilon=0\}=\\Gamma\.Thus, the curveε↦Retrvρ​\(ε​Γ\)\\varepsilon\\mapsto\\mathrm\{Retr\}\_\{v^\{\\rho\}\}\(\\varepsilon\\Gamma\)starts atvρ\{v^\{\\rho\}\}with initial velocityΓ\\Gamma, even though its finite\-stepsize trajectory need not coincide with the straight linevρ\+ε​Γ\{v^\{\\rho\}\}\+\\varepsilon\\Gammain the velocity space𝒱\\mathscr\{V\}\.

When the exponential map of a selected affine connection is locally well\-defined, it provides a natural geometric example of a retraction, i\.e\., one follows the geodesic starting fromvρ\{v^\{\\rho\}\}with initial velocityΓ\\Gamma\. On the canonical manifold, however, evaluating such an update would require solving a nonlinear infinite\-dimensional geodesic equation\. This does not expose the conditional\-expectation structure needed for scalable regression and is generally computationally intractable\.

We instead exploit two structures already available from Section[4](https://arxiv.org/html/2609.05727#S4), namely, the linear structure of𝒱\\mathscr\{V\}and the canonical projectionℛ:𝒱→𝒱can\\mathcal\{R\}:\\mathscr\{V\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. Starting fromvρ∈𝒱canv^\{\\rho\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}, we first perform the tangential update, obtaining the ambient fieldv¯=vρ\+Γ\\bar\{v\}=v^\{\\rho\}\+\\Gamma, and then project the resulting field back onto the canonical manifold:

v\+=ℛ⁡\(v¯\)=𝒞⁡\(𝒯⁡\(v¯\)\)∈𝒱can\.v^\{\+\}=\\mathcal\{R\}\(\\bar\{v\}\)=\\mathcal\{C\}\(\\mathcal\{T\}\(\\bar\{v\}\)\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.We refer to this projection step ascanonicalization\. Intuitively, the terminal\-density map𝒯\\mathcal\{T\}first forgets the particular velocity representation and retains only its terminal density\. The canonical\-velocity map𝒞\\mathcal\{C\}then reconstructs the canonical representative associated with the terminal density\. Since𝒯∘ℛ=𝒯\\mathcal\{T\}\\circ\\mathcal\{R\}=\\mathcal\{T\},v\+v^\{\+\}andv¯\\bar\{v\}share the same terminal densityq=𝒯⁡\(v¯\)=𝒯⁡\(v\+\)q=\\mathcal\{T\}\(\\bar\{v\}\)=\\mathcal\{T\}\(v^\{\+\}\)\. Motivated by this two\-stage construction, define the map

Retrvρcan:Tvρ​𝒱can→𝒱can,Γ↦ℛ⁡\(vρ\+Γ\)\.\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}:T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\},\\qquad\\Gamma\\mapsto\\mathcal\{R\}\(v^\{\\rho\}\+\\Gamma\)\.\(19\)This two\-stage construction is illustrated in Figure[5](https://arxiv.org/html/2609.05727#S5.F5)\.

𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}vρ=𝒞⁡\(ρ\)v^\{\\rho\}=\\mathcal\{C\}\(\\rho\)Tvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}Γρ,f\\Gamma^\{\\rho,f\}v¯=vρ\+η​Γρ,f\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}ℛ\\mathcal\{R\}ℛ⁡\(v¯\)=𝒞⁡\(q\)\\mathcal\{R\}\(\\bar\{v\}\)=\\mathcal\{C\}\(q\)𝔼q​\[f\]≥𝔼ρ​\[f\]\\mathbb\{E\}\_\{q\}\[f\]\\geq\\mathbb\{E\}\_\{\\rho\}\[f\]high𝔼𝒯⁡\(v\)​\[f\]\\mathbb\{E\}\_\{\\mathcal\{T\}\(v\)\}\[f\]lowFigure 5:Canonical retraction and its value\-ascent certificate\. Starting from the canonical velocity fieldvρ=𝒞⁡\(ρ\)v^\{\\rho\}=\\mathcal\{C\}\(\\rho\), the finite\-stepsize tangential updatev¯=vρ\+η​Γρ,f\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}generally leaves the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. Then,v¯\\bar\{v\}is projected back to the canonical manifold asℛ⁡\(v¯\)\\mathcal\{R\}\(\\bar\{v\}\)while preserving the terminal densityq=𝒯⁡\(v¯\)q=\\mathcal\{T\}\(\\bar\{v\}\)\. The induced update satisfies the value\-ascent certificate𝔼X∼q​\[f⁡\(X\)\]≥𝔼X∼ρ​\[f⁡\(X\)\]\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]\\geq\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\.The following proposition states that the map defined in \([19](https://arxiv.org/html/2609.05727#S5.E19)\) is a valid retraction\.

###### Proposition 9\.

Fix the terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}and its canonical velocity fieldvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. The map \([19](https://arxiv.org/html/2609.05727#S5.E19)\) is well\-defined onTvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}and smooth\. Moreover, we have

Retrvρcan​\(0\)=vρ,d0​Retrvρcan=IdTvρ​𝒱can\.\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}\(0\)=\{v^\{\\rho\}\},\\qquad\\mathrm\{d\}\_\{0\}\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}=\\mathrm\{Id\}\_\{T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\.Therefore, the map \([19](https://arxiv.org/html/2609.05727#S5.E19)\) is a retraction, which is called the canonical retraction\.

###### Proof\.

We first prove that the map \([19](https://arxiv.org/html/2609.05727#S5.E19)\) is well\-defined onTvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. According to Definition[1](https://arxiv.org/html/2609.05727#Thmtheorem1),𝒱\\mathscr\{V\}is a Banach space\. Since𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a split embedded manifold of the Banach space𝒱\\mathscr\{V\}, all tangent vectors inTvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}can be identified as elements in𝒱\\mathscr\{V\}\. Therefore, for everyΓ∈Tvρ​𝒱can\\Gamma\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\},vρ\+Γ∈𝒱\{v^\{\\rho\}\}\+\\Gamma\\in\\mathscr\{V\}; hence,ℛ⁡\(vρ\+Γ\)\\mathcal\{R\}\(\{v^\{\\rho\}\}\+\\Gamma\)is well\-defined\.

According to Theorem[3](https://arxiv.org/html/2609.05727#Thmtheorem3), the canonical projectionℛ:𝒱→𝒱can\\mathcal\{R\}:\\mathscr\{V\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}is smooth and satisfiesℛ\|𝒱can=Id𝒱can\\left\.\\mathcal\{R\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathrm\{Id\}\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\. Therefore,Retrvρcan\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}is smooth and satisfiesRetrvρcan​\(0\)=ℛ⁡\(vρ\+0\)=vρ\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}\(0\)=\\mathcal\{R\}\(\{v^\{\\rho\}\}\+0\)=\{v^\{\\rho\}\}\.

Finally, we considerd0​Retrvρcan:Tvρ​𝒱can→Tvρ​𝒱can\\mathrm\{d\}\_\{0\}\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}:T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. To evaluate this differential, consider any path\(vε\)ε∈\(−δ,δ\)⊂𝒱can\(v^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{V\}\_\{\\mathrm\{can\}\}which passes throughvρ\{v^\{\\rho\}\}atε=0\\varepsilon=0\. Then, we haveℛ⁡\(vε\)≡vε\\mathcal\{R\}\(v^\{\\varepsilon\}\)\\equiv v^\{\\varepsilon\}\. ForΓ=dd​ε​vε\|ε=0∈Tvρ​𝒱can\\Gamma=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}v^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}, differentiating atε=0\\varepsilon=0gives

dvρ​ℛ:Tvρ​𝒱→Tvρ​𝒱can,dvρ​ℛ​\[Γ\]=Γ\.\\mathrm\{d\}\_\{v^\{\\rho\}\}\\mathcal\{R\}:T\_\{v^\{\\rho\}\}\\mathscr\{V\}\\to T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\},\\qquad\\mathrm\{d\}\_\{v^\{\\rho\}\}\\mathcal\{R\}\[\\Gamma\]=\\Gamma\.The chain rule impliesd0​Retrvρcan=IdTvρ​𝒱can\\mathrm\{d\}\_\{0\}\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}=\\mathrm\{Id\}\_\{T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\. ∎

As shown in Figure[4](https://arxiv.org/html/2609.05727#S4.F4), the canonical retractionRetrcan\\mathrm\{Retr\}^\{\\mathrm\{can\}\}can be transported back to the density manifold𝒫\\mathscr\{P\}through the inverse diffeomorphism𝒞−1=𝒯\|𝒱can:𝒱can→∼𝒫\\mathcal\{C\}^\{\-1\}=\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}:\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\xrightarrow\{\\sim\}\\mathscr\{P\}\. Forρ∈𝒫\\rho\\in\\mathscr\{P\},vρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}, define

Retrρden:=𝒞−1∘Retrvρcan∘dρ​𝒞:Tρ​𝒫→𝒫\.\\mathrm\{Retr\}\_\{\\rho\}^\{\\mathrm\{den\}\}:=\\mathcal\{C\}^\{\-1\}\\circ\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}\\circ\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}:T\_\{\\rho\}\\mathscr\{P\}\\to\\mathscr\{P\}\.\(20\)The following proposition states that the map \([20](https://arxiv.org/html/2609.05727#S5.E20)\) is indeed a retraction on the density manifold𝒫\\mathscr\{P\}\.

###### Proposition 10\.

The map \([20](https://arxiv.org/html/2609.05727#S5.E20)\) is a retraction on the density manifold𝒫\\mathscr\{P\}\. In particular, for a tangent vectorξρ,f=ρ⋅\(f−𝔼ρ​\[f\]\)∈Tρ​𝒫\\xi^\{\\rho,f\}=\\rho\\cdot\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\)\\in T\_\{\\rho\}\\mathscr\{P\}, the updated density is

Retrρden​\(ξρ,f\)=𝒯⁡\(vρ\+Γρ,f\)\.\\mathrm\{Retr\}\_\{\\rho\}^\{\\mathrm\{den\}\}\(\\xi^\{\\rho,f\}\)=\\mathcal\{T\}\(v^\{\\rho\}\+\\Gamma^\{\\rho,f\}\)\.

###### Proof\.

Since𝒞−1\\mathcal\{C\}^\{\-1\},Retrvρcan\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}, anddρ​𝒞\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}are smooth,Retrρden\\mathrm\{Retr\}\_\{\\rho\}^\{\\mathrm\{den\}\}is smooth\. For a tangent vectorξρ,f=ρ⋅\(f−𝔼ρ​\[f\]\)∈Tρ​𝒫\\xi^\{\\rho,f\}=\\rho\\cdot\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\)\\in T\_\{\\rho\}\\mathscr\{P\}, we have

Retrρden​\(ξρ,f\)=𝒞−1​\(Retrvρcan​\(dρ​𝒞​\[ξρ,f\]\)\)=𝒯⁡\(Retrvρcan​\(Γρ,f\)\)=𝒯⁡\(vρ\+Γρ,f\)\.\\mathrm\{Retr\}\_\{\\rho\}^\{\\mathrm\{den\}\}\(\\xi^\{\\rho,f\}\)=\\mathcal\{C\}^\{\-1\}\\left\(\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}\(\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\xi^\{\\rho,f\}\]\)\\right\)=\\mathcal\{T\}\\left\(\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}\(\\Gamma^\{\\rho,f\}\)\\right\)=\\mathcal\{T\}\(v^\{\\rho\}\+\\Gamma^\{\\rho,f\}\)\.In particular, forξρ,f=0\\xi^\{\\rho,f\}=0, i\.e\.,Γρ,f=dρ​𝒞​\[ξρ,f\]=0\\Gamma^\{\\rho,f\}=\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\xi^\{\\rho,f\}\]=0, we have

Retrρden​\(0\)=𝒯⁡\(vρ\+0\)=ρ\.\\mathrm\{Retr\}\_\{\\rho\}^\{\\mathrm\{den\}\}\(0\)=\\mathcal\{T\}\(v^\{\\rho\}\+0\)=\\rho\.For the differential, the chain rule implies

d0​Retrρden=dvρ​𝒞−1∘d0​Retrvρcan∘dρ​𝒞=dvρ​𝒞−1∘IdTvρ​𝒱can∘dρ​𝒞=IdTρ​𝒫\.\\mathrm\{d\}\_\{0\}\\mathrm\{Retr\}\_\{\\rho\}^\{\\mathrm\{den\}\}=\\mathrm\{d\}\_\{v^\{\\rho\}\}\\mathcal\{C\}^\{\-1\}\\circ\\mathrm\{d\}\_\{0\}\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}\\circ\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}=\\mathrm\{d\}\_\{v^\{\\rho\}\}\\mathcal\{C\}^\{\-1\}\\circ\\mathrm\{Id\}\_\{T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\\circ\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}=\\mathrm\{Id\}\_\{T\_\{\\rho\}\\mathscr\{P\}\}\.Therefore,Retrρden\\mathrm\{Retr\}\_\{\\rho\}^\{\\mathrm\{den\}\}is a retraction on𝒫\\mathscr\{P\}\. ∎

For a stepsizeη\>0\\eta\>0, we evaluate the canonical and density retractions along the scaled tangent directionη​Γρ,f=Γρ,η​f\\eta\\Gamma^\{\\rho,f\}=\\Gamma^\{\\rho,\\eta f\}:

v¯=vρ\+η​Γρ,f∈𝒱,\\displaystyle\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\\in\\mathscr\{V\},\(21a\)q=𝒯⁡\(v¯\)=𝒯⁡\(vρ\+η​Γρ,f\)∈𝒫,\\displaystyle q=\\mathcal\{T\}\(\\bar\{v\}\)=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\)\\in\\mathscr\{P\},\(21b\)vq=𝒞⁡\(q\)=ℛ⁡\(vρ\+η​Γρ,f\)∈𝒱can\.\\displaystyle v^\{q\}=\\mathcal\{C\}\(q\)=\\mathcal\{R\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.\(21c\)The stepsizeη\\etacontrols the displacement magnitude\. As summarized in Figure[5](https://arxiv.org/html/2609.05727#S5.F5), the canonical retraction consists of two steps: tangential update \([21a](https://arxiv.org/html/2609.05727#S5.E21.1)\) and canonicalization \([21b](https://arxiv.org/html/2609.05727#S5.E21.2)\)–\([21c](https://arxiv.org/html/2609.05727#S5.E21.3)\)\.

### 5\.2Value\-Ascent Certificate

Section[5\.1](https://arxiv.org/html/2609.05727#S5.SS1)constructs a geometrically feasible and CFM\-compatible finite\-stepsize update rule\. The retraction properties determine where the updated iterate lives, but they do not yet characterize how the finite\-stepsize velocity\-space perturbation changes the induced probability path or terminal density\. In this subsection, we show that canonical tangent directions generated by posterior values admit an exact dynamical characterization\.

Let\(νt\)t∈\[0,1\]\(\\nu\_\{t\}\)\_\{t\\in\[0,1\]\}denote the density path generated byv¯\\bar\{v\}, and recall that\(ptρ\)t∈\[0,1\]\(p\_\{t\}^\{\\rho\}\)\_\{t\\in\[0,1\]\}is the density path generated byvρv^\{\\rho\}\. These two paths satisfy

ν0=p0ρ=p0,ν1=q,p1ρ=ρ\.\\nu\_\{0\}=p\_\{0\}^\{\\rho\}=p\_\{0\},\\qquad\\nu\_\{1\}=q,\\qquad p\_\{1\}^\{\\rho\}=\\rho\.The following proposition gives the central dynamics governing their difference\.

###### Proposition 11\.

Consider the canonical retraction \([21](https://arxiv.org/html/2609.05727#S5.E21)\) with a stepsizeη\>0\\eta\>0\. Along every characteristic

d​Ytd​t=v¯t​\(Yt\),\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=\\bar\{v\}\_\{t\}\(Y\_\{t\}\),we have the trajectory\-wise identity

dd​t​\[log⁡ptρ​\(Yt\)νt​\(Yt\)\+η⁡\(Vtρ​\[f\]​\(Yt\)−𝔼ρ​\[f\]\)\]=η2​κt​‖∇Vtρ​\[f\]​\(Yt\)‖22≥0\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\[\\log\\frac\{p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\}\{\\nu\_\{t\}\(Y\_\{t\}\)\}\+\\eta\\left\(V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\)\\right\]=\\eta^\{2\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\geq 0\.\(22\)Consequently, we have the density\-level identity

dd​t​\(𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼ρ​\[f\]−1η​KL⁡\(νt∥ptρ\)\)=η​κt​𝔼X∼νt​\[‖∇Vtρ​\[f\]​\(X\)‖22\]≥0\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]\-\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)\\right\)=\\eta\\kappa\_\{t\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]\\geq 0\.\(23\)In other words,

𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼ρ​\[f\]=1η​KL⁡\(νt∥ptρ\)\+η​∫0tκs​𝔼X∼νs​\[‖∇Vsρ​\[f\]​\(X\)‖22\]​𝑑s≥0\.\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]=\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)\+\\eta\\int\_\{0\}^\{t\}\\kappa\_\{s\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{s\}\}\\left\[\\left\\\|\\nabla V\_\{s\}^\{\\rho\}\[f\]\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}s\\geq 0\.\(24\)

###### Proof\.

Along the characteristicd​Ytd​t=v¯t​\(Yt\)\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=\\bar\{v\}\_\{t\}\(Y\_\{t\}\), the continuity equation of\(νt,v¯t\)\(\\nu\_\{t\},\\bar\{v\}\_\{t\}\)gives

dd​tlogνt\(Yt\)=∂tlogνt\(Yt\)\+v¯t\(Yt\)⋅∇logνt\(Yt\)=−∇⋅v¯t\(Yt\)\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log\\nu\_\{t\}\(Y\_\{t\}\)=\\partial\_\{t\}\\log\\nu\_\{t\}\(Y\_\{t\}\)\+\\bar\{v\}\_\{t\}\(Y\_\{t\}\)\\cdot\\nabla\\log\\nu\_\{t\}\(Y\_\{t\}\)=\-\\nabla\\cdot\\bar\{v\}\_\{t\}\(Y\_\{t\}\)\.The continuity equation of\(ptρ,vtρ\)\(p\_\{t\}^\{\\rho\},v\_\{t\}^\{\\rho\}\)gives

dd​t​log⁡ptρ​\(Yt\)=\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)=∂tlogptρ\(Yt\)\+\(vtρ\(Yt\)\+ηκt∇Vtρ\[f\]\(Yt\)\)⋅∇logptρ\(Yt\)\\displaystyle\\partial\_\{t\}\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\left\(v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\eta\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\)\\cdot\\nabla\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)=\\displaystyle=−∇⋅vtρ\(Yt\)\+ηκt∇Vtρ\[f\]\(Yt\)⋅∇logptρ\(Yt\)\.\\displaystyle\-\\nabla\\cdot v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\eta\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\cdot\\nabla\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\.Therefore,

dd​tlogptρ​\(Yt\)νt​\(Yt\)=ηκt∇Vtρ\[f\]\(Yt\)⋅∇logptρ\(Yt\)\+ηκtΔVtρ\[f\]\(Yt\)\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log\\frac\{p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\}\{\\nu\_\{t\}\(Y\_\{t\}\)\}=\\eta\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\cdot\\nabla\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\eta\\kappa\_\{t\}\\Delta V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\.
As provided in Appendix[D\.2](https://arxiv.org/html/2609.05727#A4.SS2), the backward equation \([172](https://arxiv.org/html/2609.05727#A4.E172)\) for the posterior value is

∂tVtρ\[f\]\(Yt\)\+\(vtρ\(Yt\)\+κt∇logptρ\(Yt\)\)⋅∇Vtρ\[f\]\(Yt\)\+κtΔVtρ\[f\]\(Yt\)=0;\\partial\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\+\\left\(v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\kappa\_\{t\}\\nabla\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\\right\)\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\+\\kappa\_\{t\}\\Delta V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)=0;hence, along the characteristicd​Ytd​t=v¯t​\(Yt\)\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=\\bar\{v\}\_\{t\}\(Y\_\{t\}\), the posterior value satisfies

dd​t​Vtρ​\[f\]​\(Yt\)=\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)=∂tVtρ\[f\]\(Yt\)\+\(vtρ\(Yt\)\+ηκt∇Vtρ\[f\]\(Yt\)\)⋅∇Vtρ\[f\]\(Yt\)\\displaystyle\\partial\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\+\\left\(v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\eta\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\)\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)=\\displaystyle=∂tVtρ​\[f\]​\(Yt\)\+vtρ​\(Yt\)⋅∇Vtρ​\[f\]​\(Yt\)\+η​κt​‖∇Vtρ​\[f\]​\(Yt\)‖22\\displaystyle\\partial\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\+v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\+\\eta\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=−κt∇logptρ\(Yt\)⋅∇Vtρ\[f\]\(Yt\)−κtΔVtρ\[f\]\(Yt\)\+ηκt‖∇Vtρ\[f\]\(Yt\)‖22\.\\displaystyle\-\\kappa\_\{t\}\\nabla\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\-\\kappa\_\{t\}\\Delta V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\+\\eta\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\.Therefore, \([22](https://arxiv.org/html/2609.05727#S5.E22)\) holds\. Taking expectation overY0∼p0Y\_\{0\}\\sim p\_\{0\}, which impliesYt∼νtY\_\{t\}\\sim\\nu\_\{t\}, gives \([23](https://arxiv.org/html/2609.05727#S5.E23)\)\. Integrating from00tottgives \([24](https://arxiv.org/html/2609.05727#S5.E24)\)\. ∎

The density\-level identity \([23](https://arxiv.org/html/2609.05727#S5.E23)\) provides an intuitive dynamical picture of how posterior\-value improvement and marginal KL divergence evolve along the updated probability path\. By the definition ofVtρ​\[f\]V\_\{t\}^\{\\rho\}\[f\]and the law of total expectation, its expectation under the marginalptρp\_\{t\}^\{\\rho\}equals𝔼ρ​\[f\]\\mathbb\{E\}\_\{\\rho\}\[f\]at every timett:

𝔼X∼ptρ\[Vtρ\[f\]\(X\)\]=𝔼Xt∼ptρ\[𝔼X1∼p1\|tρ\(⋅\|Xt\)\[f\(X1\)\]\]=𝔼ρ\[f\]\.\\mathbb\{E\}\_\{X\\sim p\_\{t\}^\{\\rho\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]=\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\}\}\\left\[\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[f\(X\_\{1\}\)\\right\]\\right\]=\\mathbb\{E\}\_\{\\rho\}\[f\]\.Therefore,

𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼ρ​\[f\]=𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼X∼ptρ​\[Vtρ​\[f\]​\(X\)\]\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]=\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{X\\sim p\_\{t\}^\{\\rho\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]measures the expected posterior\-value gain of the updated marginalνt\\nu\_\{t\}relative to the marginalptρp\_\{t\}^\{\\rho\}at the same timett\. This gain starts from zero,

\(𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼ρ​\[f\]\)\|t=0=0,\\left\.\\left\(\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\)\\right\|\_\{t=0\}=0,and terminates at the terminal\-value improvement

\(𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼ρ​\[f\]\)\|t=1=𝔼X∼q​\[f⁡\(X\)\]−𝔼X∼ρ​\[f⁡\(X\)\]\.\\left\.\\left\(\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\)\\right\|\_\{t=1\}=\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\.Equation \([24](https://arxiv.org/html/2609.05727#S5.E24)\) decomposes this posterior\-value gain into two nonnegative terms: the scaled marginal KL divergence1η​KL⁡\(νt∥ptρ\)\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)and the integral term

η​∫0tκs​𝔼X∼νs​\[‖∇Vsρ​\[f\]​\(X\)‖22\]​𝑑s\.\\eta\\int\_\{0\}^\{t\}\\kappa\_\{s\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{s\}\}\\left\[\\left\\\|\\nabla V\_\{s\}^\{\\rho\}\[f\]\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}s\.
ttOO11𝔼q​\[f\]−𝔼ρ​\[f\]\\mathbb\{E\}\_\{q\}\[f\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]1η​KL⁡\(q∥ρ\)\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)𝔼νt​\[Vtρ​\[f\]\]−𝔼ptρ​\[Vtρ​\[f\]\]\\mathbb\{E\}\_\{\\nu\_\{t\}\}\\\!\\left\[V\_\{t\}^\{\\rho\}\[f\]\\right\]\-\\mathbb\{E\}\_\{p\_\{t\}^\{\\rho\}\}\\\!\\left\[V\_\{t\}^\{\\rho\}\[f\]\\right\]1η​KL⁡\(νt∥ptρ\)\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)η​∫0tκs​𝔼νs​\[‖∇Vsρ​\[f\]‖22\]​𝑑s≥0\\eta\\int\_\{0\}^\{t\}\\kappa\_\{s\}\\,\\mathbb\{E\}\_\{\\nu\_\{s\}\}\\\!\\left\[\\left\\\|\\nabla V\_\{s\}^\{\\rho\}\[f\]\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}s\\geq 0Figure 6:Mechanism of finite\-stepsize value ascent\. Along the updated density path\(νt\)t∈\[0,1\]\(\\nu\_\{t\}\)\_\{t\\in\[0,1\]\}generated byv¯=vρ\+η​Γρ,f\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}, the expected posterior\-value improvement relative to the density path\(ptρ\)t∈\[0,1\]\(p\_\{t\}^\{\\rho\}\)\_\{t\\in\[0,1\]\}decomposes into the scaled marginal divergence1η​KL⁡\(νt∥ptρ\)\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)and an accumulated nonnegative quadratic termη​∫0tκs​𝔼X∼νs​\[‖∇Vsρ​\[f\]​\(X\)‖22\]​𝑑s\\eta\\int\_\{0\}^\{t\}\\kappa\_\{s\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{s\}\}\\left\[\\left\\\|\\nabla V\_\{s\}^\{\\rho\}\[f\]\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}s\. Att=1t=1, this decomposition yields the terminal value improvement𝔼X∼q​\[f⁡\(X\)\]−𝔼X∼ρ​\[f⁡\(X\)\]≥0\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\geq 0\.As illustrated in Figure[6](https://arxiv.org/html/2609.05727#S5.F6), neither the posterior\-value gain nor the scaled marginal KL divergence is required to be monotone intt\. Their difference, however, is exactly the integral term and is therefore nondecreasing\. Equivalently, at every timet∈\[0,1\]t\\in\[0,1\],

𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼ρ​\[f\]≥1η​KL⁡\(νt∥ptρ\)≥0,\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\geq\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)\\geq 0,and the excess of the posterior\-value gain over the scaled KL divergence accumulates monotonically along the updated probability path\. At the terminal timet=1t=1, the fully accumulated gain becomes𝔼q​\[f\]−𝔼ρ​\[f\]\\mathbb\{E\}\_\{q\}\[f\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]\.

The subsequent canonicalization step \([21c](https://arxiv.org/html/2609.05727#S5.E21.3)\) forgets both the particular noncanonical velocity representationv¯\\bar\{v\}and its intermediate marginal path\(νt\)t∈\(0,1\)\(\\nu\_\{t\}\)\_\{t\\in\(0,1\)\}, while retaining the terminal densityq=𝒯⁡\(v¯\)q=\\mathcal\{T\}\(\\bar\{v\}\)\. Consequently, the complete terminal\-value improvement accumulated along the updated path is preserved, since𝔼q​\[f\]−𝔼ρ​\[f\]\\mathbb\{E\}\_\{q\}\[f\]\-\\mathbb\{E\}\_\{\\rho\}\[f\]depends only on the retained terminal densityqq\. At the same time, the projection replacesv¯\\bar\{v\}by the canonical representativevq=𝒞⁡\(q\)v^\{q\}=\\mathcal\{C\}\(q\), thereby providing a new canonical marginal path\(ptq\)t∈\[0,1\]\(p\_\{t\}^\{q\}\)\_\{t\\in\[0,1\]\}and the corresponding posterior\-value structures required by the next update\. In general, the new canonical pathptqp\_\{t\}^\{q\}need not coincide with the intermediate updated pathνt\\nu\_\{t\}fort∈\(0,1\)t\\in\(0,1\), even though both share the same source densityp0p\_\{0\}and terminal densityqq\. The above results are summarized in the following theorem\.

###### Theorem 12\.

Define the path dissipation𝒟ρ,η,f:ℝd→\[0,∞\)\\mathcal\{D\}\_\{\\rho,\\eta,f\}:\\mathbb\{R\}^\{d\}\\to\[0,\\infty\)by

𝒟ρ,η,f​\(x\):=η2​∫01κt​‖∇Vtρ​\[f\]​\(Φ1→tρ,η,f​\(x\)\)‖22​𝑑t≥0,\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(x\):=\\eta^\{2\}\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,f\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\geq 0,\(25\)whereΦρ,η,f\\Phi^\{\\rho,\\eta,f\}is the flow map generated by the velocity fieldv¯=vρ\+η​Γρ,f\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\. The updated terminal density \([21b](https://arxiv.org/html/2609.05727#S5.E21.2)\) is

q⁡\(x\)=ρ⁡\(x\)​exp⁡\(η⁡\(f⁡\(x\)−𝔼X∼ρ​\[f⁡\(X\)\]\)−𝒟ρ,η,f​\(x\)\),q\(x\)=\\rho\(x\)\\exp\\left\(\\eta\\left\(f\(x\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(x\)\\right\),\(26\)which, forη\>0\\eta\>0, enjoys the value\-ascent certificate

𝔼X∼q​\[f⁡\(X\)\]−𝔼X∼ρ​\[f⁡\(X\)\]=1η​KL⁡\(q∥ρ\)\+1η​𝔼X∼q​\[𝒟ρ,η,f​\(X\)\]≥0\.\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]=\\frac\{1\}\{\\eta\}\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)\+\\frac\{1\}\{\\eta\}\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]\\geq 0\.\(27\)Furthermore, we have

𝔼X∼q\[f\(X\)\]=𝔼X∼ρ\[f\(X\)\]⇔ρ=q⇔f≡const\.\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]=\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\quad\\Leftrightarrow\\quad\\rho=q\\quad\\Leftrightarrow\\quad f\\equiv\\mathrm\{const\}\.

###### Proof\.

Integrating \([22](https://arxiv.org/html/2609.05727#S5.E22)\) fromt=0t=0tot=1t=1gives \([26](https://arxiv.org/html/2609.05727#S5.E26)\)\. Substitutingt=1t=1into \([24](https://arxiv.org/html/2609.05727#S5.E24)\) gives \([27](https://arxiv.org/html/2609.05727#S5.E27)\)\. The remaining task is to prove the necessity and sufficiency of the equality\. First, if𝔼X∼q​\[f⁡\(X\)\]=𝔼X∼ρ​\[f⁡\(X\)\]\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]=\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]holds, then \([27](https://arxiv.org/html/2609.05727#S5.E27)\) impliesKL⁡\(q∥ρ\)=0\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)=0\. By the continuity ofρ,q\\rho,q, we haveρ=q\\rho=q\. Second,ρ=q\\rho=qimplies𝔼X∼q​\[𝒟ρ,η,f​\(X\)\]=0\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]=0\. By \([25](https://arxiv.org/html/2609.05727#S5.E25)\), we have∇Vtρ​\[f\]​\(Φ1→tρ,η,f​\(x\)\)a\.e\.0\\nabla V\_\{t\}^\{\\rho\}\[f\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,f\}\(x\)\\right\)\\mathrel\{\\mathop\{\\vbox\{\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\}\\kern 1\.50696pt\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\} \}\}\\limits^\{\\mathrm\{a\.e\.\}\}\}0inℝd\\mathbb\{R\}^\{d\}\. Since the flow map is a diffeomorphism, we have∇Vtρ​\[f\]​\(x\)=0\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\)=0fort∈\(0,1\)t\\in\(0,1\)andx∈ℝdx\\in\\mathbb\{R\}^\{d\}almost everywhere \(a\.e\.\)\. Since𝔼ρ​\[\|f\|\]<∞\\mathbb\{E\}\_\{\\rho\}\[\\left\|f\\right\|\]<\\inftyand the convolution kernel is a nondegenerate Gaussian,Vtρ​\[f\]V\_\{t\}^\{\\rho\}\[f\]is smooth; hence,Vtρ​\[f\]≡ctV\_\{t\}^\{\\rho\}\[f\]\\equiv c\_\{t\}is constant\. By Proposition[7](https://arxiv.org/html/2609.05727#Thmtheorem7),f≡constf\\equiv\\mathrm\{const\}\. Third, iff≡constf\\equiv\\mathrm\{const\}, then𝔼X∼q​\[f⁡\(X\)\]=𝔼X∼ρ​\[f⁡\(X\)\]\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]=\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\. ∎

Remarkably, Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12)provides a finite\-stepsize guarantee, rather than merely a first\-order ascent result for infinitesimal perturbations\. The increase in the expectation offfis decomposed exactly into a KL displacement ofqqfromρ\\rhoand a nonnegative path\-dissipation term\. This decomposition provides the stability mechanism behind the finite\-stepsize KL descent of Newton Matching in Section[7\.1\.1](https://arxiv.org/html/2609.05727#S7.SS1.SSS1)\. Extensions with a time\-dependent step scheduleηt\\eta\_\{t\}and a non\-tangent update direction are provided in Appendix[G](https://arxiv.org/html/2609.05727#A7)\.

## 6Newton Matching on the Canonical Manifold

Section[3\.2](https://arxiv.org/html/2609.05727#S3.SS2)motivates restricting the search to canonical velocity fields\. With the canonical manifold and its update mechanism now established in Sections[4](https://arxiv.org/html/2609.05727#S4)and[5](https://arxiv.org/html/2609.05727#S5), we can state this optimization problem rigorously\. The optimization problem on the density manifold𝒫\\mathscr\{P\}is

minρ∈𝒫⁡J⁡\(ρ\),J⁡\(ρ\):=KL⁡\(ρ∥πμ,τ,r\)\.\\min\_\{\\rho\\in\\mathscr\{P\}\}J\(\\rho\),\\qquad J\(\\rho\):=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\_\{\\mu,\\tau,r\}\\right\)\.By Theorem[3](https://arxiv.org/html/2609.05727#Thmtheorem3), the canonical\-velocity map𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a diffeomorphism\. Therefore, the density\-coordinate problem is equivalent to the velocity\-coordinate problem

minv∈𝒱can⁡Jcan​\(v\),Jcan​\(v\):=KL⁡\(𝒯⁡\(v\)∥πμ,τ,r\)\.\\min\_\{v\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}J^\{\\mathrm\{can\}\}\(v\),\\qquad J^\{\\mathrm\{can\}\}\(v\):=\\operatorname\{KL\}\\left\(\\mathcal\{T\}\(v\)\\middle\\\|\\pi\_\{\\mu,\\tau,r\}\\right\)\.Under the assumptionπμ,τ,r∈𝒫\\pi\_\{\\mu,\\tau,r\}\\in\\mathscr\{P\}, the unique minimizers are

ρ⋆=πμ,τ,r,v⋆=𝒞⁡\(πμ,τ,r\)\.\\rho^\{\\star\}=\\pi\_\{\\mu,\\tau,r\},\\qquad v^\{\\star\}=\\mathcal\{C\}\(\\pi\_\{\\mu,\\tau,r\}\)\.
These equivalent optimization problems specify their respective objectives and minimizers, but they do not yet specify the tangent direction used to approach them\. Recall from Section[5](https://arxiv.org/html/2609.05727#S5)that, at a current canonical velocity fieldvρ=𝒞⁡\(ρ\)v^\{\\rho\}=\\mathcal\{C\}\(\\rho\), the canonical retraction maps any tangent vectorΓ∈Tvρ​𝒱can\\Gamma\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}to a finite\-stepsize feasible update:

Retrvρcan​\(Γ\)=ℛ⁡\(vρ\+Γ\)∈𝒱can\.\\mathrm\{Retr\}\_\{v^\{\\rho\}\}^\{\\mathrm\{can\}\}\(\\Gamma\)=\\mathcal\{R\}\(v^\{\\rho\}\+\\Gamma\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.The remaining question is therefore which tangent vectorΓ\\Gammashould be supplied to this retraction for the reverse\-KL objective\. By Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5), every canonical tangent vector can be represented asΓρ,f\\Gamma^\{\\rho,f\}for a terminal observableff, withΓtρ,f=κt∇Vtρ\[f\]\\Gamma\_\{t\}^\{\\rho,f\}=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\. Thus, we need to identify the density\-dependent terminal observablefρf^\{\\rho\}whose canonical liftΓρ,fρ\\Gamma^\{\\rho,f^\{\\rho\}\}gives the appropriate search direction\.

Since the problem is formulated as an optimization problem on a Riemannian manifold, two standard choices are the negative Riemannian gradient, which uses first\-order information, and the Newton direction, which uses the Hessian defined by an affine connection\. For the Fisher–Rao metric and mixture connection introduced in Section[4](https://arxiv.org/html/2609.05727#S4), we will prove that these two directions coincide\. This tangent direction then instantiates the objective\-agnostic canonical retraction of Section[5](https://arxiv.org/html/2609.05727#S5)\.

Section[6\.1](https://arxiv.org/html/2609.05727#S6.SS1)derives the negative Fisher–Rao gradient and transports it from the density manifold𝒫\\mathscr\{P\}to the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. Section[6\.2](https://arxiv.org/html/2609.05727#S6.SS2)derives the Newton direction under the mixture connection and proves that it coincides with the negative Fisher–Rao gradient\. Finally, Section[6\.3](https://arxiv.org/html/2609.05727#S6.SS3)applies the canonical retraction to this tangent direction, thereby defining the Newton Matching iteration and deriving the exact formula for the terminal\-density update, which is important for the convergence analysis in Section[7](https://arxiv.org/html/2609.05727#S7)\.

### 6\.1Fisher–Rao Gradient

We first derive the first\-order choice of the tangent direction for the canonical retraction\. Since the reverse\-KL objectiveJJis naturally defined on the density manifold𝒫\\mathscr\{P\}, it is convenient to derive its gradient in density coordinates and then transport the resulting tangent vector to the canonical manifold throughdρ​𝒞\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\. The derivation proceeds in three steps\. We first compute the differentialdρ​J∈Tρ∗​𝒫\\mathrm\{d\}\_\{\\rho\}J\\in T\_\{\\rho\}^\{\*\}\\mathscr\{P\}\. Then, we identify its tangent\-vector representation under the Fisher–Rao metric\. Finally, we canonically lift this density tangent vector toTvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}, obtaining the negative\-gradient direction that can be supplied to the canonical retraction\.

For the first step, we compute the differentialdρ​J\\mathrm\{d\}\_\{\\rho\}J\. Consider a smooth curve\(ρε\)ε∈\(−δ,δ\)⊂𝒫\(\\rho^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{P\}which passes throughρ\\rhoatε=0\\varepsilon=0, whereξ=dd​ε​ρε\|ε=0∈Tρ​𝒫\\xi=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}\\in T\_\{\\rho\}\\mathscr\{P\}denotes the tangent vector\. According to Proposition[4](https://arxiv.org/html/2609.05727#Thmtheorem4), we have

∫ℝdξ⁡\(x\)​𝑑x=0\.\\int\_\{\\mathbb\{R\}^\{d\}\}\\xi\(x\)\\,\\mathrm\{d\}x=0\.Along the tangent vectorξ∈Tρ​𝒫\\xi\\in T\_\{\\rho\}\\mathscr\{P\}, the first variation ofJJis

dd​ε​J​\(ρε\)\|ε=0=∫ℝd\(log⁡ρ⁡\(x\)πμ,τ,r​\(x\)\+1\)​ξ​\(x\)​𝑑x=∫ℝdlog⁡ρ⁡\(x\)πμ,τ,r​\(x\)​ξ​\(x\)​𝑑x\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}J\(\\rho^\{\\varepsilon\}\)\\right\|\_\{\\varepsilon=0\}=\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\(\\log\\frac\{\\rho\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\+1\\right\)\\xi\(x\)\\,\\mathrm\{d\}x=\\int\_\{\\mathbb\{R\}^\{d\}\}\\log\\frac\{\\rho\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\\xi\(x\)\\,\\mathrm\{d\}x\.By definition, the differential ofJJatρ\\rhois a linear functional ofξ\\xi:

dρ​J∈Tρ∗​𝒫,dρ​J:Tρ​𝒫→ℝ,ξ↦dρ​J​\[ξ\]:=∫ℝdlog⁡ρ⁡\(x\)πμ,τ,r​\(x\)​ξ​\(x\)​𝑑x\.\\mathrm\{d\}\_\{\\rho\}J\\in T\_\{\\rho\}^\{\*\}\\mathscr\{P\},\\qquad\\mathrm\{d\}\_\{\\rho\}J:T\_\{\\rho\}\\mathscr\{P\}\\to\\mathbb\{R\},\\qquad\\xi\\mapsto\\mathrm\{d\}\_\{\\rho\}J\[\\xi\]:=\\int\_\{\\mathbb\{R\}^\{d\}\}\\log\\frac\{\\rho\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\\xi\(x\)\\,\\mathrm\{d\}x\.
For the second step, to obtain an admissible update direction inTρ​𝒫T\_\{\\rho\}\\mathscr\{P\}, we compute the metric dual ofdρ​J\\mathrm\{d\}\_\{\\rho\}Junder the Fisher–Rao metric, i\.e\., the Fisher–Rao gradient\. Recall from Section[4\.2\.3](https://arxiv.org/html/2609.05727#S4.SS2.SSS3)that𝒫\\mathscr\{P\}is equipped with the Fisher–Rao metricgρFRg\_\{\\rho\}^\{\\mathrm\{FR\}\}in \([15](https://arxiv.org/html/2609.05727#S4.E15)\)\. The Fisher–Rao gradientgradFR⁡Jρ∈Tρ​𝒫\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\\in T\_\{\\rho\}\\mathscr\{P\}ofJJatρ\\rhois a tangent vector which satisfies

∀ξ∈Tρ​𝒫​,​gρFR​\(gradFR⁡Jρ,ξ\)=dρ​J​\[ξ\]\.\\forall\\xi\\in T\_\{\\rho\}\\mathscr\{P\}\\text\{, \}g\_\{\\rho\}^\{\\mathrm\{FR\}\}\(\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\},\\xi\)=\\mathrm\{d\}\_\{\\rho\}J\[\\xi\]\.In other words, we need to find the unique tangent vectorgradFR⁡Jρ∈Tρ​𝒫\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\\in T\_\{\\rho\}\\mathscr\{P\}such that

∀ξ∈Tρ​𝒫,∫ℝd\(gradFR⁡Jρ​\(x\)ρ⁡\(x\)−log⁡ρ⁡\(x\)πμ,τ,r​\(x\)\)​ξ​\(x\)​𝑑x=0\.\\forall\\xi\\in T\_\{\\rho\}\\mathscr\{P\},\\,\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\(\\frac\{\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\(x\)\}\{\\rho\(x\)\}\-\\log\\frac\{\\rho\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\\right\)\\xi\(x\)\\,\\mathrm\{d\}x=0\.\(28\)Corollary[73](https://arxiv.org/html/2609.05727#Thmtheorem73)shows that

ξρ,log⁡\(ρ/πμ,τ,r\)=ρ⋅\(log⁡ρπμ,τ,r−𝔼X∼ρ​\[log⁡ρ⁡\(X\)πμ,τ,r​\(X\)\]\)\\xi^\{\\rho,\\log\(\\rho/\\pi\_\{\\mu,\\tau,r\}\)\}=\\rho\\cdot\\left\(\\log\\frac\{\\rho\}\{\\pi\_\{\\mu,\\tau,r\}\}\-\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\log\\frac\{\\rho\(X\)\}\{\\pi\_\{\\mu,\\tau,r\}\(X\)\}\\right\]\\right\)is a tangent vector inTρ​𝒫T\_\{\\rho\}\\mathscr\{P\}\. The detailed proof is deferred to Appendix[A\.3\.2](https://arxiv.org/html/2609.05727#A1.SS3.SSS2)\. This tangent vector satisfies condition \([28](https://arxiv.org/html/2609.05727#S6.E28)\)\. Therefore, the Fisher–Rao gradient is

gradFR⁡Jρ​\(x\)=ρ⁡\(x\)⋅\(log⁡ρ⁡\(x\)πμ,τ,r​\(x\)−𝔼X∼ρ​\[log⁡ρ⁡\(X\)πμ,τ,r​\(X\)\]\)\.\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\(x\)=\\rho\(x\)\\cdot\\left\(\\log\\frac\{\\rho\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\-\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\log\\frac\{\\rho\(X\)\}\{\\pi\_\{\\mu,\\tau,r\}\(X\)\}\\right\]\\right\)\.\(29\)
Finally, we compute the canonical liftdρ​𝒞​\[gradFR⁡Jρ\]∈Tvρ​𝒱can\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\]\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}to obtain a directly learnable direction\. Sinceπμ,τ,r​\(x\)∝μ⁡\(x\)​eτ​r​\(x\)\\pi\_\{\\mu,\\tau,r\}\(x\)\\propto\\mu\(x\)e^\{\\tau r\(x\)\}, we define the regularized reward as

r~ρ​\(x\):=r⁡\(x\)−1τ​log⁡ρ⁡\(x\)μ⁡\(x\),\\tilde\{r\}^\{\\rho\}\(x\):=r\(x\)\-\\frac\{1\}\{\\tau\}\\log\\frac\{\\rho\(x\)\}\{\\mu\(x\)\},which satisfies

log⁡ρ⁡\(x\)πμ,τ,r​\(x\)=−τ​r~ρ​\(x\)\+const\.\\log\\frac\{\\rho\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}=\-\\tau\\tilde\{r\}^\{\\rho\}\(x\)\+\\mathrm\{const\}\.In other words,

gradFR⁡Jρ​\(x\)=ξρ,−τ​r~ρ=−τ​ξρ,r~ρ∈Tρ​𝒫\.\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\(x\)=\\xi^\{\\rho,\-\\tau\\tilde\{r\}^\{\\rho\}\}=\-\\tau\\xi^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\in T\_\{\\rho\}\\mathscr\{P\}\.According to Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5), the canonical lift is

dρ​𝒞​\[gradFR⁡Jρ\]=Γρ,−τ​r~ρ=−τ​Γρ,r~ρ∈Tvρ​𝒱can,\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\]=\\Gamma^\{\\rho,\-\\tau\\tilde\{r\}^\{\\rho\}\}=\-\\tau\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\},where

Γtρ,r~ρ\(xt\):=κt∇Vtρ\[r~ρ\]\(xt\)=CovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),r~ρ\(X1\)\)\.\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\):=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\(x\_\{t\}\)=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\)\.\(30\)Thus, the density\-dependent signalr~ρ\\tilde\{r\}^\{\\rho\}specializes the objective\-agnostic canonical retraction of Section[5](https://arxiv.org/html/2609.05727#S5)to the first\-order descent direction of the reverse\-KL objective\.

### 6\.2Newton Direction

Section[6\.1](https://arxiv.org/html/2609.05727#S6.SS1)identifies the first\-order choice for the input to the canonical retraction:

−gradcan⁡Jvρcan=τ​Γρ,r~ρ\.\-\\operatorname\{grad\}\_\{\\mathrm\{can\}\}J^\{\\mathrm\{can\}\}\_\{v^\{\\rho\}\}=\\tau\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.We now derive the corresponding second\-order choice\. The Newton direction is obtained by linearizing the stationarity conditiondρ​J=0\\mathrm\{d\}\_\{\\rho\}J=0, where the Hessian is defined using the mixture connection introduced in Section[4\.3](https://arxiv.org/html/2609.05727#S4.SS3)\. The central result is the bilinear\-form identity

Hessmix​Jρ=gρFR\.\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}=g\_\{\\rho\}^\{\\mathrm\{FR\}\}\.Consequently, after the Fisher–Rao metric identifies tangent and cotangent vectors, the Hessian acts as the identity, and the Newton direction coincides with the negative gradient\.

For the first step, we compute the Hessian ofJJ\. Consider the mixture connection defined in \([17](https://arxiv.org/html/2609.05727#S4.E17)\) and recall the definition of the Hessian in Section[2\.2\.7](https://arxiv.org/html/2609.05727#S2.SS2.SSS7)\. For anyρ∈𝒫\\rho\\in\\mathscr\{P\}and two tangent vectorsξ1,ξ2∈Tρ​𝒫\\xi\_\{1\},\\xi\_\{2\}\\in T\_\{\\rho\}\\mathscr\{P\}, we extend them to smooth tangent vector fieldsP,Q∈𝔛⁡\(𝒫\)P,Q\\in\\mathfrak\{X\}\(\\mathscr\{P\}\)which satisfyPρ=ξ1P\_\{\\rho\}=\\xi\_\{1\},Qρ=ξ2Q\_\{\\rho\}=\\xi\_\{2\}\. By definition, the Hessian takes a bilinear form

Hessmix​Jρ:Tρ​𝒫×Tρ​𝒫→ℝ,Hessmix​Jρ​\[ξ1,ξ2\]:=P⁡\[Q⁡\[J\]\]​\(ρ\)−dρ​J​\[\(∇PmixQ\)ρ\]\.\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}:T\_\{\\rho\}\\mathscr\{P\}\\times T\_\{\\rho\}\\mathscr\{P\}\\to\\mathbb\{R\},\\qquad\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}\[\\xi\_\{1\},\\xi\_\{2\}\]:=P\[Q\[J\]\]\(\\rho\)\-\\mathrm\{d\}\_\{\\rho\}J\\left\[\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\}\\right\]\.Consider a smooth curve\(ρε\)ε∈\(−δ,δ\)⊂𝒫\(\\rho^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{P\}which passes throughρ\\rhoatε=0\\varepsilon=0and satisfiesξ1=Pρ=dd​ε​ρε\|ε=0\\xi\_\{1\}=P\_\{\\rho\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}\. According to \([17](https://arxiv.org/html/2609.05727#S4.E17)\), we have

dρ​J​\[\(∇PmixQ\)ρ\]=gρFR​\(\(∇PmixQ\)ρ,ξρ,log⁡\(ρ/πμ,τ,r\)\)=P⁡\[gFR​\(Q,ξ⋅,log⁡\(ρ/πμ,τ,r\)\)\]​\(ρ\)\.\\mathrm\{d\}\_\{\\rho\}J\\left\[\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\}\\right\]=g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\\left\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Q\\right\)\_\{\\rho\},\\xi^\{\\rho,\\log\(\\rho/\\pi\_\{\\mu,\\tau,r\}\)\}\\right\)=P\\left\[g^\{\\mathrm\{FR\}\}\\left\(Q,\\xi^\{\\cdot,\\log\(\\rho/\\pi\_\{\\mu,\\tau,r\}\)\}\\right\)\\right\]\(\\rho\)\.Hereξ⋅,log⁡\(ρ/πμ,τ,r\)∈𝔛⁡\(𝒫\)\\xi^\{\\cdot,\\log\(\\rho/\\pi\_\{\\mu,\\tau,r\}\)\}\\in\\mathfrak\{X\}\(\\mathscr\{P\}\)is the fixed\-observable extension,

ξ⋅,log⁡\(ρ/πμ,τ,r\):q↦ξq,log⁡\(ρ/πμ,τ,r\):=q⋅\(log⁡ρπμ,τ,r−𝔼q​\[log⁡ρπμ,τ,r\]\)\.\\xi^\{\\cdot,\\log\(\\rho/\\pi\_\{\\mu,\\tau,r\}\)\}:q\\mapsto\\xi^\{q,\\log\(\\rho/\\pi\_\{\\mu,\\tau,r\}\)\}:=q\\cdot\\left\(\\log\\frac\{\\rho\}\{\\pi\_\{\\mu,\\tau,r\}\}\-\\mathbb\{E\}\_\{q\}\\left\[\\log\\frac\{\\rho\}\{\\pi\_\{\\mu,\\tau,r\}\}\\right\]\\right\)\.Therefore, we have

Hessmix​Jρ​\[ξ1,ξ2\]=\\displaystyle\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}\[\\xi\_\{1\},\\xi\_\{2\}\]=dd​ε​∫ℝdlog⁡ρε​\(x\)πμ,τ,r​\(x\)​Qρε​\(x\)​𝑑x\|ε=0−dd​ε​∫ℝdlog⁡ρ⁡\(x\)πμ,τ,r​\(x\)​Qρε​\(x\)​𝑑x\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\log\\frac\{\\rho^\{\\varepsilon\}\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}Q\_\{\\rho^\{\\varepsilon\}\}\(x\)\\,\\mathrm\{d\}x\\right\|\_\{\\varepsilon=0\}\-\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\log\\frac\{\\rho\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}Q\_\{\\rho^\{\\varepsilon\}\}\(x\)\\,\\mathrm\{d\}x\\right\|\_\{\\varepsilon=0\}=\\displaystyle=∫ℝddd​ε​log⁡ρε​\(x\)πμ,τ,r​\(x\)\|ε=0​Qρ​\(x\)​𝑑x\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\log\\frac\{\\rho^\{\\varepsilon\}\(x\)\}\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\\right\|\_\{\\varepsilon=0\}Q\_\{\\rho\}\(x\)\\,\\mathrm\{d\}x=\\displaystyle=∫ℝdξ1​\(x\)​ξ2​\(x\)ρ⁡\(x\)​𝑑x\.\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\frac\{\\xi\_\{1\}\(x\)\\xi\_\{2\}\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}x\.In other words, we have the bilinear\-form identity

Hessmix​Jρ=gρFR\.\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}=g\_\{\\rho\}^\{\\mathrm\{FR\}\}\.\(31\)Thus, the Hessian is exactly the same bilinear form that represents the differential as a Fisher–Rao gradient\. As illustrated in Remark[16](https://arxiv.org/html/2609.05727#Thmtheorem16), this identity corresponds to the Euclidean case in which the classical Hessian matrix is the identity matrix\.

For the second step, we state the Newton equation and solve the Newton directionnρmix∈Tρ​𝒫n\_\{\\rho\}^\{\\mathrm\{mix\}\}\\in T\_\{\\rho\}\\mathscr\{P\}\. We regard the Hessian as a map from the tangent spaceTρ​𝒫T\_\{\\rho\}\\mathscr\{P\}to the cotangent spaceTρ∗​𝒫T\_\{\\rho\}^\{\*\}\\mathscr\{P\}:

Hessmix​Jρ:Tρ​𝒫→Tρ∗​𝒫,\(Hessmix​Jρ​\[ξ1\]\)​\[ξ2\]:=Hessmix​Jρ​\[ξ1,ξ2\]\.\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}:T\_\{\\rho\}\\mathscr\{P\}\\to T\_\{\\rho\}^\{\*\}\\mathscr\{P\},\\qquad\\left\(\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}\[\\xi\_\{1\}\]\\right\)\[\\xi\_\{2\}\]:=\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}\[\\xi\_\{1\},\\xi\_\{2\}\]\.The Newton equation is the linearized equation ofdρ​J=0\\mathrm\{d\}\_\{\\rho\}J=0, i\.e\.,

dρ​J\+Hessmix​Jρ​\[nρmix\]=0∈Tρ∗​𝒫\.\\mathrm\{d\}\_\{\\rho\}J\+\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}\[n\_\{\\rho\}^\{\\mathrm\{mix\}\}\]=0\\in T\_\{\\rho\}^\{\*\}\\mathscr\{P\}\.In this equation, the right\-hand side denotes the zero functional\. In other words,∀ξ∈Tρ​𝒫\\forall\\xi\\in T\_\{\\rho\}\\mathscr\{P\},

dρ​J​\[ξ\]\+Hessmix​Jρ​\[nρmix,ξ\]=gρFR​\(gradFR⁡Jρ\+nρmix,ξ\)=0\.\\mathrm\{d\}\_\{\\rho\}J\[\\xi\]\+\\mathrm\{Hess\}^\{\\mathrm\{mix\}\}J\_\{\\rho\}\[n\_\{\\rho\}^\{\\mathrm\{mix\}\},\\xi\]=g\_\{\\rho\}^\{\\mathrm\{FR\}\}\(\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\+n\_\{\\rho\}^\{\\mathrm\{mix\}\},\\xi\)=0\.Here the Fisher–Rao gradient is provided in \([29](https://arxiv.org/html/2609.05727#S6.E29)\)\. Takeξ=gradFR⁡Jρ\+nρmix\\xi=\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\+n\_\{\\rho\}^\{\\mathrm\{mix\}\}\. By the positive definiteness of the Fisher–Rao metric, the Newton direction is the unique solution

nρmix=−gradFR⁡Jρ=τ​ξρ,r~ρ\.n\_\{\\rho\}^\{\\mathrm\{mix\}\}=\-\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}=\\tau\\xi^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.
Finally, the canonical lift of the Newton directionnρmixn\_\{\\rho\}^\{\\mathrm\{mix\}\}is

nρcan:=dρ​𝒞​\[nρmix\]=−dρ​𝒞​\[gradFR⁡Jρ\]=τ​Γρ,r~ρ\.n\_\{\\rho\}^\{\\mathrm\{can\}\}:=\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[n\_\{\\rho\}^\{\\mathrm\{mix\}\}\]=\-\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\[\\operatorname\{grad\}\_\{\\mathrm\{FR\}\}J\_\{\\rho\}\]=\\tau\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.By the same argument as in Remark[15](https://arxiv.org/html/2609.05727#Thmtheorem15),nρcann\_\{\\rho\}^\{\\mathrm\{can\}\}is the Newton direction ofJcan=J∘𝒯\|𝒱canJ^\{\\mathrm\{can\}\}=J\\circ\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}under the transported Fisher–Rao metricgcang^\{\\mathrm\{can\}\}and the transported connection∇can\\nabla^\{\\mathrm\{can\}\}\.

We now turn the interior\-time identifiability of the canonical lift in Proposition[7](https://arxiv.org/html/2609.05727#Thmtheorem7)into an exact stationarity criterion for the Newton direction\.

###### Proposition 18\.

For any terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}, the following statements are equivalent:

ρ=πμ,τ,r⇔r~ρ≡const⇔∀t∈\(0,1\),Γρ,r~ρ≡0⇔∃t0∈\(0,1\),Γt0ρ,r~ρ≡0\.\\rho=\\pi\_\{\\mu,\\tau,r\}\\quad\\Leftrightarrow\\quad\\tilde\{r\}^\{\\rho\}\\equiv\\mathrm\{const\}\\quad\\Leftrightarrow\\quad\\forall t\\in\(0,1\),\\,\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\equiv 0\\quad\\Leftrightarrow\\quad\\exists t\_\{0\}\\in\(0,1\),\\,\\Gamma\_\{t\_\{0\}\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\equiv 0\.

###### Proof\.

Sinceρ\\rhoandπμ,τ,r\\pi\_\{\\mu,\\tau,r\}are both normalized densities, we haveρ=πμ,τ,r⇔r~ρ≡const\\rho=\\pi\_\{\\mu,\\tau,r\}\\,\\Leftrightarrow\\,\\tilde\{r\}^\{\\rho\}\\equiv\\mathrm\{const\}\. Substitutingf=r~ρf=\\tilde\{r\}^\{\\rho\}into Proposition[7](https://arxiv.org/html/2609.05727#Thmtheorem7)finishes the proof\. ∎

Thus, vanishing at any single interior time characterizes the target density: wheneverρ≠πμ,τ,r\\rho\\neq\\pi\_\{\\mu,\\tau,r\},

Γtρ,r~ρ≢0for every​t∈\(0,1\)\.\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\not\\equiv 0\\qquad\\text\{for every \}t\\in\(0,1\)\.Having identified the direction and its stationarity criterion, we next insert it into the canonical retraction to define the Newton Matching iteration\.

### 6\.3Newton’s Method with Canonical Retraction

We now substitute the tangent vector \([30](https://arxiv.org/html/2609.05727#S6.E30)\) into the canonical retraction \([21](https://arxiv.org/html/2609.05727#S5.E21)\) of Section[5](https://arxiv.org/html/2609.05727#S5)\. Letvk=vρk=𝒞⁡\(ρk\)∈𝒱canv^\{k\}=v^\{\\rho\_\{k\}\}=\\mathcal\{C\}\(\\rho\_\{k\}\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}be the current canonical iterate, with terminal densityρk=𝒯⁡\(vk\)\\rho\_\{k\}=\\mathcal\{T\}\(v^\{k\}\)\. Since the full Newton direction is

nρkcan=τ​Γρk,r~ρk,n\_\{\\rho\_\{k\}\}^\{\\mathrm\{can\}\}=\\tau\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\},a stepsizeηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]corresponds to the standard Newton damping factorηkτ∈\(0,1\]\\frac\{\\eta\_\{k\}\}\{\\tau\}\\in\(0,1\]\. Applying the canonical retraction \([21](https://arxiv.org/html/2609.05727#S5.E21)\) gives

v¯k\+1:=vk\+ηk​Γρk,r~ρk∈𝒱,\\displaystyle\\bar\{v\}^\{k\+1\}:=v^\{k\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\\in\\mathscr\{V\},\(32a\)ρk\+1:=𝒯⁡\(v¯k\+1\)∈𝒫,\\displaystyle\\rho\_\{k\+1\}:=\\mathcal\{T\}\\left\(\\bar\{v\}^\{k\+1\}\\right\)\\in\\mathscr\{P\},\(32b\)vk\+1:=𝒞⁡\(ρk\+1\)=ℛ⁡\(v¯k\+1\)∈𝒱can\.\\displaystyle v^\{k\+1\}:=\\mathcal\{C\}\(\\rho\_\{k\+1\}\)=\\mathcal\{R\}\(\\bar\{v\}^\{k\+1\}\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.\(32c\)In particular, the regularized reward at the current iterate is

r~ρk​\(X1\):=r⁡\(X1\)−1τ​log⁡ρk​\(X1\)μ⁡\(X1\),\\tilde\{r\}^\{\\rho\_\{k\}\}\(X\_\{1\}\):=r\(X\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\},and the associated tangent vector is given by

Γtρk,r~ρk\(xt\)=κt∇Vtρk\[r~ρk\]\(xt\)=CovX1∼p1\|tρk\(⋅\|xt\)\(vt\|1\(xt\|X1\),r~ρk\(X1\)\)\.\\Gamma\_\{t\}^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\(x\_\{t\}\)=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\_\{k\}\}\[\\tilde\{r\}^\{\\rho\_\{k\}\}\]\(x\_\{t\}\)=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\tilde\{r\}^\{\\rho\_\{k\}\}\(X\_\{1\}\)\\right\)\.Whenηk=τ\\eta\_\{k\}=\\tau, \([32](https://arxiv.org/html/2609.05727#S6.E32)\) corresponds to a full Newton step\. Whenηk∈\(0,τ\)\\eta\_\{k\}\\in\(0,\\tau\), \([32](https://arxiv.org/html/2609.05727#S6.E32)\) is a damped Newton step\. For later use, one generic Newton Matching stage \([32](https://arxiv.org/html/2609.05727#S6.E32)\) can be written compactly as

q=𝒯⁡\(vρ\+η​Γρ,r~ρ\)\.q=\\mathcal\{T\}\\left\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\right\)\.\(33\)The settingq=ρk\+1q=\\rho\_\{k\+1\},ρ=ρk\\rho=\\rho\_\{k\}, andη=ηk\\eta=\\eta\_\{k\}corresponds to an ideal Newton update \([32](https://arxiv.org/html/2609.05727#S6.E32)\) at iterationkk\. An ideal Newton Matching step \([32](https://arxiv.org/html/2609.05727#S6.E32)\) is illustrated in Figure[7](https://arxiv.org/html/2609.05727#S6.F7)\. The Newton Matching iteration is provided in Algorithm[1](https://arxiv.org/html/2609.05727#alg1)\.

Newton\-direction field𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}vρ=𝒞⁡\(ρ\)v^\{\\rho\}=\\mathcal\{C\}\(\\rho\)Tvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}Γρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}η∈\(0,τ\]\\eta\\in\(0,\\tau\]v¯=vρ\+η​Γρ,r~ρ\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}ℛ\\mathcal\{R\}ℛ⁡\(v¯\)=𝒞⁡\(q\)\\mathcal\{R\}\(\\bar\{v\}\)=\\mathcal\{C\}\(q\)vπ=𝒞⁡\(π\)v^\{\\pi\}=\\mathcal\{C\}\(\\pi\)KL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)↑∞\\uparrow\\inftyKL⁡\(𝒯⁡\(v\)∥π\)\\operatorname\{KL\}\\left\(\\mathcal\{T\}\(v\)\\middle\\\|\\pi\\right\)0Figure 7:One Newton Matching step\. The negative Fisher–Rao gradient and the mixture\-connection Newton direction coincide\. The canonical retraction with this direction produces a canonical\-velocity update that decreases the reverse KL divergence\.Algorithm 1Newton Matching\.0:A reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; a stepsize sequence

\{ηk\}k=0∞⊂\(0,τ\]\\\{\\eta\_\{k\}\\\}\_\{k=0\}^\{\\infty\}\\subset\(0,\\tau\]; and a convergence criterion\.

0:The target canonical velocity field

vπ=𝒞⁡\(πμ,τ,r\)∈𝒱canv^\{\\pi\}=\\mathcal\{C\}\(\\pi\_\{\\mu,\\tau,r\}\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

1:Initialize

k←0k\\leftarrow 0and choose an arbitrary canonical velocity

vρ0∈𝒱canv^\{\\rho\_\{0\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

⊳\\trianglerightFor fine\-tuning, it is convenient to setvρ0v^\{\\rho\_\{0\}\}asvbasev^\{\\mathrm\{base\}\}; for sampling, it is convenient to setvρ0v^\{\\rho\_\{0\}\}as the analytical canonical velocity associated with a Gaussianρ0\\rho\_\{0\}\.

2:repeat

3:Perform the tangential update

v¯k\+1←vρk\+ηk​Γρk,r~ρk\\bar\{v\}^\{k\+1\}\\leftarrow v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

4:Perform the canonicalization

vρk\+1←ℛ⁡\(v¯k\+1\)v^\{\\rho\_\{k\+1\}\}\\leftarrow\\mathcal\{R\}\(\\bar\{v\}^\{k\+1\}\)\.

5:Set

k←k\+1k\\leftarrow k\+1\.

6:untilthe convergence criterion is satisfied\.

7:Return

vρkv^\{\\rho\_\{k\}\}\.

To analyze objective descent and convergence in density coordinates, we specialize Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12)to the density\-dependent observablef=r~ρf=\\tilde\{r\}^\{\\rho\}\. This yields the following exact formula for the terminal\-density update of one ideal Newton Matching stage\.

###### Proposition 19\.

For a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}and a stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\], we define the path dissipation𝒟ρ,η,r~ρ:ℝd→\[0,∞\)\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}:\\mathbb\{R\}^\{d\}\\to\[0,\\infty\)by

𝒟ρ,η,r~ρ​\(x\):=η2​∫01κt​‖∇Vtρ​\[r~ρ\]​\(Φ1→tρ,η,r~ρ​\(x\)\)‖22​𝑑t,\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\):=\\eta^\{2\}\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t,whereΦρ,η,r~ρ\\Phi^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}is the flow map generated by the velocity fieldvρ\+η​Γρ,r~ρv^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\. Then, for an ideal Newton step \([33](https://arxiv.org/html/2609.05727#S6.E33)\), we have

q⁡\(x\)=\\displaystyle q\(x\)=ρ​\(x\)1−η/τ​πμ,τ,r​\(x\)η/τ​exp⁡\(ητ​KL⁡\(ρ∥πμ,τ,r\)−𝒟ρ,η,r~ρ​\(x\)\)\.\\displaystyle\\rho\(x\)^\{1\-\\eta/\\tau\}\\pi\_\{\\mu,\\tau,r\}\(x\)^\{\\eta/\\tau\}\\exp\\left\(\\frac\{\\eta\}\{\\tau\}\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\_\{\\mu,\\tau,r\}\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)\.\(34\)

###### Proof\.

By Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12), lettingf=r~ρf=\\tilde\{r\}^\{\\rho\}gives

q⁡\(x\)=\\displaystyle q\(x\)=ρ⁡\(x\)​exp⁡\(η⁡\(r~ρ​\(x\)−𝔼X∼ρ​\[r~ρ​\(X\)\]\)−𝒟ρ,η,r~ρ​\(x\)\)\\displaystyle\\rho\(x\)\\exp\\left\(\\eta\\left\(\\tilde\{r\}^\{\\rho\}\(x\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[\\tilde\{r\}^\{\\rho\}\(X\)\]\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)=\\displaystyle=ρ⁡\(x\)​exp⁡\(ητ​\(log⁡πμ,τ,r​\(x\)ρ⁡\(x\)−𝔼X∼ρ​\[log⁡πμ,τ,r​\(X\)ρ⁡\(X\)\]\)−𝒟ρ,η,r~ρ​\(x\)\),\\displaystyle\\rho\(x\)\\exp\\left\(\\frac\{\\eta\}\{\\tau\}\\left\(\\log\\frac\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\{\\rho\(x\)\}\-\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\log\\frac\{\\pi\_\{\\mu,\\tau,r\}\(X\)\}\{\\rho\(X\)\}\\right\]\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\),which proves \([34](https://arxiv.org/html/2609.05727#S6.E34)\)\. ∎

In \([34](https://arxiv.org/html/2609.05727#S6.E34)\), the factorρ1−η/τ​πη/τ\\rho^\{1\-\\eta/\\tau\}\\pi^\{\\eta/\\tau\}is the linear interpolation between the current and target densities in log\-density coordinates, while the path\-dissipation term𝒟ρ,η,r~ρ\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}is the state\-dependent correction induced by the finite\-stepsize update in velocity space\. In particular, whenη=τ\\eta=\\tau, the path\-dissipation term is the only state\-dependent deviation of the updated terminal density from the target density\.

The geometric construction of Newton Matching is now complete\. At the intrinsic level, one stage follows the Newton direction on the canonical manifold\. At the algorithmic level, this direction is implemented by a velocity\-space correction followed by canonicalization\. Finally, at the density level, the same stage satisfies the exact terminal\-density formula \([34](https://arxiv.org/html/2609.05727#S6.E34)\)\. The next section makes use of this density\-level representation to analyze the convergence properties of Newton Matching\.

## 7Convergence Guarantees of Newton Matching

Section[6](https://arxiv.org/html/2609.05727#S6)identifies the canonical tangentτ​Γρ,r~ρ\\tau\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}as both the negative Fisher–Rao gradient and the mixture\-connection Newton direction, and combines it with the canonical retraction to define Newton Matching\. This section studies the convergence guarantees of the Newton Matching iteration\. Section[7\.1](https://arxiv.org/html/2609.05727#S7.SS1)first establishes finite\-stepsize reverse\-KL descent for every stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\], and then proves global convergence to the target in reverse KL under mild conditions\. Furthermore, reflecting the characteristic local behavior of Newton’s method, Section[7\.2](https://arxiv.org/html/2609.05727#S7.SS2)establishes local quadratic convergence of the full\-step iteration \(η=τ\\eta=\\tau\) for both terminal densities and canonical velocity fields\. Motivated by this local result, we further provide an optional continuation scheme along the inverse\-temperature path as a means of exploiting the quadratic\-convergence regime\. Under a sufficiently fine temperature grid, each continuation stage is initialized within the local quadratic\-convergence neighborhood of its target and hence converges quadratically\. Detailed proofs of the global and local convergence results are collected in Appendix[B](https://arxiv.org/html/2609.05727#A2)\.

A time\-dependent extension with a stepsize scheduleηt\\eta\_\{t\}is developed in Appendix[G\.2\.2](https://arxiv.org/html/2609.05727#A7.SS2.SSS2)\. To complement the general infinite\-dimensional theory with a concrete, analytically tractable validation, Appendix[C](https://arxiv.org/html/2609.05727#A3)studies Newton Matching on the isotropic Gaussian family\. This family is finite\-dimensional and closed under Newton Matching, so the iteration reduces to an explicit parameter recursion\. Direct analysis of this recursion shows that both forward and reverse KL divergences converge to zero and, when the initial density is sufficiently close to the target, exhibit local quadratic convergence\.

### 7\.1KL Descent and Global Convergence

This subsection is organized into three parts\. Section[7\.1\.1](https://arxiv.org/html/2609.05727#S7.SS1.SSS1)first records the infinitesimal descent property of the Newton direction and then strengthens it to a finite\-stepsize result through a dissipation\-corrected three\-point identity\. Consequently, the reverse KL decreases at every ideal Newton Matching stage for any finite stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\], while a separate sufficient condition yields forward\-KL descent\. Section[7\.1\.2](https://arxiv.org/html/2609.05727#S7.SS1.SSS2)studies the asymptotic consequences of this stagewise monotonicity\. Under non\-summable stepsizes, the reverse\-KL values converge, and the log\-density residuallog⁡ρkπ\\log\\frac\{\\rho\_\{k\}\}\{\\pi\}becomes asymptotically constant inL1​\(ρk\)L^\{1\}\(\\rho\_\{k\}\)\. Since this on\-policy property does not by itself rule out loss of target mass or identify the limiting reverse KL value as zero, Section[7\.1\.3](https://arxiv.org/html/2609.05727#S7.SS1.SSS3)introduces two additional mild conditions, under either of which convergence to the target is established\. All proofs in Sections[7\.1\.2](https://arxiv.org/html/2609.05727#S7.SS1.SSS2)and[7\.1\.3](https://arxiv.org/html/2609.05727#S7.SS1.SSS3)are deferred to Appendix[B\.1](https://arxiv.org/html/2609.05727#A2.SS1)\.

For notational simplicity, we writeπ:=πμ,τ,r\\pi:=\\pi\_\{\\mu,\\tau,r\}throughout this subsection\.

#### 7\.1\.1Finite\-Stepsize KL Descent

Section[6](https://arxiv.org/html/2609.05727#S6)identifies the tangent vector used in the canonical retraction as both the negative Fisher–Rao gradient and the Newton direction\. The remaining question is finite\-stepsize stability: a Newton direction is defined through local differential information, whereas Newton Matching applies a prescribed damped or full canonical retraction\. We first record the infinitesimal descent property as a baseline\.

###### Proposition 21\.

Denoteqη:=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q^\{\\eta\}:=\\mathcal\{T\}\\left\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\right\)\. Assume that for everyx∈ℝdx\\in\\mathbb\{R\}^\{d\}, the map

η↦∫01κt​‖∇Vtρ​\[r~ρ\]​\(Φ1→tρ,η,r~ρ​\(x\)\)‖22​𝑑t\\eta\\mapsto\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}tis continuous in a neighborhood of00\. Then, we have

dd​η​KL⁡\(qη∥π\)\|η=0=−τ​VarX∼ρ⁡\(r~ρ​\(X\)\)≤0\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\eta\}\\operatorname\{KL\}\\left\(q^\{\\eta\}\\middle\\\|\\pi\\right\)\\right\|\_\{\\eta=0\}=\-\\tau\\operatorname\{Var\}\_\{X\\sim\\rho\}\\left\(\\tilde\{r\}^\{\\rho\}\(X\)\\right\)\\leq 0\.In particular,

VarX∼ρ\(r~ρ\(X\)\)=0⇔ρ=π\.\\operatorname\{Var\}\_\{X\\sim\\rho\}\\left\(\\tilde\{r\}^\{\\rho\}\(X\)\\right\)=0\\quad\\Leftrightarrow\\quad\\rho=\\pi\.

###### Proof\.

According to Proposition[19](https://arxiv.org/html/2609.05727#Thmtheorem19), we have

qη​\(x\)=ρ⁡\(x\)​\(1\+η⁡\(r~ρ​\(x\)−𝔼ρ​\[r~ρ\]\)\+O⁡\(η2\)\)\.q^\{\\eta\}\(x\)=\\rho\(x\)\\left\(1\+\\eta\\left\(\\tilde\{r\}^\{\\rho\}\(x\)\-\\mathbb\{E\}\_\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\right\)\+O\(\\eta^\{2\}\)\\right\)\.Therefore,

dd​η​KL⁡\(qη∥π\)\|η=0=\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\eta\}\\operatorname\{KL\}\\left\(q^\{\\eta\}\\middle\\\|\\pi\\right\)\\right\|\_\{\\eta=0\}=∫ℝd\(log⁡ρ⁡\(x\)π⁡\(x\)\+1\)​dd​η​qη​\(x\)\|η=0​𝑑x\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\(\\log\\frac\{\\rho\(x\)\}\{\\pi\(x\)\}\+1\\right\)\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\eta\}q^\{\\eta\}\(x\)\\right\|\_\{\\eta=0\}\\mathrm\{d\}x=\\displaystyle=∫ℝd\(log⁡ρ⁡\(x\)π⁡\(x\)\+1\)​ρ​\(x\)​\(r~ρ​\(x\)−𝔼ρ​\[r~ρ\]\)​𝑑x\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\(\\log\\frac\{\\rho\(x\)\}\{\\pi\(x\)\}\+1\\right\)\\rho\(x\)\\left\(\\tilde\{r\}^\{\\rho\}\(x\)\-\\mathbb\{E\}\_\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\right\)\\mathrm\{d\}x=\\displaystyle=−τ​VarX∼ρ⁡\(r~ρ​\(X\)\)≤0\.\\displaystyle\-\\tau\\operatorname\{Var\}\_\{X\\sim\\rho\}\\left\(\\tilde\{r\}^\{\\rho\}\(X\)\\right\)\\leq 0\.In particular,VarX∼ρ⁡\(r~ρ​\(X\)\)=0\\operatorname\{Var\}\_\{X\\sim\\rho\}\\left\(\\tilde\{r\}^\{\\rho\}\(X\)\\right\)=0if and only ifr~ρ=const\\tilde\{r\}^\{\\rho\}=\\mathrm\{const\}, i\.e\.,ρ=π\\rho=\\pi\. ∎

Proposition[21](https://arxiv.org/html/2609.05727#Thmtheorem21)records the guarantee available from generic first\-order analysis\. Away from the target, the directional derivative atη=0\\eta=0is strictly negative and therefore ensures descent for sufficiently small positive steps\. Without quantitative control of the nonlinear remainder along the retraction, however, this result neither specifies how small the stepsize must be nor certifies any prescribed finite\-stepsize update\.

The following theorem goes beyond this infinitesimal result by exploiting the specific structure of Newton Matching\. The posterior\-value\-gradient representation of the canonical tangent makes the density path generated by the velocity\-space update exactly analyzable, while canonicalization preserves its terminal density\. This structure yields a dissipation\-corrected three\-point identity that controls the complete nonlinear update rather than only its linearization atη=0\\eta=0\. Consequently, the reverse KL decreases throughout the full admissible stepsize rangeη∈\(0,τ\]\\eta\\in\(0,\\tau\]\.

###### Theorem 22\.

Letq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\)be the terminal density generated by one ideal Newton Matching stage \([33](https://arxiv.org/html/2609.05727#S6.E33)\)\. For any densityp∈𝒫p\\in\\mathscr\{P\}such thatKL⁡\(p∥q\)\\operatorname\{KL\}\\left\(p\\middle\\\|q\\right\),KL⁡\(p∥ρ\)\\operatorname\{KL\}\\left\(p\\middle\\\|\\rho\\right\),KL⁡\(p∥π\)\\operatorname\{KL\}\\left\(p\\middle\\\|\\pi\\right\), and𝔼X∼p​\[𝒟ρ,η,r~ρ​\(X\)\]\\mathbb\{E\}\_\{X\\sim p\}\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\]are finite, we haveKL⁡\(p∥q\)=\(1−ητ\)​KL⁡\(p∥ρ\)\+ητ​\(KL⁡\(p∥π\)−KL⁡\(ρ∥π\)\)\+𝔼X∼p​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(p\\middle\\\|q\\right\)=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)\\operatorname\{KL\}\\left\(p\\middle\\\|\\rho\\right\)\+\\frac\{\\eta\}\{\\tau\}\\left\(\\operatorname\{KL\}\\left\(p\\middle\\\|\\pi\\right\)\-\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\right\)\+\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.\(35a\)Consequently, applyingp=qp=qgivesKL⁡\(ρ∥π\)−KL⁡\(q∥π\)=\(τη−1\)​KL⁡\(q∥ρ\)\+τη​𝔼X∼q​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\-\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\left\(\\frac\{\\tau\}\{\\eta\}\-1\\right\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)\+\\frac\{\\tau\}\{\\eta\}\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.\(35b\)Applyingp=πp=\\pigivesKL⁡\(π∥ρ\)−KL⁡\(π∥q\)=ητ​\(KL⁡\(π∥ρ\)\+KL⁡\(ρ∥π\)\)−𝔼X∼π​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\-\\operatorname\{KL\}\\left\(\\pi\\middle\\\|q\\right\)=\\frac\{\\eta\}\{\\tau\}\(\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\+\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\)\-\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.\(35c\)Applyingp=ρp=\\rhogivesKL⁡\(ρ∥q\)=𝔼X∼ρ​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(\\rho\\middle\\\|q\\right\)=\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.\(35d\)

###### Proof\.

By Proposition[19](https://arxiv.org/html/2609.05727#Thmtheorem19), we have

log⁡q⁡\(x\)=\(1−ητ\)​log⁡ρ⁡\(x\)\+ητ​log⁡π⁡\(x\)\+ητ​KL​\(ρ∥π\)−𝒟ρ,η,r~ρ​\(x\)\.\\log q\(x\)=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)\\log\\rho\(x\)\+\\frac\{\\eta\}\{\\tau\}\\log\\pi\(x\)\+\\frac\{\\eta\}\{\\tau\}\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\.For anyp∈𝒫p\\in\\mathscr\{P\}, taking the expectation gives

KL⁡\(p∥q\)\\displaystyle\\operatorname\{KL\}\\left\(p\\middle\\\|q\\right\)=\\displaystyle=𝔼X∼p​\[log⁡p⁡\(X\)q⁡\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\log\\frac\{p\(X\)\}\{q\(X\)\}\\right\]=\\displaystyle=\(1−ητ\)​𝔼X∼p​\[log⁡p⁡\(X\)ρ⁡\(X\)\]\+ητ​𝔼X∼p​\[log⁡p⁡\(X\)π⁡\(X\)\]−ητ​KL⁡\(ρ∥π\)\+𝔼X∼p​\[𝒟ρ,η,r~ρ​\(X\)\],\\displaystyle\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\log\\frac\{p\(X\)\}\{\\rho\(X\)\}\\right\]\+\\frac\{\\eta\}\{\\tau\}\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\log\\frac\{p\(X\)\}\{\\pi\(X\)\}\\right\]\-\\frac\{\\eta\}\{\\tau\}\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\+\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\],which gives \([35a](https://arxiv.org/html/2609.05727#S7.E35.1)\)\. Applyingp=ρ,π,qp=\\rho,\\pi,qfinishes the proof\. ∎

The finite\-stepsize descent theorem below follows directly from \([35b](https://arxiv.org/html/2609.05727#S7.E35.2)\)\.

###### Theorem 24\.

Letq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\)be the terminal density generated by one ideal Newton Matching stage \([33](https://arxiv.org/html/2609.05727#S6.E33)\)\. Then, for any stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\], the reverse KL decreases:

KL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.Furthermore, sinceπ,ρ,q\\pi,\\rho,qare positive densities onℝd\\mathbb\{R\}^\{d\}, we have

KL\(q∥π\)=KL\(ρ∥π\)⇔ρ=q=π\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\quad\\Leftrightarrow\\quad\\rho=q=\\pi\.

###### Proof\.

Since𝔼X∼q​\[𝒟ρ,η,r~ρ​\(X\)\]≥0\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\\geq 0andη∈\(0,τ\]\\eta\\in\(0,\\tau\], \([35b](https://arxiv.org/html/2609.05727#S7.E35.2)\) impliesKL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.

ρ=q=π\\rho=q=\\pievidently impliesKL⁡\(q∥π\)=KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\. For the opposite direction, assume thatKL⁡\(q∥π\)=KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\. By \([35b](https://arxiv.org/html/2609.05727#S7.E35.2)\), we have\(τη−1\)​KL⁡\(q∥ρ\)=0\\left\(\\frac\{\\tau\}\{\\eta\}\-1\\right\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)=0and𝔼X∼q​\[𝒟ρ,η,r~ρ​\(X\)\]=0\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]=0; hence,𝒟ρ,η,r~ρa\.e\.0\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\\mathrel\{\\mathop\{\\vbox\{\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\}\\kern 1\.50696pt\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\} \}\}\\limits^\{\\mathrm\{a\.e\.\}\}\}0\. By Proposition[19](https://arxiv.org/html/2609.05727#Thmtheorem19), we have∇Vtρ​\[r~ρ\]a\.e\.0\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\mathrel\{\\mathop\{\\vbox\{\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\}\\kern 1\.50696pt\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\} \}\}\\limits^\{\\mathrm\{a\.e\.\}\}\}0\. Since𝔼ρ​\[\|r~ρ\|\]<∞\\mathbb\{E\}\_\{\\rho\}\[\\left\|\\tilde\{r\}^\{\\rho\}\\right\|\]<\\inftyand the convolution kernel is a nondegenerate Gaussian,Vtρ​\[r~ρ\]V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]is smooth; hence,Vtρ​\[r~ρ\]≡constV\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\equiv\\mathrm\{const\}for a\.e\.tt\. According to Proposition[18](https://arxiv.org/html/2609.05727#Thmtheorem18), we haveρ=π\\rho=\\piandΓρ,r~ρ=0\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}=0; hence,q=ρ=πq=\\rho=\\pi\. ∎

Note that the reverse\-KL descent in Theorem[24](https://arxiv.org/html/2609.05727#Thmtheorem24)is stronger than the value ascent for general tangent vectors in Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12)\. According to Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12), the value\-ascent certificate \([27](https://arxiv.org/html/2609.05727#S5.E27)\) gives

𝔼X∼q\[r~ρ\(X\)\]≥𝔼X∼ρ\[r~ρ\(X\)\]⇔KL\(q∥π\)−KL\(q∥ρ\)≤KL\(ρ∥π\)\.\\mathbb\{E\}\_\{X\\sim q\}\[\\tilde\{r\}^\{\\rho\}\(X\)\]\\geq\\mathbb\{E\}\_\{X\\sim\\rho\}\[\\tilde\{r\}^\{\\rho\}\(X\)\]\\quad\\Leftrightarrow\\quad\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\-\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.In contrast, Theorem[24](https://arxiv.org/html/2609.05727#Thmtheorem24)directly gives

KL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.
Therefore, for the ideal iteration

ρk\+1=𝒯⁡\(vρk\+ηk​Γρk,r~ρk\),ηk∈\(0,τ\],\\rho\_\{k\+1\}=\\mathcal\{T\}\\left\(v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\\right\),\\qquad\\eta\_\{k\}\\in\(0,\\tau\],the sequence\{KL⁡\(ρk∥π\)\}k=0∞\\\{\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\\\}\_\{k=0\}^\{\\infty\}is nonincreasing\. Furthermore, the reverse KL strictly decreases at each Newton Matching stage, unless the current densityρk\\rho\_\{k\}reaches the target densityπ\\pi\. This establishes the finite\-stepsize descent depicted in Figure[7](https://arxiv.org/html/2609.05727#S6.F7)throughout the full admissible stepsize range, including the full Newton step\.

Before proceeding to the global convergence of Newton Matching, we state an auxiliary forward\-KL criterion\. This result is not needed for reverse KL descent, but it later provides an ideal anti\-collapse safeguard in Remark[29](https://arxiv.org/html/2609.05727#Thmtheorem29)\.

###### Corollary 25\.

Letq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\)be the terminal density generated by one ideal Newton Matching stage \([33](https://arxiv.org/html/2609.05727#S6.E33)\)\. Consider a stepsizeη\>0\\eta\>0\. If

τ​η​𝔼X∼π​\[∫01κt​‖∇Vtρ​\[r~ρ\]​\(Φ1→tρ,η,r~ρ​\(X\)\)‖22​𝑑t\]≤KL⁡\(π∥ρ\)\+KL⁡\(ρ∥π\),\\tau\\eta\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\right\]\\leq\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\+\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\),\(36\)then the forward KL decreases:

KL⁡\(π∥q\)≤KL⁡\(π∥ρ\)\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|q\\right\)\\leq\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\.

###### Proof\.

By \([35c](https://arxiv.org/html/2609.05727#S7.E35.3)\),KL⁡\(π∥q\)≤KL⁡\(π∥ρ\)\\operatorname\{KL\}\\left\(\\pi\\middle\\\|q\\right\)\\leq\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)holds if and only if

τη​𝔼X∼π​\[𝒟ρ,η,r~ρ​\(X\)\]≤KL⁡\(π∥ρ\)\+KL⁡\(ρ∥π\),\\frac\{\\tau\}\{\\eta\}\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\\leq\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\+\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\),which finishes the proof\. ∎

In Corollary[25](https://arxiv.org/html/2609.05727#Thmtheorem25), if the map

η↦𝔼X∼π​\[∫01κt​‖∇Vtρ​\[r~ρ\]​\(Φ1→tρ,η,r~ρ​\(X\)\)‖22​𝑑t\]\\eta\\mapsto\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\right\]is continuous near00, the left\-hand side of \([36](https://arxiv.org/html/2609.05727#S7.E36)\) isO⁡\(η\)O\(\\eta\)asη→0\\eta\\to 0\. In this case, the forward KL also decreases with a sufficiently small stepsizeη\>0\\eta\>0\.

#### 7\.1\.2Self\-Calibration

Finite\-stepsize descent guarantees a monotone sequence of objective values, but it does not yet describe the asymptotic shape of the density residual\. The following theorem identifies the stronger, unconditional statement supplied by the exact Newton Matching iteration: under non\-summable stepsizes, the log\-density ratiolog⁡ρkπ\\log\\frac\{\\rho\_\{k\}\}\{\\pi\}becomes asymptotically flat under the current densityρk\\rho\_\{k\}\.

###### Theorem 26\.

Consider the ideal Newton Matching stage \([32](https://arxiv.org/html/2609.05727#S6.E32)\), where the sequences of densities and stepsizes are\{ρk\}k=0∞⊂𝒫\\\{\\rho\_\{k\}\\\}\_\{k=0\}^\{\\infty\}\\subset\\mathscr\{P\}and\{ηk\}k=0∞⊂\(0,τ\]\\\{\\eta\_\{k\}\\\}\_\{k=0\}^\{\\infty\}\\subset\(0,\\tau\], respectively\. Then, the reverse KL converges:

K∞:=limk→∞KL⁡\(ρk∥π\)∈\[0,KL⁡\(ρ0∥π\)\]\.K\_\{\\infty\}:=\\lim\_\{k\\to\\infty\}\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\\in\[0,\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\]\.\(37\)If the stepsizes are non\-summable, i\.e\.,∑k=0∞ηk=∞\\sum\_\{k=0\}^\{\\infty\}\\eta\_\{k\}=\\infty, then, ask→∞k\\to\\infty,

𝔼X∼ρk​\[\|log⁡ρk​\(X\)π⁡\(X\)−K∞\|\]→0\.\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[\\left\|\\log\\frac\{\\rho\_\{k\}\(X\)\}\{\\pi\(X\)\}\-K\_\{\\infty\}\\right\|\\right\]\\to 0\.\(38\)Equivalently,

log⁡ρkπ→L1​\(ρk\)K∞\.\\log\\frac\{\\rho\_\{k\}\}\{\\pi\}\\xlongrightarrow\{L^\{1\}\(\\rho\_\{k\}\)\}K\_\{\\infty\}\.

Theorem[26](https://arxiv.org/html/2609.05727#Thmtheorem26)shows more than convergence of the scalar reverse\-KL values\. It states that Newton Matching becomes asymptotically self\-consistent on the regions visited by the current sampler\. This is a genuine stationarity property, but it is on\-policy: it does not by itself certify that the current density continues to cover every region carrying target mass\. The following remark separates these two notions\.

#### 7\.1\.3Global KL Convergence under Mild Conditions

Self\-calibration identifies the unconditional asymptotic stationarity of Newton Matching under the current density\. To identify this stationary limit with the target, it remains to rule out loss of target mass\. We give two complementary mild conditions\. The first is an anti\-collapse condition expressed through bounded forward KL\. The second requires the target\-averaged path dissipation to vanish asymptotically\. Either condition forces the limiting reverse KL value to be zero, while the second additionally yields convergence of the forward KL\. Before stating these results, we summarize the corresponding notions of convergence for terminal densities and canonical velocity fields\.

Consider two densitiesρ,q∈𝒫\\rho,q\\in\\mathscr\{P\}\. Their total variation distance is

TV⁡\(ρ,q\):=12​∫ℝd\|ρ⁡\(x\)−q⁡\(x\)\|​𝑑x,\\operatorname\{TV\}\\left\(\\rho,q\\right\):=\\frac\{1\}\{2\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\|\\rho\(x\)\-q\(x\)\\right\|\\,\\mathrm\{d\}x,and Pinsker’s inequality\[[30](https://arxiv.org/html/2609.05727#bib.bib58)\]gives

TV⁡\(ρ,q\)≤12​min⁡\{KL⁡\(ρ∥q\),KL⁡\(q∥ρ\)\}\.\\operatorname\{TV\}\\left\(\\rho,q\\right\)\\leq\\sqrt\{\\frac\{1\}\{2\}\\min\\left\\\{\\operatorname\{KL\}\\left\(\\rho\\middle\\\|q\\right\),\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)\\right\\\}\}\.
Now consider a sequence\{ρk\}k=0∞⊂𝒫\\\{\\rho\_\{k\}\\\}\_\{k=0\}^\{\\infty\}\\subset\\mathscr\{P\}and a targetπ∈𝒫\\pi\\in\\mathscr\{P\}\. Since convergence in total variation implies weak convergence, convergence of either the reverse or forward KL divergence to zero already impliesρk→wπ\\rho\_\{k\}\\xrightarrow\{\\mathrm\{w\}\}\\pi\. Moreover, Corollary[107](https://arxiv.org/html/2609.05727#Thmtheorem107)in Appendix[D\.5](https://arxiv.org/html/2609.05727#A4.SS5)identifies the following properties:

KL⁡\(ρk∥π\)=∫011κt​𝔼Xt∼ptρk​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​𝑑t,\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)=\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\_\{k\}\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t,and

KL⁡\(π∥ρk\)=∫011κt​𝔼Xt∼ptπ​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​𝑑t\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)=\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\pi\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\.Thus, reverse\-KL convergence yields vanishing velocity discrepancy averaged along the current marginal pathsptρkp\_\{t\}^\{\\rho\_\{k\}\}, whereas forward\-KL convergence yields the analogous statement along the target marginal pathptπp\_\{t\}^\{\\pi\}\. In either case, it also implies convergence of the terminal densities in total variation and hence weakly\. The KL divergences therefore simultaneously control distributional convergence and a corresponding integrated notion of canonical\-velocity convergence\.

In the following, we consider the ideal Newton Matching iteration \([32](https://arxiv.org/html/2609.05727#S6.E32)\), with density sequence\{ρk\}k=0∞⊂𝒫\\\{\\rho\_\{k\}\\\}\_\{k=0\}^\{\\infty\}\\subset\\mathscr\{P\}and stepsizes\{ηk\}k=0∞⊂\(0,τ\]\\\{\\eta\_\{k\}\\\}\_\{k=0\}^\{\\infty\}\\subset\(0,\\tau\]\. We assume that the stepsizes are non\-summable:

∑k=0∞ηk=∞\.\\sum\_\{k=0\}^\{\\infty\}\\eta\_\{k\}=\\infty\.Accordingly, we use the forward and reverse KL divergences as the primary convergence quantities; whenever either is shown to converge to zero, the corresponding velocity\-field and distributional convergence follow from the identities above\.

The following theorem proves convergence of the reverse KL under the anti\-collapse condition \([39](https://arxiv.org/html/2609.05727#S7.E39)\)\.

###### Theorem 28\.

Using the notation and setting of Theorem[26](https://arxiv.org/html/2609.05727#Thmtheorem26), assume that the forward KL is bounded, i\.e\.,

supkKL⁡\(π∥ρk\)<∞\.\\sup\_\{k\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)<\\infty\.\(39\)Then, ask→∞k\\to\\infty, both the terminal densities and the canonical velocity fields converge:

KL⁡\(ρk∥π\)=∫011κt​𝔼Xt∼ptρk​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​𝑑t→0\.\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)=\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\_\{k\}\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\\to 0\.

Inspired by Remark[20](https://arxiv.org/html/2609.05727#Thmtheorem20), we develop the second condition \([40](https://arxiv.org/html/2609.05727#S7.E40)\) to control the nonlinear path dissipation introduced by the finite\-stepsize canonical retraction\. It is complementary to the anti\-collapse condition \([39](https://arxiv.org/html/2609.05727#S7.E39)\): rather than assuming a uniform bound on the forward KL sequence, it assumes finite initial forward KL and requires the target\-averaged dissipation to vanish asymptotically\. Under this condition, both forward and reverse KL divergences converge to zero\.

###### Theorem 30\.

Using the notation and setting of Theorem[26](https://arxiv.org/html/2609.05727#Thmtheorem26), assume thatKL⁡\(π∥ρ0\)<∞\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{0\}\\right\)<\\inftyand the dissipation vanishes, i\.e\.,

limk→∞ηk​𝔼X∼π​\[∫01κt​‖∇Vtρk​\[r~ρk\]​\(Φ1→tρk,ηk,r~ρk​\(X\)\)‖22​𝑑t\]=0\.\\lim\_\{k\\to\\infty\}\\eta\_\{k\}\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\_\{k\}\}\[\\tilde\{r\}^\{\\rho\_\{k\}\}\]\\left\(\\Phi\_\{1\\to t\}^\{\{\\rho\_\{k\}\},\\eta\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\(X\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\right\]=0\.\(40\)Then, ask→∞k\\to\\infty, both the terminal densities and the canonical velocity fields converge:

KL⁡\(ρk∥π\)=∫011κt​𝔼Xt∼ptρk​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​dt→0,\\displaystyle\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)=\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\_\{k\}\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\\to 0,KL⁡\(π∥ρk\)=∫011κt​𝔼Xt∼ptπ​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​dt→0\.\\displaystyle\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)=\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\pi\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\\to 0\.

In summary, Theorems[28](https://arxiv.org/html/2609.05727#Thmtheorem28)and[30](https://arxiv.org/html/2609.05727#Thmtheorem30)provide two complementary routes from self\-calibration to KL convergence\. Bounded forward KL rules out collapse and yields reverse\-KL convergence, whereas vanishing target\-averaged dissipation yields convergence in both forward and reverse KL\.

### 7\.2Local Quadratic Convergence and Continuation Method

The global convergence results above describe the long\-run behavior of Newton Matching from general initial points under mild target\-identification conditions\. We now turn to the complementary local regime\. As expected for Newton’s method, the full\-step iteration converges quadratically once the current density lies sufficiently close to the target\. Under the compatible smooth realization in Assumption[67](https://arxiv.org/html/2609.05727#Thmtheorem67), Section[7\.2\.1](https://arxiv.org/html/2609.05727#S7.SS2.SSS1)states the local quadratic convergence rate for both terminal densities and canonical velocity fields\. As an optional way to exploit this rapid local convergence, Section[7\.2\.2](https://arxiv.org/html/2609.05727#S7.SS2.SSS2)develops a continuation scheme designed to keep successive stages within the quadratic\-convergence regime as the inverse temperature increases\.

#### 7\.2\.1Local Quadratic Convergence

Here, we denoteπ:=πμ,τ,r\\pi:=\\pi\_\{\\mu,\\tau,r\}and take the full stepsizeηk=τ\\eta\_\{k\}=\\tauin \([34](https://arxiv.org/html/2609.05727#S6.E34)\)\. All proofs are deferred to Appendix[B\.2\.1](https://arxiv.org/html/2609.05727#A2.SS2.SSS1)\.

For the local convergence analysis, we equip the density manifold𝒫\\mathscr\{P\}with a metric

d𝒫:𝒫×𝒫→\[0,∞\)d\_\{\\mathscr\{P\}\}:\\mathscr\{P\}\\times\\mathscr\{P\}\\to\[0,\\infty\)induced by the norm of its model Banach space through a global chart\. The detailed construction is deferred to Appendix[B\.2\.1](https://arxiv.org/html/2609.05727#A2.SS2.SSS1)\. We also transportd𝒫d\_\{\\mathscr\{P\}\}to the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}by

d𝒱can:𝒱can×𝒱can→\[0,∞\),d𝒱can​\(v\(1\),v\(2\)\):=d𝒫​\(𝒯⁡\(v\(1\)\),𝒯⁡\(v\(2\)\)\)\.d\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}:\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\times\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to\[0,\\infty\),\\qquad d\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\(v^\{\(1\)\},v^\{\(2\)\}\):=d\_\{\\mathscr\{P\}\}\(\\mathcal\{T\}\(v^\{\(1\)\}\),\\mathcal\{T\}\(v^\{\(2\)\}\)\)\.\(41\)In particular, we verify thatd𝒫d\_\{\\mathscr\{P\}\}andd𝒱cand\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}are valid metrics on𝒫\\mathscr\{P\}and𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}, respectively\.

###### Proposition 31\.

The mapd𝒫:𝒫×𝒫→\[0,∞\)d\_\{\\mathscr\{P\}\}:\\mathscr\{P\}\\times\\mathscr\{P\}\\to\[0,\\infty\)is symmetric and definite, and satisfies the triangle inequality\. In other words,d𝒫d\_\{\\mathscr\{P\}\}is a metric on the density manifold𝒫\\mathscr\{P\}\. Furthermore,d𝒫d\_\{\\mathscr\{P\}\}induces exactly the Banach\-manifold topology on𝒫\\mathscr\{P\}\.

Consequently,d𝒱can:𝒱can×𝒱can→\[0,∞\)d\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}:\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\times\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to\[0,\\infty\)is a metric on the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}and induces exactly the Banach\-manifold topology on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

Under these metrics, full\-step Newton Matching exhibits the characteristic local behavior of Newton’s method: once the current density lies within a sufficiently small neighborhood of the target, a single ideal stage reduces the error to at most a fixed constant times its square\. Since the metric on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is transported from𝒫\\mathscr\{P\}through the canonical diffeomorphism, the same quadratic rate holds for the corresponding canonical velocity fields\.

###### Theorem 32\.

We take the full stepsizeη=τ\\eta=\\tau\. Then, there exist a radiusRπ∈\(0,∞\)R\_\{\\pi\}\\in\(0,\\infty\)and a constantCπ∈\(0,∞\)C\_\{\\pi\}\\in\(0,\\infty\)such that for allρ∈𝒫\\rho\\in\\mathscr\{P\}satisfyingd𝒫​\(ρ,π\)<Rπd\_\{\\mathscr\{P\}\}\(\\rho,\\pi\)<R\_\{\\pi\}, we have

d𝒫​\(q,π\)≤Cπ​d𝒫​\(ρ,π\)2,d\_\{\\mathscr\{P\}\}\(q,\\pi\)\\leq C\_\{\\pi\}d\_\{\\mathscr\{P\}\}\(\\rho,\\pi\)^\{2\},\(42\)whereq=𝒯⁡\(vρ\+τ​Γρ,r~ρ\)∈𝒫q=\\mathcal\{T\}\\left\(v^\{\\rho\}\+\\tau\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\right\)\\in\\mathscr\{P\}is the updated density obtained fromρ\\rhoafter one step with the full stepsizeη=τ\\eta=\\tau\.

Consequently, if the initial pointρ0\\rho\_\{0\}satisfiesd𝒫​\(ρ0,π\)<min⁡\{Rπ,1Cπ\}d\_\{\\mathscr\{P\}\}\(\\rho\_\{0\},\\pi\)<\\min\\left\\\{R\_\{\\pi\},\\frac\{1\}\{C\_\{\\pi\}\}\\right\\\}, then under the full stepsizeηk=τ\\eta\_\{k\}=\\taufor allkk, we always have

d𝒫​\(ρk,π\)<min⁡\{Rπ,1Cπ\},d𝒫​\(ρk\+1,π\)≤Cπ​d𝒫​\(ρk,π\)2\.d\_\{\\mathscr\{P\}\}\(\\rho\_\{k\},\\pi\)<\\min\\left\\\{R\_\{\\pi\},\\frac\{1\}\{C\_\{\\pi\}\}\\right\\\},\\qquad d\_\{\\mathscr\{P\}\}\(\\rho\_\{k\+1\},\\pi\)\\leq C\_\{\\pi\}d\_\{\\mathscr\{P\}\}\(\\rho\_\{k\},\\pi\)^\{2\}\.\(43\)Therefore, both the terminal densitiesρk\\rho\_\{k\}and the canonical velocity fieldsvρk:=𝒞⁡\(ρk\)v^\{\\rho\_\{k\}\}:=\\mathcal\{C\}\(\\rho\_\{k\}\)converge quadratically to their targets, i\.e\.,π\\piandvπ:=𝒞⁡\(π\)v^\{\\pi\}:=\\mathcal\{C\}\(\\pi\), respectively:

Cπ​d𝒫​\(ρk,π\)=Cπ​d𝒱can​\(vρk,vπ\)≤\(Cπ​d𝒫​\(ρ0,π\)\)2k→0\.C\_\{\\pi\}d\_\{\\mathscr\{P\}\}\(\\rho\_\{k\},\\pi\)=C\_\{\\pi\}d\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\(v^\{\\rho\_\{k\}\},v^\{\\pi\}\)\\leq\\left\(C\_\{\\pi\}d\_\{\\mathscr\{P\}\}\(\\rho\_\{0\},\\pi\)\\right\)^\{2^\{k\}\}\\to 0\.

#### 7\.2\.2Continuation Method

Motivated by the local quadratic convergence established above, we next develop an optional continuation method along the inverse\-temperature path\. Note that, under the mild target\-identification conditions of Section[7\.1](https://arxiv.org/html/2609.05727#S7.SS1), Newton Matching can be applied directly to the final targetπμ,τ,r\\pi\_\{\\mu,\\tau,r\}and converges globally without introducing intermediate temperatures\. The continuation method provides an optional way to exploit this rapid local convergence by choosing successive targets close enough that each stage begins within the quadratic\-convergence neighborhood of its target\.

Let

πs​\(x\)∝μ⁡\(x\)​es​r​\(x\)\\pi\_\{s\}\(x\)\\propto\\mu\(x\)e^\{sr\(x\)\}and choose a finite grid of the inverse temperature

0≤τ0<τ1<⋯<τn=τ\.0\\leq\\tau\_\{0\}<\\tau\_\{1\}<\\cdots<\\tau\_\{n\}=\\tau\.Rather than solving directly forπτ\\pi\_\{\\tau\}from an arbitrary initialization, theiith continuation stage uses the previous targetπτi−1\\pi\_\{\\tau\_\{i\-1\}\}as a warm start for the nearby targetπτi\\pi\_\{\\tau\_\{i\}\}\. Since

πτi​\(x\)∝πτi−1​\(x\)​e\(τi−τi−1\)​r​\(x\),\\pi\_\{\\tau\_\{i\}\}\(x\)\\propto\\pi\_\{\\tau\_\{i\-1\}\}\(x\)e^\{\(\\tau\_\{i\}\-\\tau\_\{i\-1\}\)r\(x\)\},this stage is itself a Newton Matching problem with reference densityπτi−1\\pi\_\{\\tau\_\{i\-1\}\}, targetπτi\\pi\_\{\\tau\_\{i\}\}, inverse\-temperature increment

Δi:=τi−τi−1,\\Delta\_\{i\}:=\\tau\_\{i\}\-\\tau\_\{i\-1\},and full stepsizeηk\(i\)=Δi\\eta\_\{k\}^\{\(i\)\}=\\Delta\_\{i\}\. It is expected that a sufficiently fine grid ensures that each warm start lies inside the local quadratic\-convergence neighborhood of the next target\. Ifμ\\muis a density andπ0=μ∈𝒫\\pi\_\{0\}=\\mu\\in\\mathscr\{P\}, we may takeτ0=0\\tau\_\{0\}=0\. Ifμ=1\\mu=1, we instead choose a positiveτ0\\tau\_\{0\}such thatπs∈𝒫\\pi\_\{s\}\\in\\mathscr\{P\}for everys∈\[τ0,τ\]s\\in\[\\tau\_\{0\},\\tau\]\.

The following proposition shows that a single quadratic\-convergence constant and a single upper bound on the grid spacing suffice for all continuation stages along the inverse\-temperature interval\. The proof is deferred to Appendix[B\.2\.2](https://arxiv.org/html/2609.05727#A2.SS2.SSS2)\.

###### Proposition 33\.

Assume that∀s∈\[τ0,τ\]\\forall s\\in\[\\tau\_\{0\},\\tau\],πs:=πμ,s,r∈𝒫\\pi\_\{s\}:=\\pi\_\{\\mu,s,r\}\\in\\mathscr\{P\}, whereτ0=0\\tau\_\{0\}=0is allowed whenμ\\muis a normalized density andτ0\>0\\tau\_\{0\}\>0is required whenμ=1\\mu=1\. There exist a positive constantC∈\(0,∞\)C\\in\(0,\\infty\)and an admissible grid sizeΔmax∈\(0,∞\)\\Delta\_\{\\max\}\\in\(0,\\infty\), both depending onμ,τ0,τ,r\\mu,\\tau\_\{0\},\\tau,r, such that for any finite grid0≤τ0<τ1<⋯<τn=τ0\\leq\\tau\_\{0\}<\\tau\_\{1\}<\\cdots<\\tau\_\{n\}=\\tausatisfyingΔi:=τi−τi−1≤Δmax\\Delta\_\{i\}:=\\tau\_\{i\}\-\\tau\_\{i\-1\}\\leq\\Delta\_\{\\max\}for allii, Newton Matching with full stepsizeηk\(i\)=Δi\\eta\_\{k\}^\{\(i\)\}=\\Delta\_\{i\}satisfies

d𝒫​\(ρk\+1\(i\),π\(i\)\)≤C​d𝒫​\(ρk\(i\),π\(i\)\)2,d𝒫​\(ρ0\(i\),π\(i\)\)<1C\.d\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{k\+1\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)\\leq Cd\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{k\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)^\{2\},\\qquad d\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{0\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)<\\frac\{1\}\{C\}\.Here, the target isπ\(i\):=πτi\\pi^\{\(i\)\}:=\\pi\_\{\\tau\_\{i\}\}and the initial density isρ0\(i\):=π\(i−1\)\\rho\_\{0\}^\{\(i\)\}:=\\pi^\{\(i\-1\)\}\.

Consequently, for every continuation stageii, both the terminal densitiesρk\(i\)\\rho\_\{k\}^\{\(i\)\}and the canonical velocity fieldsvρk\(i\):=𝒞⁡\(ρk\(i\)\)v^\{\\rho\_\{k\}^\{\(i\)\}\}:=\\mathcal\{C\}\\left\(\\rho\_\{k\}^\{\(i\)\}\\right\)converge quadratically to their targets, i\.e\.,π\(i\)\\pi^\{\(i\)\}andvπ\(i\):=𝒞⁡\(π\(i\)\)v^\{\\pi^\{\(i\)\}\}:=\\mathcal\{C\}\\left\(\\pi^\{\(i\)\}\\right\), respectively:

C​d𝒫​\(ρk\(i\),π\(i\)\)=C​d𝒱can​\(vρk\(i\),vπ\(i\)\)≤\(C​d𝒫​\(ρ0\(i\),π\(i\)\)\)2k→0\.Cd\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{k\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)=Cd\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\\left\(v^\{\\rho\_\{k\}^\{\(i\)\}\},v^\{\\pi^\{\(i\)\}\}\\right\)\\leq\\left\(Cd\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{0\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)\\right\)^\{2^\{k\}\}\\to 0\.

Together, Sections[7\.1](https://arxiv.org/html/2609.05727#S7.SS1)and[7\.2](https://arxiv.org/html/2609.05727#S7.SS2)establish global and local convergence guarantees for Newton Matching\. Section[8](https://arxiv.org/html/2609.05727#S8)next turns from the abstract iteration \([32](https://arxiv.org/html/2609.05727#S6.E32)\) to its computational realization by deriving population\-exact regression objectives for the tangential update and canonicalization that retain the scalable matching structure of diffusion and flow training\.

## 8Exact Newton Matching

The preceding sections characterize the abstract Newton Matching iteration and establish its convergence properties\. Given a canonical velocity fieldvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}and a stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\], one exact stage of Newton Matching consists of two operations\. First, the tangential update constructs the ambient velocity field

v¯t​\(xt\)=vtρ​\(xt\)\+η​Γtρ,r~ρ​\(xt\),\\bar\{v\}\_\{t\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\),\(44\)where

r~ρ\(x\)=r\(x\)−1τlogρ⁡\(x\)μ⁡\(x\),Γtρ,r~ρ\(xt\)=CovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),r~ρ\(X1\)\)\.\\tilde\{r\}^\{\\rho\}\(x\)=r\(x\)\-\\frac\{1\}\{\\tau\}\\log\\frac\{\\rho\(x\)\}\{\\mu\(x\)\},\\qquad\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\)\.Second, the canonicalization produces a new canonical velocity fieldvq=𝒞⁡\(q\)∈𝒱canv^\{q\}=\\mathcal\{C\}\(q\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}, whose terminal density coincides with that induced byv¯\\bar\{v\}, i\.e\.,q=𝒯⁡\(vq\)=𝒯⁡\(v¯\)q=\\mathcal\{T\}\(v^\{q\}\)=\\mathcal\{T\}\(\\bar\{v\}\)\. The next Newton Matching stage then proceeds withvqv^\{q\}\.

In this section, we present population\-exact implementations of both the tangential update and canonicalization\. We denote the neural velocity field being trained byvθv^\{\\theta\}, whereθ\\thetarepresents its network parameters\. To distinguish it from the trainable field, we refer tovρv^\{\\rho\}as the*anchor*, emphasizing thatvρv^\{\\rho\}remains fixed throughout each tangential\-update stage and is used to construct the training objectives, including generating samples, estimating the required quantities, and evaluating the relevant gradients\. Consequently, all anchor\-dependent objects, such asptρp\_\{t\}^\{\\rho\}andp1\|tρp\_\{1\|t\}^\{\\rho\}, are also fixed during the training ofvθv^\{\\theta\}at one tangential\-update stage\. In practice, the anchorvρv^\{\\rho\}is also represented by a neural network\. It is therefore natural to initializevθv^\{\\theta\}with the parameters ofvρv^\{\\rho\}and subsequently optimize it using the matching losses introduced later\. Throughout this optimization, the losses are differentiated only with respect to the trainable parametersθ\\theta\.

Sections[8\.1](https://arxiv.org/html/2609.05727#S8.SS1)–[8\.5](https://arxiv.org/html/2609.05727#S8.SS5)develop population\-exact realizations of the tangential update\. First, Section[8\.1](https://arxiv.org/html/2609.05727#S8.SS1)presents two equivalent conditional\-expectation representations of the tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}: a covariance form and a gradient form\. These representations form the basis of the regression targets used in the subsequent losses\. Then, Section[8\.2](https://arxiv.org/html/2609.05727#S8.SS2)studies the sampling order used to construct the regression pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)\. The forward construction first samplesX1X\_\{1\}from the anchor density and then applies the noising kernel to obtainXtX\_\{t\}, whereas the reverse construction first samplesXtX\_\{t\}and then drawsX1X\_\{1\}from the corresponding anchor posterior\. Both constructions satisfy the posterior requirementX1\|Xt∼p1\|tρ\(⋅\|Xt\)X\_\{1\}\|X\_\{t\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)and therefore support objectives for scalable regression with the same minimizerv¯=vρ\+η​Γρ,r~ρ\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\. Together, Sections[8\.1](https://arxiv.org/html/2609.05727#S8.SS1)and[8\.2](https://arxiv.org/html/2609.05727#S8.SS2)separate two independent design choices: the tangent representation, which may be covariance or gradient, and the sampling construction, which may be forward or reverse\. Sections[8\.3](https://arxiv.org/html/2609.05727#S8.SS3)and[8\.4](https://arxiv.org/html/2609.05727#S8.SS4)then develop the covariance\-form and gradient\-form losses under both constructions, respectively, with particular attention to the nontrivial computations required to evaluater~ρ\\tilde\{r\}^\{\\rho\}or its gradient∇r~ρ\\nabla\\tilde\{r\}^\{\\rho\}\. Section[8\.5](https://arxiv.org/html/2609.05727#S8.SS5)unifies these forms through Stein control variates: all resulting targets have the same conditional mean and hence the same population minimizer, while differing in their sample\-wise variance\. Table[1](https://arxiv.org/html/2609.05727#S8.T1)summarizes representative losses for the tangential update\. To explicitly illustrate the usage of multi\-time supervision for enhancing sample efficiency, Section[8\.6](https://arxiv.org/html/2609.05727#S8.SS6)shows how regression pairs at multiple supervision times can be constructed from a single endpoint or path sample, with the associated endpoint\-level or path\-level computations shared across those pairs\. Finally, Section[8\.7](https://arxiv.org/html/2609.05727#S8.SS7)presents the training procedures for canonicalization, completing one exact Newton Matching stage\.

### 8\.1Tangent Representations

The tangential update in \([44](https://arxiv.org/html/2609.05727#S8.E44)\) is determined by the tangent vector

Γtρ,r~ρ\(xt\)=CovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),r~ρ\(X1\)\)\.\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\)\.Since the canonical anchor satisfiesvtρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|X1\)\]v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\], we have

Γtρ,r~ρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(vt\|1\(xt\|X1\)−vtρ\(xt\)\)\(r~ρ\(X1\)−Bt\(xt\)\)\],\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\\left\(\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\\right\)\\right\],\(45\)whereBt​\(xt\)∈ℝB\_\{t\}\(x\_\{t\}\)\\in\\mathbb\{R\}is any regular baseline that depends only on\(t,xt\)\(t,x\_\{t\}\)\. This conditional\-expectation structure naturally suggests a CFM\-style loss\. As we show later, exploiting this structure yields the*covariance form*of the tangential\-update loss for exact Newton Matching\.

On the other hand, recall from Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5)that the tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}can also be written as the gradient of a posterior value:

Γtρ,r~ρ​\(xt\)=κt​∇xtVtρ​\[r~ρ\]​\(xt\),\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=\\kappa\_\{t\}\\nabla\_\{x\_\{t\}\}V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\(x\_\{t\}\),where

Vtρ\[r~ρ\]\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[r~ρ\(X1\)\]V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\bigl\[\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\bigr\]denotes the expected regularized reward under the anchor posterior\. However, the gradient lies outside the conditional expectation and is therefore not directly amenable to a CFM\-style objective\. Ideally, we would move the gradient inside the expectation, thereby recovering the conditional\-expectation structure needed to construct such an objective\. This interchange is more subtle than it may initially appear, since the anchor posterior itself depends onxtx\_\{t\}\. In particular, differentiating the conditional expectation with respect toxtx\_\{t\}does not simply yield the conditional expectation of the regularized\-reward gradient:

∇xt𝔼X1∼p1\|tρ\(⋅\|xt\)\[r~ρ\(X1\)\]≠𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇X1r~ρ\(X1\)\]in general\.\\nabla\_\{x\_\{t\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\bigl\[\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\bigr\]\\neq\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\bigl\[\\nabla\_\{X\_\{1\}\}\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\bigr\]\\text\{ in general\}\.
Fortunately, it is still possible to move the gradient operator inside the conditional expectation through a careful reformulation\. To motivate the construction, we first recall the Gaussian Stein identity\. LetX∼𝒩⁡\(m,Σ\)X\\sim\\mathcal\{N\}\(m,\\Sigma\)be add\-dimensional Gaussian random variable\. Then, for every sufficiently regular test functionff,

CovX∼𝒩⁡\(m,Σ\)\(X,f\(X\)\)=𝔼X∼𝒩⁡\(m,Σ\)\[Σ∇f\(X\)\]\.\\operatorname\{Cov\}\_\{X\\sim\\mathcal\{N\}\(m,\\Sigma\)\}\\left\(X,f\(X\)\\right\)=\\mathbb\{E\}\_\{X\\sim\\mathcal\{N\}\(m,\\Sigma\)\}\\left\[\\Sigma\\nabla f\(X\)\\right\]\.This identity can be obtained via integration by parts\.

Our setting is more involved, as the anchor posteriorp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)is generally non\-Gaussian\. Nevertheless, an analogous integration\-by\-parts identity can be obtained via the*Langevin Stein operator*\. To tailor our construction to the posterior\-covariance structure fromΓtρ,r~ρ\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}, we further introduce a*posterior Stein kernel*based on Langevin Stein operators\. For fixed\(t,xt\)\(t,x\_\{t\}\), let the matrix field

Λtρ\(⋅\|xt\):ℝd→ℝd×d\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\colon\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}be a posterior Stein kernel associated withp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\. Then, for every sufficiently regular scalar\-valued functionff, it satisfies the covariance–gradient identity

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇f\(X1\)\]\.\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]\.\(46\)Detailed results on Langevin Stein operators and posterior Stein kernels are provided in Appendix[E](https://arxiv.org/html/2609.05727#A5)\.

To apply this identity to the tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}, first observe that the conditional velocityvt\|1​\(xt\|x1\)v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)is affine in the endpointx1x\_\{1\}:

vt\|1​\(xt\|x1\)=β˙tβt​xt\+αt​κtβt2​x1\.v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\+\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}x\_\{1\}\.Conditioning onxtx\_\{t\}, the first term is independent ofx1x\_\{1\}and therefore does not contribute to the covariance\. It follows that

Γtρ,r~ρ​\(xt\)\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=αt​κtβt2CovX1∼p1\|tρ\(⋅\|xt\)\(X1,r~ρ\(X1\)\)\.\\displaystyle=\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\)\.\(47\)Applying \([46](https://arxiv.org/html/2609.05727#S8.E46)\) withf=r~ρf=\\tilde\{r\}^\{\\rho\}to \([47](https://arxiv.org/html/2609.05727#S8.E47)\) yields

Γtρ,r~ρ​\(xt\)\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=αt​κtβt2𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇r~ρ\(X1\)\]\.\\displaystyle=\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\]\.\(48\)This recovers the desired conditional\-expectation structure in terms of the regularized\-reward gradient\. As we show later, this representation leads to the*gradient form*of the tangential\-update loss for exact Newton Matching\.

Equations \([45](https://arxiv.org/html/2609.05727#S8.E45)\) and \([48](https://arxiv.org/html/2609.05727#S8.E48)\) provide two exact representations of the same tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\. The former requires a scalar evaluation ofr~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\), whereas the latter requires evaluating the action of a posterior Stein kernel on∇r~ρ​\(X1\)\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\. As the following subsections show, these two representations give rise to distinct loss functions for tangential\-update training and require different computational techniques for their implementation\.

### 8\.2Forward and Reverse Constructions

The conditional\-expectation representations of the tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}are particularly useful for scalable training, since squared\-loss regression allows conditional expectations to be recovered from sample\-wise targets\. Specifically, let\(X,Y\)\(X,Y\)be a pair of random variables, and suppose that the goal is to learn a functionfθf^\{\\theta\}approximating𝔼Y∼pY\|X\(⋅\|x\)\[g\(x,Y\)\]\\mathbb\{E\}\_\{Y\\sim p\_\{Y\|X\}\(\\cdot\|x\)\}\\\!\\left\[g\(x,Y\)\\right\]\. Rather than evaluating this conditional expectation explicitly, we can useg⁡\(X,Y\)g\(X,Y\)directly as a regression target and minimize

𝔼\(X,Y\)∼pX,Y​\[‖fθ​\(X\)−g⁡\(X,Y\)‖22\],\\mathbb\{E\}\_\{\(X,Y\)\\sim p\_\{X,Y\}\}\\\!\\left\[\\left\\\|f^\{\\theta\}\(X\)\-g\(X,Y\)\\right\\\|\_\{2\}^\{2\}\\right\],provided that the joint densitypX,Yp\_\{X,Y\}used to generate the training samples induces the required conditional densitypY\|X​\(y\|x\)p\_\{Y\|X\}\(y\|x\)\.

In our case,X=XtX=X\_\{t\}andY=X1Y=X\_\{1\}\. We therefore do not need to evaluate the tangent vector explicitly during training, provided that we can generate pairs\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)from a joint distributionΞt,1ρ\\Xi\_\{t,1\}^\{\\rho\}satisfying

\(Xt,X1\)∼Ξt,1ρ⇒X1\|Xt∼p1\|tρ\(⋅\|Xt\)\.\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\quad\\Rightarrow\\quad X\_\{1\}\|X\_\{t\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\.\(49\)We useΞt,1ρ\\Xi\_\{t,1\}^\{\\rho\}as the generic notation for any joint distribution satisfying \([49](https://arxiv.org/html/2609.05727#S8.E49)\)\. ItsXtX\_\{t\}\-marginal is positive, and may be implicitly defined by the chosen construction\.

Before presenting the matching losses for the tangential update, we describe two complementary constructions ofΞt,1ρ\\Xi\_\{t,1\}^\{\\rho\}\. The*forward construction*first generates the endpointX1X\_\{1\}and then produces the noisy stateXtX\_\{t\}, following the order used in the standard CFM loss\. In contrast, the*reverse construction*first generatesXtX\_\{t\}and then samplesX1X\_\{1\}conditionally\. This construction has a Bayesian flavor:XtX\_\{t\}is treated as observed evidence, andX1X\_\{1\}is inferred from the anchor posterior\.

#### 8\.2\.1Forward Construction

The forward approach proceeds by

X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0,X\_\{1\}\\sim\\rho,\\qquad X\_\{0\}\\sim p\_\{0\},\\qquad X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\},\(50\)whereX0X\_\{0\}andX1X\_\{1\}are sampled independently\.

Conditioning onX1=x1X\_\{1\}=x\_\{1\}, the interpolant in \([50](https://arxiv.org/html/2609.05727#S8.E50)\) follows the conditional probability path

Xt\|X1∼pt\|1\(⋅\|X1\)\.X\_\{t\}\|X\_\{1\}\\sim p\_\{t\|1\}\(\\cdot\|X\_\{1\}\)\.Combining this conditional density with the endpoint marginalX1∼ρX\_\{1\}\\sim\\rho, the joint density of\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)is

pt,1ρ​\(xt,x1\)=ρ⁡\(x1\)​pt\|1​\(xt\|x1\)\.p\_\{t,1\}^\{\\rho\}\(x\_\{t\},x\_\{1\}\)=\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\.Marginalizing overx1x\_\{1\}and applying Bayes’ rule therefore gives

Xt∼ptρ,X1\|Xt∼p1\|tρ\(⋅\|Xt\)\.X\_\{t\}\\sim p\_\{t\}^\{\\rho\},\\qquad X\_\{1\}\|X\_\{t\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\.\(51\)Thus,Ξt,1ρ\\Xi\_\{t,1\}^\{\\rho\}satisfies \([49](https://arxiv.org/html/2609.05727#S8.E49)\) and is given by

Ξt,1ρ​\(xt,x1\)=ptρ​\(xt\)​p1\|tρ​\(x1\|xt\)under the forward construction\.\\Xi\_\{t,1\}^\{\\rho\}\(x\_\{t\},x\_\{1\}\)=p\_\{t\}^\{\\rho\}\(x\_\{t\}\)p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\qquad\\text\{under the forward construction\}\.
The endpointX1X\_\{1\}may be generated with any exact sampler whose terminal density isρ\\rho\. A broad family of such samplers is given by

dYt=\(vtρ\(Yt\)\+σt22∇logptρ\(Yt\)\)dt\+σtdWt,Y0∼p0,\\mathrm\{d\}Y\_\{t\}=\\left\(v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\\nabla\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\\right\)\\mathrm\{d\}t\+\\sigma\_\{t\}\\mathrm\{d\}W\_\{t\},\\qquad Y\_\{0\}\\sim p\_\{0\},which has marginal densityYt∼ptρY\_\{t\}\\sim p\_\{t\}^\{\\rho\}for everyt∈\[0,1\]t\\in\[0,1\], and hence producesX1:=Y1∼ρX\_\{1\}:=Y\_\{1\}\\sim\\rhofor any noise scheduleσt≥0\\sigma\_\{t\}\\geq 0\. Using the score–velocity relation

∇log⁡ptρ​\(x\)=1κt​\(vtρ​\(x\)−α˙tαt​x\),\\nabla\\log p\_\{t\}^\{\\rho\}\(x\)=\\frac\{1\}\{\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\\right\),we can equivalently write

d​Yt=\[\(1\+σt22​κt\)​vtρ​\(Yt\)−σt22​κt​α˙tαt​Yt\]​d​t\+σt​d​Wt,Y0∼p0\.\\mathrm\{d\}Y\_\{t\}=\\left\[\\left\(1\+\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\\kappa\_\{t\}\}\\right\)v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\-\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\\kappa\_\{t\}\}\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\]\\mathrm\{d\}t\+\\sigma\_\{t\}\\mathrm\{d\}W\_\{t\},\\qquad Y\_\{0\}\\sim p\_\{0\}\.\(52\)Whenσt=0\\sigma\_\{t\}=0, this SDE reduces to the standard probability flow ODE\. Therefore, one key advantage of the forward construction is its compatibility with a wide range of samplers\. For example, one may use an ODE sampler to potentially reduce the number of function evaluations \(NFEs\)\. Alternatively, since the neural\-network anchorvρv^\{\\rho\}is generally imperfect in practice, one may use an SDE sampler with an empirically tuned noise scheduleσt\\sigma\_\{t\}to encourage better exploration of the state space\.

#### 8\.2\.2Reverse Construction

The reverse approach is motivated by the observation that, for training, the only requirement on the pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)is that it satisfies \([49](https://arxiv.org/html/2609.05727#S8.E49)\); the marginal density ofXtX\_\{t\}does not affect the global minimizer of the squared\-loss regression\. Therefore, we can sample in the reverse order:

Xt∼p^t,X1∼p1\|tρ\(⋅\|Xt\),X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\qquad X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\),wherep^t\\hat\{p\}\_\{t\}is any positive proposal density onℝd\\mathbb\{R\}^\{d\}\.

The key remaining challenge is to determine how, given a noisy stateXtX\_\{t\}, to sampleX1∼p1\|tρ\(⋅\|Xt\)X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)in practice\. To highlight the subtlety, we first note that the SDE \([52](https://arxiv.org/html/2609.05727#S8.E52)\), with an arbitrary noise scheduleσt≥0\\sigma\_\{t\}\\geq 0, does not in general preserve the anchor posterior\. Specifically, if we run the SDE over the time interval\[t,1\]\[t,1\],

d​Ys=\[\(1\+σs22​κs\)​vsρ​\(Ys\)−σs22​κs​α˙sαs​Ys\]​d​s\+σs​d​Ws,s∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{s\}=\\left\[\\left\(1\+\\frac\{\\sigma\_\{s\}^\{2\}\}\{2\\kappa\_\{s\}\}\\right\)v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-\\frac\{\\sigma\_\{s\}^\{2\}\}\{2\\kappa\_\{s\}\}\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}\\right\]\\mathrm\{d\}s\+\\sigma\_\{s\}\\mathrm\{d\}W\_\{s\},\\qquad s\\in\[t,1\],Yt=xt,\\displaystyle Y\_\{t\}=x\_\{t\},the resulting terminal stateY1Y\_\{1\}does not generally satisfypY1\|Yt\(⋅\|xt\)=p1\|tρ\(⋅\|xt\)p\_\{Y\_\{1\}\|Y\_\{t\}\}\(\\cdot\|x\_\{t\}\)=p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\. The SDE preserves the prescribed marginal densities only when initialized withY0∼p0Y\_\{0\}\\sim p\_\{0\}, or equivalently, when initialized at timettwithYt∼ptρY\_\{t\}\\sim p\_\{t\}^\{\\rho\}\. It does not, however, preserve the posterior density when started from an arbitrary fixed stateYt=xtY\_\{t\}=x\_\{t\}\.

Fortunately, within the family of SDEs in \([52](https://arxiv.org/html/2609.05727#S8.E52)\), there is a unique SDE that preserves the posteriorpY1\|Yt\(⋅\|xt\)=p1\|tρ\(⋅\|xt\)p\_\{Y\_\{1\}\|Y\_\{t\}\}\(\\cdot\|x\_\{t\}\)=p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)fort∈\(0,1\)t\\in\(0,1\)\. It corresponds to the specific choice of noise scheduleσt=2​κt\\sigma\_\{t\}=\\sqrt\{2\\kappa\_\{t\}\}\. Given a canonical modelvρv^\{\\rho\}, we define the*posterior\-preserving SDE*by

d​Ys=bsρ​\(Ys\)​d​s\+2​κs​d​Ws,bsρ​\(x\):=2​vsρ​\(x\)−α˙sαs​x,s∈\[t,1\]\.\\mathrm\{d\}Y\_\{s\}=b\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\\mathrm\{d\}s\+\\sqrt\{2\\kappa\_\{s\}\}\\mathrm\{d\}W\_\{s\},\\qquad b\_\{s\}^\{\\rho\}\(x\):=2v\_\{s\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x,\\qquad s\\in\[t,1\]\.\(53\)With this SDE, the resultingΞt,1ρ\\Xi\_\{t,1\}^\{\\rho\}satisfies \([49](https://arxiv.org/html/2609.05727#S8.E49)\) and is given by

Ξt,1ρ​\(xt,x1\)=p^t​\(xt\)​p1\|tρ​\(x1\|xt\)under the reverse construction\.\\Xi\_\{t,1\}^\{\\rho\}\(x\_\{t\},x\_\{1\}\)=\\hat\{p\}\_\{t\}\(x\_\{t\}\)p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\qquad\\text\{under the reverse construction\}\.The posterior\-preserving SDE is an important tool for constructing Newton Matching losses and has several properties useful for both theoretical analysis and computational implementation\. We summarize these results in Appendix[D\.3](https://arxiv.org/html/2609.05727#A4.SS3)\.

The forward construction producesXt∼ptρX\_\{t\}\\sim p\_\{t\}^\{\\rho\}by design\. In contrast, the reverse construction does not have to choosep^t=ptρ\\hat\{p\}\_\{t\}=p\_\{t\}^\{\\rho\}\. Changing the proposalp^t\\hat\{p\}\_\{t\}affects the weighting of the regression problem but not its pointwise conditional target\. This allows the proposal to be chosen for practical convenience\. For example, one may use an “off\-policy” approach to obtainXtX\_\{t\}: sampleX1X\_\{1\}from a buffer, drawX0∼p0X\_\{0\}\\sim p\_\{0\}, and setXt=αt​X1\+βt​X0X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\. Whenever the sampling order and theXtX\_\{t\}\-marginal are immaterial, we use the generic notation\(Xt,X1\)∼Ξt,1ρ\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}to indicate only the posterior requirementX1\|Xt∼p1\|tρ\(⋅\|Xt\)X\_\{1\}\|X\_\{t\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\.

Another advantage of the reverse construction is that multiple endpoints can be generated from the same noisy state\. By running \([53](https://arxiv.org/html/2609.05727#S8.E53)\) multiple times at a fixedXtX\_\{t\}, we obtain

X1\(1\),⋯,X1\(N\)∼i\.i\.d\.p1\|tρ\(⋅\|Xt\),X\_\{1\}^\{\(1\)\},\\cdots,X\_\{1\}^\{\(N\)\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\),wherei\.i\.d\.\\mathrm\{i\.i\.d\.\}denotes independent and identically distributed\. Averaging the corresponding targets potentially reduces the variance and improves training stability\. In contrast, the forward construction cannot achieve this, sinceX1X\_\{1\}is sampled first andXtX\_\{t\}is subsequently constructed from it\.

### 8\.3Covariance Form

The covariance form is derived from the covariance representation of the tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}in \([45](https://arxiv.org/html/2609.05727#S8.E45)\)\. In particular, define the sample\-wise target

𝒴ρ,η,Bcov​\(t,Xt,X1\):=\\displaystyle\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,X\_\{t\},X\_\{1\}\):=vt\|1​\(Xt\|X1\)\+η⁡\(r~ρ​\(X1\)−Bt​\(Xt\)\)​\(vt\|1​\(Xt\|X1\)−vtρ​\(Xt\)\),\\displaystyle v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\eta\\left\(\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\),\(54\)whereBt​\(xt\)B\_\{t\}\(x\_\{t\}\)is any scalar baseline that depends only on\(t,xt\)\(t,x\_\{t\}\)\. Since

𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|X1\)\]=vtρ\(xt\),\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]=v\_\{t\}^\{\\rho\}\(x\_\{t\}\),the target satisfies

𝔼X1∼p1\|tρ\(⋅\|xt\)\[𝒴ρ,η,Bcov\(t,xt,X1\)\]=vtρ\(xt\)\+ηΓtρ,r~ρ\(xt\)\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\]=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)\.\(55\)
Therefore, under the forward construction, the covariance form of the tangential\-update loss is given by

ℒρ,η,Bcov,fwd​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\},\\mathrm\{fwd\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0​\[‖vtθ​\(Xt\)−𝒴ρ,η,Bcov​\(t,Xt,X1\)‖22\],\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\],\(56\)and under the reverse construction,

ℒρ,η,Bcov,rev​\(θ\):=\\displaystyle\\hskip\-2\.84526pt\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\},\\mathrm\{rev\}\}\(\\theta\):=𝔼t∼U\(0,1\),Xt∼p^t,X1∼p1\|tρ\(⋅\|Xt\)​\[‖vtθ​\(Xt\)−𝒴ρ,η,Bcov​\(t,Xt,X1\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\,X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\(57a\)=\\displaystyle=𝔼t∼U⁡\(0,1\),Xt∼p^t\[‖vtθ\(Xt\)−𝔼X1∼p1\|tρ\(⋅\|Xt\)\[𝒴ρ,η,Bcov\(t,Xt,X1\)\]‖22\]\+const\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\+\\mathrm\{const\}\.\(57b\)In \([57b](https://arxiv.org/html/2609.05727#S8.E57.2)\), we push the expectation overX1X\_\{1\}inside the squared loss\. Since the reverse construction can generate multiple i\.i\.d\. posterior samples ofX1X\_\{1\}for a fixedXtX\_\{t\}, this expectation can be estimated by averaging over these samples, thereby reducing the variance\.

The remaining task is to specify how to compute the regularized reward

r~ρ​\(X1\)=r⁡\(X1\)−1τ​log⁡ρ⁡\(X1\)μ⁡\(X1\),\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)=r\(X\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\},where thelog⁡ρ⁡\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}term is generally nontrivial to evaluate\. In the sampling setting, whereμ=1\\mu=1, this term reduces to the log densitylog⁡ρ⁡\(X1\)\\log\\rho\(X\_\{1\}\)\. In the fine\-tuning setting, whereμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}, it is the log\-density ratiolog⁡ρ⁡\(X1\)ρbase​\(X1\)\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\rho^\{\\mathrm\{base\}\}\(X\_\{1\}\)\}\.

We present two methods for computing it exactly: an ODE\-based approach that leverages the instantaneous change\-of\-variables formula for the probability flow ODE, and an SDE\-based approach that exploits the stochastic path associated with the posterior\-preserving SDE\. Note that “ODE\-based” and “SDE\-based” refer to the calculus used to compute the termlog⁡ρ⁡\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}, not to the sampling method used to generate the endpointX1X\_\{1\}\. Each calculus is compatible with both forward and reverse constructions, thereby yielding a2×22\\times 2matrix of possible combinations\.

#### 8\.3\.1ODE\-Based Estimation

The key idea is to evaluate the log density or log\-density ratio at the sampled endpoint via the instantaneous change\-of\-variables formula\. LetΦvρ\\Phi^\{v^\{\\rho\}\}denote the ODE flow generated by the anchorvρv^\{\\rho\}\. We have

dd​slogpsρ\(Φt→svρ\(xt\)\)=−∇⋅vsρ\(Φt→svρ\(xt\)\)\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\log p\_\{s\}^\{\\rho\}\\left\(\\Phi\_\{t\\to s\}^\{v^\{\\rho\}\}\(x\_\{t\}\)\\right\)=\-\\nabla\\cdot v\_\{s\}^\{\\rho\}\\left\(\\Phi\_\{t\\to s\}^\{v^\{\\rho\}\}\(x\_\{t\}\)\\right\)\.\(58\)For simplicity, in the case ofμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}, letvtbasev\_\{t\}^\{\\mathrm\{base\}\}denote the base canonical velocity andptbasep\_\{t\}^\{\\mathrm\{base\}\}its associated marginal probability path\. The identity \([58](https://arxiv.org/html/2609.05727#S8.E58)\) holds for the base model upon replacingptρp\_\{t\}^\{\\rho\}andvtρv\_\{t\}^\{\\rho\}withptbasep\_\{t\}^\{\\mathrm\{base\}\}andvtbasev\_\{t\}^\{\\mathrm\{base\}\}, respectively\.

IfX1X\_\{1\}is sampled using an ODE \(under forward construction\), thenlog⁡ρ⁡\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}can be computed alongside the forward integration by augmenting the ODE with an additional equation for the log density or log\-density ratio\. IfX1X\_\{1\}is sampled using an SDE, under either the forward or reverse construction, thenlog⁡ρ⁡\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}can be computed by integrating a backward ODE for the log density or log\-density ratio from the endpointX1X\_\{1\}to the initial timet=0t=0\.

##### Augmented forward integration\.

Forμ=1\\mu=1, the sampling ODE

d​Ytd​t=vtρ​\(Yt\),\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=v\_\{t\}^\{\\rho\}\(Y\_\{t\}\),can be augmented directly with

dd​tlogptρ\(Yt\)=−∇⋅vtρ\(Yt\),logp0ρ\(Y0\)=logp0\(Y0\)\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)=\-\\nabla\\cdot v\_\{t\}^\{\\rho\}\(Y\_\{t\}\),\\qquad\\log p\_\{0\}^\{\\rho\}\(Y\_\{0\}\)=\\log p\_\{0\}\(Y\_\{0\}\)\.\(59\)LetX1:=Y1X\_\{1\}:=Y\_\{1\}\. Then, the terminal value is

log⁡ρ⁡\(X1\)=log⁡p1ρ​\(Y1\)\.\\log\\rho\(X\_\{1\}\)=\\log p\_\{1\}^\{\\rho\}\(Y\_\{1\}\)\.
Forμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}, the sampling ODE can be augmented with

dd​t​log⁡ptρ​\(Yt\)ptbase​\(Yt\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log\\frac\{p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\}\{p\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\}=\\displaystyle=∇⋅vtbase\(Yt\)−∇⋅vtρ\(Yt\)−\(vtρ\(Yt\)−vtbase\(Yt\)\)⊤∇logptbase\(Yt\)\\displaystyle\\nabla\\cdot v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\-\\nabla\\cdot v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\-\\left\(v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\\right\)^\{\\top\}\\nabla\\log p\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)=\\displaystyle=∇⋅\(vtbase−vtρ\)​\(Yt\)−1κt​\(vtρ​\(Yt\)−vtbase​\(Yt\)\)⊤​\(vtbase​\(Yt\)−α˙tαt​Yt\),\\displaystyle\\nabla\\cdot\\left\(v\_\{t\}^\{\\mathrm\{base\}\}\-v\_\{t\}^\{\\rho\}\\right\)\(Y\_\{t\}\)\-\\frac\{1\}\{\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\\right\)^\{\\top\}\\left\(v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\),\(60\)with initial condition

log⁡p0ρ​\(Y0\)p0base​\(Y0\)=0\.\\log\\frac\{p\_\{0\}^\{\\rho\}\(Y\_\{0\}\)\}\{p\_\{0\}^\{\\mathrm\{base\}\}\(Y\_\{0\}\)\}=0\.Att=1t=1, settingX1:=Y1X\_\{1\}:=Y\_\{1\}gives

log⁡ρ⁡\(X1\)ρbase​\(X1\)=log⁡p1ρ​\(Y1\)p1base​\(Y1\)\.\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\rho^\{\\mathrm\{base\}\}\(X\_\{1\}\)\}=\\log\\frac\{p\_\{1\}^\{\\rho\}\(Y\_\{1\}\)\}\{p\_\{1\}^\{\\mathrm\{base\}\}\(Y\_\{1\}\)\}\.

##### Backward integration from a given endpoint\.

IfX1=x1X\_\{1\}=x\_\{1\}has already been generated by an SDE,log⁡ρ⁡\(x1\)\\log\\rho\(x\_\{1\}\)orlog⁡ρ⁡\(x1\)ρbase​\(x1\)\\log\\frac\{\\rho\(x\_\{1\}\)\}\{\\rho^\{\\mathrm\{base\}\}\(x\_\{1\}\)\}can be evaluated by integrating backward fromx1x\_\{1\}\. Forμ=1\\mu=1, we have

log⁡ρ⁡\(x1\)=log⁡p0​\(Φ1→0vρ​\(x1\)\)−∫01∇⋅vsρ​\(Φ1→svρ​\(x1\)\)​𝑑s\.\\log\\rho\(x\_\{1\}\)=\\log p\_\{0\}\\left\(\\Phi\_\{1\\to 0\}^\{v^\{\\rho\}\}\(x\_\{1\}\)\\right\)\-\\int\_\{0\}^\{1\}\\nabla\\cdot v\_\{s\}^\{\\rho\}\\left\(\\Phi\_\{1\\to s\}^\{v^\{\\rho\}\}\(x\_\{1\}\)\\right\)\\mathrm\{d\}s\.\(61\)
For simplicity, in the case ofμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}, we use the shorthandΦbase:=Φvbase\\Phi^\{\\mathrm\{base\}\}:=\\Phi^\{v^\{\\mathrm\{base\}\}\}for the ODE flow generated byvbasev^\{\\mathrm\{base\}\}\. In this case, we have

log⁡ρ⁡\(x1\)ρbase​\(x1\)=log⁡p0​\(Φ1→0vρ​\(x1\)\)p0​\(Φ1→0base​\(x1\)\)−∫01\[∇⋅vsρ​\(Φ1→svρ​\(x1\)\)−∇⋅vsbase​\(Φ1→sbase​\(x1\)\)\]​𝑑s\.\\displaystyle\\log\\frac\{\\rho\(x\_\{1\}\)\}\{\\rho^\{\\mathrm\{base\}\}\(x\_\{1\}\)\}=\\log\\frac\{p\_\{0\}\\left\(\\Phi\_\{1\\to 0\}^\{v^\{\\rho\}\}\(x\_\{1\}\)\\right\)\}\{p\_\{0\}\\left\(\\Phi\_\{1\\to 0\}^\{\\mathrm\{base\}\}\(x\_\{1\}\)\\right\)\}\-\\int\_\{0\}^\{1\}\\left\[\\nabla\\cdot v\_\{s\}^\{\\rho\}\\left\(\\Phi\_\{1\\to s\}^\{v^\{\\rho\}\}\(x\_\{1\}\)\\right\)\-\\nabla\\cdot v\_\{s\}^\{\\mathrm\{base\}\}\\left\(\\Phi\_\{1\\to s\}^\{\\mathrm\{base\}\}\(x\_\{1\}\)\\right\)\\right\]\\mathrm\{d\}s\.\(62\)Unlike the augmented forward integration in \([60](https://arxiv.org/html/2609.05727#S8.E60)\), the above evaluation follows two backward trajectories, one under the anchor ODE and one under the base ODE\.

The required divergences may be evaluated exactly or estimated using Hutchinson’s trace estimator\[[27](https://arxiv.org/html/2609.05727#bib.bib54)\]:

∇⋅vs\(x\)=𝔼ε\[ε⊤∇vs\(x\)ε\],\\nabla\\cdot v\_\{s\}\(x\)=\\mathbb\{E\}\_\{\\varepsilon\}\\left\[\\varepsilon^\{\\top\}\\nabla v\_\{s\}\(x\)\\varepsilon\\right\],where the probe vectorε\\varepsilonis a random variable inℝd\\mathbb\{R\}^\{d\}and satisfies

𝔼ε​\[ε​ε⊤\]=I\.\\mathbb\{E\}\_\{\\varepsilon\}\\left\[\\varepsilon\\varepsilon^\{\\top\}\\right\]=I\.For example,ε∼𝒩⁡\(0,I\)\\varepsilon\\sim\\mathcal\{N\}\(0,I\)is a valid choice\.

#### 8\.3\.2SDE\-Based Estimation

The SDE\-based method exploits two special properties of the posterior\-preserving SDE\. First, its generator yields exact pathwise representations of the terminal log density and log\-density ratio through Itô’s formula\. Second, conditioning on the two endpointsXtX\_\{t\}andX1X\_\{1\}, the intervening stochastic path has a universal distribution independent of the drift, allowing the same representations to be used regardless of the sampler employed to generateX1X\_\{1\}\. The pathwise equalities below use the convention of Appendix[D\.1](https://arxiv.org/html/2609.05727#A4.SS1)\.

Fixt∈\(0,1\)t\\in\(0,1\)andxt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}\. Recall from \([53](https://arxiv.org/html/2609.05727#S8.E53)\) that the posterior\-preserving SDE initialized atYt=xtY\_\{t\}=x\_\{t\}is

d​Ys=bsρ​\(Ys\)​d​s\+2​κs​d​Ws,bsρ​\(x\)=2​vsρ​\(x\)−α˙sαs​x,s∈\[t,1\]\.\\mathrm\{d\}Y\_\{s\}=b\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\\mathrm\{d\}s\+\\sqrt\{2\\kappa\_\{s\}\}\\mathrm\{d\}W\_\{s\},\\qquad b\_\{s\}^\{\\rho\}\(x\)=2v\_\{s\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x,\\qquad s\\in\[t,1\]\.Let

𝒀\[t,1\]=\(Ys\)s∈\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}=\(Y\_\{s\}\)\_\{s\\in\[t,1\]\}denote the resulting random path\. We write

𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\(63\)to indicate that𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}is generated by the posterior\-preserving SDE starting fromYt=xtY\_\{t\}=x\_\{t\}\. In particular, its endpoint satisfies

Y1∼p1\|tρ\(⋅\|xt\)\.Y\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\.
For aC2C^\{2\}functionf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}, the generator of the posterior\-preserving SDE is

\(ℒsρf\)\(x\):=bsρ\(x\)⊤∇f\(x\)\+κsΔf\(x\)\.\\left\(\\mathcal\{L\}\_\{s\}^\{\\rho\}f\\right\)\(x\):=b\_\{s\}^\{\\rho\}\(x\)^\{\\top\}\\nabla f\(x\)\+\\kappa\_\{s\}\\Delta f\(x\)\.Accordingly, for a sufficiently regular time\-dependent functionhs:ℝd→ℝh\_\{s\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}, Itô’s formula gives

dhs\(Ys\)=\[∂shs\+ℒsρhs\]\(Ys\)ds\+2​κs∇hs\(Ys\)⊤dWs\.\\mathrm\{d\}h\_\{s\}\(Y\_\{s\}\)=\\left\[\\partial\_\{s\}h\_\{s\}\+\\mathcal\{L\}\_\{s\}^\{\\rho\}h\_\{s\}\\right\]\(Y\_\{s\}\)\\mathrm\{d\}s\+\\sqrt\{2\\kappa\_\{s\}\}\\nabla h\_\{s\}\(Y\_\{s\}\)^\{\\top\}\\mathrm\{d\}W\_\{s\}\.\(64\)We now apply this formula separately to the two cases considered in this paper\.

Whenμ=1\\mu=1, consider

hs​\(x\)=log⁡\(αsd​psρ​\(x\)\)\.h\_\{s\}\(x\)=\\log\\left\(\\alpha\_\{s\}^\{d\}p\_\{s\}^\{\\rho\}\(x\)\\right\)\.Sinceα1=1\\alpha\_\{1\}=1andp1ρ=ρp\_\{1\}^\{\\rho\}=\\rho, its terminal value is

h1​\(x\)=log⁡ρ⁡\(x\)\.h\_\{1\}\(x\)=\\log\\rho\(x\)\.The generator of the posterior\-preserving SDE forhs=log⁡\(αsd​psρ\)h\_\{s\}=\\log\(\\alpha\_\{s\}^\{d\}p\_\{s\}^\{\\rho\}\)admits an analytic expression:

\[∂shs\+ℒsρ​hs\]​\(x\)\\displaystyle\\left\[\\partial\_\{s\}h\_\{s\}\+\\mathcal\{L\}\_\{s\}^\{\\rho\}h\_\{s\}\\right\]\(x\)=1κs​‖vsρ​\(x\)−α˙sαs​x‖22\.\\displaystyle=\\frac\{1\}\{\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x\\right\\\|\_\{2\}^\{2\}\.Also, the canonical score–velocity relation gives

∇hs​\(x\)\\displaystyle\\nabla h\_\{s\}\(x\)=1κs​\(vsρ​\(x\)−α˙sαs​x\)\.\\displaystyle=\\frac\{1\}\{\\kappa\_\{s\}\}\\left\(v\_\{s\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x\\right\)\.Substituting these identities into \([64](https://arxiv.org/html/2609.05727#S8.E64)\) and integrating fromttto11gives

log⁡ρ⁡\(Y1\)=\\displaystyle\\log\\rho\(Y\_\{1\}\)=log⁡\(αtd​ptρ​\(xt\)\)\+∫t11κs​‖vsρ​\(Ys\)−α˙sαs​Ys‖22​𝑑s\+∫t12κs​\(vsρ​\(Ys\)−α˙sαs​Ys\)⊤​d​Ws\.\\displaystyle\\log\\left\(\\alpha\_\{t\}^\{d\}p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\+\\int\_\{t\}^\{1\}\\frac\{1\}\{\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}s\+\\int\_\{t\}^\{1\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}\.\(65\)The detailed derivation is given in Proposition[108](https://arxiv.org/html/2609.05727#Thmtheorem108)\.

Whenμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}, consider

hs​\(x\)=log⁡psρ​\(x\)psbase​\(x\)\.h\_\{s\}\(x\)=\\log\\frac\{p\_\{s\}^\{\\rho\}\(x\)\}\{p\_\{s\}^\{\\mathrm\{base\}\}\(x\)\}\.Its terminal value is

h1​\(x\)=log⁡ρ⁡\(x\)ρbase​\(x\)\.h\_\{1\}\(x\)=\\log\\frac\{\\rho\(x\)\}\{\\rho^\{\\mathrm\{base\}\}\(x\)\}\.Using the canonical score–velocity relations for the anchor and base models, we have

∇hs​\(x\)\\displaystyle\\nabla h\_\{s\}\(x\)=1κs​\(vsρ​\(x\)−vsbase​\(x\)\)\.\\displaystyle=\\frac\{1\}\{\\kappa\_\{s\}\}\\left\(v\_\{s\}^\{\\rho\}\(x\)\-v\_\{s\}^\{\\mathrm\{base\}\}\(x\)\\right\)\.The generator of the posterior\-preserving SDE forhs=log⁡psρpsbaseh\_\{s\}=\\log\\frac\{p\_\{s\}^\{\\rho\}\}\{p\_\{s\}^\{\\mathrm\{base\}\}\}admits an analytic expression:

\(∂shs\+ℒsρ​hs\)​\(x\)\\displaystyle\\left\(\\partial\_\{s\}h\_\{s\}\+\\mathcal\{L\}\_\{s\}^\{\\rho\}h\_\{s\}\\right\)\(x\)=1κs​‖vsρ​\(x\)−vsbase​\(x\)‖22\.\\displaystyle=\\frac\{1\}\{\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\}\(x\)\-v\_\{s\}^\{\\mathrm\{base\}\}\(x\)\\right\\\|\_\{2\}^\{2\}\.Applying \([64](https://arxiv.org/html/2609.05727#S8.E64)\) and integrating fromttto11therefore yields

log⁡ρ⁡\(Y1\)ρbase​\(Y1\)=\\displaystyle\\log\\frac\{\\rho\(Y\_\{1\}\)\}\{\\rho^\{\\mathrm\{base\}\}\(Y\_\{1\}\)\}=log⁡ptρ​\(xt\)ptbase​\(xt\)\+∫t1‖vsρ​\(Ys\)−vsbase​\(Ys\)‖22κs​𝑑s\\displaystyle\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\}\+\\int\_\{t\}^\{1\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mathrm\{base\}\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\+∫t12κs\(vsρ\(Ys\)−vsbase\(Ys\)\)⊤dWs\.\\displaystyle\+\\int\_\{t\}^\{1\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mathrm\{base\}\}\(Y\_\{s\}\)\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}\.\(66\)The detailed derivation is given in Proposition[104](https://arxiv.org/html/2609.05727#Thmtheorem104)\.

For simplicity, we write

vtμ​\(xt\):=\{α˙tαt​xt,μ=1,vtbase​\(xt\),μ=ρbase,ptμ​\(xt\):=\{αt−d,μ=1,ptbase​\(xt\),μ=ρbase\.v\_\{t\}^\{\\mu\}\(x\_\{t\}\):=\\begin\{cases\}\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\},&\\mu=1,\\\\ v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\),&\\mu=\\rho^\{\\mathrm\{base\}\},\\end\{cases\}\\qquad p\_\{t\}^\{\\mu\}\(x\_\{t\}\):=\\begin\{cases\}\\alpha\_\{t\}^\{\-d\},&\\mu=1,\\\\ p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\),&\\mu=\\rho^\{\\mathrm\{base\}\}\.\\end\{cases\}Then, \([65](https://arxiv.org/html/2609.05727#S8.E65)\) and \([66](https://arxiv.org/html/2609.05727#S8.E66)\) can be unified as

log⁡ρ⁡\(Y1\)μ⁡\(Y1\)=\\displaystyle\\log\\frac\{\\rho\(Y\_\{1\}\)\}\{\\mu\(Y\_\{1\}\)\}=log⁡ptρ​\(xt\)ptμ​\(xt\)\+∫t1‖vsρ​\(Ys\)−vsμ​\(Ys\)‖22κs​𝑑s\+∫t12κs​\(vsρ​\(Ys\)−vsμ​\(Ys\)\)⊤​d​Ws\.\\displaystyle\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mu\}\(x\_\{t\}\)\}\+\\int\_\{t\}^\{1\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mu\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\+\\int\_\{t\}^\{1\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mu\}\(Y\_\{s\}\)\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}\.\(67\)
Observe that thelog⁡ptρ​\(xt\)ptμ​\(xt\)\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mu\}\(x\_\{t\}\)\}term in \([67](https://arxiv.org/html/2609.05727#S8.E67)\) depends only on\(t,xt\)\(t,x\_\{t\}\)\. Since the covariance representation is unchanged when any function of\(t,xt\)\(t,x\_\{t\}\)is added to the baseline, these terms do not need to be evaluated\. After absorbing

1τ​log⁡ptρ​\(xt\)ptμ​\(xt\)\\frac\{1\}\{\\tau\}\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mu\}\(x\_\{t\}\)\}intoBt​\(xt\)B\_\{t\}\(x\_\{t\}\), the same scalar factor can be evaluated as

r⁡\(Y1\)−Bt​\(xt\)−1τ​∫t1‖vsρ​\(Ys\)−vsμ​\(Ys\)‖22κs​𝑑s−1τ​∫t12κs​\(vsρ​\(Ys\)−vsμ​\(Ys\)\)⊤​d​Ws\.\\displaystyle r\(Y\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\-\\frac\{1\}\{\\tau\}\\int\_\{t\}^\{1\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mu\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\-\\frac\{1\}\{\\tau\}\\int\_\{t\}^\{1\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mu\}\(Y\_\{s\}\)\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}\.\(68\)
It remains to demonstrate how the stochastic path \([63](https://arxiv.org/html/2609.05727#S8.E63)\) is generated for the forward and reverse constructions\. Under the reverse construction, this path is obtained directly\. Starting from the given noisy stateXtX\_\{t\}, the posterior\-preserving SDE directly generates

𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|Xt\),Yt=Xt\.\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\),\\qquad Y\_\{t\}=X\_\{t\}\.The path and Brownian increments required by \([68](https://arxiv.org/html/2609.05727#S8.E68)\) are therefore obtained from the same simulation\.

Algorithm 2Tangential Update: Exact Realization with Covariance Form\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v¯k\+1=vρk\+ηk​Γρk,r~ρk\\bar\{v\}^\{k\+1\}=v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

2:foreach gradient stepdo

3:

⊳\\trianglerightThe following operations are applied to a batch\.

4:ifforward construction is usedthen

5:ifODE\-based sampling is usedthen

6:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vρkv^\{\\rho\_\{k\}\}\.

7:ifODE\-based estimation is usedthen

8:Compute

log⁡ρk​\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}by augmenting \([1](https://arxiv.org/html/2609.05727#S2.E1)\) with \([59](https://arxiv.org/html/2609.05727#S8.E59)\) or \([60](https://arxiv.org/html/2609.05727#S8.E60)\)\.

9:endif

10:elseifSDE\-based sampling is usedthen

11:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

12:ifODE\-based estimation is usedthen

13:Compute

log⁡ρk​\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}by the backward integration \([61](https://arxiv.org/html/2609.05727#S8.E61)\) or \([62](https://arxiv.org/html/2609.05727#S8.E62)\)\.

14:endif

15:endif

16:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

17:ifSDE\-based estimation is usedthen

18:Sample

𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|Xt,X1\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)based on Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\.

19:endif

20:elseifreverse construction is usedthen

21:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

22:Sample

𝒀\[t,1\]∼ℙ\[t,1\]\|tρk\(⋅\|Xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

23:Set

X1←Y1X\_\{1\}\\leftarrow Y\_\{1\}\.

24:ifODE\-based estimation is usedthen

25:Compute

log⁡ρk​\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}by the backward integration \([61](https://arxiv.org/html/2609.05727#S8.E61)\) or \([62](https://arxiv.org/html/2609.05727#S8.E62)\)\.

26:endif

27:endif

28:ifSDE\-based estimation is usedthen

29:Compute

log⁡ρk​\(X1\)ρμ​\(X1\)−log⁡ptρk​\(Xt\)ptμ​\(Xt\)\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\rho^\{\\mu\}\(X\_\{1\}\)\}\-\\log\\frac\{p\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\}\{p\_\{t\}^\{\\mu\}\(X\_\{t\}\)\}by \([67](https://arxiv.org/html/2609.05727#S8.E67)\)\.

30:endif

31:Compute the target

𝒴\\mathcal\{Y\}by \([54](https://arxiv.org/html/2609.05727#S8.E54)\), where

log⁡ptρk​\(Xt\)ptμ​\(Xt\)\\log\\frac\{p\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\}\{p\_\{t\}^\{\\mu\}\(X\_\{t\}\)\}is merged into

Bt​\(Xt\)B\_\{t\}\(X\_\{t\}\)in SDE\-based estimation\.

32:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

33:endfor

34:Set

v¯k\+1←vθ\\bar\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

Under the forward construction, the pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)is constructed and satisfies

X1\|Xt∼p1\|tρ\(⋅\|Xt\),X\_\{1\}\|X\_\{t\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\),but this construction does not provide a bridge of the posterior\-preserving SDE between the two states\. For fixed endpoints\(xt,xs\)\(x\_\{t\},x\_\{s\}\),0<t<s≤10<t<s\\leq 1, let

ℙ\[t,s\]\|t,sρ\(⋅\|xt,xs\)\\mathbb\{P\}\_\{\[t,s\]\|t,s\}^\{\\rho\}\(\\cdot\|x\_\{t\},x\_\{s\}\)denote the conditional path distribution of the posterior\-preserving SDE givenYt=xtY\_\{t\}=x\_\{t\}andYs=xsY\_\{s\}=x\_\{s\}\. In particular, fors=1s=1, the full posterior path distribution can be reproduced by the hierarchical sampling rule

X1∼p1\|tρ\(⋅\|xt\),𝒀\[t,1\]∼ℙ\[t,1\]\|t,1ρ\(⋅\|xt,X1\)⇒𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\.X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\),\\quad\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{\\rho\}\(\\cdot\|x\_\{t\},X\_\{1\}\)\\quad\\Rightarrow\\quad\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\.\(69\)Since the forward construction has already produced the first sample in \([69](https://arxiv.org/html/2609.05727#S8.E69)\), it remains only to draw

𝒀\[t,1\]∼ℙ\[t,1\]\|t,1ρ\(⋅\|Xt,X1\)\.\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{\\rho\}\(\\cdot\|X\_\{t\},X\_\{1\}\)\.
An interesting property of the posterior\-preserving SDE is that its two\-sided conditional distributionℙ\[t,s\]\|t,sρ\(⋅\|xt,xs\)\\mathbb\{P\}\_\{\[t,s\]\|t,s\}^\{\\rho\}\(\\cdot\|x\_\{t\},x\_\{s\}\)is independent of the terminal densityρ\\rhoand depends only on the schedulesαt\\alpha\_\{t\}andβt\\beta\_\{t\}\. We refer to this property as*bridge universality*, and denote the bridge distribution by

ℙ\[t,s\]uni\(⋅\|xt,xs\)\.\\mathbb\{P\}\_\{\[t,s\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\)\.It enables efficient generation of a stochastic path𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}that is consistent with the posterior\-preserving SDE conditioning on the two endpoints\(Xt,X1\)\(X\_\{t\},X\_\{1\}\), thereby allowing the hierarchical sampling scheme in \([69](https://arxiv.org/html/2609.05727#S8.E69)\) to be realized efficiently\. A detailed derivation is provided in Appendix[D\.4](https://arxiv.org/html/2609.05727#A4.SS4)\. Specifically, for any two terminal densitiesρ1,ρ2\\rho\_\{1\},\\rho\_\{2\}and any0<t<s≤10<t<s\\leq 1,

ℙ\[t,s\]\|t,sρ1\(⋅\|xt,xs\)=ℙ\[t,s\]uni\(⋅\|xt,xs\)=ℙ\[t,s\]\|t,sρ2\(⋅\|xt,xs\)\.\\mathbb\{P\}\_\{\[t,s\]\|t,s\}^\{\\rho\_\{1\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\)=\\mathbb\{P\}\_\{\[t,s\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\)=\\mathbb\{P\}\_\{\[t,s\]\|t,s\}^\{\\rho\_\{2\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\)\.
To give the explicit construction, defineγs:=βs2αs2\\gamma\_\{s\}:=\\frac\{\\beta\_\{s\}^\{2\}\}\{\\alpha\_\{s\}^\{2\}\}\. Fort<u<st<u<s, conditioning on the two endpoints, the state at timeuufollows

Yu\|\(Yt,Ys\)∼𝒩⁡\(mu\|t,s​\(Yt,Ys\),Σu\|t,s\),Y\_\{u\}\|\\left\(Y\_\{t\},Y\_\{s\}\\right\)\\sim\\mathcal\{N\}\\left\(m\_\{u\|t,s\}\(Y\_\{t\},Y\_\{s\}\),\\Sigma\_\{u\|t,s\}\\right\),\(70\)where

mu\|t,s​\(xt,xs\)\\displaystyle m\_\{u\|t,s\}\(x\_\{t\},x\_\{s\}\):=αu​\[γu−γsγt−γs​xtαt\+γt−γuγt−γs​xsαs\],Σu\|t,s:=αu2​\(γt−γu\)​\(γu−γs\)γt−γs​I\.\\displaystyle:=\\alpha\_\{u\}\\left\[\\frac\{\\gamma\_\{u\}\-\\gamma\_\{s\}\}\{\\gamma\_\{t\}\-\\gamma\_\{s\}\}\\frac\{x\_\{t\}\}\{\\alpha\_\{t\}\}\+\\frac\{\\gamma\_\{t\}\-\\gamma\_\{u\}\}\{\\gamma\_\{t\}\-\\gamma\_\{s\}\}\\frac\{x\_\{s\}\}\{\\alpha\_\{s\}\}\\right\],\\qquad\\Sigma\_\{u\|t,s\}:=\\alpha\_\{u\}^\{2\}\\frac\{\\left\(\\gamma\_\{t\}\-\\gamma\_\{u\}\\right\)\\left\(\\gamma\_\{u\}\-\\gamma\_\{s\}\\right\)\}\{\\gamma\_\{t\}\-\\gamma\_\{s\}\}I\.Equation \([70](https://arxiv.org/html/2609.05727#S8.E70)\) can be applied recursively to sample the entire path on a discrete time grid\.

The universal bridge directly generates the state path, whereas the stochastic integral in \([68](https://arxiv.org/html/2609.05727#S8.E68)\) is written in terms of the Brownian motion driving the posterior\-preserving SDE\. Since the diffusion coefficient is state\-independent, the corresponding increments can be recovered from the sampled state path through

d​Ws=d​Ys−bsρ​\(Ys\)​d​s2​κs\.\\mathrm\{d\}W\_\{s\}=\\frac\{\\mathrm\{d\}Y\_\{s\}\-b\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\\mathrm\{d\}s\}\{\\sqrt\{2\\kappa\_\{s\}\}\}\.
The exact tangential update under the covariance form is summarized in Algorithm[2](https://arxiv.org/html/2609.05727#alg2)\.

### 8\.4Gradient Form

Whereas \([45](https://arxiv.org/html/2609.05727#S8.E45)\) leads to the covariance form, \([48](https://arxiv.org/html/2609.05727#S8.E48)\) expresses the tangent vector through the action of a posterior Stein kernel on the regularized\-reward gradient\. We now present the derivation of the gradient form based on the posterior Stein kernelΛtρ\\Lambda\_\{t\}^\{\\rho\}\.

Suppose first that an exact posterior Stein kernelΛtρ\(⋅\|xt\)\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)is available\. The gradient representation \([48](https://arxiv.org/html/2609.05727#S8.E48)\) then suggests the following target

𝒴ρ,ηgrad,ker\(t,Xt,X1\):=vt\|1\(Xt\|X1\)\+ηαt​κtβt2Λtρ\(X1\|Xt\)⊤∇r~ρ\(X1\)\.\\displaystyle\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,X\_\{t\},X\_\{1\}\):=v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|X\_\{t\}\)^\{\\top\}\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\.\(71\)By \([48](https://arxiv.org/html/2609.05727#S8.E48)\),

𝔼X1∼p1\|tρ\(⋅\|xt\)\[𝒴ρ,ηgrad,ker\(t,xt,X1\)\]=vtρ\(xt\)\+ηΓtρ,r~ρ\(xt\)\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\]=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)\.\(72\)
Therefore, under the forward construction, the ideal kernel\-based gradient loss is given by

ℒρ,ηgrad,ker,fwd​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\},\\mathrm\{fwd\}\}\(\\theta\):=𝔼t∼U\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0​\[‖vtθ​\(Xt\)−𝒴ρ,ηgrad,ker​\(t,Xt,X1\)‖22\],\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\\\ X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\],\(73\)and under the reverse construction,

ℒρ,ηgrad,ker,rev​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\},\\mathrm\{rev\}\}\(\\theta\):⁣=\\displaystyle:=𝔼t∼U\(0,1\),Xt∼p^t,X1∼p1\|tρ\(⋅\|Xt\)​\[‖vtθ​\(Xt\)−𝒴ρ,ηgrad,ker​\(t,Xt,X1\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\,X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\(74a\)=\\displaystyle=𝔼t∼U⁡\(0,1\),Xt∼p^t\[‖vtθ\(Xt\)−𝔼X1∼p1\|tρ\(⋅\|Xt\)\[𝒴ρ,ηgrad,ker\(t,Xt,X1\)\]‖22\]\+const\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\+\\mathrm\{const\}\.\(74b\)These losses exactly parallel the covariance\-form losses in \([56](https://arxiv.org/html/2609.05727#S8.E56)\) and \([57b](https://arxiv.org/html/2609.05727#S8.E57.2)\)\. The key question is how to construct a posterior Stein kernel and evaluateΛtρ\(X1\|Xt\)⊤∇r~ρ\(X1\)\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|X\_\{t\}\)^\{\\top\}\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\.

For a general non\-Gaussian posterior, obtaining such a kernel is a nontrivial problem\. For fixed\(t,xt\)\(t,x\_\{t\}\), a posterior Stein kernel must satisfy

∇x1⋅\(p1\|tρ​\(x1\|xt\)​Λtρ​\(x1\|xt\)\)=−p1\|tρ​\(x1\|xt\)​\(x1−Mtρ​\(xt\)\),\\nabla\_\{x\_\{1\}\}\\cdot\\left\(p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\)=\-p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\left\(x\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\),\(75\)whereMtρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[X1\]M\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[X\_\{1\}\]denotes the posterior mean\. The details are presented in Appendix[E\.3](https://arxiv.org/html/2609.05727#A5.SS3)\. Solving \([75](https://arxiv.org/html/2609.05727#S8.E75)\) directly is in general intractable\.

Fortunately, the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) provides a constructive way to obtain a particular posterior Stein kernel\. Its terminal transition density is exactlyp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\. Hence, for every sufficiently regularff,

𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[f\(Y1\)\]=𝔼X1∼p1\|tρ\(⋅\|xt\)\[f\(X1\)\]=Vtρ\[f\]\(xt\)\.\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(Y\_\{1\}\)\\right\]=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(X\_\{1\}\)\\right\]=V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\.Forf=r~ρf=\\tilde\{r\}^\{\\rho\}, the initial\-state sensitivity of this SDE therefore yields the posterior\-value gradient∇xtVtρ​\[r~ρ\]​\(xt\)\\nabla\_\{x\_\{t\}\}V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\(x\_\{t\}\)\. This observation motivates using sensitivity analysis of the posterior\-preserving SDE to construct a posterior Stein kernel\. Indeed, conditioning the resulting state\-transition matrix on the endpoints\(xt,x1\)\(x\_\{t\},x\_\{1\}\)and applying the schedule\-dependent rescaling below yields an exact posterior Stein kernel, while a backward adjoint ODE provides a matrix\-free Monte Carlo evaluation of its action on a vector\. We call this special kernel the*posterior sensitivity kernel*\.

##### Posterior sensitivity kernel\.

Fixt∈\(0,1\)t\\in\(0,1\)andxt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}, and recall the posterior\-preserving SDE

d​Ys=bsρ​\(Ys\)​d​s\+2​κs​d​Ws,s∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{s\}=b\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\\mathrm\{d\}s\+\\sqrt\{2\\kappa\_\{s\}\}\\mathrm\{d\}W\_\{s\},\\qquad s\\in\[t,1\],\(76a\)Yt=xt\.\\displaystyle Y\_\{t\}=x\_\{t\}\.\(76b\)Its path distribution isℙ\[t,1\]\|tρ\(⋅\|xt\)\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\), and its endpoint satisfiesY1∼p1\|tρ\(⋅\|xt\)Y\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\.

For a realized continuous path𝒀\[t,1\]=\(Ys\)s∈\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}=\(Y\_\{s\}\)\_\{s\\in\[t,1\]\}, define the pathwise state\-transition matrix of the SDE \([76](https://arxiv.org/html/2609.05727#S8.E76)\) by

dd​s​Gt→sbρ​\(𝒀\[t,1\]\)=∇bsρ​\(Ys\)​Gt→sbρ​\(𝒀\[t,1\]\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}G\_\{t\\to s\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla b\_\{s\}^\{\\rho\}\(Y\_\{s\}\)G\_\{t\\to s\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\),\\qquad s\\in\[t,1\],\(77a\)Gt→tbρ​\(𝒀\[t,1\]\)=I\.\\displaystyle G\_\{t\\to t\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=I\.\(77b\)When the path𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}is generated by the same SDE \([76](https://arxiv.org/html/2609.05727#S8.E76)\),Gt→sbρG\_\{t\\to s\}^\{b^\{\\rho\}\}coincides with the Jacobian ofYsY\_\{s\}with respect to the initial statextx\_\{t\}\. Throughout this paper, whenever derivatives with respect to the initial statextx\_\{t\}are considered, the same Brownian motion\(Ws\)s∈\[t,1\]\(W\_\{s\}\)\_\{s\\in\[t,1\]\}is used for all initial states\.

A posterior Stein kernel must depend only on the endpoint pair\(xt,x1\)\(x\_\{t\},x\_\{1\}\), not on the intervening realized path\. We therefore define the endpoint\-conditioned state\-transition matrix

K1\|tρ\(x1\|xt\):=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|t,1ρ\(⋅\|xt,x1\)\[Gt→1bρ\(𝒀\[t,1\]\)\]=𝔼𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|xt,x1\)\[Gt→1bρ\(𝒀\[t,1\]\)\]\.K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\):=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{\\rho\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[G\_\{t\\to 1\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[G\_\{t\\to 1\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.\(78\)In Appendix[E\.4](https://arxiv.org/html/2609.05727#A5.SS4), we show that

Λtρ​\(x1\|xt\)=βt2αt​K1\|tρ​\(x1\|xt\)\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\(79\)is a posterior Stein kernel forp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\), called the posterior sensitivity kernel\. Therefore, sensitivity analysis of the posterior\-preserving SDE provides both a construction of a posterior Stein kernel and a practical representation in terms of the state\-transition matrix dynamics in \([77](https://arxiv.org/html/2609.05727#S8.E77)\)\.

However, explicitly forming the matrixGt→sbρ∈ℝd×dG\_\{t\\to s\}^\{b^\{\\rho\}\}\\in\\mathbb\{R\}^\{d\\times d\}can be computationally expensive for high\-dimensional data such as images\. A standard technique from optimal control, known as the adjoint method, allows the relevant matrix\-vector products to be computed without explicitly forming the matrix\. Next, we state the adjoint formulation needed for the posterior\-preserving SDE \([76](https://arxiv.org/html/2609.05727#S8.E76)\)\. In Appendix[D\.6](https://arxiv.org/html/2609.05727#A4.SS6), we develop the pathwise sensitivity and adjoint identities for a general SDE; thus, the adjoint construction itself is not specific to the posterior\-preserving SDE \([76](https://arxiv.org/html/2609.05727#S8.E76)\)\.

##### Matrix\-free representation via adjoint\.

For a terminal objectiveφ:ℝd→ℝ\\varphi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}and a running objectivels:ℝd→ℝl\_\{s\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}, define the pathwise objective

Jtφ,l​\(𝒀\[t,1\]\):=φ⁡\(Y1\)\+∫t1ls​\(Ys\)​𝑑s\.J\_\{t\}^\{\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\):=\\varphi\(Y\_\{1\}\)\+\\int\_\{t\}^\{1\}l\_\{s\}\(Y\_\{s\}\)\\mathrm\{d\}s\.Its pathwise adjoint is defined by

dd​sλsbρ,φ,l\(𝒀\[t,1\]\)=−∇bsρ\(Ys\)⊤λsbρ,φ,l\(𝒀\[t,1\]\)−∇ls\(Ys\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{b^\{\\rho\},\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla b\_\{s\}^\{\\rho\}\(Y\_\{s\}\)^\{\\top\}\\lambda\_\{s\}^\{b^\{\\rho\},\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\-\\nabla l\_\{s\}\(Y\_\{s\}\),\\qquad s\\in\[t,1\],λ1bρ,φ,l​\(𝒀\[t,1\]\)=∇φ​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b^\{\\rho\},\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla\\varphi\(Y\_\{1\}\)\.In Appendix[D\.6\.1](https://arxiv.org/html/2609.05727#A4.SS6.SSS1), we show that the adjoint and the state\-transition matrix are related by

λtbρ,φ,l\(𝒀\[t,1\]\)=Gt→1bρ\(𝒀\[t,1\]\)⊤∇φ\(Y1\)\+∫t1Gt→sbρ\(𝒀\[t,1\]\)⊤∇ls\(Ys\)ds\.\\displaystyle\\lambda\_\{t\}^\{b^\{\\rho\},\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=G\_\{t\\to 1\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla\\varphi\(Y\_\{1\}\)\+\\int\_\{t\}^\{1\}G\_\{t\\to s\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla l\_\{s\}\(Y\_\{s\}\)\\mathrm\{d\}s\.\(80\)For a terminal observableff, takeφ=f\\varphi=fandls=0l\_\{s\}=0\. Equations \([78](https://arxiv.org/html/2609.05727#S8.E78)\), \([79](https://arxiv.org/html/2609.05727#S8.E79)\), and \([80](https://arxiv.org/html/2609.05727#S8.E80)\) imply

αt​κtβt2Λtρ\(x1\|xt\)⊤∇f\(x1\)=κt𝔼𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|xt,x1\)\[λtbρ,f,0\(𝒀\[t,1\]\)\]\.\\displaystyle\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)=\\kappa\_\{t\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},f,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.\(81\)Thus, a single pathwise adjoint provides a matrix\-free Monte Carlo estimator of the kernel\-based target\.

The covariance\-form target itself is defined only by the endpoint pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\), with any ODE or SDE path used only to evaluater~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\. In contrast, the adjoint\-based targets below are intrinsically path\-dependent and therefore require a path distribution\. Whenever the specific path construction and theYtY\_\{t\}\-marginal are immaterial, we use the generic notation

𝒀\[t,1\]∼Ξ\[t,1\]ρ⇒𝒀\[t,1\]\|Yt∼ℙ\[t,1\]\|tρ\(⋅\|Yt\)\.\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\rho\}\\quad\\Rightarrow\\quad\\bm\{Y\}\_\{\[t,1\]\}\|Y\_\{t\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|Y\_\{t\}\)\.\(82\)TheYtY\_\{t\}\-marginal is required to have a positive density\.

We next present two exact adjoint\-based realizations for the gradient form\. The first retains the full regularized reward in the terminal objective and includes no running objective, corresponding to the Mayer form in optimal\-control terminology\. The second expresses the same posterior value using the terminal rewardrrtogether with a running regularization objective, corresponding to the Bolza form\. Both yield targets whose conditional mean, givenYt=xtY\_\{t\}=x\_\{t\}, isvtρ​\(xt\)\+η​Γtρ,r~ρ​\(xt\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)\.

##### Mayer realization\.

In \([81](https://arxiv.org/html/2609.05727#S8.E81)\), we can directly setf=r~ρf=\\tilde\{r\}^\{\\rho\}\. The adjoint satisfies

dd​sλsbρ,r~ρ,0\(𝒀\[t,1\]\)=−\(2∇vsρ\(Ys\)−α˙sαsI\)⊤λsbρ,r~ρ,0\(𝒀\[t,1\]\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\left\(2\\nabla v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}I\\right\)^\{\\top\}\\lambda\_\{s\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\(\\bm\{Y\}\_\{\[t,1\]\}\),\\qquad s\\in\[t,1\],\(83a\)λ1bρ,r~ρ,0​\(𝒀\[t,1\]\)=∇r~ρ​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla\\tilde\{r\}^\{\\rho\}\(Y\_\{1\}\)\.\(83b\)Conditioning on an endpoint pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\), let𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}denote a draw from the endpoint\-conditioned path distributionℙ\[t,1\]\|t,1ρ\(⋅\|Xt,X1\)=ℙ\[t,1\]uni\(⋅\|Xt,X1\)\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{\\rho\}\(\\cdot\|X\_\{t\},X\_\{1\}\)=\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)\. The corresponding pathwise Mayer target is written as

𝒴ρ,ηgrad,Mayer​\(t,𝒀\[t,1\]\):=\\displaystyle\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\):=vt\|1​\(Yt\|Y1\)\+η​κt​λtbρ,r~ρ,0​\(𝒀\[t,1\]\)\.\\displaystyle v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)\+\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\.\(84\)By \([81](https://arxiv.org/html/2609.05727#S8.E81)\),

𝔼𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|xt,x1\)\[𝒴ρ,ηgrad,Mayer\(t,𝒀\[t,1\]\)\]=𝒴ρ,ηgrad,ker\(t,xt,x1\)\.\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]=\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,x\_\{t\},x\_\{1\}\)\.Thus, the Mayer target is a bridge\-level Monte Carlo estimation of the ideal kernel\-based target\.

Whenμ=1\\mu=1, the boundary condition is

λ1bρ,r~ρ,0\(𝒀\[t,1\]\)=∇r\(Y1\)−1τ∇logρ\(Y1\)\.\\lambda\_\{1\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\nabla\\log\\rho\(Y\_\{1\}\)\.Whenμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}, it is

λ1bρ,r~ρ,0\(𝒀\[t,1\]\)=∇r\(Y1\)−1τ∇logρ⁡\(Y1\)ρbase​\(Y1\)\.\\lambda\_\{1\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\nabla\\log\\frac\{\\rho\(Y\_\{1\}\)\}\{\\rho^\{\\mathrm\{base\}\}\(Y\_\{1\}\)\}\.The Mayer realization involves no running objective, but its terminal condition requires the exact gradient of the log density or log\-density ratio\. Computing this terminal condition can be challenging in practice whenρ\\rhoandρbase\\rho^\{\\mathrm\{base\}\}are terminal densities induced by neural\-network velocity fields and may therefore require careful numerical treatment\.

Under the forward construction, the Mayer realization of the gradient\-form loss is

ℒρ,ηgrad,Mayer,fwd​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\},\\mathrm\{fwd\}\}\(\\theta\):=𝔼t∼U\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0,𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|Xt,X1\)​\[‖vtθ​\(Xt\)−𝒴ρ,ηgrad,Mayer​\(t,𝒀\[t,1\]\)‖22\],\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\\\ X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\},\\\\ \\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\],\(85\)and under the reverse construction,

ℒρ,ηgrad,Mayer,rev​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\},\\mathrm\{rev\}\}\(\\theta\):⁣=\\displaystyle:=𝔼t∼U\(0,1\),Xt∼p^t,𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|Xt\)​\[‖vtθ​\(Xt\)−𝒴ρ,ηgrad,Mayer​\(t,𝒀\[t,1\]\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\(86a\)=\\displaystyle=𝔼t∼U⁡\(0,1\),Xt∼p^t\[‖vtθ\(Xt\)−𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|Xt\)\[𝒴ρ,ηgrad,Mayer\(t,𝒀\[t,1\]\)\]‖22\]\+const\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\+\\mathrm\{const\}\.\(86b\)

##### Bolza realization\.

The density regularization can instead be moved from the terminal objective to a running objective along the paths compatible with the posterior\-preserving SDE\. This gives a Bolza realization with terminal objectiverr\.

Forμ∈\{1,ρbase\}\\mu\\in\\\{1,\\rho^\{\\mathrm\{base\}\}\\\}, define the running objective

lsρ,μ​\(x\):=−1τ​κs​‖vsρ​\(x\)−vsμ​\(x\)‖22\.l\_\{s\}^\{\\rho,\\mu\}\(x\):=\-\\frac\{1\}\{\\tau\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\}\(x\)\-v\_\{s\}^\{\\mu\}\(x\)\\right\\\|\_\{2\}^\{2\}\.\(87\)Whenμ=1\\mu=1,vs1​\(x\):=α˙sαs​xv\_\{s\}^\{1\}\(x\):=\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x\. Whenμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\},vsbase​\(x\)v\_\{s\}^\{\\mathrm\{base\}\}\(x\)denotes the canonical velocity field associated with the base densityρbase\\rho^\{\\mathrm\{base\}\}\. For either case, letλbρ,r,lρ,μ\\lambda^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}denote the adjoint associated with the terminal objectiverrand the running objectivelρ,μl^\{\\rho,\\mu\}:

dd​sλsbρ,r,lρ,μ\(𝒀\[t,1\]\)=−\(2∇vsρ\(Ys\)−α˙sαsI\)⊤λsbρ,r,lρ,μ\(𝒀\[t,1\]\)−∇lsρ,μ\(Ys\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\left\(2\\nabla v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}I\\right\)^\{\\top\}\\lambda\_\{s\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\-\\nabla l\_\{s\}^\{\\rho,\\mu\}\(Y\_\{s\}\),\\quad s\\in\[t,1\],\(88a\)λ1bρ,r,lρ,μ​\(𝒀\[t,1\]\)=∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\.\(88b\)
For theμ=1\\mu=1case, we can derive

Vtρ\[r~ρ\]\(xt\)=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[r\(Y1\)\+∫t1lsρ,1\(Ys\)ds\]−1τlog\(αtdptρ\(xt\)\)\.\\displaystyle V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\(x\_\{t\}\)=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[r\(Y\_\{1\}\)\+\\int\_\{t\}^\{1\}l\_\{s\}^\{\\rho,1\}\(Y\_\{s\}\)\\mathrm\{d\}s\\right\]\-\\frac\{1\}\{\\tau\}\\log\\left\(\\alpha\_\{t\}^\{d\}p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.\(89\)The proof is provided in Appendix[D\.5](https://arxiv.org/html/2609.05727#A4.SS5)\. We also have

∇xt​log​\(αtd​ptρ​\(xt\)\)=1κt​\(vtρ​\(xt\)−vt1​\(xt\)\)\.\\nabla\_\{x\_\{t\}\}\\log\\left\(\\alpha\_\{t\}^\{d\}p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)=\\frac\{1\}\{\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{1\}\(x\_\{t\}\)\\right\)\.Differentiating \([89](https://arxiv.org/html/2609.05727#S8.E89)\) and applying the adjoint representation therefore gives

Γtρ,r~ρ\(xt\)=κt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[λtbρ,r,lρ,1\(𝒀\[t,1\]\)\]−1τ\(vtρ\(xt\)−vt1\(xt\)\)\.\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=\\kappa\_\{t\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},r,l^\{\\rho,1\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\-\\frac\{1\}\{\\tau\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{1\}\(x\_\{t\}\)\\right\)\.\(90\)
Algorithm 3Tangential Update: Exact Realization with Gradient Form\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v¯k\+1=vρk\+ηk​Γρk,r~ρk\\bar\{v\}^\{k\+1\}=v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

2:foreach gradient stepdo

3:

⊳\\trianglerightThe following operations are applied to a batch\.

4:ifforward construction is usedthen

5:ifODE\-based sampling is usedthen

6:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vρkv^\{\\rho\_\{k\}\}\.

7:elseifSDE\-based sampling is usedthen

8:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

9:endif

10:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

11:Sample

𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|Xt,X1\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)based on Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\.

12:elseifreverse construction is usedthen

13:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

14:Sample

𝒀\[t,1\]∼ℙ\[t,1\]\|tρk\(⋅\|Xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

15:endif

16:ifMayer realization is usedthen

17:Compute the pathwise adjoint

λtbρk,r~ρk,0​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\rho\_\{k\}\},\\tilde\{r\}^\{\\rho\_\{k\}\},0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)by ODE \([83](https://arxiv.org/html/2609.05727#S8.E83)\)\.

18:Compute the target

𝒴\\mathcal\{Y\}by \([84](https://arxiv.org/html/2609.05727#S8.E84)\)\.

19:elseifBolza realization is usedthen

20:Compute the pathwise adjoint

λtbρk,r,lρk,μ​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\rho\_\{k\}\},r,l^\{\\rho\_\{k\},\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)by ODE \([88](https://arxiv.org/html/2609.05727#S8.E88)\)\.

21:Compute the target

𝒴\\mathcal\{Y\}by \([93](https://arxiv.org/html/2609.05727#S8.E93)\)\.

22:endif

23:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

24:endfor

25:Set

v¯k\+1←vθ\\bar\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

For theμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}case, we can derive

Vtρ\[r~ρ\]\(xt\)=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[r\(Y1\)\+∫t1lsρ,ρbase\(Ys\)ds\]−1τlogptρ​\(xt\)ptbase​\(xt\)\.\\displaystyle V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\(x\_\{t\}\)=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[r\(Y\_\{1\}\)\+\\int\_\{t\}^\{1\}l\_\{s\}^\{\\rho,\\rho^\{\\mathrm\{base\}\}\}\(Y\_\{s\}\)\\mathrm\{d\}s\\right\]\-\\frac\{1\}\{\\tau\}\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\}\.\(91\)The proof is provided in Appendix[D\.5](https://arxiv.org/html/2609.05727#A4.SS5)\. We also have

∇xt​log​ptρ​\(xt\)ptbase​\(xt\)=1κt​\(vtρ​\(xt\)−vtbase​\(xt\)\)\.\\nabla\_\{x\_\{t\}\}\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\}=\\frac\{1\}\{\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)\.Differentiating \([91](https://arxiv.org/html/2609.05727#S8.E91)\) and applying the adjoint representation therefore gives

Γtρ,r~ρ\(xt\)=κt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[λtbρ,r,lρ,ρbase\(𝒀\[t,1\]\)\]−1τ\(vtρ\(xt\)−vtbase\(xt\)\)\.\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=\\kappa\_\{t\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},r,l^\{\\rho,\\rho^\{\\mathrm\{base\}\}\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\-\\frac\{1\}\{\\tau\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)\.\(92\)
For eitherμ∈\{1,ρbase\}\\mu\\in\\\{1,\\rho^\{\\mathrm\{base\}\}\\\}, the pathwise Bolza target is written as

𝒴ρ,ηgrad,Bolza​\(t,𝒀\[t,1\]\):=\(1−ητ\)​vt\|1​\(Yt\|Y1\)\+ητ​vtμ​\(Yt\)\+η​κt​λtbρ,r,lρ,μ​\(𝒀\[t,1\]\)\.\\displaystyle\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\):=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\+\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\.\(93\)Equations \([90](https://arxiv.org/html/2609.05727#S8.E90)\) and \([92](https://arxiv.org/html/2609.05727#S8.E92)\) imply

𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[𝒴ρ,ηgrad,Bolza\(t,𝒀\[t,1\]\)\]=vtρ\(xt\)\+ηΓtρ,r~ρ\(xt\)\.\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)\.
Under the forward construction, the Bolza realization of the gradient\-form loss is

ℒρ,ηgrad,Bolza,fwd​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{fwd\}\}\(\\theta\):=𝔼t∼U\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0,𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|Xt,X1\)​\[‖vtθ​\(Xt\)−𝒴ρ,ηgrad,Bolza​\(t,𝒀\[t,1\]\)‖22\],\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\\\ X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\},\\\\ \\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\],\(94\)and under the reverse construction,

ℒρ,ηgrad,Bolza,rev​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{rev\}\}\(\\theta\):⁣=\\displaystyle:=𝔼t∼U\(0,1\),Xt∼p^t,𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|Xt\)​\[‖vtθ​\(Xt\)−𝒴ρ,ηgrad,Bolza​\(t,𝒀\[t,1\]\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\(95a\)=\\displaystyle=𝔼t∼U⁡\(0,1\),Xt∼p^t\[‖vtθ\(Xt\)−𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|Xt\)\[𝒴ρ,ηgrad,Bolza\(t,𝒀\[t,1\]\)\]‖22\]\+const\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\+\\mathrm\{const\}\.\(95b\)
The exact tangential update under the gradient form is summarized in Algorithm[3](https://arxiv.org/html/2609.05727#alg3)\.

Table 1:Exact Newton Matching: summary of losses for the tangential updatevtθ​\(xt\)=vtρ​\(xt\)\+η​Γtρ,r~ρ​\(xt\)v\_\{t\}^\{\\theta\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)\.

### 8\.5Unification via Control Variates

The covariance and gradient forms developed earlier involve different sample\-wise quantities, but both recover the same tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}in conditional expectation\. This equivalence can be understood more deeply through zero\-mean control variates induced by Langevin Stein operators\. In Appendix[E](https://arxiv.org/html/2609.05727#A5), we present the corresponding Stein identity and the product rule for the Langevin Stein operator\. We now specialize these results to the regularized reward, thereby constructing a unified family of targets that interpolates between the covariance and gradient forms\.

By the definitions of the covariance target in \([54](https://arxiv.org/html/2609.05727#S8.E54)\) and the kernel\-based gradient target in \([71](https://arxiv.org/html/2609.05727#S8.E71)\), their sample\-wise difference is

𝒴ρ,ηgrad,ker​\(t,xt,X1\)−𝒴ρ,η,Bcov​\(t,xt,X1\)\\displaystyle\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,x\_\{t\},X\_\{1\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)=\\displaystyle=η\[αt​κtβt2Λtρ\(X1\|xt\)⊤∇r~ρ\(X1\)−\(r~ρ\(X1\)−Bt\(xt\)\)\(vt\|1\(xt\|X1\)−vtρ\(xt\)\)\]\.\\displaystyle\\eta\\left\[\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-\\left\(\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\\right\]\.By \([55](https://arxiv.org/html/2609.05727#S8.E55)\) and \([72](https://arxiv.org/html/2609.05727#S8.E72)\), the posterior means of both targets arevtρ​\(xt\)\+η​Γtρ,r~ρ​\(xt\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\); hence,

𝔼X1∼p1\|tρ\(⋅\|xt\)\[𝒴ρ,ηgrad,ker\(t,xt,X1\)−𝒴ρ,η,Bcov\(t,xt,X1\)\]=0\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,x\_\{t\},X\_\{1\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\]=0\.\(96\)This demonstrates that the term𝒴ρ,ηgrad,ker​\(t,xt,X1\)−𝒴ρ,η,Bcov​\(t,xt,X1\)\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,x\_\{t\},X\_\{1\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)is a zero\-mean control variate obtained from the Stein identity, with the corresponding matrix test function given byη​αt​κtβt2​\(r~ρ​\(x1\)−Bt​\(xt\)\)​Λtρ​\(x1\|xt\)\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\(\\tilde\{r\}^\{\\rho\}\(x\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\)\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\. The proof is provided in Appendix[E](https://arxiv.org/html/2609.05727#A5)\.

LetΩt​\(xt\)∈ℝd×d\\Omega\_\{t\}\(x\_\{t\}\)\\in\\mathbb\{R\}^\{d\\times d\}be any matrix field that depends only on\(t,xt\)\(t,x\_\{t\}\)\. Left\-multiplying \([96](https://arxiv.org/html/2609.05727#S8.E96)\) byΩt​\(xt\)\\Omega\_\{t\}\(x\_\{t\}\)preserves its zero conditional mean\. Starting from the covariance target, we therefore have

𝔼X1∼p1\|tρ\(⋅\|xt\)\[𝒴ρ,η,Bcov\(t,xt,X1\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\]=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[𝒴ρ,η,Bcov\(t,xt,X1\)\+Ωt\(xt\)\(𝒴ρ,ηgrad,ker\(t,xt,X1\)−𝒴ρ,η,Bcov\(t,xt,X1\)\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)\+\\Omega\_\{t\}\(x\_\{t\}\)\\left\(\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,x\_\{t\},X\_\{1\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\)\\right\]=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(I−Ωt\(xt\)\)𝒴ρ,η,Bcov\(t,xt,X1\)\+Ωt\(xt\)𝒴ρ,ηgrad,ker\(t,xt,X1\)\]\.\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\left\(I\-\\Omega\_\{t\}\(x\_\{t\}\)\\right\)\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,x\_\{t\},X\_\{1\}\)\+\\Omega\_\{t\}\(x\_\{t\}\)\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\]\.This identity motivates the unified target

𝒴ρ,η,B,Ωuni​\(t,Xt,X1\):=\(I−Ωt​\(Xt\)\)​𝒴ρ,η,Bcov​\(t,Xt,X1\)\+Ωt​\(Xt\)​𝒴ρ,ηgrad,ker​\(t,Xt,X1\)\.\\displaystyle\\mathcal\{Y\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\}\}\(t,X\_\{t\},X\_\{1\}\):=\\left\(I\-\\Omega\_\{t\}\(X\_\{t\}\)\\right\)\\mathcal\{Y\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{cov\}\}\(t,X\_\{t\},X\_\{1\}\)\+\\Omega\_\{t\}\(X\_\{t\}\)\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}\(t,X\_\{t\},X\_\{1\}\)\.\(97\)Substituting the definitions of the targets gives

𝒴ρ,η,B,Ωuni​\(t,Xt,X1\)=\\displaystyle\\mathcal\{Y\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\}\}\(t,X\_\{t\},X\_\{1\}\)=vt\|1\(Xt\|X1\)\+ηΩt\(Xt\)αt​κtβt2Λtρ\(X1\|Xt\)⊤∇r~ρ\(X1\)\\displaystyle v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\eta\\Omega\_\{t\}\(X\_\{t\}\)\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|X\_\{t\}\)^\{\\top\}\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\+η⁡\(I−Ωt​\(Xt\)\)​\(r~ρ​\(X1\)−Bt​\(Xt\)\)​\(vt\|1​\(Xt\|X1\)−vtρ​\(Xt\)\)\.\\displaystyle\+\\eta\\left\(I\-\\Omega\_\{t\}\(X\_\{t\}\)\\right\)\\left\(\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\)\.The choicesΩt≡0\\Omega\_\{t\}\\equiv 0andΩt≡I\\Omega\_\{t\}\\equiv Irecover the covariance target and kernel\-based gradient target, respectively\. We also have the unified representation of the tangent vector

𝔼X1∼p1\|tρ\(⋅\|xt\)\[𝒴ρ,η,B,Ωuni\(t,xt,X1\)\]=vtρ\(xt\)\+ηΓtρ,r~ρ\(xt\)\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\]=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)\.
The construction above uses the kernel\-based gradient target since it makes the role of the posterior Stein kernel explicit\. The preceding subsection also develops the Mayer and Bolza targets as exact pathwise realizations of the same gradient form\. We have

vtρ​\(xt\)\+η​Γtρ,r~ρ​\(xt\)=\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[𝒴ρ,ηgrad,Mayer\(t,𝒀\[t,1\]\)\]\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Mayer\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]=\\displaystyle=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[𝒴ρ,ηgrad,Bolza\(t,𝒀\[t,1\]\)\]\.\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,Bolza\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.Therefore, the kernel\-based component𝒴ρ,ηgrad,ker\\mathcal\{Y\}\_\{\\rho,\\eta\}^\{\\mathrm\{grad,ker\}\}in \([97](https://arxiv.org/html/2609.05727#S8.E97)\) may be replaced by either pathwise target\. For simplicity, we omit these versions here\.

Under the forward construction, the unified tangential\-update loss is

ℒρ,η,B,Ωuni,fwd​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\},\\mathrm\{fwd\}\}\(\\theta\):=𝔼t∼U\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0​\[‖vtθ​\(Xt\)−𝒴ρ,η,B,Ωuni​\(t,Xt,X1\)‖22\],\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\\\ X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\],\(98\)and under the reverse construction,

ℒρ,η,B,Ωuni,rev​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\},\\mathrm\{rev\}\}\(\\theta\):⁣=\\displaystyle:=𝔼t∼U\(0,1\),Xt∼p^t,X1∼p1\|tρ\(⋅\|Xt\)​\[‖vtθ​\(Xt\)−𝒴ρ,η,B,Ωuni​\(t,Xt,X1\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\,X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\(99a\)=\\displaystyle=𝔼t∼U⁡\(0,1\),Xt∼p^t\[‖vtθ\(Xt\)−𝔼X1∼p1\|tρ\(⋅\|Xt\)\[𝒴ρ,η,B,Ωuni\(t,Xt,X1\)\]‖22\]\+const\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\_\{\\rho,\\eta,B,\\Omega\}^\{\\mathrm\{uni\}\}\(t,X\_\{t\},X\_\{1\}\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\+\\mathrm\{const\}\.\(99b\)In principle,Ωt​\(xt\)\\Omega\_\{t\}\(x\_\{t\}\)can be chosen according to an optimality criterion, such as minimizing the variance of the unified tangential\-update loss\. We provide a relevant result in Appendix[E\.2](https://arxiv.org/html/2609.05727#A5.SS2)\.

The preceding conditional\-mean identities yield the following theorem\. Throughout this paper, population minimizers and stationary points are understood up to a\.e\. equality under the corresponding regression distribution\. The displayed formulas are taken as representatives\.

###### Theorem 34\.

Fix a canonical anchorvρv^\{\\rho\}, a scalar stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\], and a positive proposalp^t\\hat\{p\}\_\{t\}wherever a reverse construction is used\. Assume that the sample\-wise targets are square\-integrable\. Viewed as regression over square\-integrable velocity fields, all exact tangential\-update losses introduced in Sections[8\.3](https://arxiv.org/html/2609.05727#S8.SS3)–[8\.5](https://arxiv.org/html/2609.05727#S8.SS5)—whether based on the forward or reverse construction and whether expressed in covariance, gradient, or unified form—share the same population minimizer,

vtρ​\(xt\)\+η​Γtρ,r~ρ​\(xt\),v^\{\\rho\}\_\{t\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\),unique up to a\.e\. equality\. This statement applies to the losses defined in \([56](https://arxiv.org/html/2609.05727#S8.E56)\), \([57](https://arxiv.org/html/2609.05727#S8.E57)\), \([73](https://arxiv.org/html/2609.05727#S8.E73)\), \([74](https://arxiv.org/html/2609.05727#S8.E74)\), \([85](https://arxiv.org/html/2609.05727#S8.E85)\), \([86](https://arxiv.org/html/2609.05727#S8.E86)\), \([94](https://arxiv.org/html/2609.05727#S8.E94)\), \([95](https://arxiv.org/html/2609.05727#S8.E95)\), \([98](https://arxiv.org/html/2609.05727#S8.E98)\), and \([99](https://arxiv.org/html/2609.05727#S8.E99)\)\.

###### Proof\.

For every loss covered by this theorem, the conditional mean of its sample\-wise target, including any auxiliary path randomness, isvtρ​\(x\)\+η​Γtρ,r~ρ​\(x\)v^\{\\rho\}\_\{t\}\(x\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\)by the identities established in Sections[8\.3](https://arxiv.org/html/2609.05727#S8.SS3)–[8\.5](https://arxiv.org/html/2609.05727#S8.SS5)\. Since an expected squared\-error loss is uniquely minimized up to a\.e\. equality by the conditional mean of its target, the stated conclusion follows\. ∎

Theorem[34](https://arxiv.org/html/2609.05727#Thmtheorem34)shows that the population minimizer is invariant across several implementation choices\. These choices can be organized along three distinct design dimensions:

- •Target realization, which may take the covariance form, the gradient form, or a control\-variate combination of the two;
- •Sampling construction, which may be forward or reverse and determines how the noisy state and posterior endpoint are generated;
- •Representation\-specific computational realization, namely ODE\-based or SDE\-based evaluation oflog⁡ρμ\\log\\frac\{\\rho\}\{\\mu\}for the covariance form, and kernel\-based or adjoint\-based \(Mayer or Bolza\) calculation for the gradient form\.

Representative exact Newton Matching configurations are summarized in Table[1](https://arxiv.org/html/2609.05727#S8.T1)\.

### 8\.6Multi\-Time Supervision from Shared Samples

Sections[8\.3](https://arxiv.org/html/2609.05727#S8.SS3)–[8\.5](https://arxiv.org/html/2609.05727#S8.SS5)primarily develop the exact tangential\-update losses from the population perspective\. Accordingly, both the text and the algorithms describe each loss in terms of the basic sample\-wise objects associated with a givenXtX\_\{t\}: an endpoint pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\), or, when the target is path\-dependent, a path pair\(Xt,𝒀\[t,1\]\)\(X\_\{t\},\\bm\{Y\}\_\{\[t,1\]\}\)\. This sample\-wise presentation does not imply that a separate endpoint or path must be generated for each supervision timett\. In practice, both endpoint generation and target computation can be relatively expensive\. Under the forward construction, samplingX1∼ρX\_\{1\}\\sim\\rhorequires solving the ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) or SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) associated withρ\\rho\. Under the reverse construction, samplingX1∼p1\|tρ\(⋅\|Xt\)X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)requires solving the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\)\. The covariance form further requires ODE\-based or SDE\-based estimation of the regularized rewardr~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\), whereas the gradient form requires solving an adjoint ODE\. Repeating these operations independently for every supervision timettand its corresponding stateXtX\_\{t\}would discard information already available from the same endpoint or path\.

Given the practical importance of sample efficiency, we now make explicit how a single endpointX1X\_\{1\}, or a single posterior\-compatible path ending atX1X\_\{1\}, can provide supervision at multiple times\. Consider an ordered collection of supervision times

0<t0<t1<⋯<tNtime−1<tNtime:=1\.0<t\_\{0\}<t\_\{1\}<\\cdots<t\_\{N\_\{\\mathrm\{time\}\}\-1\}<t\_\{N\_\{\\mathrm\{time\}\}\}:=1\.\(100\)For an endpoint\-based target, it is sufficient that, for everyj=0,1,…,Ntime−1j=0,1,\\ldots,N\_\{\\mathrm\{time\}\}\-1,

X1\|Xtj∼p1\|tjρ\(⋅\|Xtj\)\.X\_\{1\}\|X\_\{t\_\{j\}\}\\sim p\_\{1\|t\_\{j\}\}^\{\\rho\}\(\\cdot\|X\_\{t\_\{j\}\}\)\.\(101\)For a pathwise target, it is sufficient that

𝒀\[tj,1\]\|Ytj=Xtj∼ℙ\[tj,1\]\|tjρ\(⋅\|Xtj\)\.\\bm\{Y\}\_\{\[t\_\{j\},1\]\}\|Y\_\{t\_\{j\}\}=X\_\{t\_\{j\}\}\\sim\\mathbb\{P\}\_\{\[t\_\{j\},1\]\|t\_\{j\}\}^\{\\rho\}\(\\cdot\|X\_\{t\_\{j\}\}\)\.\(102\)These are precisely the posterior and conditional\-path requirements in \([49](https://arxiv.org/html/2609.05727#S8.E49)\) and \([82](https://arxiv.org/html/2609.05727#S8.E82)\), imposed separately at each supervision time\. Sharing an endpoint or path generally induces correlation among the resulting regression pairs across supervision times, but this correlation does not affect their validity: exactness requires the appropriate conditional law at each time, not independence across times\.

This subsection develops representative shared\-sample constructions tailored to the sampling and target\-computation procedures of the covariance and gradient forms\. Each construction produces regression pairs at multiple supervision times from a single endpoint or path sample, with the associated endpoint\-level or path\-level computations shared across those pairs\.

#### 8\.6\.1Multi\-Time Supervision in Covariance Form

##### ODE\-based estimation\.

Suppose first that the ODE\-based estimation method in Section[8\.3](https://arxiv.org/html/2609.05727#S8.SS3)is used to evaluatelog⁡ρ⁡\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}\. IfX1∼ρX\_\{1\}\\sim\\rhois generated by the anchor ODE under the forward construction, the log density or log\-density ratio can be computed together with the endpoint through the augmented integrations \([59](https://arxiv.org/html/2609.05727#S8.E59)\) or \([60](https://arxiv.org/html/2609.05727#S8.E60)\)\. IfX1X\_\{1\}is generated by an SDE, either under the forward construction or by the posterior\-preserving SDE under the reverse construction, the same quantity can be computed from the obtained endpoint through the backward integrations \([61](https://arxiv.org/html/2609.05727#S8.E61)\) or \([62](https://arxiv.org/html/2609.05727#S8.E62)\)\. In either case, this gives one exact endpoint value

r~ρ​\(X1\)=r⁡\(X1\)−1τ​log⁡ρ⁡\(X1\)μ⁡\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)=r\(X\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}for the covariance target \([54](https://arxiv.org/html/2609.05727#S8.E54)\)\. The remaining task is therefore to construct as many inexpensive noisy states\(t,Xt\)\(t,X\_\{t\}\)as possible, each of which can reuse the same valuer~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)for supervision\.

Under the forward construction, one direct choice is to sampleX0∼p0X\_\{0\}\\sim p\_\{0\}once and set

Xtj=αtj​X1\+βtj​X0\.X\_\{t\_\{j\}\}=\\alpha\_\{t\_\{j\}\}X\_\{1\}\+\\beta\_\{t\_\{j\}\}X\_\{0\}\.Alternatively, one may sample several source variablesX0\(j\)​∼i\.i\.d\.​p0X\_\{0\}^\{\(j\)\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}p\_\{0\}, and set

Xtj=αtj​X1\+βtj​X0\(j\)\.X\_\{t\_\{j\}\}=\\alpha\_\{t\_\{j\}\}X\_\{1\}\+\\beta\_\{t\_\{j\}\}X\_\{0\}^\{\(j\)\}\.For every fixedjj, both choices reproduce \([51](https://arxiv.org/html/2609.05727#S8.E51)\), so the posterior requirement \([101](https://arxiv.org/html/2609.05727#S8.E101)\) is satisfied\. The supervision times can be sampled independently

t0,t1,⋯,tNtime−1​∼i\.i\.d\.​U⁡\(0,1\),t\_\{0\},\\,t\_\{1\},\\,\\cdots,\\,t\_\{N\_\{\\mathrm\{time\}\}\-1\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\operatorname\{U\}\(0,1\),or

tj∼U\(jNtime,j\+1Ntime\),j=0,1,…,Ntime−1\.t\_\{j\}\\sim\\operatorname\{U\}\\left\(\\frac\{j\}\{N\_\{\\mathrm\{time\}\}\},\\frac\{j\+1\}\{N\_\{\\mathrm\{time\}\}\}\\right\),\\qquad j=0,1,\\ldots,N\_\{\\mathrm\{time\}\}\-1\.\(103\)In the former case, the notation means thatNtimeN\_\{\\mathrm\{time\}\}times are first sampled i\.i\.d\. fromU⁡\(0,1\)\\operatorname\{U\}\(0,1\)and then relabeled to comply with \([100](https://arxiv.org/html/2609.05727#S8.E100)\)\.

Algorithm 4Tangential Update: Multi\-Time Supervision with Shared Samples\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; a batch size

NbatchN\_\{\\mathrm\{batch\}\}; and a number of supervision times

NtimeN\_\{\\mathrm\{time\}\}\.

0:The updated velocity model

v¯k\+1=vρk\+ηk​Γρk,r~ρk\\bar\{v\}^\{k\+1\}=v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

2:foreach gradient stepdo

3:

⊳\\trianglerightThe following operations are applied to a batch\.

4:Sample

0<t0<t1<⋯<tNtime=10<t\_\{0\}<t\_\{1\}<\\cdots<t\_\{N\_\{\\mathrm\{time\}\}\}=1\.

5:ifforward construction is usedthen

6:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) or SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

7:Sample

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\)\.

8:ifcovariance form and ODE\-based estimation are usedthen

9:For

j←0,1,⋯,Ntime−1j\\leftarrow 0,1,\\cdots,N\_\{\\mathrm\{time\}\}\-1, set

Xtj←αtj​X1\+βtj​X0X\_\{t\_\{j\}\}\\leftarrow\\alpha\_\{t\_\{j\}\}X\_\{1\}\+\\beta\_\{t\_\{j\}\}X\_\{0\}\.

10:else

11:Set

Xt0←αt0​X1\+βt0​X0X\_\{t\_\{0\}\}\\leftarrow\\alpha\_\{t\_\{0\}\}X\_\{1\}\+\\beta\_\{t\_\{0\}\}X\_\{0\}\.

12:Sample

𝒀\[t0,1\]∼ℙ\[t0,1\]uni\(⋅\|Xt0,X1\)\\bm\{Y\}\_\{\[t\_\{0\},1\]\}\\sim\\mathbb\{P\}\_\{\[t\_\{0\},1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\_\{0\}\},X\_\{1\}\)based on Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\.

13:For

j←1,2,⋯,Ntime−1j\\leftarrow 1,2,\\cdots,N\_\{\\mathrm\{time\}\}\-1, set

Xtj←YtjX\_\{t\_\{j\}\}\\leftarrow Y\_\{t\_\{j\}\}\.

14:endif

15:elseifreverse construction is usedthen

16:Sample

Xt0∼p^t0X\_\{t\_\{0\}\}\\sim\\hat\{p\}\_\{t\_\{0\}\}, and sample

𝒀\[t0,1\]∼ℙ\[t0,1\]\|t0ρk\(⋅\|Xt0\)\\bm\{Y\}\_\{\[\{t\_\{0\}\},1\]\}\\sim\\mathbb\{P\}\_\{\[\{t\_\{0\}\},1\]\|\{t\_\{0\}\}\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\_\{0\}\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

17:For

j←1,2,⋯,Ntimej\\leftarrow 1,2,\\cdots,N\_\{\\mathrm\{time\}\}, set

Xtj←YtjX\_\{t\_\{j\}\}\\leftarrow Y\_\{t\_\{j\}\}\.

18:endif

19:ifcovariance form is usedthen

20:ifODE\-based estimation is usedthen

21:Compute

log⁡ρk​\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}using applicable branches in Lines[8](https://arxiv.org/html/2609.05727#alg2.l8),[13](https://arxiv.org/html/2609.05727#alg2.l13), and[25](https://arxiv.org/html/2609.05727#alg2.l25)of Algorithm[2](https://arxiv.org/html/2609.05727#alg2)\.

22:Based on the same value

log⁡ρk​\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}, for

j←0,1,⋯,Ntime−1j\\leftarrow 0,1,\\cdots,N\_\{\\mathrm\{time\}\}\-1, compute the target

𝒴j\\mathcal\{Y\}\_\{j\}at

XtjX\_\{t\_\{j\}\}using Line[31](https://arxiv.org/html/2609.05727#alg2.l31)of Algorithm[2](https://arxiv.org/html/2609.05727#alg2)\.

23:elseifSDE\-based estimation is usedthen

24:For

j←Ntime−1,Ntime−2,⋯,0j\\leftarrow N\_\{\\mathrm\{time\}\}\-1,N\_\{\\mathrm\{time\}\}\-2,\\cdots,0, recursively compute

log⁡ρk​\(X1\)ρμ​\(X1\)−log⁡ptρk​\(Xt\)ptμ​\(Xt\)\\log\\frac\{\\rho\_\{k\}\(X\_\{1\}\)\}\{\\rho^\{\\mu\}\(X\_\{1\}\)\}\-\\log\\frac\{p\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\}\{p\_\{t\}^\{\\mu\}\(X\_\{t\}\)\}and the target

𝒴j\\mathcal\{Y\}\_\{j\}at

XtjX\_\{t\_\{j\}\}backward along

𝒀\[t0,1\]\\bm\{Y\}\_\{\[\{t\_\{0\}\},1\]\}using Lines[29](https://arxiv.org/html/2609.05727#alg2.l29)and[31](https://arxiv.org/html/2609.05727#alg2.l31)of Algorithm[2](https://arxiv.org/html/2609.05727#alg2)\.

25:endif

26:elseifgradient form is usedthen

27:For

j←Ntime−1,Ntime−2,⋯,0j\\leftarrow N\_\{\\mathrm\{time\}\}\-1,N\_\{\\mathrm\{time\}\}\-2,\\cdots,0, recursively compute the pathwise adjoint

λtj​\(𝒀\[t0,1\]\)\\lambda\_\{t\_\{j\}\}\(\\bm\{Y\}\_\{\[\{t\_\{0\}\},1\]\}\)and the target

𝒴j\\mathcal\{Y\}\_\{j\}at

XtjX\_\{t\_\{j\}\}backward along

𝒀\[t0,1\]\\bm\{Y\}\_\{\[\{t\_\{0\}\},1\]\}using applicable branches in Lines[16](https://arxiv.org/html/2609.05727#alg3.l16)–[22](https://arxiv.org/html/2609.05727#alg3.l22)of Algorithm[3](https://arxiv.org/html/2609.05727#alg3)\.

28:endif

29:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​Ntime​∑\(tj,Xtj,𝒴j\)‖vtjθ​\(Xtj\)−𝒴j‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}N\_\{\\mathrm\{time\}\}\}\\sum\_\{\(t\_\{j\},X\_\{t\_\{j\}\},\\mathcal\{Y\}\_\{j\}\)\}\\left\\\|v\_\{t\_\{j\}\}^\{\\theta\}\(X\_\{t\_\{j\}\}\)\-\\mathcal\{Y\}\_\{j\}\\right\\\|\_\{2\}^\{2\}\.

30:endfor

31:Set

v¯k\+1←vθ\\bar\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

Under the reverse construction, we first samplet0∼U⁡\(0,1\)t\_\{0\}\\sim\\operatorname\{U\}\(0,1\)andXt0∼p^t0X\_\{t\_\{0\}\}\\sim\\hat\{p\}\_\{t\_\{0\}\}\. We then generate

𝒀\[t0,1\]∼ℙ\[t0,1\]\|t0ρ\(⋅\|Xt0\)\\bm\{Y\}\_\{\[t\_\{0\},1\]\}\\sim\\mathbb\{P\}\_\{\[t\_\{0\},1\]\|t\_\{0\}\}^\{\\rho\}\(\\cdot\|X\_\{t\_\{0\}\}\)using the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\), and setX1:=Y1X\_\{1\}:=Y\_\{1\}\.

To avoid restricting supervision to a fixed time grid, the intermediate supervision timestjt\_\{j\}may be sampled according to \([103](https://arxiv.org/html/2609.05727#S8.E103)\), with the SDE solution evaluated at the resulting times\. In practice, we may impose a small lower cutoffε\>0\\varepsilon\>0and requiret0≥εt\_\{0\}\\geq\\varepsilonto avoid the singularity of the posterior\-preserving SDE att=0t=0\.

The endpoint valuer~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)is computed once using either backward integration \([61](https://arxiv.org/html/2609.05727#S8.E61)\) or \([62](https://arxiv.org/html/2609.05727#S8.E62)\)\. Since the complete path𝒀\[t0,1\]\\bm\{Y\}\_\{\[t\_\{0\},1\]\}has already been generated in obtainingX1X\_\{1\}, the states at the supervision times can be taken directly from this path:

Xtj:=Ytj\.X\_\{t\_\{j\}\}:=Y\_\{t\_\{j\}\}\.By the Markov property, each tail path𝒀\[tj,1\]\\bm\{Y\}\_\{\[t\_\{j\},1\]\}satisfies \([102](https://arxiv.org/html/2609.05727#S8.E102)\), which in turn implies \([101](https://arxiv.org/html/2609.05727#S8.E101)\)\. Thus, a single posterior\-preserving SDE trajectory𝒀\[t0,1\]\\bm\{Y\}\_\{\[t\_\{0\},1\]\}, together with a single backward ODE\-based evaluation ofr~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\), provides multiple exact covariance targets \([54](https://arxiv.org/html/2609.05727#S8.E54)\), one at each supervision time\.

##### SDE\-based estimation\.

Although the regularized rewardr~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)in the covariance target \([54](https://arxiv.org/html/2609.05727#S8.E54)\) depends only on the endpointX1X\_\{1\}, its SDE\-based estimation requires a path\. Specifically, in \([68](https://arxiv.org/html/2609.05727#S8.E68)\), the quantity

r~ρ​\(X1\)\+1τ​log⁡ptρ​\(Xt\)ptμ​\(Xt\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\log\\frac\{p\_\{t\}^\{\\rho\}\(X\_\{t\}\)\}\{p\_\{t\}^\{\\mu\}\(X\_\{t\}\)\}is evaluated along a path𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}satisfyingYt=XtY\_\{t\}=X\_\{t\}andY1=X1Y\_\{1\}=X\_\{1\}\. Thus, independently generating multiple endpoint pairs\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)does not allow the pathwise computation to be shared across supervision times\. Instead, the statesXtjX\_\{t\_\{j\}\}should be taken from a single posterior\-compatible path\. The supervision timestjt\_\{j\}should be sampled according to \([103](https://arxiv.org/html/2609.05727#S8.E103)\)\.

Under the forward construction, after samplingX1∼ρX\_\{1\}\\sim\\rho,X0∼p0X\_\{0\}\\sim p\_\{0\}, and the earliest supervision timet0t\_\{0\}, we construct the initial stateXt0X\_\{t\_\{0\}\}and then sample a single universal bridge using \([70](https://arxiv.org/html/2609.05727#S8.E70)\):

Xt0=αt0X1\+βt0X0,𝒀\[t0,1\]∼ℙ\[t0,1\]uni\(⋅\|Xt0,X1\)\.X\_\{t\_\{0\}\}=\\alpha\_\{t\_\{0\}\}X\_\{1\}\+\\beta\_\{t\_\{0\}\}X\_\{0\},\\qquad\\bm\{Y\}\_\{\[t\_\{0\},1\]\}\\sim\\mathbb\{P\}\_\{\[t\_\{0\},1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\_\{0\}\},X\_\{1\}\)\.\(104\)The forward construction of\(Xt0,X1\)\(X\_\{t\_\{0\}\},X\_\{1\}\), together with the sampling decomposition \([69](https://arxiv.org/html/2609.05727#S8.E69)\), ensures that the resulting path satisfies the requirement \([102](https://arxiv.org/html/2609.05727#S8.E102)\) att0t\_\{0\}\. Under the reverse construction, a posterior\-compatible path is obtained directly by sampling

Xt0∼p^t0,𝒀\[t0,1\]∼ℙ\[t0,1\]\|t0ρ\(⋅\|Xt0\)X\_\{t\_\{0\}\}\\sim\\hat\{p\}\_\{t\_\{0\}\},\\qquad\\bm\{Y\}\_\{\[t\_\{0\},1\]\}\\sim\\mathbb\{P\}\_\{\[t\_\{0\},1\]\|t\_\{0\}\}^\{\\rho\}\(\\cdot\|X\_\{t\_\{0\}\}\)\(105\)using the posterior\-preserving SDE\. Under either construction, we set

Xtj:=Ytj,j=0,1,…,Ntime−1\.X\_\{t\_\{j\}\}:=Y\_\{t\_\{j\}\},\\qquad j=0,1,\\ldots,N\_\{\\mathrm\{time\}\}\-1\.The Markov property then guarantees \([102](https://arxiv.org/html/2609.05727#S8.E102)\), and hence \([101](https://arxiv.org/html/2609.05727#S8.E101)\), at every supervision timetjt\_\{j\}\. Again, in practice, we may impose a lower cutoffε\>0\\varepsilon\>0by requiringt0≥εt\_\{0\}\\geq\\varepsilon\.

The SDE\-based evaluation in \([68](https://arxiv.org/html/2609.05727#S8.E68)\) can be carried out at all supervision times through a single backward recursion along the shared path\. At the terminal timetNtime=1t\_\{N\_\{\\mathrm\{time\}\}\}=1, the recursion is initialized with

r⁡\(X1\)−B1​\(X1\)\.r\(X\_\{1\}\)\-B\_\{1\}\(X\_\{1\}\)\.Proceeding backward, forj=Ntime−1,…,0j=N\_\{\\mathrm\{time\}\}\-1,\\ldots,0, the value attjt\_\{j\}is obtained from its value attj\+1t\_\{j\+1\}by adding

Btj\+1​\(Xtj\+1\)−Btj​\(Xtj\)−1τ​∫tjtj\+1‖vsρ​\(Ys\)−vsμ​\(Ys\)‖22κs​𝑑s−1τ​∫tjtj\+12κs​\(vsρ​\(Ys\)−vsμ​\(Ys\)\)⊤​d​Ws\.B\_\{t\_\{j\+1\}\}\(X\_\{t\_\{j\+1\}\}\)\-B\_\{t\_\{j\}\}\(X\_\{t\_\{j\}\}\)\-\\frac\{1\}\{\\tau\}\\int\_\{t\_\{j\}\}^\{t\_\{j\+1\}\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mu\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\-\\frac\{1\}\{\\tau\}\\int\_\{t\_\{j\}\}^\{t\_\{j\+1\}\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mu\}\(Y\_\{s\}\)\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}\.Consequently, the covariance targets at all supervision times can be evaluated through a single backward recursion along𝒀\[t0,1\]\\bm\{Y\}\_\{\[t\_\{0\},1\]\}\.

#### 8\.6\.2Multi\-Time Supervision in Gradient Form

Under both the Mayer and Bolza realizations, the gradient form requires a pathwise adjoint to evaluate the sample\-wise target\. For both forward and reverse constructions, the supervision timestjt\_\{j\}, the statesXtjX\_\{t\_\{j\}\}, and the shared posterior\-compatible path𝒀\[t0,1\]\\bm\{Y\}\_\{\[t\_\{0\},1\]\}should be sampled and constructed in the same manner as in the covariance form with SDE\-based estimation\. Notably, a single backward solution of the Mayer adjoint ODE \([83](https://arxiv.org/html/2609.05727#S8.E83)\) or the Bolza adjoint ODE \([88](https://arxiv.org/html/2609.05727#S8.E88)\) along the full trajectory, fromt=1t=1tot=t0t=t\_\{0\}, yields the adjoint value at every supervision timetjt\_\{j\}\. More precisely, the value obtained attjt\_\{j\}from the full backward solve coincides with that obtained by solving the adjoint separately along the tail path𝒀\[tj,1\]\\bm\{Y\}\_\{\[t\_\{j\},1\]\}:

λtjbρ,r~ρ,0​\(𝒀\[t0,1\]\)=λtjbρ,r~ρ,0​\(𝒀\[tj,1\]\),λtjbρ,r,lρ,μ​\(𝒀\[t0,1\]\)=λtjbρ,r,lρ,μ​\(𝒀\[tj,1\]\)\.\\displaystyle\\lambda\_\{t\_\{j\}\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\\left\(\\bm\{Y\}\_\{\[t\_\{0\},1\]\}\\right\)=\\lambda\_\{t\_\{j\}\}^\{b^\{\\rho\},\\tilde\{r\}^\{\\rho\},0\}\\left\(\\bm\{Y\}\_\{\[t\_\{j\},1\]\}\\right\),\\qquad\\lambda\_\{t\_\{j\}\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\\left\(\\bm\{Y\}\_\{\[t\_\{0\},1\]\}\\right\)=\\lambda\_\{t\_\{j\}\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\\left\(\\bm\{Y\}\_\{\[t\_\{j\},1\]\}\\right\)\.Indeed, for eachjj, the restriction of the full adjoint solution to\[tj,1\]\[t\_\{j\},1\]satisfies the same terminal condition and the same backward ODE along the same tail path as the adjoint solved separately on\[tj,1\]\[t\_\{j\},1\]\. The identities above therefore follow from uniqueness of the terminal\-value problem\. Hence, the adjoint values obtained from a single full\-trajectory solve can be inserted directly into the Mayer target \([84](https://arxiv.org/html/2609.05727#S8.E84)\) or the Bolza target \([93](https://arxiv.org/html/2609.05727#S8.E93)\) at every supervision time, without solving a separate adjoint ODE for eachtjt\_\{j\}\. Consequently, the gradient\-form targets at all supervision times can be evaluated from a single backward solution of the corresponding adjoint ODE along𝒀\[t0,1\]\\bm\{Y\}\_\{\[t\_\{0\},1\]\}, with the pathwise adjoint computation shared across the resulting regression pairs\.

The multi\-time supervision schemes developed in this subsection are representative rather than exhaustive and provide natural ways to improve sample efficiency\. Other valid schemes may also exist\. Algorithm[4](https://arxiv.org/html/2609.05727#alg4)summarizes one representative scheme for each configuration considered above\.

### 8\.7Canonicalization

The preceding subsections present several exact realizations of the tangential updatev¯=vρ\+η​Γρ,r~ρ\\bar\{v\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}at a canonical anchorvρv^\{\\rho\}\. However,v¯\\bar\{v\}is not canonical in general\. To return to the canonical manifold and proceed to the next stage of Newton Matching, we need to canonicalizev¯\\bar\{v\}, that is, to find a canonical velocity fieldvqv^\{q\}whose terminal density matches that induced byv¯\\bar\{v\}\. In this subsection, we first present a straightforward canonicalization procedure based on the standard CFM loss, which we call*explicit canonicalization*\. We then discuss an interesting idea, called*implicit canonicalization*, which sidesteps this need by carefully constructing the tangential\-update loss so that the noncanonical fieldv¯\\bar\{v\}can serve as the anchor\.

##### Explicit canonicalization via CFM\.

Letq=𝒯⁡\(v¯\)q=\\mathcal\{T\}\(\\bar\{v\}\)denote the terminal density induced byv¯\\bar\{v\}\. We can generate endpoint samplesX1∼qX\_\{1\}\\sim qusing the ODE driven byv¯\\bar\{v\}\. To construct a canonical velocity fieldvqv^\{q\}with terminal densityqq, we can utilize the standard CFM loss\. Specifically, after samplingX1∼qX\_\{1\}\\sim q, we independently drawX0∼p0X\_\{0\}\\sim p\_\{0\}, setXt=αt​X1\+βt​X0X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}, and train a parameterized velocity fieldvθv^\{\\theta\}using the loss

ℒcan​\(θ\):=𝔼t∼U⁡\(0,1\),X1∼q,X0∼p0,Xt=αt​X1\+βt​X0​\[‖vtθ​\(Xt\)−vt\|1​\(Xt\|X1\)‖22\]\.\\mathcal\{L\}^\{\\mathrm\{can\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim q,\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.\(106\)In practice,vθv^\{\\theta\}can be warm\-started from the parameters representingv¯\\bar\{v\}\. The population minimizer of \([106](https://arxiv.org/html/2609.05727#S8.E106)\) is the canonical velocity fieldvq=𝒞⁡\(q\)=ℛ⁡\(v¯\)v^\{q\}=\\mathcal\{C\}\(q\)=\\mathcal\{R\}\(\\bar\{v\}\)\. Once training is complete, we replace the current anchorvρv^\{\\rho\}withvqv^\{q\}and proceed to the next stage of Newton Matching\. The explicit canonicalization is summarized in Algorithm[5](https://arxiv.org/html/2609.05727#alg5)\.

Algorithm 5Explicit Canonicalization\.0:At stage

kk, the updated velocity model

v¯k\+1∈𝒱\\bar\{v\}^\{k\+1\}\\in\\mathscr\{V\}; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The canonical velocity model

vρk\+1=ℛ⁡\(v¯k\+1\)v^\{\\rho\_\{k\+1\}\}=\\mathcal\{R\}\\left\(\\bar\{v\}^\{k\+1\}\\right\)\.

1:Warm\-start

θ\\thetawith the network parameters of

v¯k\+1\\bar\{v\}^\{k\+1\}\.

2:foreach gradient stepdo

3:

⊳\\trianglerightThe following operations are applied to a batch\.

4:Sample

X1∼ρk\+1=𝒯⁡\(v¯k\+1\)X\_\{1\}\\sim\\rho\_\{k\+1\}=\\mathcal\{T\}\\left\(\\bar\{v\}^\{k\+1\}\\right\)through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

v¯k\+1\\bar\{v\}^\{k\+1\}\.

5:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

6:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,X1\)‖vtθ​\(Xt\)−vt\|1​\(Xt\|X1\)‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},X\_\{1\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\.

7:endfor

8:Set

vρk\+1←vθv^\{\\rho\_\{k\+1\}\}\\leftarrow v^\{\\theta\}\.

##### Implicit canonicalization\.

Sometimes, we may prefer to avoid the explicit canonicalization procedure\. Suppose thatv¯\\bar\{v\}is a noncanonical velocity field with terminal densityq=𝒯⁡\(v¯\)q=\\mathcal\{T\}\(\\bar\{v\}\)\. At the next stage of Newton Matching, our goal is to perform a tangential update at the canonical anchorvqv^\{q\}, yielding the updated velocity fieldvq\+η​Γq,r~qv^\{q\}\+\\eta\\Gamma^\{q,\\tilde\{r\}^\{q\}\}\. The question is whether this goal can be achieved usingv¯\\bar\{v\}as the endpoint\-generation field, without first explicitly constructingvqv^\{q\}\. The answer is yes, subject to additional restrictions\.

The forward construction provides a way to sidestepvqv^\{q\}\. We first sampleX1∼qX\_\{1\}\\sim qusing the ODE driven byv¯\\bar\{v\}\. Applying \([51](https://arxiv.org/html/2609.05727#S8.E51)\) withρ=q\\rho=qyields

X1\|Xt∼p1\|tq\(⋅\|Xt\)\.X\_\{1\}\|X\_\{t\}\\sim p\_\{1\|t\}^\{q\}\(\\cdot\|X\_\{t\}\)\.Thus, even thoughv¯\\bar\{v\}may be noncanonical, the\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)pair satisfies the posterior requirement\.

For the covariance form, replace the canonical anchorvtqv\_\{t\}^\{q\}in \([54](https://arxiv.org/html/2609.05727#S8.E54)\) byv¯t\\bar\{v\}\_\{t\}, giving

vt\|1​\(Xt\|X1\)\+η⁡\(r~q​\(X1\)−Bt​\(Xt\)\)​\(vt\|1​\(Xt\|X1\)−v¯t​\(Xt\)\)\.\\displaystyle v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\eta\\left\(\\tilde\{r\}^\{q\}\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\bar\{v\}\_\{t\}\(X\_\{t\}\)\\right\)\.For every fixed\(t,xt\)\(t,x\_\{t\}\), its conditional mean is

𝔼X1∼p1\|tq\(⋅\|xt\)\[vt\|1\(xt\|X1\)\+η\(r~q\(X1\)−Bt\(xt\)\)\(vt\|1\(xt\|X1\)−v¯t\(xt\)\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{q\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\+\\eta\\left\(\\tilde\{r\}^\{q\}\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-\\bar\{v\}\_\{t\}\(x\_\{t\}\)\\right\)\\right\]=\\displaystyle=vtq​\(xt\)\+η​Γtq,r~q​\(xt\)\+η⁡\(Vtq​\[r~q\]​\(xt\)−Bt​\(xt\)\)​\(vtq​\(xt\)−v¯t​\(xt\)\)\.\\displaystyle v\_\{t\}^\{q\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{q,\\tilde\{r\}^\{q\}\}\(x\_\{t\}\)\+\\eta\\left\(V\_\{t\}^\{q\}\[\\tilde\{r\}^\{q\}\]\(x\_\{t\}\)\-B\_\{t\}\(x\_\{t\}\)\\right\)\\left\(v\_\{t\}^\{q\}\(x\_\{t\}\)\-\\bar\{v\}\_\{t\}\(x\_\{t\}\)\\right\)\.In particular, the exact value baseline

Bt​\(xt\)=Vtq​\[r~q\]​\(xt\)B\_\{t\}\(x\_\{t\}\)=V\_\{t\}^\{q\}\[\\tilde\{r\}^\{q\}\]\(x\_\{t\}\)removes the dependence of the conditional mean on the noncanonicalv¯t\\bar\{v\}\_\{t\}, yieldingvtq​\(xt\)\+η​Γtq,r~q​\(xt\)v\_\{t\}^\{q\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{q,\\tilde\{r\}^\{q\}\}\(x\_\{t\}\)\. Moreover, the termlog⁡q⁡\(X1\)μ⁡\(X1\)\\log\\frac\{q\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}inr~q\\tilde\{r\}^\{q\}can be evaluated using the ODE\-based method, which does not requirev¯\\bar\{v\}to be canonical\.

We note that obtaining the exact value baseline may be challenging and may require training a separate neural network\. With an approximate baseline, the conditional\-mean error is

η⁡\(Vtq​\[r~q\]​\(xt\)−Bt​\(xt\)\)​\(vtq​\(xt\)−v¯t​\(xt\)\)\.\\eta\\left\(V\_\{t\}^\{q\}\[\\tilde\{r\}^\{q\}\]\(x\_\{t\}\)\-B\_\{t\}\(x\_\{t\}\)\\right\)\\left\(v\_\{t\}^\{q\}\(x\_\{t\}\)\-\\bar\{v\}\_\{t\}\(x\_\{t\}\)\\right\)\.Thus, the error in the tangential update arising from the use of the noncanonicalv¯\\bar\{v\}is given by the product of the value\-function error and the canonicality defect\.

## 9Approximate Newton Matching

At the population level, the exact realizations developed in Section[8](https://arxiv.org/html/2609.05727#S8)recover the ideal Newton Matching stage introduced in Section[6\.3](https://arxiv.org/html/2609.05727#S6.SS3)\. As established in Section[7](https://arxiv.org/html/2609.05727#S7), they retain the finite\-stepsize reverse\-KL descent, global convergence under mild conditions, and local quadratic convergence of the full\-step iteration\. These exact realizations, however, involve several nontrivial computations\. The regularized reward

r~ρ​\(x\)=r⁡\(x\)−1τ​log⁡ρ⁡\(x\)μ⁡\(x\)\\tilde\{r\}^\{\\rho\}\(x\)=r\(x\)\-\\frac\{1\}\{\\tau\}\\log\\frac\{\\rho\(x\)\}\{\\mu\(x\)\}contains either the log\-density termlog⁡ρ\\log\\rhowhenμ=1\\mu=1, or the log\-density\-ratio termlog⁡ρρbase\\log\\frac\{\\rho\}\{\\rho^\{\\mathrm\{base\}\}\}whenμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}\. In the covariance form, this correction term must be evaluated at the sampled endpoints, whereas in the gradient form, its gradient is computed through adjoint equations\. When computational resources are limited, it is therefore natural to trade population\-level exactness at each stage for reduced computational cost\.

In this section, we develop approximate Newton Matching, making this trade\-off explicit\. Rather than requiring the matching objective to recover the exact tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}, we replace it with a computationally cheaper surrogate

Γ^ρ≈Γρ,r~ρ\.\\widehat\{\\Gamma\}^\{\\rho\}\\approx\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.Throughout Sections[9\.1](https://arxiv.org/html/2609.05727#S9.SS1)–[9\.3](https://arxiv.org/html/2609.05727#S9.SS3), we develop principled choices ofΓ^ρ\\widehat\{\\Gamma\}^\{\\rho\}\. We define the resulting approximate Newton Matching stage by

v^:=vρ\+η​Γ^ρ,\\displaystyle\\hat\{v\}:=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\},\(107a\)q=𝒯⁡\(v^\),vq=ℛ⁡\(v^\)=𝒞⁡\(q\)\.\\displaystyle q=\\mathcal\{T\}\(\\hat\{v\}\),\\,v^\{q\}=\\mathcal\{R\}\(\\hat\{v\}\)=\\mathcal\{C\}\(q\)\.\(107b\)Thus, only the tangential update is approximated, while canonicalization remains exact\. Exact Newton Matching is recovered by settingΓ^ρ=Γρ,r~ρ\\widehat\{\\Gamma\}^\{\\rho\}=\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\. For notational simplicity, we writeπ:=πμ,τ,r\\pi:=\\pi\_\{\\mu,\\tau,r\}\. At a given stage, we fix a canonical anchorvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}and choose a scalar stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\]\.

As expected, an approximate direction need not coincide with the exact Newton tangent vector at a generic anchor\. As a target\-correctness requirement, however, it should preserve the intended critical point\. This motivates the following definition\.

###### Definition 36\.

A method \([107](https://arxiv.org/html/2609.05727#S9.E107)\) is*critical\-point consistent*forπ\\piif

Γ^ρ=0⇔ρ=π\.\\widehat\{\\Gamma\}^\{\\rho\}=0\\quad\\Leftrightarrow\\quad\\rho=\\pi\.

Sinceη\>0\\eta\>0, critical\-point consistency is equivalently expressed asv^=vρ\\hat\{v\}=v^\{\\rho\}if and only ifρ=π\\rho=\\pifor the ambient update \([107a](https://arxiv.org/html/2609.05727#S9.E107.1)\)\. We impose this condition before canonicalization\. Requiring the displacementΓ^ρ\\widehat\{\\Gamma\}^\{\\rho\}itself to vanish at the target densityπ\\piis stronger than requiring the complete stage \([107](https://arxiv.org/html/2609.05727#S9.E107)\) to preserve the terminal density there, since the terminal\-density map𝒯\\mathcal\{T\}is many\-to\-one\. According to Proposition[18](https://arxiv.org/html/2609.05727#Thmtheorem18), the exact Newton Matching is critical\-point consistent\.

Sections[9\.1](https://arxiv.org/html/2609.05727#S9.SS1)and[9\.2](https://arxiv.org/html/2609.05727#S9.SS2)develop two covariance approximations and two gradient approximations, respectively\. These four methods preserve critical\-point consistency\. Since the density\-ratio correction in the regularized rewardr~ρ\\tilde\{r\}^\{\\rho\}is a major source of computational challenge, Section[9\.3](https://arxiv.org/html/2609.05727#S9.SS3)examines the trade\-offs that arise when this term is approximated or removed entirely\. In the fully unregularized case, the update is driven by the raw rewardrrand admits an exact finite\-stepsize reward\-ascent certificate\. Throughout these three subsections, the prescribed update direction is realized stagewise and followed by exact canonicalization\. Method\-specific proofs are collected in Appendix[F](https://arxiv.org/html/2609.05727#A6)\.

Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4)then develops a complementary viewpoint\. Rather than realizing the additive update \([107a](https://arxiv.org/html/2609.05727#S9.E107.1)\) stage by stage, we consider its fixed\-point condition directly\. Forcing the displacement to vanish gives

vρ=vρ\+η​Γ^ρ\.v^\{\\rho\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}\.Since the right\-hand side admits a conditional\-expectation representation, we can construct a matching\-style stop\-gradient objective whose population\-stationary points are exactly the fixed points of the additive update rule\.

The constructions in this section are representative rather than exhaustive\. The four critical\-point\-consistent examples illustrate only a subset of the possible choices\. Additionally, if computational efficiency is strongly prioritized, the ideas in this section can be taken further to develop more aggressive approximations\.

Some approximate constructions requireμ\\muto be a normalized reference density, since they draw samples from the corresponding posteriorp1\|tμ\(⋅\|xt\)p\_\{1\|t\}^\{\\mu\}\(\\cdot\|x\_\{t\}\)\. Whenμ=1\\mu=1, this requirement can be satisfied through the followingauxiliary\-reference factorization\. Choose any normalized auxiliary reference densityμ^∈𝒫\\hat\{\\mu\}\\in\\mathscr\{P\}, such as a Gaussian\. Define the auxiliary reward

r^​\(x\):=r⁡\(x\)−1τ​log⁡μ^​\(x\)\.\\hat\{r\}\(x\):=r\(x\)\-\\frac\{1\}\{\\tau\}\\log\\hat\{\\mu\}\(x\)\.\(108\)The unnormalized target factor is preserved pointwise:

μ^​\(x\)​eτ​r^​\(x\)=eτ​r​\(x\)\.\\hat\{\\mu\}\(x\)e^\{\\tau\\hat\{r\}\(x\)\}=e^\{\\tau r\(x\)\}\.Consequently, the pairs\(1,r\)\(1,r\)and\(μ^,r^\)\(\\hat\{\\mu\},\\hat\{r\}\)define exactly the same target density

π1,τ,r=πμ^,τ,r^,\\pi\_\{1,\\tau,r\}=\\pi\_\{\\hat\{\\mu\},\\tau,\\hat\{r\}\},while the latter provides a normalized reference density whose posterior can be readily sampled from\. Any construction requiring such a reference density may therefore be applied to\(μ^,τ,r^\)\(\\hat\{\\mu\},\\tau,\\hat\{r\}\)without changing the sampling target\. Ifμ^\\hat\{\\mu\}is Gaussian, Proposition[85](https://arxiv.org/html/2609.05727#Thmtheorem85)gives a closed\-form expression for the canonical velocity fieldvμ^v^\{\\hat\{\\mu\}\}, which can be computed efficiently\. Whenever this auxiliary\-reference factorization is used later, we relabel\(μ^,r^\)\(\\hat\{\\mu\},\\hat\{r\}\)as\(μ,r\)\(\\mu,r\)for notational simplicity\.

### 9\.1Approximate Covariance Forms with Critical\-Point Consistency

The exact covariance form represents the Newton tangential update as

v¯t\(xt\)=vtρ\(xt\)\+ηCovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),r~ρ\(X1\)\)\.\\bar\{v\}\_\{t\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\)\.In this representation, the computationally challenging quantity is the logarithmic componentlog⁡ρ⁡\(X1\)μ⁡\(X1\)\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}in the regularized rewardr~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\. Both approximations below first rewrite this logarithmic component through posterior density ratios and then replace it by a linearized form\. Both constructions preserve critical\-point consistency\. Their proofs are given in Appendix[F\.1](https://arxiv.org/html/2609.05727#A6.SS1)\.

#### 9\.1\.1Direct Linearization

The first construction rewrites the exact Newton tangent vector through the posterior log\-density ratio between the target and the current anchor\. Since

r~ρ​\(x\)=1τ​log⁡π⁡\(x\)ρ⁡\(x\)\+const,\\tilde\{r\}^\{\\rho\}\(x\)=\\frac\{1\}\{\\tau\}\\log\\frac\{\\pi\(x\)\}\{\\rho\(x\)\}\+\\mathrm\{const\},the additive constant does not contribute to the posterior covariance\. Moreover, by Bayes’ rule,

p1\|tπ​\(x1\|xt\)p1\|tρ​\(x1\|xt\)=π⁡\(x1\)​pt\|1​\(xt\|x1\)ptπ​\(xt\)​ptρ​\(xt\)ρ⁡\(x1\)​pt\|1​\(xt\|x1\)=ptρ​\(xt\)​π​\(x1\)ptπ​\(xt\)​ρ​\(x1\)\.\\frac\{p\_\{1\|t\}^\{\\pi\}\(x\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\}=\\frac\{\\pi\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{p\_\{t\}^\{\\pi\}\(x\_\{t\}\)\}\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}=\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\pi\(x\_\{1\}\)\}\{p\_\{t\}^\{\\pi\}\(x\_\{t\}\)\\rho\(x\_\{1\}\)\}\.The factor depending only on\(t,xt\)\(t,x\_\{t\}\)disappears inside the posterior covariance\. Consequently, we have

Γtρ,r~ρ​\(xt\)=\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=1τCovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),logπ⁡\(X1\)ρ⁡\(X1\)\)\\displaystyle\\frac\{1\}\{\\tau\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\log\\frac\{\\pi\(X\_\{1\}\)\}\{\\rho\(X\_\{1\}\)\}\\right\)=\\displaystyle=1τCovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),logp1\|tπ​\(X1\|xt\)p1\|tρ​\(X1\|xt\)\)\.\\displaystyle\\frac\{1\}\{\\tau\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\log\\frac\{p\_\{1\|t\}^\{\\pi\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\right\)\.\(109\)
The posterior ratio equals one whenρ=π\\rho=\\pi\. We linearize the logarithmic term:

log⁡p1\|tπ​\(X1\|xt\)p1\|tρ​\(X1\|xt\)≈p1\|tπ​\(X1\|xt\)p1\|tρ​\(X1\|xt\)−1\.\\log\\frac\{p\_\{1\|t\}^\{\\pi\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\approx\\frac\{p\_\{1\|t\}^\{\\pi\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\-1\.This linearization is exact at the target and retains the first\-order variation of the posterior log\-density ratio nearby\. Substituting it into \([109](https://arxiv.org/html/2609.05727#S9.E109)\) gives

v¯t​\(xt\)≈\\displaystyle\\bar\{v\}\_\{t\}\(x\_\{t\}\)\\approxvtρ\(xt\)\+ητCovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),p1\|tπ​\(X1\|xt\)p1\|tρ​\(X1\|xt\)\)\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\frac\{p\_\{1\|t\}^\{\\pi\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\right\)=\\displaystyle=vtρ\(xt\)\+ητ𝔼X1∼p1\|tρ\(⋅\|xt\)\[p1\|tπ​\(X1\|xt\)p1\|tρ​\(X1\|xt\)\(vt\|1\(xt\|X1\)−vtρ\(xt\)\)\]\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\frac\{p\_\{1\|t\}^\{\\pi\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\\right\]=\\displaystyle=vtρ\(xt\)\+ητ𝔼X1∼p1\|tπ\(⋅\|xt\)\[vt\|1\(xt\|X1\)−vtρ\(xt\)\]\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\pi\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\]=\\displaystyle=\(1−ητ\)​vtρ​\(xt\)\+ητ​vtπ​\(xt\)\.\\displaystyle\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\pi\}\(x\_\{t\}\)\.Therefore, we approximate the tangential update through direct linearization:

Γ^tρ,dir​\-​lin​\(xt\):=1τ​\(vtπ​\(xt\)−vtρ​\(xt\)\),\\displaystyle\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}\(x\_\{t\}\):=\\frac\{1\}\{\\tau\}\\left\(v\_\{t\}^\{\\pi\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\),\(110a\)v^tdir​\-​lin​\(xt\):=vtρ​\(xt\)\+η​Γ^tρ,dir​\-​lin​\(xt\)=\(1−ητ\)​vtρ​\(xt\)\+ητ​vtπ​\(xt\)\.\\displaystyle\\hat\{v\}\_\{t\}^\{\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}\(x\_\{t\}\):=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}\(x\_\{t\}\)=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\pi\}\(x\_\{t\}\)\.\(110b\)Thus, the posterior\-ratio linearization produces a convex interpolation in canonical\-velocity coordinates\. It changes the exact Newton tangent vector at a generic anchor, while the full stepη=τ\\eta=\\taudirectly targetsvπv^\{\\pi\}\.

Algorithm 6Tangential Update: Approximate Realization with Direct Linearization\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v^k\+1≈vρk\+ηk​Γρk,r~ρk\\hat\{v\}^\{k\+1\}\\approx v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:if

μ=1\\mu=1then

2:Set

ρbase←𝒩⁡\(m,σ2​I\)\\rho^\{\\mathrm\{base\}\}\\leftarrow\\mathcal\{N\}\(m,\\sigma^\{2\}I\)and its analytical velocity field

vbasev^\{\\mathrm\{base\}\}based on the auxiliary\-reference factorization \([108](https://arxiv.org/html/2609.05727#S9.E108)\)\.

3:Set

r←r−1τ​log⁡ρbaser\\leftarrow r\-\\frac\{1\}\{\\tau\}\\log\\rho^\{\\mathrm\{base\}\}\.

4:endif

5:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

6:foreach gradient stepdo

7:

⊳\\trianglerightThe following operations are applied to a batch\.

8:ifforward construction is usedthen

9:if

μ=1\\mu=1then

10:Sample

X1∼ρbase=𝒩⁡\(m,σ2​I\)X\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\}=\\mathcal\{N\}\(m,\\sigma^\{2\}I\)\.

11:elseifODE\-based sampling is usedthen

12:Sample

X1∼ρbaseX\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vbasev^\{\\mathrm\{base\}\}\.

13:elseifSDE\-based sampling is usedthen

14:Sample

X1∼ρbaseX\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vbasev^\{\\mathrm\{base\}\}\.

15:endif

16:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

17:elseifreverse construction is usedthen

18:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

19:if

μ=1\\mu=1then

20:Sample

X1∼p1\|tbase\(⋅\|Xt\)=𝒩\(αt​σ2αt2​σ2\+βt2Xt\+βt2αt2​σ2\+βt2m,βt2​σ2αt2​σ2\+βt2I\)X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|X\_\{t\}\)=\\mathcal\{N\}\\left\(\\frac\{\\alpha\_\{t\}\\sigma^\{2\}\}\{\\alpha\_\{t\}^\{2\}\\sigma^\{2\}\+\\beta\_\{t\}^\{2\}\}X\_\{t\}\+\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\\sigma^\{2\}\+\\beta\_\{t\}^\{2\}\}m,\\frac\{\\beta\_\{t\}^\{2\}\\sigma^\{2\}\}\{\\alpha\_\{t\}^\{2\}\\sigma^\{2\}\+\\beta\_\{t\}^\{2\}\}I\\right\)\.

21:elseif

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}then

22:Sample

X1∼p1\|tbase\(⋅\|Xt\)X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρbase\\rho^\{\\mathrm\{base\}\}\.

23:endif

24:endif

25:Compute the target

𝒴\\mathcal\{Y\}by \([112](https://arxiv.org/html/2609.05727#S9.E112)\)\.

26:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

27:endfor

28:Set

v^k\+1←vθ\\hat\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

The fieldvπv^\{\\pi\}is not directly available, so the remaining task is to realize \([110](https://arxiv.org/html/2609.05727#S9.E110)\) through matching\-style regression\. Assume thatμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}is normalized; when the original reference isμ=1\\mu=1, we apply the auxiliary\-reference factorization \([108](https://arxiv.org/html/2609.05727#S9.E108)\)\. The target posterior is then an exponential tilt of the reference posterior:

p1\|tπ​\(x1\|xt\)∝eτ​r​\(x1\)​p1\|tbase​\(x1\|xt\)\.p\_\{1\|t\}^\{\\pi\}\(x\_\{1\}\|x\_\{t\}\)\\propto e^\{\\tau r\(x\_\{1\}\)\}p\_\{1\|t\}^\{\\mathrm\{base\}\}\(x\_\{1\}\|x\_\{t\}\)\.\(111\)The conditional normalizing constant in \([111](https://arxiv.org/html/2609.05727#S9.E111)\) depends on\(t,xt\)\(t,x\_\{t\}\)and is generally unavailable\. Nevertheless, the approximate tangential update in \([110](https://arxiv.org/html/2609.05727#S9.E110)\) admits the following stop\-gradient realization under the joint distribution from the reference:

ℒρ,η,Bdir​\-​lin\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1base\[∥vtθ\(Xt\)−\[\(1−ητ\)vtρ\(Xt\)\+ητvtbase\(Xt\)\]\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\mathrm\{base\}\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\[\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\]−\(eτ⁡\(r⁡\(X1\)−Bt​\(Xt\)\)−1\)\[\(1−ητ\)vtρ\(Xt\)\+ητvt\|1\(Xt\|X1\)−sg\(vtθ\(Xt\)\)\]∥22\]\.\\displaystyle\\quad\-\\left\(e^\{\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\)\}\-1\\right\)\\Big\[\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\\Big\]\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(112\)Here,sg\\operatorname\{sg\}denotes the stop\-gradient operator, andBt​\(Xt\)B\_\{t\}\(X\_\{t\}\)is an arbitrary scalar baseline\. We adopt the notation in \([49](https://arxiv.org/html/2609.05727#S8.E49)\) and use the shorthandΞt,1base:=Ξt,1ρbase\\Xi\_\{t,1\}^\{\\mathrm\{base\}\}:=\\Xi\_\{t,1\}^\{\\rho^\{\\mathrm\{base\}\}\}for simplicity\. At a fixed noisy state, subtractingBt​\(Xt\)B\_\{t\}\(X\_\{t\}\)rescales all exponential weights by the same factor and therefore does not change the population\-stationary point\. It can nevertheless improve numerical conditioning and reduce variance\. A useful choice is

Bt\(xt\)=𝔼X1∼p1\|tbase\(⋅\|xt\)\[r\(X1\)\]\.B\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\}\[r\(X\_\{1\}\)\]\.
The computation of approximate tangential update with direct linearization is summarized in Algorithm[6](https://arxiv.org/html/2609.05727#alg6)\. Section[9\.4\.2](https://arxiv.org/html/2609.05727#S9.SS4.SSS2)presents a general formula for deriving the population\-stationary points of stop\-gradient objectives, and Appendix[F\.1](https://arxiv.org/html/2609.05727#A6.SS1)applies the formula to prove the following proposition\.

###### Proposition 37\.

Assume thatμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}is a normalized reference density and the loss \([112](https://arxiv.org/html/2609.05727#S9.E112)\) is integrable\. The unique population\-stationary point of the loss \([112](https://arxiv.org/html/2609.05727#S9.E112)\), up to a\.e\. equality, isv^dir​\-​lin\\hat\{v\}^\{\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}in \([110](https://arxiv.org/html/2609.05727#S9.E110)\)\. Furthermore, the direct linearization is critical\-point consistent forπ\\pi, i\.e\.,Γ^ρ,dir​\-​lin=0\\widehat\{\\Gamma\}^\{\\rho,\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}=0if and only ifρ=π\\rho=\\pi\.

#### 9\.1\.2Split Linearization

The preceding construction realizes the target posterior through a positive exponential tilt of the reference posterior\. A complementary construction starts from the current posterior and introduces the normalized negative\-tilt density

ρ−​\(x\):=ρ⁡\(x\)​e−τ​r​\(x\)∫ℝdρ⁡\(z\)​e−τ​r​\(z\)​𝑑z\.\\rho^\{\-\}\(x\):=\\frac\{\\rho\(x\)e^\{\-\\tau r\(x\)\}\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(z\)e^\{\-\\tau r\(z\)\}\\mathrm\{d\}z\}\.We assume thatρ\\rhois a normalizable density in𝒫\\mathscr\{P\}, satisfying

∫ℝdρ⁡\(z\)​e−τ​r​\(z\)​𝑑z<∞\.\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(z\)e^\{\-\\tau r\(z\)\}\\mathrm\{d\}z<\\infty\.We also assume thatμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}is normalized, applying the auxiliary\-reference factorization \([108](https://arxiv.org/html/2609.05727#S9.E108)\) when the original reference isμ=1\\mu=1\.

The role ofρ−\\rho^\{\-\}is to express the target\-to\-current posterior ratio through two ratios centered at the current posterior\. By Bayes’ rule,

p1\|tπ​\(X1\|xt\)p1\|tbase​\(X1\|xt\)=π⁡\(x1\)​pt\|1​\(xt\|x1\)ptπ​\(xt\)​ptbase​\(xt\)ρbase​\(x1\)​pt\|1​\(xt\|x1\)=ptbase​\(xt\)​eτ​r​\(x1\)ptπ​\(xt\)​𝔼X1∼ρbase​\[eτ​r​\(X1\)\],\\displaystyle\\frac\{p\_\{1\|t\}^\{\\pi\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\mathrm\{base\}\}\(X\_\{1\}\|x\_\{t\}\)\}=\\frac\{\\pi\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{p\_\{t\}^\{\\pi\}\(x\_\{t\}\)\}\\frac\{p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\}\{\\rho^\{\\mathrm\{base\}\}\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}=\\frac\{p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)e^\{\\tau r\(x\_\{1\}\)\}\}\{p\_\{t\}^\{\\pi\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\}\}\\left\[e^\{\\tau r\(X\_\{1\}\)\}\\right\]\},\(113a\)p1\|tρ​\(X1\|xt\)p1\|tρ−​\(X1\|xt\)=ρ⁡\(x1\)​pt\|1​\(xt\|x1\)ptρ​\(xt\)​ptρ−​\(xt\)ρ−​\(x1\)​pt\|1​\(xt\|x1\)=ptρ−​\(xt\)​eτ​r​\(x1\)ptρ​\(xt\)​𝔼X1∼ρ−​\[eτ​r​\(X1\)\]\.\\displaystyle\\frac\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho^\{\-\}\}\(X\_\{1\}\|x\_\{t\}\)\}=\\frac\{\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\\frac\{p\_\{t\}^\{\\rho^\{\-\}\}\(x\_\{t\}\)\}\{\\rho^\{\-\}\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}=\\frac\{p\_\{t\}^\{\\rho^\{\-\}\}\(x\_\{t\}\)e^\{\\tau r\(x\_\{1\}\)\}\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim\\rho^\{\-\}\}\\left\[e^\{\\tau r\(X\_\{1\}\)\}\\right\]\}\.\(113b\)The factors depending only on\(t,xt\)\(t,x\_\{t\}\)disappear inside the posterior covariance\. Combining \([113a](https://arxiv.org/html/2609.05727#S9.E113.1)\) and \([113b](https://arxiv.org/html/2609.05727#S9.E113.2)\) with \([109](https://arxiv.org/html/2609.05727#S9.E109)\) gives

Γtρ,r~ρ​\(xt\)=\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=1τCovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),logp1\|tbase​\(X1\|xt\)p1\|tρ−​\(X1\|xt\)\)\\displaystyle\\frac\{1\}\{\\tau\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\log\\frac\{p\_\{1\|t\}^\{\\mathrm\{base\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho^\{\-\}\}\(X\_\{1\}\|x\_\{t\}\)\}\\right\)=\\displaystyle=1τCovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),logp1\|tbase​\(X1\|xt\)p1\|tρ​\(X1\|xt\)−logp1\|tρ−​\(X1\|xt\)p1\|tρ​\(X1\|xt\)\)\.\\displaystyle\\frac\{1\}\{\\tau\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\log\\frac\{p\_\{1\|t\}^\{\\mathrm\{base\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\-\\log\\frac\{p\_\{1\|t\}^\{\\rho^\{\-\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\right\)\.
Algorithm 7Tangential Update: Approximate Realization with Split Linearization\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v^k\+1≈vρk\+ηk​Γρk,r~ρk\\hat\{v\}^\{k\+1\}\\approx v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:if

μ=1\\mu=1then

2:Set

ρbase←𝒩⁡\(m,σ2​I\)\\rho^\{\\mathrm\{base\}\}\\leftarrow\\mathcal\{N\}\(m,\\sigma^\{2\}I\)and its analytical velocity field

vbasev^\{\\mathrm\{base\}\}based on the auxiliary\-reference factorization \([108](https://arxiv.org/html/2609.05727#S9.E108)\)\.

3:Set

r←r−1τ​log⁡ρbaser\\leftarrow r\-\\frac\{1\}\{\\tau\}\\log\\rho^\{\\mathrm\{base\}\}\.

4:endif

5:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

6:foreach gradient stepdo

7:

⊳\\trianglerightThe following operations are applied to a batch\.

8:ifforward construction is usedthen

9:ifODE\-based sampling is usedthen

10:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vρkv^\{\\rho\_\{k\}\}\.

11:elseifSDE\-based sampling is usedthen

12:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

13:endif

14:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

15:elseifreverse construction is usedthen

16:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

17:Sample

X1∼p1\|tρk\(⋅\|Xt\)X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

18:endif

19:Compute the target

𝒴\\mathcal\{Y\}by \([115](https://arxiv.org/html/2609.05727#S9.E115)\)\.

20:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

21:endfor

22:Set

v^k\+1←vθ\\hat\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

We linearize both logarithmic terms:

log⁡p1\|tbase​\(X1\|xt\)p1\|tρ​\(X1\|xt\)≈p1\|tbase​\(X1\|xt\)p1\|tρ​\(X1\|xt\)−1,log⁡p1\|tρ−​\(X1\|xt\)p1\|tρ​\(X1\|xt\)≈p1\|tρ−​\(X1\|xt\)p1\|tρ​\(X1\|xt\)−1\.\\log\\frac\{p\_\{1\|t\}^\{\\mathrm\{base\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\approx\\frac\{p\_\{1\|t\}^\{\\mathrm\{base\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\-1,\\qquad\\log\\frac\{p\_\{1\|t\}^\{\\rho^\{\-\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\approx\\frac\{p\_\{1\|t\}^\{\\rho^\{\-\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\-1\.Substituting the linearizations into the covariance representation yields

v¯t​\(xt\)≈\\displaystyle\\bar\{v\}\_\{t\}\(x\_\{t\}\)\\approxvtρ\(xt\)\+ητCovX1∼p1\|tρ\(⋅\|xt\)\(vt\|1\(xt\|X1\),p1\|tbase​\(X1\|xt\)p1\|tρ​\(X1\|xt\)−p1\|tρ−​\(X1\|xt\)p1\|tρ​\(X1\|xt\)\)\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),\\frac\{p\_\{1\|t\}^\{\\mathrm\{base\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\-\\frac\{p\_\{1\|t\}^\{\\rho^\{\-\}\}\(X\_\{1\}\|x\_\{t\}\)\}\{p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)\}\\right\)=\\displaystyle=vtρ\(xt\)\+ητ\(𝔼X1∼p1\|tbase\(⋅\|xt\)\[vt\|1\(xt\|X1\)−vtρ\(xt\)\]−𝔼X1∼p1\|tρ−\(⋅\|xt\)\[vt\|1\(xt\|X1\)−vtρ\(xt\)\]\)\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}\\left\(\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\}\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\]\-\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho^\{\-\}\}\(\\cdot\|x\_\{t\}\)\}\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\]\\right\)=\\displaystyle=vtρ​\(xt\)\+ητ​\(vtbase​\(xt\)−vtρ−​\(xt\)\)\.\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}\\left\(v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho^\{\-\}\}\(x\_\{t\}\)\\right\)\.Accordingly, we approximate the tangential update through split linearization:

Γ^tρ,spl​\-​lin​\(xt\):=1τ​\(vtbase​\(xt\)−vtρ−​\(xt\)\),\\displaystyle\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{spl\}\\text\{\-\}\\mathrm\{lin\}\}\(x\_\{t\}\):=\\frac\{1\}\{\\tau\}\\left\(v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho^\{\-\}\}\(x\_\{t\}\)\\right\),\(114a\)v^tspl​\-​lin​\(xt\):=vtρ​\(xt\)\+η​Γ^tρ,spl​\-​lin​\(xt\)=vtρ​\(xt\)\+ητ​\(vtbase​\(xt\)−vtρ−​\(xt\)\)\.\\displaystyle\\hat\{v\}\_\{t\}^\{\\mathrm\{spl\}\\text\{\-\}\\mathrm\{lin\}\}\(x\_\{t\}\):=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{spl\}\\text\{\-\}\\mathrm\{lin\}\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}\\left\(v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho^\{\-\}\}\(x\_\{t\}\)\\right\)\.\(114b\)Unlike \([110](https://arxiv.org/html/2609.05727#S9.E110)\), this approximation is expressed as a correction to the current anchor rather than an interpolation withvπv^\{\\pi\}\. It involves the canonical velocity of the negative\-tilt density\.

The approximate tangential update in \([114](https://arxiv.org/html/2609.05727#S9.E114)\) admits the following stop\-gradient realization under the joint distribution from the current anchor:

ℒρ,η,Bspl​\-​lin\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[∥vtθ\(Xt\)−\[\(1−ητ\)vtρ\(Xt\)\+ητvtbase\(Xt\)\]\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{spl\}\\text\{\-\}\\mathrm\{lin\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\[\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\]−\(e−τ⁡\(r⁡\(X1\)−Bt​\(Xt\)\)−1\)\[vtρ\(Xt\)−ητ\(vt\|1\(Xt\|X1\)−vtbase\(Xt\)\)−sg\(vtθ\(Xt\)\)\]∥22\]\.\\displaystyle\\quad\-\\left\(e^\{\-\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\)\}\-1\\right\)\\Big\[v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\)\-\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\\Big\]\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(115\)Similar to the direct\-linearization case, we adopt the notation in \([49](https://arxiv.org/html/2609.05727#S8.E49)\)\.Bt​\(Xt\)B\_\{t\}\(X\_\{t\}\)is an arbitrary scalar baseline, with a useful choice

Bt\(Xt\)=𝔼X1∼p1\|tρ\(⋅\|Xt\)\[r\(X1\)\]\.B\_\{t\}\(X\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\[r\(X\_\{1\}\)\]\.
The computation of the approximate tangential update with split linearization is summarized in Algorithm[7](https://arxiv.org/html/2609.05727#alg7)\. Section[9\.4\.2](https://arxiv.org/html/2609.05727#S9.SS4.SSS2)presents a general formula for deriving the population\-stationary points of stop\-gradient objectives, and Appendix[F\.1](https://arxiv.org/html/2609.05727#A6.SS1)applies the formula to prove the following proposition\.

###### Proposition 38\.

Assume thatρ−∈𝒫\\rho^\{\-\}\\in\\mathscr\{P\}is normalizable,μ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}is a normalized reference density, and the loss \([115](https://arxiv.org/html/2609.05727#S9.E115)\) is integrable\. The unique population\-stationary point of the loss \([115](https://arxiv.org/html/2609.05727#S9.E115)\), up to a\.e\. equality, isv^spl​\-​lin\\hat\{v\}^\{\\mathrm\{spl\}\\text\{\-\}\\mathrm\{lin\}\}in \([114](https://arxiv.org/html/2609.05727#S9.E114)\)\. Furthermore, the split linearization is critical\-point consistent forπ\\pi, i\.e\.,Γ^ρ,spl​\-​lin=0\\widehat\{\\Gamma\}^\{\\rho,\\mathrm\{spl\}\\text\{\-\}\\mathrm\{lin\}\}=0if and only ifρ=π\\rho=\\pi\.

The two covariance approximations therefore use complementary exponential reweightings\. Direct linearization samples from the reference posterior and applies a positive reward tilt, whereas split linearization samples from the current posterior and applies a negative reward tilt\. Their updates may differ given the same anchorvρv^\{\\rho\}, but they are both critical\-point consistent\.

### 9\.2Approximate Gradient Forms with Critical\-Point Consistency

The exact gradient form represents the Newton tangential update as

v¯t\(xt\)=vtρ\(xt\)\+ηαt​κtβt2𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇r~ρ\(X1\)\]\.\\bar\{v\}\_\{t\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\]\.This form avoids direct evaluation of the scalar log\-density ratiolog⁡ρμ\\log\\frac\{\\rho\}\{\\mu\}, but it requires the action of an exact posterior Stein kernel on the regularized\-reward gradient\. Section[8](https://arxiv.org/html/2609.05727#S8)provides exact realizations through adjoint calculus of the posterior\-preserving SDE\. The approximations below reduce the computational cost in two different ways\. Section[9\.2\.1](https://arxiv.org/html/2609.05727#S9.SS2.SSS1)uses the adjoint ODE associated with the reference in place of that associated with the current anchor\. Section[9\.2\.2](https://arxiv.org/html/2609.05727#S9.SS2.SSS2)approximates the posterior Stein kernel by a positive\-definite matrix field, which corresponds to treating the anchor posterior as Gaussian\. These two approximations use samples from the current densityρ\\rho, which is essential to the critical\-point consistency\.

#### 9\.2\.1Reference\-Adjoint Approximation

The exact Bolza realization in Section[8\.4](https://arxiv.org/html/2609.05727#S8.SS4)uses the path distributionℙ\[t,1\]\|tρ\(⋅\|xt\)\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)of the posterior\-preserving SDE associated with the current anchor, together with the pathwise adjointλsbρ,r,lρ,μ\\lambda\_\{s\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\. Its adjoint ODE \([88a](https://arxiv.org/html/2609.05727#S8.E88.1)\) contains two terms: the first arises from the Jacobian of the drift in the posterior\-preserving SDE associated with the current anchor and therefore involves∇vsρ\\nabla v\_\{s\}^\{\\rho\}; the second involves the gradient∇lsρ,μ\\nabla l\_\{s\}^\{\\rho,\\mu\}of the running regularization objective\.

In the reference\-adjoint approximation, every occurrence of the current anchorvsρv\_\{s\}^\{\\rho\}in the adjoint ODE is replaced by the reference velocity fieldvsμv\_\{s\}^\{\\mu\}\. Recall that, whenμ=1\\mu=1,vs1​\(x\)=α˙sαs​xv\_\{s\}^\{1\}\(x\)=\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x, whereas, whenμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\},vsμ=vsbasev\_\{s\}^\{\\mu\}=v\_\{s\}^\{\\mathrm\{base\}\}is the pretrained velocity field\. The adjoint calculation is therefore frozen at the reference model:

λsbρ,r,lρ,μ≈λsbμ,r,lμ,μ=λsbμ,r,0,\\lambda\_\{s\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\\approx\\lambda\_\{s\}^\{b^\{\\mu\},r,l^\{\\mu,\\mu\}\}=\\lambda\_\{s\}^\{b^\{\\mu\},r,0\},where

bsμ​\(x\)=2​vsμ​\(x\)−α˙sαs​xb\_\{s\}^\{\\mu\}\(x\)=2v\_\{s\}^\{\\mu\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}xis the reference drift\. Under this approximation, the running\-objective term vanishes sincelsμ,μ≡0l\_\{s\}^\{\\mu,\\mu\}\\equiv 0\. Moreover, the adjoint ODE no longer depends on the current anchor and therefore remains the same across stages; only the path along which it is evaluated changes\.

For a prescribed path𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}, the reference adjoint is solved backward through

dd​sλsbμ,r,0\(𝒀\[t,1\]\)=−∇bsμ\(Ys\)⊤λsbμ,r,0\(𝒀\[t,1\]\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla b\_\{s\}^\{\\mu\}\(Y\_\{s\}\)^\{\\top\}\\lambda\_\{s\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\),\\qquad s\\in\[t,1\],\(116a\)λ1bμ,r,0​\(𝒀\[t,1\]\)=∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\.\(116b\)The adjoint ODE is defined along the prescribed path and does not determine how that path is sampled\. It is important to retain the path distributionℙ\[t,1\]\|tρ\(⋅\|xt\)\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)of the posterior\-preserving SDE associated with the current anchor and replace only the adjoint dynamics\.

Substituting the reference\-adjoint approximation into the exact Bolza realization \([90](https://arxiv.org/html/2609.05727#S8.E90)\) and \([92](https://arxiv.org/html/2609.05727#S8.E92)\), we obtain

Γtρ,r~ρ\(xt\)≈κt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[λtbμ,r,0\(𝒀\[t,1\]\)\]−1τ\(vtρ\(xt\)−vtμ\(xt\)\)\.\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)\\approx\\kappa\_\{t\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\-\\frac\{1\}\{\\tau\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mu\}\(x\_\{t\}\)\\right\)\.Accordingly, the reference\-adjoint approximation for the tangential update is given by:

Γ^tρ,ref​\-​adj\(xt\):=κt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[λtbμ,r,0\(𝒀\[t,1\]\)\]−1τ\(vtρ\(xt\)−vtμ\(xt\)\),\\displaystyle\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}\(x\_\{t\}\):=\\kappa\_\{t\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\-\\frac\{1\}\{\\tau\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mu\}\(x\_\{t\}\)\\right\),\(117a\)v^tref​\-​adj​\(xt\):=vtρ​\(xt\)\+η​Γ^tρ,ref​\-​adj​\(xt\)\\displaystyle\\hat\{v\}\_\{t\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}\(x\_\{t\}\):=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}\(x\_\{t\}\)v^\_t^ref\-adj\(x\_t\)=\(1−ητ\)vtρ\(xt\)\+ητvtμ\(xt\)\+ηκt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[λtbμ,r,0\(𝒀\[t,1\]\)\]\.\\displaystyle\\hphantom\{\\hat\{v\}\_t^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}\(x\_t\)\}\\,=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mu\}\(x\_\{t\}\)\+\\eta\\kappa\_\{t\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.\(117b\)The corresponding loss is

ℒρ,ηref​\-​adj\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ\[∥\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\rho\}\\end\{subarray\}\}\\bigg\[\\Big\\\|vtθ​\(Yt\)−\(1−ητ\)​vt\|1​\(Yt\|Y1\)\\displaystyle v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητvtμ\(Yt\)−ηκtλtbμ,r,0\(𝒀\[t,1\]\)∥22\],\\displaystyle\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\],\(118\)where the path distributionΞ\[t,1\]ρ\\Xi\_\{\[t,1\]\}^\{\\rho\}satisfies \([82](https://arxiv.org/html/2609.05727#S8.E82)\)\. The reference\-adjoint approximation deliberately pairs the path distribution associated with the current anchor with adjoint dynamics determined by the reference velocity field\. This reduces the computational cost of the adjoint calculation, but, when the current terminal density differs from the target, the resulting population minimizer generally differs from that of exact Newton Matching\. The following proposition shows that this discrepancy vanishes at the target and that the reference\-adjoint approximation is critical\-point consistent\.

Algorithm 8Tangential Update: Approximate Realization with Reference Adjoint\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v^k\+1≈vρk\+ηk​Γρk,r~ρk\\hat\{v\}^\{k\+1\}\\approx v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

2:foreach gradient stepdo

3:

⊳\\trianglerightThe following operations are applied to a batch\.

4:ifforward construction is usedthen

5:ifODE\-based sampling is usedthen

6:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vρkv^\{\\rho\_\{k\}\}\.

7:elseifSDE\-based sampling is usedthen

8:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

9:endif

10:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

11:if

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}then

12:Sample

𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|Xt,X1\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)based on Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\.

13:endif

14:elseifreverse construction is usedthen

15:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

16:Sample

𝒀\[t,1\]∼ℙ\[t,1\]\|tρk\(⋅\|Xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

17:Set

X1←Y1X\_\{1\}\\leftarrow Y\_\{1\}\.

18:endif

19:if

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}then

20:Compute the pathwise adjoint

λtbμ,r,0​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)by ODE \([116](https://arxiv.org/html/2609.05727#S9.E116)\)\.

21:Compute the target

𝒴\\mathcal\{Y\}by \([118](https://arxiv.org/html/2609.05727#S9.E118)\)\.

22:elseif

μ=1\\mu=1then

23:Compute the target

𝒴\\mathcal\{Y\}by \([119](https://arxiv.org/html/2609.05727#S9.E119)\)\.

24:endif

25:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

26:endfor

27:Set

v^k\+1←vθ\\hat\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

###### Proposition 39\.

Assume that the loss \([118](https://arxiv.org/html/2609.05727#S9.E118)\) is integrable\. The unique population minimizer of the loss \([118](https://arxiv.org/html/2609.05727#S9.E118)\), up to a\.e\. equality, isv^ref​\-​adj\\hat\{v\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}in \([117](https://arxiv.org/html/2609.05727#S9.E117)\)\. Furthermore, the reference\-adjoint approximation is critical\-point consistent forπ\\pi, i\.e\.,Γ^ρ,ref​\-​adj=0\\widehat\{\\Gamma\}^\{\\rho,\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}=0if and only ifρ=π\\rho=\\pi\.

The reference\-adjoint approximation applies to bothμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}andμ=1\\mu=1\. The caseμ=1\\mu=1admits a further closed\-form simplification\. Since

vs1​\(x\)=α˙sαs​x,bs1​\(x\)=α˙sαs​x,v\_\{s\}^\{1\}\(x\)=\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x,\\qquad b\_\{s\}^\{1\}\(x\)=\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x,the reference\-adjoint equation \([116a](https://arxiv.org/html/2609.05727#S9.E116.1)\) reduces to

dd​sλsb1,r,0\(𝒀\[t,1\]\)=−α˙sαsλsb1,r,0\(𝒀\[t,1\]\)⇔dd​s\(αsλsb1,r,0\(𝒀\[t,1\]\)\)≡0\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{b^\{1\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}\\lambda\_\{s\}^\{b^\{1\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\quad\\Leftrightarrow\\quad\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\left\(\\alpha\_\{s\}\\lambda\_\{s\}^\{b^\{1\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)\\equiv 0\.Usingα1=1\\alpha\_\{1\}=1and the terminal conditionλ1b1,r,0​\(𝒀\[t,1\]\)=∇r​\(Y1\)\\lambda\_\{1\}^\{b^\{1\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\), the pathwise adjoint therefore admits the closed\-form expression

λtb1,r,0\(𝒀\[t,1\]\)=1αt∇r\(Y1\)\.\\lambda\_\{t\}^\{b^\{1\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\frac\{1\}\{\\alpha\_\{t\}\}\\nabla r\(Y\_\{1\}\)\.Consequently, no backward adjoint integration is required in this case\. Substituting this expression into \([118](https://arxiv.org/html/2609.05727#S9.E118)\) yields

ℒρ,ηref​\-​adj\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[‖vtθ\(Xt\)−\(1−ητ\)vt\|1\(Xt\|X1\)−η​α˙tτ​αtXt−η​κtαt∇r\(X1\)‖22\]\.\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\\\ \(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\frac\{\\eta\\dot\{\\alpha\}\_\{t\}\}\{\\tau\\alpha\_\{t\}\}X\_\{t\}\-\\frac\{\\eta\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\nabla r\(X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.\(119\)
The reference\-adjoint approximation is summarized in Algorithm[8](https://arxiv.org/html/2609.05727#alg8)\.

#### 9\.2\.2Gaussian\-Kernel Approximation

The Mayer and Bolza realizations of the gradient form in Section[8\.4](https://arxiv.org/html/2609.05727#S8.SS4)arise from the posterior sensitivity kernel, a particular exact posterior Stein kernelΛtρ​\(x1\|xt\)\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)constructed from the initial\-state sensitivity of the posterior\-preserving SDE\. The reference\-adjoint approximation modifies the adjoint dynamics used to evaluate this particular kernel\. A complementary approach is to work directly with the general posterior Stein kernel and approximate it without relying on the sensitivity\-based construction\.

For each fixed\(t,xt\)\(t,x\_\{t\}\), an exact posterior Stein kernel must satisfy condition \([75](https://arxiv.org/html/2609.05727#S8.E75)\)\. Solving this equation is generally intractable for a generic, potentially complex posterior densityp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\. A convenient simplification is to approximate this posterior by a Gaussian distribution:

p^1\|tρ\(⋅\|xt\):=𝒩\(M^tρ\(xt\),Σ^tρ\(xt\)\)\.\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\):=\\mathcal\{N\}\\left\(\\widehat\{M\}\_\{t\}^\{\\rho\}\(x\_\{t\}\),\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.\(120\)For this Gaussian surrogate, replacingp1\|tρp\_\{1\|t\}^\{\\rho\}withp^1\|tρ\\hat\{p\}\_\{1\|t\}^\{\\rho\}in \([75](https://arxiv.org/html/2609.05727#S8.E75)\) yields an equation with an explicit solutionΣ^tρ​\(xt\)\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\. This motivates the direct approximation

Λtρ​\(x1\|xt\)≈Σ^tρ​\(xt\),\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\approx\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\),whereΣ^tρ​\(xt\)∈𝕊\+⁣\+d\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}is the covariance matrix of the Gaussian surrogate\. Properties related to Gaussian Stein kernels and posterior moments are collected in Appendix[E\.5](https://arxiv.org/html/2609.05727#A5.SS5)\.

The matrix fieldΣ^tρ​\(xt\)\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)may be learned by a separate model or prescribed analytically\. Since it does not depend onx1x\_\{1\}, it can be pulled out of the conditional expectation overX1X\_\{1\}:

𝔼X1∼p1\|tρ\(⋅\|xt\)\[Σ^tρ\(xt\)∇r~ρ\(X1\)\]=Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r~ρ\(X1\)\]\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\]=\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\]\.Therefore, the tangential update with the Gaussian\-kernel approximation is given by:

Γ^tρ,Gau​\-​ker\(xt\):=αt​κtβt2Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r~ρ\(X1\)\],\\displaystyle\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(x\_\{t\}\):=\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\],\(121a\)v^tGau​\-​ker​\(xt\):=vtρ​\(xt\)\+η​Γ^tρ,Gau​\-​ker​\(xt\)\.\\displaystyle\\hat\{v\}\_\{t\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(x\_\{t\}\):=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(x\_\{t\}\)\.\(121b\)
Expanding the regularized\-reward gradient in \([121b](https://arxiv.org/html/2609.05727#S9.E121.2)\) gives

v^tGau​\-​ker​\(xt\)=\\displaystyle\\hat\{v\}\_\{t\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(x\_\{t\}\)=vtρ\(xt\)\+ηαt​κtβt2Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r\(X1\)−1τ∇logρ⁡\(X1\)μ⁡\(X1\)\]\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla r\(X\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\nabla\\log\\frac\{\\rho\(X\_\{1\}\)\}\{\\mu\(X\_\{1\}\)\}\\right\]=\\displaystyle=vtρ\(xt\)\+ηαt​κtβt2Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r\(X1\)\+1τ∇logμ\(X1\)−1τ∇logρ\(X1\)\]\.\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\nabla\\log\\rho\(X\_\{1\}\)\\right\]\.The endpoint score∇log⁡ρ​\(X1\)\\nabla\\log\\rho\(X\_\{1\}\)need not be evaluated\. The posterior score has zero conditional mean, which yields the endpoint\-score identity

𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇logρ\(X1\)\+αtβt2\(xt−αtX1\)\]=0\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla\\log\\rho\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\]=0\.Then the approximate update can be rewritten as

v^tGau​\-​ker​\(xt\)\\displaystyle\\hat\{v\}\_\{t\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(x\_\{t\}\)=\\displaystyle=vtρ\(xt\)\+ηαt​κtβt2Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(xt−αtX1\)\]\.\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\]\.Consequently, \([121b](https://arxiv.org/html/2609.05727#S9.E121.2)\) is realized by the regression loss

ℒρ,η,Σ^Gau​\-​ker​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[∥vtθ\(Xt\)−vt\|1\(Xt\|X1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−ηαt​κtβt2Σ^tρ\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\-\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.\(122\)The following proposition shows that the Gaussian\-kernel approximation is critical\-point consistent\. Its procedure is summarized in Algorithm[9](https://arxiv.org/html/2609.05727#alg9)\.

Algorithm 9Tangential Update: Approximate Realization with Gaussian Kernel\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; a matrix field

Σ^ρ:\(0,1\)×ℝd→𝕊\+⁣\+d\\widehat\{\\Sigma\}^\{\\rho\}:\(0,1\)\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{S\}\_\{\+\+\}^\{d\}; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v^k\+1≈vρk\+ηk​Γρk,r~ρk\\hat\{v\}^\{k\+1\}\\approx v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

2:foreach gradient stepdo

3:

⊳\\trianglerightThe following operations are applied to a batch\.

4:ifforward construction is usedthen

5:ifODE\-based sampling is usedthen

6:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vρkv^\{\\rho\_\{k\}\}\.

7:elseifSDE\-based sampling is usedthen

8:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

9:endif

10:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

11:elseifreverse construction is usedthen

12:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

13:Sample

X1∼p1\|tρk\(⋅\|Xt\)X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

14:endif

15:Compute the target

𝒴\\mathcal\{Y\}by \([122](https://arxiv.org/html/2609.05727#S9.E122)\)\.

16:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

17:endfor

18:Set

v^k\+1←vθ\\hat\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

###### Proposition 41\.

Assume thatΣ^tρ​\(xt\)\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)is positive definite for every interior\(t,xt\)\(t,x\_\{t\}\)and the loss \([122](https://arxiv.org/html/2609.05727#S9.E122)\) is integrable\. The unique population minimizer of the loss \([122](https://arxiv.org/html/2609.05727#S9.E122)\), up to a\.e\. equality, isv^Gau​\-​ker\\hat\{v\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}in \([121b](https://arxiv.org/html/2609.05727#S9.E121.2)\)\. Furthermore, the Gaussian\-kernel approximation is critical\-point consistent forπ\\pi, i\.e\.,Γ^ρ,Gau​\-​ker=0\\widehat\{\\Gamma\}^\{\\rho,\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}=0if and only ifρ=π\\rho=\\pi\.

The posterior covariance provides a natural choice for the surrogateΣ^tρ​\(xt\)\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\. For the Gaussian source and linear interpolant, we have

VarX1∼p1\|tρ\(⋅\|xt\)\[X1\]=βt2αt2I\+βt4αt2∇xt2logptρ\(xt\)\.\\operatorname\{Var\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[X\_\{1\}\]=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\+\\frac\{\\beta\_\{t\}^\{4\}\}\{\\alpha\_\{t\}^\{2\}\}\\nabla\_\{x\_\{t\}\}^\{2\}\\log p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\.This expression motivates the efficient choice

Σ^tρ​\(xt\)=βt2αt2​I\.\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\.In this case, the resulting update simplifies to

v^tGau​\-​ker​\(xt\)\|Σ^tρ≡βt2αt2​I\\displaystyle\\left\.\\hat\{v\}\_\{t\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(x\_\{t\}\)\\right\|\_\{\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\\equiv\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\}=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|x1\)\+η​κtαt\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(xt−αtX1\)\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\+\\frac\{\\eta\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\right\]=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[β˙tβtxt\+αt​κtβt2X1\+η​κtαt\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(xt−αtX1\)\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\+\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}X\_\{1\}\+\\frac\{\\eta\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\right\]=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(β˙tβt\+η​κtτ​βt2\)xt\+\(1−ητ\)αt​κtβt2X1\+η​κtαt\(∇r\(X1\)\+1τ∇logμ\(X1\)\)\]\.\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\left\(\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}\+\\frac\{\\eta\\kappa\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\right\)x\_\{t\}\+\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}X\_\{1\}\+\\frac\{\\eta\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\\right\)\\right\]\.In particular, whenμ=1\\mu=1,

v^tGau​\-​ker\(xt\)\|μ=1,Σ^tρ≡βt2αt2​I=𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(β˙tβt\+η​κtτ​βt2\)xt\+\(1−ητ\)αt​κtβt2X1\+η​κtαt∇r\(X1\)\]\.\\left\.\\hat\{v\}\_\{t\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(x\_\{t\}\)\\right\|\_\{\\mu=1,\\,\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\\equiv\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\}=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\left\(\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}\+\\frac\{\\eta\\kappa\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\right\)x\_\{t\}\+\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}X\_\{1\}\+\\frac\{\\eta\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\nabla r\(X\_\{1\}\)\\right\]\.The corresponding loss is

ℒρ,η,Σ^Gau​\-​ker​\(θ\)\|μ=1,Σ^tρ≡βt2αt2​I\\displaystyle\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\}\}\(\\theta\)\\right\|\_\{\\mu=1,\\,\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\\equiv\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\}=\\displaystyle=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[‖vtθ\(Xt\)−\(β˙tβt\+η​κtτ​βt2\)Xt−\(1−ητ\)αt​κtβt2X1−η​κtαt∇r\(X1\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\(\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}\+\\frac\{\\eta\\kappa\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\right\)X\_\{t\}\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}X\_\{1\}\-\\frac\{\\eta\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\nabla r\(X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Usingκt=βt2​\(α˙tαt−β˙tβt\)\\kappa\_\{t\}=\\beta\_\{t\}^\{2\}\\left\(\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}\-\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}\\right\), the loss above is sample\-wise identical to the reference\-adjoint approximation loss \([119](https://arxiv.org/html/2609.05727#S9.E119)\)\. Therefore, whenμ=1\\mu=1, the reference\-adjoint approximation is a special case of the Gaussian\-kernel approximation\.

### 9\.3Regularization Trade\-Offs

In the previous two subsections, we provide several approximate tangential updates that preserve critical\-point consistency\. We now take a different perspective by focusing directly on the regularization itself\. The termlog⁡ρμ\\log\\frac\{\\rho\}\{\\mu\}is the key component that makes the iteration follow the Newton direction for the reverse\-KL objective\. It is also a major source of computational difficulty in exact Newton Matching, requiring nontrivial estimation procedures\.

In this subsection, we directly analyze the influence of regularization on the update rules and consider approximating or removing it entirely\. Section[9\.3\.1](https://arxiv.org/html/2609.05727#S9.SS3.SSS1)first decomposes the exact regularization into a marginal velocity correction and a posterior\-KL gradient\. The former is easy to compute exactly, whereas the latter admits various approximation options\. Section[9\.3\.2](https://arxiv.org/html/2609.05727#S9.SS3.SSS2)removes the regularization entirely and develops the corresponding algorithms\. The resulting update performs direct raw\-reward ascent\.

#### 9\.3\.1Approximate Regularization

Throughout the analysis below, assume thatμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}is a normalized reference density\. When the original reference isμ=1\\mu=1, we apply the auxiliary\-reference factorization \([108](https://arxiv.org/html/2609.05727#S9.E108)\)\. The canonical tangent vectorΓρ,r~ρ\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}admits the following decomposition\.

###### Proposition 42\.

Assume thatμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}\. For every canonical anchorvρv^\{\\rho\}and stepsizeη\>0\\eta\>0, we have

v¯t​\(xt\)=\\displaystyle\\bar\{v\}\_\{t\}\(x\_\{t\}\)=vtρ​\(xt\)\+η​Γtρ,r~ρ​\(xt\)\\displaystyle v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=\\displaystyle=\(1−ητ\)vtρ\(xt\)\+ητvtbase\(xt\)\+ηΓtρ,r\(xt\)−ητκt∇xtKL\(p1\|tρ\(⋅\|xt\)∥p1\|tbase\(⋅\|xt\)\)\.\\displaystyle\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,r\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}\\kappa\_\{t\}\\nabla\_\{x\_\{t\}\}\\operatorname\{KL\}\\left\(p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\middle\\\|p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\right\)\.

The decomposition consists of three conceptually distinct contributions\. The first, represented by the first two terms, interpolates between the current and reference canonical velocities\. The second is the raw\-reward tangent\. The third is the posterior\-KL gradient, which can be computationally challenging since it depends on how the current and reference posteriors vary withxtx\_\{t\}\.

We may learn this gradient in a separate model or replace it with a prescribed approximation:

∇xtKL\(p1\|tρ\(⋅\|xt\)∥p1\|tbase\(⋅\|xt\)\)≈K^t\(xt\)\.\\nabla\_\{x\_\{t\}\}\\operatorname\{KL\}\\left\(p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\middle\\\|p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\right\)\\approx\\widehat\{K\}\_\{t\}\(x\_\{t\}\)\.This gives the approximate tangential update

Γ^tρ,appr​\-​reg​\(xt\):=−1τ​\(vtρ​\(xt\)−vtbase​\(xt\)\)\+Γtρ,r​\(xt\)−1τ​κt​K^t​\(xt\),\\displaystyle\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\):=\-\\frac\{1\}\{\\tau\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)\+\\Gamma\_\{t\}^\{\\rho,r\}\(x\_\{t\}\)\-\\frac\{1\}\{\\tau\}\\kappa\_\{t\}\\widehat\{K\}\_\{t\}\(x\_\{t\}\),\(123a\)v^tappr​\-​reg​\(xt\):=vtρ​\(xt\)\+η​Γ^tρ,appr​\-​reg​\(xt\)\\displaystyle\\hat\{v\}\_\{t\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\):=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\)=\(1−ητ\)​vtρ​\(xt\)\+ητ​vtbase​\(xt\)\+η​Γtρ,r​\(xt\)−ητ​κt​K^t​\(xt\)\.\\displaystyle\\phantom\{\\hat\{v\}\_\{t\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\)\}\\,\\,=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,r\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}\\kappa\_\{t\}\\widehat\{K\}\_\{t\}\(x\_\{t\}\)\.\(123b\)The covariance\-form loss is given by

ℒρ,η,Bappr​\-​reg​\-​cov\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[∥vtθ\(Xt\)−\(1−ητ\)vt\|1\(Xt\|X1\)\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−ητvtbase\(Xt\)−η\(r\(X1\)−Bt\(Xt\)\)\(vt\|1\(Xt\|X1\)−vtρ\(Xt\)\)\+ητκtK^t\(Xt\)∥22\],\\displaystyle\\qquad\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\-\\eta\\left\(r\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\)\+\\frac\{\\eta\}\{\\tau\}\\kappa\_\{t\}\\widehat\{K\}\_\{t\}\(X\_\{t\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\],\(124\)whereBt​\(Xt\)B\_\{t\}\(X\_\{t\}\)is an arbitrary scalar baseline\. The same approximate update can be realized in gradient form by using the raw\-reward adjoint associated with the current anchor:

ℒρ,ηappr​\-​reg​\-​grad​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{grad\}\}\(\\theta\):=𝔼t∼U\(0,1\),Xt∼p^t,𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|Xt\)\[∥vtθ\(Xt\)−\(1−ητ\)vt\|1\(Xt\|Y1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(X\_\{t\}\|Y\_\{1\}\)−ητvtbase\(Xt\)−ηκtλtbρ,r,0\(𝒀\[t,1\]\)\+ητκtK^t\(Xt\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\-\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\rho\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+\\frac\{\\eta\}\{\\tau\}\\kappa\_\{t\}\\widehat\{K\}\_\{t\}\(X\_\{t\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(125\)The pathwise adjoint is computed through

dd​sλsbρ,r,0\(𝒀\[t,1\]\)=−∇bsρ\(Ys\)⊤λsbρ,r,0\(𝒀\[t,1\]\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{b^\{\\rho\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla b\_\{s\}^\{\\rho\}\(Y\_\{s\}\)^\{\\top\}\\lambda\_\{s\}^\{b^\{\\rho\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\),\\qquad s\\in\[t,1\],\(126a\)λ1bρ,r,0​\(𝒀\[t,1\]\)=∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b^\{\\rho\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\.\(126b\)For a generic approximationK^t\\widehat\{K\}\_\{t\}, there is no guarantee that the correctionΓ^ρ,appr​\-​reg\\widehat\{\\Gamma\}^\{\\rho,\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}vanishes exactly whenρ=π\\rho=\\pi\. Consequently, the resulting construction need not be critical\-point consistent\. The tangential update with approximate regularization is summarized in Algorithm[10](https://arxiv.org/html/2609.05727#alg10)\.

The most economical choice is to drop the posterior\-KL gradient entirely:

∇xtKL\(p1\|tρ\(⋅\|xt\)∥p1\|tbase\(⋅\|xt\)\)≈0\.\\nabla\_\{x\_\{t\}\}\\operatorname\{KL\}\\left\(p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\middle\\\|p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\right\)\\approx 0\.The corresponding update is

v^tappr​\-​reg​\(xt\)=\(1−ητ\)​vtρ​\(xt\)\+ητ​vtbase​\(xt\)\+η​Γtρ,r​\(xt\)\.\\hat\{v\}\_\{t\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\)=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,r\}\(x\_\{t\}\)\.The covariance\-form and gradient\-form losses are obtained by substitutingK^t=0\\widehat\{K\}\_\{t\}=0into \([124](https://arxiv.org/html/2609.05727#S9.E124)\) and \([125](https://arxiv.org/html/2609.05727#S9.E125)\)\.

Algorithm 10Tangential Update: Approximate Regularization\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk∈\(0,τ\]\\eta\_\{k\}\\in\(0,\\tau\]; a reference factor

μ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}for fine\-tuning \(with the pretrained model

vbasev^\{\\mathrm\{base\}\}\) or

μ=1\\mu=1for sampling; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; an inverse temperature

τ\>0\\tau\>0; a vector field

K^:\(0,1\)×ℝd→ℝd\\widehat\{K\}:\(0,1\)\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v^k\+1≈vρk\+ηk​Γρk,r~ρk\\hat\{v\}^\{k\+1\}\\approx v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\.

1:if

μ=1\\mu=1then

2:Set

ρbase←𝒩⁡\(m,σ2​I\)\\rho^\{\\mathrm\{base\}\}\\leftarrow\\mathcal\{N\}\(m,\\sigma^\{2\}I\)and its analytical velocity field

vbasev^\{\\mathrm\{base\}\}based on the auxiliary\-reference factorization \([108](https://arxiv.org/html/2609.05727#S9.E108)\)\.

3:Set

r←r−1τ​log⁡ρbaser\\leftarrow r\-\\frac\{1\}\{\\tau\}\\log\\rho^\{\\mathrm\{base\}\}\.

4:endif

5:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

6:foreach gradient stepdo

7:

⊳\\trianglerightThe following operations are applied to a batch\.

8:ifforward construction is usedthen

9:ifODE\-based sampling is usedthen

10:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vρkv^\{\\rho\_\{k\}\}\.

11:elseifSDE\-based sampling is usedthen

12:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

13:endif

14:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

15:ifgradient form is usedthen

16:Sample

𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|Xt,X1\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)based on Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\.

17:endif

18:elseifreverse construction is usedthen

19:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

20:Sample

𝒀\[t,1\]∼ℙ\[t,1\]\|tρk\(⋅\|Xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

21:Set

X1←Y1X\_\{1\}\\leftarrow Y\_\{1\}\.

22:endif

23:ifcovariance form is usedthen

24:Compute the target

𝒴\\mathcal\{Y\}by \([124](https://arxiv.org/html/2609.05727#S9.E124)\)\.

25:elseifgradient form is usedthen

26:Compute the pathwise adjoint

λtbρk,r,0​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\rho\_\{k\}\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)by ODE \([126](https://arxiv.org/html/2609.05727#S9.E126)\)\.

27:Compute the target

𝒴\\mathcal\{Y\}by \([125](https://arxiv.org/html/2609.05727#S9.E125)\)\.

28:endif

29:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

30:endfor

31:Set

v^k\+1←vθ\\hat\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

A less aggressive alternative approximates both posteriors by Gaussian distributions with a shared isotropic covariance:

p1\|tρ\(⋅\|xt\)≈𝒩\(Mtρ\(xt\);βt2αt2I\),p1\|tbase\(⋅\|xt\)≈𝒩\(Mtbase\(xt\);βt2αt2I\),p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\approx\\mathcal\{N\}\\left\(M\_\{t\}^\{\\rho\}\(x\_\{t\}\);\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\\right\),\\qquad p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\approx\\mathcal\{N\}\\left\(M\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\);\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\\right\),where the posterior means are exact:

Mtρ\(xt\):=𝔼X1∼p1\|tρ\(⋅\|xt\)\[X1\]=βt2αt​κt\(vtρ\(xt\)−β˙tβtxt\),\\displaystyle M\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[X\_\{1\}\]=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\\right\),Mtbase\(xt\):=𝔼X1∼p1\|tbase\(⋅\|xt\)\[X1\]=βt2αt​κt\(vtbase\(xt\)−β˙tβtxt\)\.\\displaystyle M\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\}\[X\_\{1\}\]=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\\right\)\.For two Gaussians with the same covariance, we have

∇xtKL\(p1\|tρ\(⋅\|xt\)∥p1\|tbase\(⋅\|xt\)\)≈\\displaystyle\\nabla\_\{x\_\{t\}\}\\operatorname\{KL\}\\left\(p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\middle\\\|p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\right\)\\approx∇xt​KL​\(𝒩⁡\(Mtρ​\(xt\),βt2αt2​I\)∥𝒩⁡\(Mtbase​\(xt\),βt2αt2​I\)\)\\displaystyle\\nabla\_\{x\_\{t\}\}\\operatorname\{KL\}\\left\(\\mathcal\{N\}\\left\(M\_\{t\}^\{\\rho\}\(x\_\{t\}\);\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\\right\)\\middle\\\|\\mathcal\{N\}\\left\(M\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\);\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\\right\)\\right\)=\\displaystyle=αt22​βt2​∇xt‖Mtρ​\(xt\)−Mtbase​\(xt\)‖22\\displaystyle\\frac\{\\alpha\_\{t\}^\{2\}\}\{2\\beta\_\{t\}^\{2\}\}\\nabla\_\{x\_\{t\}\}\\left\\\|M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-M\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=βt22​κt2​∇xt‖vtρ​\(xt\)−vtbase​\(xt\)‖22\.\\displaystyle\\frac\{\\beta\_\{t\}^\{2\}\}\{2\\kappa\_\{t\}^\{2\}\}\\nabla\_\{x\_\{t\}\}\\left\\\|v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\.The Gaussian KL calculation is given in Lemma[88](https://arxiv.org/html/2609.05727#Thmtheorem88)of Appendix[C\.1](https://arxiv.org/html/2609.05727#A3.SS1)\. The corresponding approximate tangential update is

v^tappr​\-​reg​\(xt\)=\(1−ητ\)​vtρ​\(xt\)\+ητ​vtbase​\(xt\)\+η​Γtρ,r​\(xt\)−η​βt22​τ​κt​∇xt‖vtρ​\(xt\)−vtbase​\(xt\)‖22\.\\hat\{v\}\_\{t\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\)=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,r\}\(x\_\{t\}\)\-\\frac\{\\eta\\beta\_\{t\}^\{2\}\}\{2\\tau\\kappa\_\{t\}\}\\nabla\_\{x\_\{t\}\}\\left\\\|v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\.The corresponding losses are obtained by setting

K^t​\(xt\)=βt22​κt2​∇xt‖vtρ​\(xt\)−vtbase​\(xt\)‖22\\widehat\{K\}\_\{t\}\(x\_\{t\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{2\\kappa\_\{t\}^\{2\}\}\\nabla\_\{x\_\{t\}\}\\left\\\|v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\\\|\_\{2\}^\{2\}in \([124](https://arxiv.org/html/2609.05727#S9.E124)\) and \([125](https://arxiv.org/html/2609.05727#S9.E125)\)\.

#### 9\.3\.2No Regularization

The most aggressive approximation is to remove the regularization entirely:

r~ρ≈r\.\\tilde\{r\}^\{\\rho\}\\approx r\.The resulting tangential update is

v^tno​\-​reg​\(xt\)=vtρ​\(xt\)\+η​Γtρ,r​\(xt\)\.\\hat\{v\}\_\{t\}^\{\\mathrm\{no\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,r\}\(x\_\{t\}\)\.\(127\)At this level of approximation, the update direction is governed by the raw rewardrrand therefore admits a different optimization interpretation, namely, direct ascent of the expected reward\.

The corresponding covariance\-form loss is

ℒρ,η,Bno​\-​reg​\-​cov​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{no\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[∥vtθ\(Xt\)−vt\|1\(Xt\|X1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\Big\[\\big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−η\(r\(X1\)−Bt\(Xt\)\)\(vt\|1\(Xt\|X1\)−vtρ\(Xt\)\)∥22\],\\displaystyle\\qquad\-\\eta\\left\(r\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\)\\big\\\|\_\{2\}^\{2\}\\Big\],\(128\)whereBt​\(Xt\)B\_\{t\}\(X\_\{t\}\)is an arbitrary scalar baseline\. We can also realize the unregularized tangential update through the following gradient\-form loss:

ℒρ,ηno​\-​reg​\-​grad​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta\}^\{\\mathrm\{no\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{grad\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ​\[‖vtθ​\(Yt\)−vt\|1​\(Yt\|Y1\)−η​κt​λtbρ,r,0​\(𝒀\[t,1\]\)‖22\],\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\rho\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)\-\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\rho\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\],\(129\)where the pathwise adjointλtbρ,r,0​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\rho\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)can be computed through the ODE \([126](https://arxiv.org/html/2609.05727#S9.E126)\)\.

Algorithm 11Tangential Update: No Regularization \(Reward Ascent\)\.0:At stage

kk, the current canonical model

vρk∈𝒱canv^\{\\rho\_\{k\}\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}; a stepsize

ηk\>0\\eta\_\{k\}\>0; a reward function

r:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}; and a batch size

NbatchN\_\{\\mathrm\{batch\}\}\.

0:The updated velocity model

v^k\+1=vρk\+ηk​Γρk,r\\hat\{v\}^\{k\+1\}=v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},r\}\.

1:Warm\-start

θ\\thetawith the network parameters of

vρkv^\{\\rho\_\{k\}\}\.

2:foreach gradient stepdo

3:

⊳\\trianglerightThe following operations are applied to a batch\.

4:ifforward construction is usedthen

5:ifODE\-based sampling is usedthen

6:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through ODE \([1](https://arxiv.org/html/2609.05727#S2.E1)\) of

vρkv^\{\\rho\_\{k\}\}\.

7:elseifSDE\-based sampling is usedthen

8:Sample

X1∼ρkX\_\{1\}\\sim\\rho\_\{k\}through SDE \([5](https://arxiv.org/html/2609.05727#S2.E5)\) of

vρkv^\{\\rho\_\{k\}\}\.

9:endif

10:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

X0∼𝒩⁡\(0,I\)X\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and set

Xt←αt​X1\+βt​X0X\_\{t\}\\leftarrow\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\.

11:ifgradient form is usedthen

12:Sample

𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|Xt,X1\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|X\_\{t\},X\_\{1\}\)based on Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\.

13:endif

14:elseifreverse construction is usedthen

15:Sample

t∼U⁡\(0,1\)t\\sim\\operatorname\{U\}\(0,1\)and

Xt∼p^tX\_\{t\}\\sim\\hat\{p\}\_\{t\}\.

16:Sample

𝒀\[t,1\]∼ℙ\[t,1\]\|tρk\(⋅\|Xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\_\{k\}\}\(\\cdot\|X\_\{t\}\)through the posterior\-preserving SDE \([53](https://arxiv.org/html/2609.05727#S8.E53)\) of

ρk\\rho\_\{k\}\.

17:Set

X1←Y1X\_\{1\}\\leftarrow Y\_\{1\}\.

18:endif

19:ifcovariance form is usedthen

20:Compute the target

𝒴\\mathcal\{Y\}by \([128](https://arxiv.org/html/2609.05727#S9.E128)\)\.

21:elseifgradient form is usedthen

22:Compute the pathwise adjoint

λtbρk,r,0​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\rho\_\{k\}\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)by ODE \([126](https://arxiv.org/html/2609.05727#S9.E126)\)\.

23:Compute the target

𝒴\\mathcal\{Y\}by \([129](https://arxiv.org/html/2609.05727#S9.E129)\)\.

24:endif

25:Update

θ\\thetabased on the batch loss

ℒ←1Nbatch​∑\(t,Xt,𝒴\)‖vtθ​\(Xt\)−𝒴‖22\\mathcal\{L\}\\leftarrow\\frac\{1\}\{N\_\{\\mathrm\{batch\}\}\}\\sum\_\{\(t,X\_\{t\},\\mathcal\{Y\}\)\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\right\\\|\_\{2\}^\{2\}\.

26:endfor

27:Set

v^k\+1←vθ\\hat\{v\}^\{k\+1\}\\leftarrow v^\{\\theta\}\.

The update \([127](https://arxiv.org/html/2609.05727#S9.E127)\) is exactly the canonical retraction of Section[5\.1](https://arxiv.org/html/2609.05727#S5.SS1)with terminal observablef=rf=r\. The computation process is summarized in Algorithm[11](https://arxiv.org/html/2609.05727#alg11)\. The value\-ascent theorem therefore yields the following finite\-stepsize reward\-ascent guarantee\.

###### Proposition 43\.

Consider the terminal density induced by \([127](https://arxiv.org/html/2609.05727#S9.E127)\), i\.e\.,

q=𝒯⁡\(vρ\+η​Γρ,r\)\.q=\\mathcal\{T\}\\left\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,r\}\\right\)\.We have the reward\-ascent certificate

𝔼X∼q​\[r⁡\(X\)\]≥𝔼X∼ρ​\[r⁡\(X\)\]\.\\mathbb\{E\}\_\{X\\sim q\}\\left\[r\(X\)\\right\]\\geq\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[r\(X\)\\right\]\.Furthermore, whenever

η\>0,r≢const,\\eta\>0,\\qquad r\\not\\equiv\\mathrm\{const\},the reward increases strictly:

𝔼X∼q​\[r⁡\(X\)\]\>𝔼X∼ρ​\[r⁡\(X\)\]\.\\mathbb\{E\}\_\{X\\sim q\}\\left\[r\(X\)\\right\]\>\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[r\(X\)\\right\]\.Consequently, if the reward is nonconstant, then the correctionΓρ,r\\Gamma^\{\\rho,r\}is not critical\-point consistent forπ\\pi\.

Naturally, repeated unregularized updates with finite stepsizes continue to increase the expected reward and therefore favor the maximal\-reward region rather than a finite\-temperature target\. The following result makes this concentration behavior precise\.

###### Proposition 44\.

Letηk\>0\\eta\_\{k\}\>0be the stepsize\. Define the unregularized iteration

ρk\+1=𝒯⁡\(vρk\+ηk​Γρk,r\)\.\\rho\_\{k\+1\}=\\mathcal\{T\}\\left\(v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},r\}\\right\)\.Assume that the stepsizes are non\-summable:

∑k=0∞ηk=∞,\\sum\_\{k=0\}^\{\\infty\}\\eta\_\{k\}=\\infty,that the reward is continuous with a finite upper bound:

rmax:=supx∈ℝdr⁡\(x\)<∞,r\_\{\\max\}:=\\sup\_\{x\\in\\mathbb\{R\}^\{d\}\}r\(x\)<\\infty,and that the dissipation vanishes:

limk→∞1ηk​𝔼X∼ρ0​\[𝒟ρk,ηk,r​\(X\)\]=0\.\\lim\_\{k\\to\\infty\}\\frac\{1\}\{\\eta\_\{k\}\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\}\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},r\}\(X\)\\right\]=0\.\(130\)Then, for anyε\>0\\varepsilon\>0, we have

limk→∞ρk​\(\{x∈ℝd:r⁡\(x\)≥rmax−ε\}\)=1\.\\lim\_\{k\\to\\infty\}\\rho\_\{k\}\\left\(\\left\\\{x\\in\\mathbb\{R\}^\{d\}:r\(x\)\\geq r\_\{\\max\}\-\\varepsilon\\right\\\}\\right\)=1\.

Proposition[44](https://arxiv.org/html/2609.05727#Thmtheorem44)characterizes the limit behavior of the unregularized updates\. The following proposition provides a complementary result\. It quantifies the terminal\-density error of a finite sequence of unregularized updates relative toπ\\piin both directions of KL divergence\.

###### Proposition 45\.

Assume thatμ∈𝒫\\mu\\in\\mathscr\{P\}and the rewardrris bounded:

‖r‖∞:=supx∈ℝd\|r⁡\(x\)\|<∞\.\\left\\\|r\\right\\\|\_\{\\infty\}:=\\sup\_\{x\\in\\mathbb\{R\}^\{d\}\}\\left\|r\(x\)\\right\|<\\infty\.Consider a sequence of stepsizes\{ηk\}k=0K−1⊂\(0,∞\)\\left\\\{\\eta\_\{k\}\\right\\\}\_\{k=0\}^\{K\-1\}\\subset\(0,\\infty\)and the unregularized iteration

ρ0=μ,ρk\+1=𝒯⁡\(vρk\+ηk​Γρk,r\)\.\\rho\_\{0\}=\\mu,\\qquad\\rho\_\{k\+1\}=\\mathcal\{T\}\\left\(v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},r\}\\right\)\.Assume that there exists a constantC\>0C\>0such that for everykk,

𝔼X∼π​\[𝒟ρk,ηk,r​\(X\)\]≤C​ηk2\.\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},r\}\(X\)\\right\]\\leq C\\eta\_\{k\}^\{2\}\.\(131\)Denote

τ^:=∑k=0K−1ηk,ηmax:=max0≤k<K⁡ηk\.\\hat\{\\tau\}:=\\sum\_\{k=0\}^\{K\-1\}\\eta\_\{k\},\\qquad\\eta\_\{\\mathrm\{max\}\}:=\\max\_\{0\\leq k<K\}\\eta\_\{k\}\.Then, we have

max⁡\{KL⁡\(ρK∥π\),KL⁡\(π∥ρK\)\}≤12​‖r‖∞2​\(τ−τ^\)2\+C​τ^​ηmax\.\\max\\left\\\{\\operatorname\{KL\}\\left\(\\rho\_\{K\}\\middle\\\|\\pi\\right\),\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{K\}\\right\)\\right\\\}\\leq\\frac\{1\}\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\(\\tau\-\\hat\{\\tau\}\)^\{2\}\+C\\hat\{\\tau\}\\eta\_\{\\mathrm\{max\}\}\.

The assumption in \([131](https://arxiv.org/html/2609.05727#S9.E131)\) is natural since

𝔼X∼π​\[𝒟ρk,ηk,r​\(X\)\]=ηk2​𝔼X∼π​\[∫01κt​‖∇Vtρk​\[r\]​\(Φ1→tρk,ηk,r​\(X\)\)‖22​𝑑t\],\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},r\}\(X\)\\right\]=\\eta\_\{k\}^\{2\}\\,\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\_\{k\}\}\[r\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho\_\{k\},\\eta\_\{k\},r\}\(X\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\right\],whereΦρk,ηk,r\\Phi^\{\\rho\_\{k\},\\eta\_\{k\},r\}is the flow map generated by the velocity fieldvρk\+ηk​Γρk,rv^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},r\}\.

### 9\.4Fixed\-Point Conditions via Zero Displacement

The preceding subsections explicitly realize a velocity fieldv^=vρ\+η​Γ^ρ\\hat\{v\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}and then canonicalize it\. The resulting canonical field is then used for the next stage of the update\.

Complementary to this stage\-by\-stage construction, we can take a fixed\-point perspective on the update rulev^=vρ\+η​Γ^ρ\\hat\{v\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}itself\. Rather than performing the iterative update, we focus on when the prescribed correction produces zero displacement and hence leavesvρv^\{\\rho\}unchanged\. This perspective applies to both exact and approximate tangential updates\. As shown below, sincevρ\+η​Γ^ρv^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}often admits a conditional\-expectation representation, the corresponding fixed\-point conditions can be converted into matching\-style regression objectives using thestop\-gradient trick\. However, this trick inevitably introduces approximations, even when applied to the exact tangential\-update formula\. We therefore place this construction in the present subsection under approximate Newton Matching\.

This subsection develops the fixed\-point viewpoint in three steps\. Section[9\.4\.1](https://arxiv.org/html/2609.05727#S9.SS4.SSS1)first characterizes fixed points of an ambient additive update through the vanishing of its displacement\. Section[9\.4\.2](https://arxiv.org/html/2609.05727#S9.SS4.SSS2)then formulates a generic stop\-gradient objective and gives conditions under which its population\-stationary points coincide with the fixed points of the additive update rule\. Finally, Section[9\.4\.3](https://arxiv.org/html/2609.05727#S9.SS4.SSS3)illustrates this general method with several examples drawn from exact and approximate Newton Matching\. These examples make concrete how exact and approximate tangential updates give rise to self\-anchored stop\-gradient objectives and pave the way for demystifying existing methods in Section[10](https://arxiv.org/html/2609.05727#S10)\.

#### 9\.4\.1Fixed Points of Tangential Updates

Both exact and approximate tangential updates considered in this paper have the ambient additive form

v^=vρ\+η​Γ^ρ,\\hat\{v\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\},where the exact Newton Matching case corresponds toΓ^ρ=Γρ,r~ρ\\widehat\{\\Gamma\}^\{\\rho\}=\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.

Sinceη\>0\\eta\>0, a canonical velocity fieldvρv^\{\\rho\}is a fixed point of the ambient additive update if and only if applying the update results in zero displacement:

v^=vρ⇔vρ\+ηΓ^ρ=vρ⇔Γ^ρ=0\.\\hat\{v\}=v^\{\\rho\}\\quad\\Leftrightarrow\\quad v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}=v^\{\\rho\}\\quad\\Leftrightarrow\\quad\\widehat\{\\Gamma\}^\{\\rho\}=0\.\(132\)IfΓ^ρ\\widehat\{\\Gamma\}^\{\\rho\}is critical\-point consistent forπ\\pi, then \([132](https://arxiv.org/html/2609.05727#S9.E132)\) holds if and only ifρ=π\\rho=\\pi\. By the injectivity of the canonical\-velocity map,vπv^\{\\pi\}is therefore the unique canonical fixed point\.

The fixed\-point condition in \([132](https://arxiv.org/html/2609.05727#S9.E132)\) is imposed before canonicalization\. It is stronger than the fixed\-point condition of the complete canonicalized stage,

vρ=ℛ⁡\(vρ\+η​Γ^ρ\),v^\{\\rho\}=\\mathcal\{R\}\\left\(v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}\\right\),since the canonical projection, like the terminal\-density map, is many\-to\-one on the ambient velocity space\. A nonzero ambient displacement can therefore return to the same canonical field after projection\. Zero displacement rules out this ambiguity and is in the same spirit as the definition of critical\-point consistency\.

#### 9\.4\.2Population\-Stationary Points of Stop\-Gradient Objectives

We next formalize how to enforce the fixed\-point condition in \([132](https://arxiv.org/html/2609.05727#S9.E132)\) through a stop\-gradient objective\. LetZZcollect all auxiliary randomness beyondttandXtX\_\{t\}\. Given a terminal densityρ^∈𝒫\\hat\{\\rho\}\\in\\mathscr\{P\}, letΞt,Zρ^\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}denote the joint distribution of\(Xt,Z\)\(X\_\{t\},Z\), and letΞZ\|tρ^\(⋅\|Xt\)\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)denote the conditional distribution ofZZgivenXtX\_\{t\}\. The particular sampling construction and itsXtX\_\{t\}\-marginal are left implicit\. For endpoint sampling associated withρ^\\hat\{\\rho\}, we write

Z=X1,\(Xt,X1\)∼Ξt,Zρ^:=Ξt,1ρ^,X1\|Xt∼ΞZ\|tρ^\(⋅\|Xt\):=p1\|tρ^\(⋅\|Xt\)\.Z=X\_\{1\},\\qquad\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}:=\\Xi\_\{t,1\}^\{\\hat\{\\rho\}\},\\qquad X\_\{1\}\|X\_\{t\}\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\):=p\_\{1\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\.For path sampling associated withρ^\\hat\{\\rho\}, we write

Z=𝒀\[t,1\],\(Xt,𝒀\[t,1\]\)∼Ξt,Zρ^:=Ξ\[t,1\]ρ^,𝒀\[t,1\]\|Xt∼ΞZ\|tρ^\(⋅\|Xt\):=ℙ\[t,1\]\|tρ^\(⋅\|Xt\)\.Z=\\bm\{Y\}\_\{\[t,1\]\},\\qquad\(X\_\{t\},\\bm\{Y\}\_\{\[t,1\]\}\)\\sim\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}:=\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\},\\qquad\\bm\{Y\}\_\{\[t,1\]\}\|X\_\{t\}\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\):=\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\.Here,Ξt,Zρ^\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}provides unified notation for the posterior requirements in \([49](https://arxiv.org/html/2609.05727#S8.E49)\) and \([82](https://arxiv.org/html/2609.05727#S8.E82)\), since the fixed\-point condition may involve either endpoint\-based or path\-based sampling\. This notation specifies the required posterior\-compatible law, which may be realized through either a forward or a reverse construction\. The superscriptρ^\\hat\{\\rho\}emphasizes that the notation applies to a general terminal density, not necessarily the terminal densityρ\\rhoassociated with the current anchorvρv^\{\\rho\}\.

For a canonical velocity modelvρv^\{\\rho\}, let

𝒴⁡\(t,xt,Z,vρ\)\\mathcal\{Y\}\(t,x\_\{t\},Z;v^\{\\rho\}\)denote the corresponding sample\-wise target, which may represent the targets presented in either exact or approximate Newton Matching\. Consider the fixed\-point condition whose right\-hand side can be expressed as a conditional expectation:

vtρ\(xt\)=𝔼Z∼ΞZ\|tρ^\(⋅\|xt\)\[𝒴\(t,xt,Z;vρ\)\]for a\.e\.\(t,xt\),v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\(t,x\_\{t\},Z;v^\{\\rho\}\)\\right\]\\qquad\\text\{for a\.e\. \}\(t,x\_\{t\}\),\(133\)whereρ^\\hat\{\\rho\}is typically the current terminal densityρ\\rho, or, in some cases, the base terminal densityρbase\\rho^\{\\mathrm\{base\}\}\. A parameterized stop\-gradient objective associated with \([133](https://arxiv.org/html/2609.05727#S9.E133)\) takes the form

ℒsg​\(θ\):=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{sg\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),\(Xt,Z\)∼Ξt,Zsg⁡\(θ\)​\[‖vtθ​\(Xt\)−𝒴⁡\(t,Xt,Z,sg⁡\(vθ\)\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},Z\)\\sim\\Xi\_\{t,Z\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;\\operatorname\{sg\}\(v^\{\\theta\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\.\(134\)HereΞt,Zsg⁡\(θ\)\\Xi\_\{t,Z\}^\{\\operatorname\{sg\}\(\\theta\)\}denotes the selected sampling lawΞt,Zρ^\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}, with any dependence of its construction onθ\\thetaplaced under stop\-gradient\. Thus, the superscriptsg⁡\(θ\)\\operatorname\{sg\}\(\\theta\)specifies the differentiation convention rather than the model used for sampling: ifρ^=ρbase\\hat\{\\rho\}=\\rho^\{\\mathrm\{base\}\}, the sampling law is simply the usualΞt,Zbase\\Xi\_\{t,Z\}^\{\\mathrm\{base\}\}; ifρ^=ρ\\hat\{\\rho\}=\\rho, we treat the current modelvθv^\{\\theta\}as canonical and substitute it forvρv^\{\\rho\}in any sampling or target construction that requires a canonical model\. Allθ\\theta\-dependent quantities in the target are likewise held fixed during differentiation\. Gradients flow only through the predictionvtθ​\(Xt\)v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\. Further discussion of this implicit canonicality assumption is provided in Remark[50](https://arxiv.org/html/2609.05727#Thmtheorem50)\.

To characterize precisely what a stop\-gradient loss represents, it is important to formulate its stationarity condition directly in function space rather than through a particular finite\-dimensional parameterization\. The following definition introduces the directional derivative and the stationarity condition of the stop\-gradient loss \([134](https://arxiv.org/html/2609.05727#S9.E134)\)\.

###### Definition 46\.

Consider the stop\-gradient loss in the function space

ℒsg​\-​func​\(v\):=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{sg\}\\text\{\-\}\\mathrm\{func\}\}\(v\):=𝔼t∼U⁡\(0,1\),\(Xt,Z\)∼Ξt,Zsg⁡\(v\)​\[‖vt​\(Xt\)−𝒴⁡\(t,Xt,Z,sg⁡\(v\)\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},Z\)\\sim\\Xi\_\{t,Z\}^\{\\operatorname\{sg\}\(v\)\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;\\operatorname\{sg\}\(v\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\.\(135\)Given an admissible perturbation directionh:\(0,1\)×ℝd→ℝdh:\(0,1\)\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}, the*directional derivative*ofℒsg​\-​func\\mathcal\{L\}^\{\\mathrm\{sg\}\\text\{\-\}\\mathrm\{func\}\}atvρv^\{\\rho\}alonghhis

dd​ε​𝔼t∼U⁡\(0,1\),\(Xt,Z\)∼Ξt,Zρ^​\[‖vtρ​\(Xt\)\+ε​ht​\(Xt\)−𝒴⁡\(t,Xt,Z,vρ\)‖22\]\|ε=0\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},Z\)\\sim\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\varepsilon h\_\{t\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\\right\|\_\{\\varepsilon=0\}\.\(136\)A canonical velocity fieldvρv^\{\\rho\}is a*population\-stationary point*of the stop\-gradient loss if, for every admissible perturbation directionhh,

dd​ε​𝔼t∼U⁡\(0,1\),\(Xt,Z\)∼Ξt,Zρ^​\[‖vtρ​\(Xt\)\+ε​ht​\(Xt\)−𝒴⁡\(t,Xt,Z,vρ\)‖22\]\|ε=0=0\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},Z\)\\sim\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\varepsilon h\_\{t\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\\right\|\_\{\\varepsilon=0\}=0\.

###### Proposition 47\.

Assume that theXtX\_\{t\}\-marginal ofΞt,Zρ^\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}admits a positive densityp^t\\hat\{p\}\_\{t\}, and that the quantities in \([136](https://arxiv.org/html/2609.05727#S9.E136)\) are square\-integrable for the perturbation directions considered below\. The directional derivative in \([136](https://arxiv.org/html/2609.05727#S9.E136)\) is equal to

2𝔼t∼U⁡\(0,1\),Xt∼p^t\[ht\(Xt\)⊤\(vtρ\(Xt\)−𝔼Z∼ΞZ\|tρ^\(⋅\|Xt\)\[𝒴\(t,Xt,Z;vρ\)\]\)\]\.2\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[h\_\{t\}\(X\_\{t\}\)^\{\\top\}\\left\(v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\]\\right\)\\right\]\.\(137\)
Assume further that there exists a functionω:\(0,1\)×ℝd→\(0,∞\)\\omega:\(0,1\)\\times\\mathbb\{R\}^\{d\}\\to\(0,\\infty\)such that

hω:\(t,xt\)↦ωt\(xt\)\(vtρ\(xt\)−𝔼Z∼ΞZ\|tρ^\(⋅\|xt\)\[𝒴\(t,xt,Z;vρ\)\]\)\\displaystyle h^\{\\omega\}:\(t,x\_\{t\}\)\\mapsto\\omega\_\{t\}\(x\_\{t\}\)\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\(t,x\_\{t\},Z;v^\{\\rho\}\)\\right\]\\right\)is an admissible perturbation direction\. Then,vρv^\{\\rho\}is a population\-stationary point of the stop\-gradient loss \([135](https://arxiv.org/html/2609.05727#S9.E135)\) if and only if it satisfies the fixed\-point condition \([133](https://arxiv.org/html/2609.05727#S9.E133)\), up to a\.e\. equality\.

###### Proof\.

The directional derivative in \([136](https://arxiv.org/html/2609.05727#S9.E136)\) is equal to \([137](https://arxiv.org/html/2609.05727#S9.E137)\) since

dd​ε​𝔼t∼U⁡\(0,1\),\(Xt,Z\)∼Ξt,Zρ^​\[‖vtρ​\(Xt\)\+ε​ht​\(Xt\)−𝒴⁡\(t,Xt,Z,vρ\)‖22\]\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},Z\)\\sim\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\varepsilon h\_\{t\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\\right\|\_\{\\varepsilon=0\}=\\displaystyle=dd​ε𝔼t∼U⁡\(0,1\),Xt∼p^t\[‖vtρ\(Xt\)\+εht\(Xt\)−𝔼Z∼ΞZ\|tρ^\(⋅\|Xt\)\[𝒴\(t,Xt,Z;vρ\)\]‖22\]\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\varepsilon h\_\{t\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\\right\|\_\{\\varepsilon=0\}=\\displaystyle=2𝔼t∼U⁡\(0,1\),Xt∼p^t\[ht\(Xt\)⊤\(vtρ\(Xt\)−𝔼Z∼ΞZ\|tρ^\(⋅\|Xt\)\[𝒴\(t,Xt,Z;vρ\)\]\)\]\.\\displaystyle 2\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[h\_\{t\}\(X\_\{t\}\)^\{\\top\}\\left\(v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\]\\right\)\\right\]\.
Consequently, the directional derivative ofℒsg​\-​func\\mathcal\{L\}^\{\\mathrm\{sg\}\\text\{\-\}\\mathrm\{func\}\}atvρv^\{\\rho\}alonghωh^\{\\omega\}is

dd​ε​𝔼t∼U⁡\(0,1\),\(Xt,Z\)∼Ξt,Zρ^​\[‖vtρ​\(Xt\)\+ε​htω​\(Xt\)−𝒴⁡\(t,Xt,Z,vρ\)‖22\]\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},Z\)\\sim\\Xi\_\{t,Z\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\varepsilon h\_\{t\}^\{\\omega\}\(X\_\{t\}\)\-\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\\right\|\_\{\\varepsilon=0\}=\\displaystyle=2𝔼t∼U⁡\(0,1\),Xt∼p^t\[htω\(Xt\)⊤\(vtρ\(Xt\)−𝔼Z∼ΞZ\|tρ^\(⋅\|Xt\)\[𝒴\(t,Xt,Z;vρ\)\]\)\]\\displaystyle 2\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[h\_\{t\}^\{\\omega\}\(X\_\{t\}\)^\{\\top\}\\left\(v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\]\\right\)\\right\]=\\displaystyle=2𝔼t∼U⁡\(0,1\),Xt∼p^t\[ωt\(Xt\)‖vtρ\(Xt\)−𝔼Z∼ΞZ\|tρ^\(⋅\|Xt\)\[𝒴\(t,Xt,Z;vρ\)\]‖22\]\.\\displaystyle 2\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim\\hat\{p\}\_\{t\}\\end\{subarray\}\}\\left\[\\omega\_\{t\}\(X\_\{t\}\)\\left\\\|v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\mathcal\{Y\}\\left\(t,X\_\{t\},Z;v^\{\\rho\}\\right\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\.Ifvρv^\{\\rho\}is a population\-stationary point up to a\.e\. equality, Definition[46](https://arxiv.org/html/2609.05727#Thmtheorem46)implies that the above directional derivative is00sincehωh^\{\\omega\}is an admissible perturbation direction\. Conversely, assume that the directional derivative alonghωh^\{\\omega\}is00\. The positivity ofp^t\\hat\{p\}\_\{t\}andωt​\(xt\)\\omega\_\{t\}\(x\_\{t\}\)implies thatvρv^\{\\rho\}satisfies \([133](https://arxiv.org/html/2609.05727#S9.E133)\) a\.e\. Substituting \([133](https://arxiv.org/html/2609.05727#S9.E133)\) into \([137](https://arxiv.org/html/2609.05727#S9.E137)\) proves that the directional derivative along every admissible perturbation directionhhis00\. ∎

###### Proposition 48\.

Letη\>0\\eta\>0, and letΓ^ρ\\widehat\{\\Gamma\}^\{\\rho\}be an exact or approximate correction at the canonical fieldvρv^\{\\rho\}\. Suppose that the sample\-wise target in \([135](https://arxiv.org/html/2609.05727#S9.E135)\) has conditional mean

𝔼Z∼ΞZ\|tρ^\(⋅\|xt\)\[𝒴\(t,xt,Z;vρ\)\]=vtρ\(xt\)\+ηΓ^tρ\(xt\)for a\.e\.\(t,xt\)\.\\mathbb\{E\}\_\{Z\\sim\\Xi\_\{Z\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}\(t,x\_\{t\},Z;v^\{\\rho\}\)\\right\]=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\widehat\{\\Gamma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\qquad\\text\{for a\.e\. \}\(t,x\_\{t\}\)\.\(138\)Then, among canonical velocity fields,

vρis a population\-stationary point⇔Γ^ρ=0\.v^\{\\rho\}\\text\{ is a population\-stationary point\}\\quad\\Leftrightarrow\\quad\\widehat\{\\Gamma\}^\{\\rho\}=0\.Consequently, if the method is critical\-point consistent forπ\\pi, then

vρis a population\-stationary point⇔ρ=π\.v^\{\\rho\}\\text\{ is a population\-stationary point\}\\quad\\Leftrightarrow\\quad\\rho=\\pi\.

###### Proof\.

By \([138](https://arxiv.org/html/2609.05727#S9.E138)\) and Proposition[47](https://arxiv.org/html/2609.05727#Thmtheorem47),vρv^\{\\rho\}is population\-stationary if and only if

vρ=vρ\+η​Γ^ρ\.v^\{\\rho\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}\.Sinceη\>0\\eta\>0, the displayed equality is equivalent toΓ^ρ=0\\widehat\{\\Gamma\}^\{\\rho\}=0, which is the zero\-displacement condition \([132](https://arxiv.org/html/2609.05727#S9.E132)\)\. Critical\-point consistency gives the second equivalence\. ∎

Proposition[47](https://arxiv.org/html/2609.05727#Thmtheorem47)characterizes the population\-stationary points of the stop\-gradient loss by showing that they coincide with the fixed points of the corresponding additive update\. It does not, however, by itself establish uniqueness or convergence\. If the fixed\-point condition \([133](https://arxiv.org/html/2609.05727#S9.E133)\) has a unique solution among canonical velocity fields, then this solution is also the unique population\-stationary point of the stop\-gradient loss within the same class\. The proposition does not guarantee that optimizing a stop\-gradient loss will converge to a population\-stationary point\.

#### 9\.4\.3Examples of Stop\-Gradient Objectives for Newton Matching

The previous analysis is agnostic to both the choice of the auxiliary variableZZand whether the additive update rule arises from exact or approximate Newton Matching\. We now instantiate this general construction with three representative gradient\-form objectives\. These stop\-gradient objectives also serve as templates in Section[10](https://arxiv.org/html/2609.05727#S10), where several existing methods are recovered as special cases of exact or approximate Newton Matching\.

We begin with the Bolza realization of the gradient\-form update in \([94](https://arxiv.org/html/2609.05727#S8.E94)\)–\([95](https://arxiv.org/html/2609.05727#S8.E95)\)\. Replacing the canonical anchorvρv^\{\\rho\}by the current modelvθv^\{\\theta\}and placing its occurrences in the sampling and adjoint constructions under stop\-gradient, we have

ℒηgrad,Bolza,sg​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\eta\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{sg\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)\[∥vtθ\(Yt\)−\(1−ητ\)vt\|1\(Yt\|Y1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητvtμ\(Yt\)−ηκtsg\(λtBolza,θ\(𝒀\[t,1\]\)\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-\\eta\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(139\)Here,Ξ\[t,1\]sg⁡\(θ\)\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}denotes the same posterior requirement asΞ\[t,1\]ρ\\Xi\_\{\[t,1\]\}^\{\\rho\}in \([82](https://arxiv.org/html/2609.05727#S8.E82)\), but with the current modelvθv^\{\\theta\}treated as canonical in place ofvρv^\{\\rho\}and all resulting dependence onθ\\thetastopped during differentiation\. The pathwise adjoint in the target is computed along each sampled path through

dd​sλsBolza,θ\(𝒀\[t,1\]\)=−\(2∇vsθ\(Ys\)−α˙sαsI\)⊤λsBolza,θ\(𝒀\[t,1\]\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\left\(2\\nabla v\_\{s\}^\{\\theta\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}I\\right\)^\{\\top\}\\lambda\_\{s\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+∇Ys\(‖vsθ​\(Ys\)−vsμ​\(Ys\)‖22τ​κs\),s∈\[t,1\],\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\+\\nabla\_\{Y\_\{s\}\}\\left\(\\frac\{\\left\\\|v\_\{s\}^\{\\theta\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mu\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\tau\\kappa\_\{s\}\}\\right\),\\qquad s\\in\[t,1\],\(140a\)λ1Bolza,θ​\(𝒀\[t,1\]\)=∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\.\(140b\)
The same fixed\-point perspective and stop\-gradient template apply to the reference\-adjoint approximation developed in Section[9\.2\.1](https://arxiv.org/html/2609.05727#S9.SS2.SSS1)\. In this construction, the posterior\-compatible path is still generated from the current anchorvρv^\{\\rho\}, while the adjoint ODE is evaluated with the reference velocityvμv^\{\\mu\}\. Applying the stop\-gradient trick to \([118](https://arxiv.org/html/2609.05727#S9.E118)\) gives

ℒηref​\-​adj,sg\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)\[∥\\displaystyle\\mathcal\{L\}\_\{\\eta\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\bigg\[\\Big\\\|vtθ​\(Yt\)−\(1−ητ\)​vt\|1​\(Yt\|Y1\)\\displaystyle v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητvtμ\(Yt\)−ηκtλtbμ,r,0\(𝒀\[t,1\]\)∥22\]\.\\displaystyle\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(141\)The notationΞ\[t,1\]sg⁡\(θ\)\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}has the same interpretation as in the stop\-gradient loss for the Bolza realization\.

The Gaussian\-kernel approximation \([122](https://arxiv.org/html/2609.05727#S9.E122)\) developed in Section[9\.2\.2](https://arxiv.org/html/2609.05727#S9.SS2.SSS2)removes the pathwise adjoint altogether and replaces the exact posterior Stein kernel with a positive\-definite matrix field\. This construction requires only an endpoint pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)\. Applying the stop\-gradient trick to \([122](https://arxiv.org/html/2609.05727#S9.E122)\) gives

ℒη,Σ^Gau​\-​ker,sg\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1sg⁡\(θ\)\[∥vtθ\(Xt\)−vt\|1\(Xt\|X1\)\\displaystyle\\mathcal\{L\}\_\{\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−ηαt​κtβt2Σ^t\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\-\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}\(X\_\{t\}\)\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.\(142\)Here,Ξt,1sg⁡\(θ\)\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}represents the same posterior requirement asΞt,1ρ\\Xi\_\{t,1\}^\{\\rho\}in \([49](https://arxiv.org/html/2609.05727#S8.E49)\), but with the current modelvθv^\{\\theta\}treated as canonical in place ofvρv^\{\\rho\}, and with all resulting dependence onθ\\thetastopped during differentiation\.

Since the Bolza realization from exact Newton Matching and the reference\-adjoint and Gaussian\-kernel approximations from approximate Newton Matching are all critical\-point consistent, Proposition[48](https://arxiv.org/html/2609.05727#Thmtheorem48)implies that each corresponding stop\-gradient objective hasvπv^\{\\pi\}as its unique canonical population\-stationary point\.

Table 2:A Newton Matching taxonomy for fine\-tuning and sampling, with representative existing methods recovered as special cases\.

## 10Demystifying Existing Works

Sections[8](https://arxiv.org/html/2609.05727#S8)and[9](https://arxiv.org/html/2609.05727#S9)develop exact Newton Matching and a family of approximate schemes, offering different trade\-offs between update exactness and computational cost\. Building on these developments, this section demystifies representative recent methods for fine\-tuning and sampling\. These works approach either of the two tasks from seemingly distinct perspectives, such as stochastic optimal control \(SOC\), contrastive learning, fixed\-point iteration, and velocity evolution under exponential tilting\. By translating their notation, sampler, and parameterization into the language of Newton Matching, we show that they can be recovered as special cases of exact or approximate Newton Matching algorithms\. This unification is summarized in Table[2](https://arxiv.org/html/2609.05727#S9.T2), with detailed derivations deferred to Appendix[I](https://arxiv.org/html/2609.05727#A9)\.

An important subtlety is canonicality\. As developed throughout the paper, both the matching\-style regression objectives and the scalable methods used to compute their targets rely on the conditional\-expectation structure of a canonical model and the associated posterior calculus\. In many of the works discussed below, this requirement is left implicit: their derivations treat the current model as canonical, even though optimizing the proposed losses does not automatically preserve canonicality\.

Many of the existing works make use of stop\-gradient losses\. As shown in Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), when an additive update of the formv^=vρ\+η​Γ^ρ\\hat\{v\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}admits a conditional\-expectation representation, its fixed\-point condition can be encoded by a stop\-gradient loss\. Two points are worth emphasizing\. First, one must identify the particular additive update, and hence the particular fixed\-point condition, encoded by the loss\. Second, even for the same update rule, the stop\-gradient formulation is only one possible implementation and may not be the best choice: it encodes an additive update rule through its fixed\-point condition, while implicitly treating the current modelvθv^\{\\theta\}as canonical, as discussed in Remark[50](https://arxiv.org/html/2609.05727#Thmtheorem50)\. By contrast, Newton Matching also provides the option of implementing the updatev^=vρ\+η​Γ^ρ\\hat\{v\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho\}stage by stage, with a natural opportunity to perform canonicalization after each stage\.

### 10\.1Tilt Matching

Explicit Tilt Matching \(ETM\) and Implicit Tilt Matching \(ITM\)\[[39](https://arxiv.org/html/2609.05727#bib.bib32)\]consider fine\-tuning from a normalized reference densityμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}with associated canonical velocityvbase=𝒞⁡\(ρbase\)∈𝒱canv^\{\\mathrm\{base\}\}=\\mathcal\{C\}\(\\rho^\{\\mathrm\{base\}\}\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. They also apply to the sampling caseμ=1\\mu=1through the auxiliary\-reference factorization \([108](https://arxiv.org/html/2609.05727#S9.E108)\)\. Both methods are organized along the exponential\-tilt continuation path

πs​\(x\):=ρbase​\(x\)​es​r​\(x\)∫ℝdρbase​\(z\)​es​r​\(z\)​𝑑z\.\\pi\_\{s\}\(x\):=\\frac\{\\rho^\{\\mathrm\{base\}\}\(x\)e^\{sr\(x\)\}\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho^\{\\mathrm\{base\}\}\(z\)e^\{sr\(z\)\}\\,\\mathrm\{d\}z\}\.For a continuation increments→s\+hs\\to s\+h, the consecutive targets satisfy

πs\+h​\(x\)∝πs​\(x\)​eh​r​\(x\)\.\\pi\_\{s\+h\}\(x\)\\propto\\pi\_\{s\}\(x\)e^\{hr\(x\)\}\.Assuming that the incrementhhis sufficiently small, each continuation stage can therefore be viewed as a local fine\-tuning problem with reference densityπs\\pi\_\{s\}, canonical anchorvπsv^\{\\pi\_\{s\}\}, and inverse\-temperaturehh\. For notational simplicity, throughout the remainder of this subsection, we relabel the current continuation densityπs\\pi\_\{s\}, its canonical velocityvπsv^\{\\pi\_\{s\}\}, and the incrementhhasρbase\\rho^\{\\mathrm\{base\}\},vbasev^\{\\mathrm\{base\}\}, andτ\\tau, respectively\. It then suffices to analyze a generic stage with reference densityμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}and target densityπρbase,τ,r∈𝒫\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\\in\\mathscr\{P\}\. Detailed derivations, together with an analysis of the discretization error defined in\[[39](https://arxiv.org/html/2609.05727#bib.bib32), Section 3\.1\], are deferred to Appendix[I\.1](https://arxiv.org/html/2609.05727#A9.SS1)\.

#### 10\.1\.1Explicit Tilt Matching

At a canonical base anchorvbasev^\{\\mathrm\{base\}\}, the ETM loss can be written as

ℒETM​\(θ\)=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{ETM\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),X1∼ρbase,X0∼p0,Xt=αt​X1\+βt​X0\[∥vtθ\(Xt\)−vt\|1\(Xt\|X1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−τ\(r\(X1\)−1τ\)\(vt\|1\(Xt\|X1\)−vtbase\(Xt\)\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\-\\tau\\left\(r\(X\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.Its unique population minimizer is

v¯ETM=vbase\+τ​Γρbase,r\.\\bar\{v\}^\{\\mathrm\{ETM\}\}=v^\{\\mathrm\{base\}\}\+\\tau\\Gamma^\{\\rho^\{\\mathrm\{base\}\},r\}\.
The ETM lossℒETM\\mathcal\{L\}^\{\\mathrm\{ETM\}\}can be recovered directly from Newton Matching by making the following choices\. We take the no\-regularization approximation from Section[9\.3\.2](https://arxiv.org/html/2609.05727#S9.SS3.SSS2), namely,

r~ρ≈r,\\tilde\{r\}^\{\\rho\}\\approx r,together with the covariance form via forward construction\. Then, we set

ρ=ρbase,η=τ,Bt≡1τ\.\\rho=\\rho^\{\\mathrm\{base\}\},\\qquad\\eta=\\tau,\\qquad B\_\{t\}\\equiv\\frac\{1\}\{\\tau\}\.Comparing the two losses yields the exact sample\-wise correspondence

ℒETM​\(θ\)=ℒρ,η,Bno​\-​reg​\-​cov​\(θ\)\|ρ=ρbase,η=τ,Bt≡1τunder the forward construction\.\\mathcal\{L\}^\{\\mathrm\{ETM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{no\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\}\}\(\\theta\)\\right\|\_\{\\rho=\\rho^\{\\mathrm\{base\}\},\\,\\eta=\\tau,\\,B\_\{t\}\\equiv\\frac\{1\}\{\\tau\}\}\\qquad\\text\{under the forward construction\.\}\(143\)Therefore, ETM is the covariance form of the full\-step unregularized update at the base anchor with the forward construction and a constant baseline\. It is not critical\-point consistent wheneverrris nonconstant\.

#### 10\.1\.2Implicit Tilt Matching

ITM uses a stop\-gradient objective\. With a positive control\-variate coefficientc⁡\(xt\)\>0c\(x\_\{t\}\)\>0, its loss is

ℒcITM​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{c\}^\{\\mathrm\{ITM\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),X1∼ρbase,X0∼p0,Xt=αt​X1\+βt​X0\[∥c\(Xt\)\(vtθ\(Xt\)−vtbase\(Xt\)\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\bigg\[\\Big\\\|c\(X\_\{t\}\)\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\)\+\(eτ​r​\(X1\)−c\(Xt\)\)\(sg\(vtθ\(Xt\)\)−vt\|1\(Xt\|X1\)\)∥22\]\.\\displaystyle\\qquad\\quad\+\\left\(e^\{\\tau r\(X\_\{1\}\)\}\-c\(X\_\{t\}\)\\right\)\\left\(\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.The unique population\-stationary point is the canonical velocity field

vITM=vπρbase,τ,r\.v^\{\\mathrm\{ITM\}\}=v^\{\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\}\.
The ITM lossℒcITM\\mathcal\{L\}\_\{c\}^\{\\mathrm\{ITM\}\}is recovered from Newton Matching by choosing the direct\-linearization loss \([241](https://arxiv.org/html/2609.05727#A9.E241)\)\. Comparing the two losses yields the exact sample\-wise correspondence

ℒcITM​\(θ\)=ℒρ,η,B,ωdir​\-​lin​\(θ\)\|ρ=ρbase,η=τ,Bt=1τ​log⁡c,ωt=c2under the forward construction\.\\mathcal\{L\}\_\{c\}^\{\\mathrm\{ITM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B,\\omega\}^\{\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}\(\\theta\)\\right\|\_\{\\rho=\\rho^\{\\mathrm\{base\}\},\\,\\eta=\\tau,\\,B\_\{t\}=\\frac\{1\}\{\\tau\}\\log c,\\,\\omega\_\{t\}=c^\{2\}\}\\quad\\text\{under the forward construction\.\}\(144\)Therefore, ITM is equivalent to the full\-step update of the direct\-linearization approximation under the forward construction, with specific choices of the baseline and weighting\. It is critical\-point consistent\.

### 10\.2DiffusionNFT

DiffusionNFT\[[58](https://arxiv.org/html/2609.05727#bib.bib36)\]approaches the fine\-tuning task from a contrastive\-learning perspective\. At each data\-collection stage, DiffusionNFT transforms the raw reward into an optimality probability by subtracting its prompt\-wise mean, dividing by a positive normalizing factor, clipping the result to\[−1,1\]\[\-1,1\], and then mapping it affinely to\[0,1\]\[0,1\]\. For our analysis, we absorb this preprocessing into the definition of the reward model, suppress the prompt dependence, and denote the resulting bounded reward byr:ℝd→\[0,1\]r:\\mathbb\{R\}^\{d\}\\to\[0,1\]\.

At a canonical anchorvρv^\{\\rho\}, its loss can be written in our notation as

ℒρ,cNFT​\(θ\)=\\displaystyle\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}\(\\theta\)=c2𝔼t∼U⁡\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0\[∥vtθ\(Xt\)−vt\|1\(Xt\|X1\)\\displaystyle c^\{2\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+2c\(r\(X1\)−c\+12\)\(vtρ\(Xt\)−vt\|1\(Xt\|X1\)\)∥22\]\+const,\\displaystyle\\qquad\+\\frac\{2\}\{c\}\\left\(r\(X\_\{1\}\)\-\\frac\{c\+1\}\{2\}\\right\)\\left\(v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\+\\mathrm\{const\},wherec\>0c\>0is a scalar hyperparameter and the additive constant is independent ofθ\\theta\. The unique population minimizer is

v¯NFT=vρ\+2c​Γρ,r\.\\bar\{v\}^\{\\mathrm\{NFT\}\}=v^\{\\rho\}\+\\frac\{2\}\{c\}\\Gamma^\{\\rho,r\}\.
The DiffusionNFT lossℒρ,cNFT​\(θ\)\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}\(\\theta\)is a special case of the covariance form from the no\-regularization variant in approximate Newton Matching\. Specifically, we drop the regularization term and set

r~ρ≈r\.\\tilde\{r\}^\{\\rho\}\\approx r\.We use the forward construction and the covariance form, and then set

η=2c,Bt≡c\+12\.\\eta=\\frac\{2\}\{c\},\\qquad B\_\{t\}\\equiv\\frac\{c\+1\}\{2\}\.Under these choices, comparing the two losses yields the exact sample\-wise correspondence up to aθ\\theta\-independent additive term:

ℒρ,cNFT​\(θ\)=c2⋅ℒρ,η,Bno​\-​reg​\-​cov​\(θ\)\|η=2c,Bt≡c\+12\+constunder the forward construction\.\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}\(\\theta\)=c^\{2\}\\cdot\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{no\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\}\}\(\\theta\)\\right\|\_\{\\eta=\\frac\{2\}\{c\},\\,B\_\{t\}\\equiv\\frac\{c\+1\}\{2\}\}\+\\mathrm\{const\}\\qquad\\text\{under the forward construction\.\}\(145\)Therefore, DiffusionNFT is the covariance form of the unregularized update under the forward construction, stepsizeη=2c\\eta=\\frac\{2\}\{c\}, and constant baselineBt≡c\+12B\_\{t\}\\equiv\\frac\{c\+1\}\{2\}\. It is not critical\-point consistent wheneverrris nonconstant\.

### 10\.3Reinforce Adjoint Matching

Reinforce Adjoint Matching \(RAM\)\[[5](https://arxiv.org/html/2609.05727#bib.bib37)\]considers the fine\-tuning caseμ=ρbase\\mu=\\rho^\{\\mathrm\{base\}\}with a canonical base velocityvbasev^\{\\mathrm\{base\}\}\. Using our notation, the RAM loss can be written as

ℒRAM​\(θ\)=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{RAM\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),X1∼ρsg⁡\(θ\),X0∼p0,Xt=αt​X1\+βt​X0\[∥vtθ\(Xt\)−vtbase\(Xt\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)−r\(X1\)\(vt\|1\(Xt\|X1\)−sg\(vtθ\(Xt\)\)\)∥22\],\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\-r\(X\_\{1\}\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\],whereX1∼ρsg⁡\(θ\)X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\}denotes an endpoint generated by the ODE of the current modelvθv^\{\\theta\}, with the dependence of the sampling procedure onθ\\thetastopped during differentiation\.

To derive the RAM loss from Newton Matching, we begin with the approximate\-regularization construction of Section[9\.3\.1](https://arxiv.org/html/2609.05727#S9.SS3.SSS1)and make the crude approximation of dropping the posterior\-KL term:

∇xtKL\(p1\|tρ\(⋅\|xt\)∥p1\|tbase\(⋅\|xt\)\)≈0\.\\nabla\_\{x\_\{t\}\}\\operatorname\{KL\}\\left\(p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\middle\\\|p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\right\)\\approx 0\.This corresponds to settingK^t≡0\\widehat\{K\}\_\{t\}\\equiv 0in \([123](https://arxiv.org/html/2609.05727#S9.E123)\)\. We then take the full\-step choiceη=τ\\eta=\\tau, withτ\\tauimplicitly fixed at11in RAM, and set the baseline to zero:

η=τ=1,K^t≡0,Bt≡0\.\\eta=\\tau=1,\\qquad\\widehat\{K\}\_\{t\}\\equiv 0,\\qquad B\_\{t\}\\equiv 0\.Under these choices, the additive update in \([123](https://arxiv.org/html/2609.05727#S9.E123)\) reduces to

v¯t​\(xt\)=vtρ​\(xt\)\+Γ^tρ,appr​\-​reg​\(xt\)\|τ=1,K^t≡0=vtbase​\(xt\)\+Γtρ,r​\(xt\)\.\\bar\{v\}\_\{t\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\left\.\\widehat\{\\Gamma\}\_\{t\}^\{\\rho,\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\(x\_\{t\}\)\\right\|\_\{\\tau=1,\\,\\widehat\{K\}\_\{t\}\\equiv 0\}=v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\+\\Gamma\_\{t\}^\{\\rho,r\}\(x\_\{t\}\)\.At a canonical anchor, its fixed\-point condition is therefore

vρ=vbase\+Γρ,r\.v^\{\\rho\}=v^\{\\mathrm\{base\}\}\+\\Gamma^\{\\rho,r\}\.\(146\)By the stop\-gradient construction of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), this fixed\-point condition can be realized by the following loss:

ℒρ,η,Bappr​\-​reg​\-​cov,sg​\(θ\)\|η=τ=1,K^t≡0,Bt≡0\\displaystyle\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\eta=\\tau=1,\\,\\widehat\{K\}\_\{t\}\\equiv 0,\\,B\_\{t\}\\equiv 0\}=\\displaystyle=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1sg⁡\(θ\)​\[‖vtθ​\(Xt\)−vtbase​\(Xt\)−r⁡\(X1\)​\(vt\|1​\(Xt\|X1\)−sg⁡\(vtθ​\(Xt\)\)\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\-r\(X\_\{1\}\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.
Comparing this loss withℒRAM\\mathcal\{L\}^\{\\mathrm\{RAM\}\}yields the exact sample\-wise correspondence

ℒRAM​\(θ\)=ℒρ,η,Bappr​\-​reg​\-​cov,sg​\(θ\)\|η=τ=1,K^t≡0,Bt≡0under the forward construction\.\\mathcal\{L\}^\{\\mathrm\{RAM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\eta=\\tau=1,\\,\\widehat\{K\}\_\{t\}\\equiv 0,\\,B\_\{t\}\\equiv 0\}\\qquad\\text\{under the forward construction\.\}\(147\)Therefore, RAM is the stop\-gradient covariance\-form realization of the fixed\-point condition \([146](https://arxiv.org/html/2609.05727#S10.E146)\) associated with the full\-step approximate\-regularization update, using the forward construction and a zero baseline\. RAM is not critical\-point consistent in general\.

### 10\.4Adjoint Matching

Basic Adjoint Matching \(BAM\) and Lean Adjoint Matching \(LAM\)\[[14](https://arxiv.org/html/2609.05727#bib.bib30)\]investigate the fine\-tuning problem through SOC\. They take a normalized reference densityμ=ρbase∈𝒫\\mu=\\rho^\{\\mathrm\{base\}\}\\in\\mathscr\{P\}and assume access to its canonical velocity fieldvbase=𝒞⁡\(ρbase\)∈𝒱canv^\{\\mathrm\{base\}\}=\\mathcal\{C\}\(\\rho^\{\\mathrm\{base\}\}\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. The inverse temperatureτ\\tauis set to11, corresponding to the target densityπρbase,1,r\\pi\_\{\\rho^\{\\mathrm\{base\}\},1,r\}\.

For a current modelvθv^\{\\theta\}, the SOC problem can be written in our notation as

minθ\\displaystyle\\min\_\{\\theta\}\\,\\,𝔼⁡\[∫011κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)‖22​𝑑t−r⁡\(Y1\)\],\\displaystyle\\mathbb\{E\}\\left\[\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\-r\\left\(Y\_\{1\}\\right\)\\right\],s\.t\.\\displaystyle\\mathrm\{s\.t\.\}\\,\\,d​Yt=\(2​vtθ​\(Yt\)−α˙tαt​Yt\)​d​t\+2​κt​d​Wt,t∈\(0,1\),\\displaystyle\\mathrm\{d\}Y\_\{t\}=\\left\(2v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\)\\mathrm\{d\}t\+\\sqrt\{2\\kappa\_\{t\}\}\\mathrm\{d\}W\_\{t\},\\qquad t\\in\(0,1\),Y0∼p0=𝒩⁡\(0,I\)\.\\displaystyle Y\_\{0\}\\sim p\_\{0\}=\\mathcal\{N\}\(0,I\)\.Both BAM and LAM generate the full path𝒀\[0,1\]\\bm\{Y\}\_\{\[0,1\]\}by simulating the SDE above fromY0∼𝒩⁡\(0,I\)Y\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and evaluate their losses along this shared trajectory\. This corresponds to the multi\-time supervision scheme for the gradient\-form \(under the reverse construction\) introduced in Section[8\.6\.2](https://arxiv.org/html/2609.05727#S8.SS6.SSS2)\. For simplicity, we present the losses equivalently in single\-time form\. We letΞ\[t,1\]θ\\Xi\_\{\[t,1\]\}^\{\\theta\}denote the same posterior requirement asΞ\[t,1\]ρ\\Xi\_\{\[t,1\]\}^\{\\rho\}in \([82](https://arxiv.org/html/2609.05727#S8.E82)\), but with the current modelvθv^\{\\theta\}treated as canonical in place ofvρv^\{\\rho\}\. See also Remark[50](https://arxiv.org/html/2609.05727#Thmtheorem50)for the discussion of canonicality in stop\-gradient objectives\.

#### 10\.4\.1Basic Adjoint Matching

Using our notation, the BAM loss can be written as

ℒBAM​\(θ\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)−κt​sg⁡\(λtBolza,θ​\(𝒀\[t,1\]\)\|τ=1,μ=ρbase\)‖22\]\.\\mathcal\{L\}^\{\\mathrm\{BAM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\\\ \\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\\end\{subarray\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\-\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\left\.\\lambda\_\{t\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\|\_\{\\tau=1,\\,\\mu=\\rho^\{\\mathrm\{base\}\}\}\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Here, the pathwise adjointλtBolza,θ​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)satisfies the adjoint ODE \([140](https://arxiv.org/html/2609.05727#S9.E140)\)\.

The BAM lossℒBAM\\mathcal\{L\}^\{\\mathrm\{BAM\}\}is recovered from the Bolza realization of the gradient form in exact Newton Matching under the reverse construction \([95](https://arxiv.org/html/2609.05727#S8.E95)\)\. For the full\-step choiceη=τ=1\\eta=\\tau=1, the associated fixed\-point condition at a canonical anchor is

vρ=vρ\+Γρ,r~ρ\.v^\{\\rho\}=v^\{\\rho\}\+\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.By the stop\-gradient construction of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), this fixed\-point condition admits the stop\-gradient lossℒη,ωgrad,Bolza,sg\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{sg\}\}in \([245](https://arxiv.org/html/2609.05727#A9.E245)\)\. We then set

τ=η=1,ωt≡1κt,μ=ρbase\.\\tau=\\eta=1,\\qquad\\omega\_\{t\}\\equiv\\frac\{1\}\{\\kappa\_\{t\}\},\\qquad\\mu=\\rho^\{\\mathrm\{base\}\}\.With these choices, we obtain the exact sample\-wise correspondence

ℒBAM​\(θ\)=ℒη,ωgrad,Bolza,sg​\(θ\)\|τ=η=1,ωt≡1κt,μ=ρbaseunder the reverse construction\.\\mathcal\{L\}^\{\\mathrm\{BAM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\tau=\\eta=1,\\,\\omega\_\{t\}\\equiv\\frac\{1\}\{\\kappa\_\{t\}\},\\,\\mu=\\rho^\{\\mathrm\{base\}\}\}\\qquad\\text\{under the reverse construction\}\.\(148\)Therefore, BAM is the stop\-gradient Bolza realization of the fixed\-point condition associated with the full\-step exact tangential update, using the reverse construction and a specific weighting\. According to Propositions[18](https://arxiv.org/html/2609.05727#Thmtheorem18)and[48](https://arxiv.org/html/2609.05727#Thmtheorem48), the stop\-gradient loss has𝒞⁡\(πρbase,1,r\)\\mathcal\{C\}\(\\pi\_\{\\rho^\{\\mathrm\{base\}\},1,r\}\)as its unique canonical population\-stationary point\.

#### 10\.4\.2Lean Adjoint Matching

LAM uses the same path distribution as BAM but replaces all occurrences of the current modelvtθ​\(Yt\)v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)with the reference modelvtbase​\(Yt\)v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)in the adjoint ODE\. Using our notation, the LAM loss can be written as

ℒLAM​\(θ\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)−κt​λtbμ,r,0​\(𝒀\[t,1\]\)\|μ=ρbase‖22\]\.\\mathcal\{L\}^\{\\mathrm\{LAM\}\}\(\\theta\)=\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\-\\left\.\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\|\_\{\\mu=\\rho^\{\\mathrm\{base\}\}\}\\right\\\|\_\{2\}^\{2\}\\right\]\.Here, the pathwise adjointλtbμ,r,0​\(𝒀\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)is obtained via the reference\-adjoint ODE \([116](https://arxiv.org/html/2609.05727#S9.E116)\)\.

The LAM lossℒLAM\\mathcal\{L\}^\{\\mathrm\{LAM\}\}is recovered from Newton Matching by adopting the reference\-adjoint approximation \([118](https://arxiv.org/html/2609.05727#S9.E118)\)\. Following Section[9\.2\.1](https://arxiv.org/html/2609.05727#S9.SS2.SSS1), we approximate the pathwise adjoint by

λtbρ,r,lρ,μ≈λtbμ,r,lμ,μ=λtbμ,r,0\.\\lambda\_\{t\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\\approx\\lambda\_\{t\}^\{b^\{\\mu\},r,l^\{\\mu,\\mu\}\}=\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\.For the full\-step choiceη=τ=1\\eta=\\tau=1, the associated fixed\-point condition at a canonical anchor is

vρ=vρ\+Γ^ρ,ref​\-​adj\.v^\{\\rho\}=v^\{\\rho\}\+\\widehat\{\\Gamma\}^\{\\rho,\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}\.By the stop\-gradient construction of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), this fixed\-point condition admits the stop\-gradient lossℒη,ωref​\-​adj,sg\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\},\\mathrm\{sg\}\}in \([246](https://arxiv.org/html/2609.05727#A9.E246)\)\. We then set

τ=η=1,ωt≡1κt,μ=ρbase\.\\tau=\\eta=1,\\qquad\\omega\_\{t\}\\equiv\\frac\{1\}\{\\kappa\_\{t\}\},\\qquad\\mu=\\rho^\{\\mathrm\{base\}\}\.Consequently, we obtain the exact sample\-wise correspondence

ℒLAM​\(θ\)=ℒη,ωref​\-​adj,sg​\(θ\)\|τ=η=1,ωt≡1κt,μ=ρbaseunder the reverse construction\.\\mathcal\{L\}^\{\\mathrm\{LAM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\tau=\\eta=1,\\,\\omega\_\{t\}\\equiv\\frac\{1\}\{\\kappa\_\{t\}\},\\,\\mu=\\rho^\{\\mathrm\{base\}\}\}\\qquad\\text\{under the reverse construction\}\.\(149\)Therefore, LAM is the stop\-gradient realization of the full\-step reference\-adjoint approximation update, using the reverse construction and a specific weighting\. According to Propositions[39](https://arxiv.org/html/2609.05727#Thmtheorem39)and[48](https://arxiv.org/html/2609.05727#Thmtheorem48), the stop\-gradient loss has𝒞⁡\(πρbase,1,r\)\\mathcal\{C\}\(\\pi\_\{\\rho^\{\\mathrm\{base\}\},1,r\}\)as its unique canonical population\-stationary point\.

### 10\.5Flow Sampling

Flow Sampling \(FS\)\[[22](https://arxiv.org/html/2609.05727#bib.bib34)\]considers the sampling taskμ=1\\mu=1\. FS is motivated by fixed\-point iterations\. It trains a drift modelbθb^\{\\theta\}and uses SDE\-based sampling\. The inverse temperatureτ\\tauis set to11\.

In our notation, the FS loss can be written as

ℒFS​\(θ\):=𝔼t∼U⁡\(0,1\),X1∼ρsg⁡\(θ\),X0∼p0,Xt=αt​X1\+βt​X0\\displaystyle\\mathcal\{L\}^\{\\mathrm\{FS\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\[∥btθ\(Xt\)−bt\|1\(Xt\|X1\)\\displaystyle\\Bigg\[\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−κtαt\(∇r\(X1\)\+αtβt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\-\\frac\{\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.Here,ρsg⁡\(θ\)\\rho^\{\\operatorname\{sg\}\(\\theta\)\}denotes the terminal density induced by the current model through \([247](https://arxiv.org/html/2609.05727#A9.E247)\), with the dependence of the sampling construction onθ\\thetaplaced under stop\-gradient\.

FS can be recovered from the Gaussian\-kernel approximation of Newton Matching\. We approximate the posterior Stein kernel by

Λtρ​\(x1\|xt\)≈Σ^tρ​\(xt\),\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\approx\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\),and use the drift\-coordinate extension of the approximation loss \([122](https://arxiv.org/html/2609.05727#S9.E122)\)\. This extension is denoted byℒρ,η,Σ^Gau​\-​ker,drift\\mathcal\{L\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\}\}and is provided in \([248](https://arxiv.org/html/2609.05727#A9.E248)\)\. The associated fixed\-point condition at a canonical anchor is

btρ\(xt\)=btρ\(xt\)\+2ηαt​κtβt2Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(xt−αtX1\)\]\.b\_\{t\}^\{\\rho\}\(x\_\{t\}\)=b\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+2\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\]\.By the stop\-gradient construction of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), this fixed\-point condition admits the stop\-gradient lossℒη,Σ^Gau​\-​ker,drift,sg\\mathcal\{L\}\_\{\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}in \([249](https://arxiv.org/html/2609.05727#A9.E249)\)\.

We then use the forward construction with

τ=1,η=τ2=12,Σ^t≡βt2αt2​I\.\\tau=1,\\qquad\\eta=\\frac\{\\tau\}\{2\}=\\frac\{1\}\{2\},\\qquad\\widehat\{\\Sigma\}\_\{t\}\\equiv\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\.Under these choices, comparing the two objectives yields the exact sample\-wise correspondence

ℒFS​\(θ\)=ℒη,Σ^Gau​\-​ker,drift,sg​\(θ\)\|μ=1,τ=1,η=12,Σ^t≡βt2αt2​Iunder the forward construction\.\\mathcal\{L\}^\{\\mathrm\{FS\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\mu=1,\\,\\tau=1,\\,\\eta=\\frac\{1\}\{2\},\\,\\widehat\{\\Sigma\}\_\{t\}\\equiv\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\}\\quad\\text\{under the forward construction\}\.\(150\)Therefore, FS is the stop\-gradient realization of the damped Gaussian\-kernel approximation update with stepsizeη=12\\eta=\\frac\{1\}\{2\}, using the drift coordinate and the forward construction with SDE\-based sampling\. According to Propositions[41](https://arxiv.org/html/2609.05727#Thmtheorem41)and[48](https://arxiv.org/html/2609.05727#Thmtheorem48), the stop\-gradient loss has𝒞⁡\(π1,1,r\)\\mathcal\{C\}\(\\pi\_\{1,1,r\}\)as its unique canonical population\-stationary point\.

### 10\.6Adjoint Sampling

Adjoint Sampling \(AS\)\[[23](https://arxiv.org/html/2609.05727#bib.bib31)\]considers the sampling taskμ=1\\mu=1\. AS is motivated by the SOC perspective\. It trains a drift model, and during sampling, the SDE is initialized at the deterministic stateY0=0Y\_\{0\}=0\.

We show that, by extending Newton Matching to the one\-sided interpolant, we can sidestep all SOC\-related elements, such as the reference processes, and specify the schedule directly from the one\-sided interpolant\. This extension is introduced in Appendix[H\.2](https://arxiv.org/html/2609.05727#A8.SS2), and the detailed derivation of the correspondence between AS and Newton Matching is provided in Appendix[I\.6](https://arxiv.org/html/2609.05727#A9.SS6)\.

With a noise scheduleσt\\sigma\_\{t\}, AS adopts

γ~t:=∫0tσu2​𝑑u,α~t:=γ~tγ~1,β~t:=γ~t​\(γ~1−γ~t\)γ~1,κ~t:=β~tα~t​\(α~˙t​β~t−α~t​β~˙t\)=12​σt2\.\\widetilde\{\\gamma\}\_\{t\}:=\\int\_\{0\}^\{t\}\\sigma\_\{u\}^\{2\}\\,\\mathrm\{d\}u,\\quad\\widetilde\{\\alpha\}\_\{t\}:=\\frac\{\\widetilde\{\\gamma\}\_\{t\}\}\{\\widetilde\{\\gamma\}\_\{1\}\},\\quad\\widetilde\{\\beta\}\_\{t\}:=\\sqrt\{\\frac\{\\widetilde\{\\gamma\}\_\{t\}\(\\widetilde\{\\gamma\}\_\{1\}\-\\widetilde\{\\gamma\}\_\{t\}\)\}\{\\widetilde\{\\gamma\}\_\{1\}\}\},\\quad\\widetilde\{\\kappa\}\_\{t\}:=\\frac\{\\widetilde\{\\beta\}\_\{t\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}\\left\(\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}\\widetilde\{\\beta\}\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}\\dot\{\\widetilde\{\\beta\}\}\_\{t\}\\right\)=\\frac\{1\}\{2\}\\sigma\_\{t\}^\{2\}\.The detailed derivation is deferred to Appendix[I\.6](https://arxiv.org/html/2609.05727#A9.SS6)\.

The AS loss can be written as

ℒAS​\(θ\)=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{AS\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),X1∼ρsg⁡\(θ\),Z∼𝒩⁡\(0,I\),Xt=α~t​X1\+β~t​Z\[18​κ~t2∥btθ\(Xt\)−b~t\|1\(Xt\|X1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,Z\\sim\\mathcal\{N\}\(0,I\),\\,X\_\{t\}=\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\+\\widetilde\{\\beta\}\_\{t\}Z\\end\{subarray\}\}\\Bigg\[\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\widetilde\{b\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−2κ~tτ\(∇r\(X1\)\+α~tτ​β~t2\(Xt−α~tX1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\-2\\widetilde\{\\kappa\}\_\{t\}\\tau\\bigg\(\\nabla r\(X\_\{1\}\)\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\tau\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\\bigg\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.Here,X1∼ρsg⁡\(θ\)X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\}denotes the terminal density induced by the current model through \([250](https://arxiv.org/html/2609.05727#A9.E250)\), with the dependence of the sampling construction onθ\\thetaplaced under stop\-gradient\.

AS can be recovered from the Gaussian\-kernel approximation of Newton Matching\. At a canonical anchor, we approximate the posterior Stein kernel by

Λtρ​\(x1\|xt\)≈Σ^tρ​\(xt\),\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\approx\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\),and extend the approximation loss \([122](https://arxiv.org/html/2609.05727#S9.E122)\) to the drift coordinate and the one\-sided interpolant\. This loss is denoted byℒ~ρ,η,Σ^,ωGau​\-​ker,drift\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\}\}and is provided in \([253](https://arxiv.org/html/2609.05727#A9.E253)\)\. The associated fixed\-point condition at a canonical anchor is

b~tρ\(xt\)=b~tρ\(xt\)\+2ηα~t​κ~tβ~t2Σ^tρ\(xt\)𝔼X1∼p~1\|tρ\(⋅\|xt\)\[∇r\(X1\)\+1τ∇logμ\(X1\)\+α~tτ​β~t2\(xt−α~tX1\)\]\.\\displaystyle\\widetilde\{b\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\widetilde\{b\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+2\\eta\\frac\{\\widetilde\{\\alpha\}\_\{t\}\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\tau\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\\right\]\.By the stop\-gradient construction of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), this fixed\-point condition admits the stop\-gradient lossℒ~ρ,η,Σ^,ωGau​\-​ker,drift,sg\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}in \([254](https://arxiv.org/html/2609.05727#A9.E254)\)\.

We use the forward construction and set

η=τ,ωt≡18​κ~t2,Σ^t≡β~t2α~t​I\.\\eta=\\tau,\\qquad\\omega\_\{t\}\\equiv\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\},\\qquad\\widehat\{\\Sigma\}\_\{t\}\\equiv\\frac\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}I\.Under these choices, comparing the two objectives yields the exact sample\-wise correspondence

ℒAS​\(θ\)=ℒ~ρ,η,Σ^,ωGau​\-​ker,drift,sg​\(θ\)\|μ=1,α~t=γ~tγ~1,β~t=γ~t​\(γ~1−γ~t\)γ~1,η=τ,Σ^t≡β~t2α~t​I,ωt≡18​κ~t2\\displaystyle\\mathcal\{L\}^\{\\mathrm\{AS\}\}\(\\theta\)=\\left\.\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\mu=1,\\,\\widetilde\{\\alpha\}\_\{t\}=\\frac\{\\widetilde\{\\gamma\}\_\{t\}\}\{\\widetilde\{\\gamma\}\_\{1\}\},\\,\\widetilde\{\\beta\}\_\{t\}=\\sqrt\{\\frac\{\\widetilde\{\\gamma\}\_\{t\}\(\\widetilde\{\\gamma\}\_\{1\}\-\\widetilde\{\\gamma\}\_\{t\}\)\}\{\\widetilde\{\\gamma\}\_\{1\}\}\},\\,\\eta=\\tau,\\,\\widehat\{\\Sigma\}\_\{t\}\\equiv\\frac\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}I,\\,\\omega\_\{t\}\\equiv\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\}\(151\)under the forward construction\.

Therefore, AS is the stop\-gradient realization of the full\-step Gaussian\-kernel approximation update under the forward construction with SDE\-based sampling, using the drift coordinate and the one\-sided interpolant\. The stop\-gradient loss has𝒞⁡\(π1,τ,r\)\\mathcal\{C\}\(\\pi\_\{1,\\tau,r\}\)as its unique canonical population\-stationary point\.

## 11Conclusion

We developed Newton Matching, a unified framework for fine\-tuning and sampling when the target distribution is specified through its defining factors rather than direct samples\. The central conceptual shift is from treating methods as isolated regression losses to viewing learning as iterative optimization over canonical models—the population minimizers that standard conditional matching associates with terminal densities\. This restriction removes representational redundancy while retaining the conditional\-expectation structure that makes diffusion and flow training scalable\. Within this general formulation, we specialized to reverse\-KL minimization and proved that the mixture\-connection Newton direction coincides with the negative Fisher–Rao gradient\. The canonical retraction realizes finite steps along this direction through a tangential update followed by terminal\-density\-preserving canonicalization\. Its exact terminal\-density characterization yields strict reverse\-KL descent for every damped or full step, global convergence under mild conditions, and local quadratic convergence of the full\-step iteration\. We further derived multiple population\-exact realizations and approximate variants organized by critical\-point consistency\. By recovering representative existing methods as configurations of shared building blocks, Newton Matching demystifies their relationships and distinguishes exact Newton updates, target\-preserving approximations, and variants that alter the underlying objective\. More broadly, Newton Matching lays a foundation for principled post\-training of generative models from reward feedback, providing a common language for designing and analyzing distributional objectives, finite\-stepsize updates, and scalable regression realizations\. In doing so, it advances both theoretical understanding and principled design of reinforcement learning algorithms for generative models\.

## Appendices

The appendices are organized into five groups\.

Appendices[A](https://arxiv.org/html/2609.05727#A1)–[C](https://arxiv.org/html/2609.05727#A3)provide the geometric and analytic theory behind Newton Matching\. Appendix[A](https://arxiv.org/html/2609.05727#A1)establishes the mathematical foundations underlying the canonical geometry, the finite\-stepsize canonical retraction, and the Newton Matching construction in Sections[4](https://arxiv.org/html/2609.05727#S4)–[6](https://arxiv.org/html/2609.05727#S6)\. It specifies a compatible smooth Banach\-manifold realization and states the regularity assumptions under which the global, first\-order, and second\-order geometric arguments in the main text are well\-defined\. It also proves that the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is generally non\-affine, thereby motivating the canonical retraction used for finite\-stepsize updates\. Appendix[B](https://arxiv.org/html/2609.05727#A2)provides the technical convergence analysis underlying Section[7](https://arxiv.org/html/2609.05727#S7)\. It turns the one\-stage descent properties established in the main text into global convergence guarantees, identifies the topology\-compatible metrics in which local quadratic convergence is measured on the density and canonical manifolds, and establishes the guarantee for the continuation scheme\. Appendix[C](https://arxiv.org/html/2609.05727#A3)complements this general theory with an analytically solvable Gaussian model, for which the Newton Matching iteration admits an explicit finite\-dimensional characterization and both global convergence and local quadratic convergence can be verified directly\.

Appendices[D](https://arxiv.org/html/2609.05727#A4)–[E](https://arxiv.org/html/2609.05727#A5)provide the mathematical and computational tools underlying the exact and approximate Newton Matching developed in Sections[8](https://arxiv.org/html/2609.05727#S8)and[9](https://arxiv.org/html/2609.05727#S9)\. Appendix[D](https://arxiv.org/html/2609.05727#A4)develops stochastic representations, posterior\-preserving dynamics, universal bridges, log\-density and log\-density\-ratio identities, initial\-state sensitivity, and adjoint calculus\. These results make the posterior quantities and density\-ratio correction required by Newton Matching accessible through sampling and pathwise computation\. Appendix[E](https://arxiv.org/html/2609.05727#A5)complements these tools by developing Stein control variates and posterior Stein kernels\. In particular, it establishes the covariance–gradient identity that converts the covariance\-form target \([45](https://arxiv.org/html/2609.05727#S8.E45)\) into the gradient\-form target \([48](https://arxiv.org/html/2609.05727#S8.E48)\)\.

Appendix[F](https://arxiv.org/html/2609.05727#A6)provides the technical analysis behind approximate Newton Matching\. It collects the proofs deferred from Section[9](https://arxiv.org/html/2609.05727#S9), identifies the population\-stationary points of the approximate regression objectives, establishes the critical\-point\-consistency guarantees stated for the covariance and gradient approximations, and proves the conclusions associated with the regularization trade\-offs\.

Appendices[G](https://arxiv.org/html/2609.05727#A7)–[H](https://arxiv.org/html/2609.05727#A8)provide two complementary extensions of the Newton Matching framework\. Appendix[G](https://arxiv.org/html/2609.05727#A7)replaces the constant stepsizeη\\etain the tangential update by a time\-dependent modulationηt\\eta\_\{t\}\. Although the resulting modulated field need not remain tangent to the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}, the appendix derives the exact formula for its terminal\-density update and shows that nondecreasing modulation schedules preserve the value\-ascent certificate in Section[5\.2](https://arxiv.org/html/2609.05727#S5.SS2)and the reverse\-KL descent in Section[7\.1\.1](https://arxiv.org/html/2609.05727#S7.SS1.SSS1)\. Appendix[H](https://arxiv.org/html/2609.05727#A8)instead changes the representation used to realize the update\. It demonstrates how the exact and approximate regression objectives can be realized in score and drift coordinates\. It also illustrates a representative alternative CFM construction: the one\-sided interpolant with deterministic initial stateX0=0X\_\{0\}=0\.

Finally, Appendix[I](https://arxiv.org/html/2609.05727#A9)provides the detailed method\-specific derivations deferred from Section[10](https://arxiv.org/html/2609.05727#S10)\. By translating the notation, sampler, and parameterization of the original works into the language of this paper, we establish the stated correspondences and show how these methods arise as special cases of exact or approximate Newton Matching algorithms\. The derivations for Flow Sampling and Adjoint Sampling additionally use the drift\-coordinate and one\-sided\-interpolant constructions developed in Appendix[H](https://arxiv.org/html/2609.05727#A8)\.

## Appendix AGeometric Foundation

This appendix provides a rigorous mathematical setting for the canonical geometry, the finite\-stepsize canonical retraction, and the Newton Matching construction presented in Sections[4](https://arxiv.org/html/2609.05727#S4)–[6](https://arxiv.org/html/2609.05727#S6)\. It defines the relevant spaces and maps, equips the density and velocity classes with compatible Banach\-manifold structures, and states the regularity assumptions under which the global, first\-order, and second\-order arguments in Newton Matching, as well as the local convergence analysis of Section[7\.2](https://arxiv.org/html/2609.05727#S7.SS2), are well\-defined\.

First, Appendix[A\.1](https://arxiv.org/html/2609.05727#A1.SS1)introduces ambient velocity and density classes on which ODE flows, terminal densities, and velocities from the CFM construction are well\-defined\. Then, Appendix[A\.2](https://arxiv.org/html/2609.05727#A1.SS2)selects admissible subclasses and constructs the canonical subclass and projection at the set\-theoretic level\. Appendix[A\.3](https://arxiv.org/html/2609.05727#A1.SS3)equips these classes with compatible Banach\-manifold structures and develops the associated first\-order and second\-order differential geometry used in Newton Matching\. Finally, Appendix[A\.4](https://arxiv.org/html/2609.05727#A1.SS4)gives a Gaussian counterexample showing that𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}need not be affine, thereby motivating the canonical retraction used for finite\-stepsize updates\.

### A\.1Ambient Classes

The ambient classes provide a common regularity setting for the constructions that follow\. They are chosen so that the ODE flow is globally well posed, the densities transported by the flow have finite second moments, and the canonical velocity field associated with each terminal density is pointwise well defined\. No manifold structure is imposed at this stage\.

#### A\.1\.1Ambient Velocity Space

Here, we define a Banach space of time\-dependent velocity fieldsv=\(vt\)t∈\[0,1\]v=\(v\_\{t\}\)\_\{t\\in\[0,1\]\}that are integrable in time and globally Lipschitz in space\. These conditions are sufficient for the existence of a unique global Carathéodory flow and provide the growth estimate needed to control pushforward moments\.

###### Definition 52\.

The ambient velocity space is defined as the Banach space

𝒱^:=\{v=\(vt\)t∈\[0,1\]∈L1​\(\[0,1\],Clin1\):‖v‖𝒱^:=∫01‖vt‖Clin1​𝑑t<∞\},\\widehat\{\\mathscr\{V\}\}:=\\left\\\{v=\(v\_\{t\}\)\_\{t\\in\[0,1\]\}\\in L^\{1\}\\left\(\[0,1\];C\_\{\\mathrm\{lin\}\}^\{1\}\\right\):\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}:=\\int\_\{0\}^\{1\}\\left\\\|v\_\{t\}\\right\\\|\_\{C\_\{\\mathrm\{lin\}\}^\{1\}\}\\mathrm\{d\}t<\\infty\\right\\\},where the Lipschitz function spaceClin1C\_\{\\mathrm\{lin\}\}^\{1\}is the Banach space

Clin1:=\{v∈C1​\(ℝd,ℝd\):‖v‖Clin1:=‖v⁡\(0\)‖2\+‖∇v‖∞<∞\}\.C\_\{\\mathrm\{lin\}\}^\{1\}:=\\left\\\{v\\in C^\{1\}\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}^\{d\}\):\\left\\\|v\\right\\\|\_\{C\_\{\\mathrm\{lin\}\}^\{1\}\}:=\\left\\\|v\(0\)\\right\\\|\_\{2\}\+\\left\\\|\\nabla v\\right\\\|\_\{\\infty\}<\\infty\\right\\\}\.In other words,t↦‖vt‖Clin1t\\mapsto\\left\\\|v\_\{t\}\\right\\\|\_\{C\_\{\\mathrm\{lin\}\}^\{1\}\}is integrable on\[0,1\]\[0,1\]\. For a\.e\.t∈\[0,1\]t\\in\[0,1\], the spatial vector fieldvtv\_\{t\}isC1C^\{1\}and globally Lipschitz continuous\.

Throughout the ambient construction, the ODEY˙t=vt​\(Yt\)\\dot\{Y\}\_\{t\}=v\_\{t\}\(Y\_\{t\}\)is understood in the Carathéodory sense:t↦Ytt\\mapsto Y\_\{t\}is absolutely continuous and satisfies

Yt=Ys\+∫stvu​\(Yu\)​𝑑u\.Y\_\{t\}=Y\_\{s\}\+\\int\_\{s\}^\{t\}v\_\{u\}\(Y\_\{u\}\)\\,\\mathrm\{d\}u\.Equivalently,Y˙t=vt​\(Yt\)\\dot\{Y\}\_\{t\}=v\_\{t\}\(Y\_\{t\}\)holds for a\.e\.t∈\[0,1\]t\\in\[0,1\]\. The following lemma records the resulting global well\-posedness and the quantitative growth bound used below\.

###### Lemma 53\.

For any velocity fieldv∈𝒱^v\\in\\widehat\{\\mathscr\{V\}\}, we denote its Carathéodory ODE flow byΦ0→tv:ℝd→ℝd\\Phi\_\{0\\to t\}^\{v\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}\. Then, for any initial pointx0∈ℝdx\_\{0\}\\in\\mathbb\{R\}^\{d\}, the ODE admits a unique nonexplosive solutionxt=Φ0→tv​\(x0\)x\_\{t\}=\\Phi\_\{0\\to t\}^\{v\}\(x\_\{0\}\)that satisfies

‖xt‖2≤e‖v‖𝒱^​\(‖x0‖2\+‖v‖𝒱^\)\.\\left\\\|x\_\{t\}\\right\\\|\_\{2\}\\leq e^\{\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}\}\(\\left\\\|x\_\{0\}\\right\\\|\_\{2\}\+\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}\)\.\(152\)

###### Proof\.

The existence of the ODE flow is a classical result\[[29](https://arxiv.org/html/2609.05727#bib.bib47), Chapter 6\]\. By Grönwall’s inequality, for any initial pointx0x\_\{0\}, the ODE solution satisfies the pathwise estimate

‖xt‖2≤\(‖x0‖2\+∫0t‖vs​\(0\)‖2​𝑑s\)​exp⁡\(∫0t‖∇vs‖∞​𝑑s\)≤e‖v‖𝒱^​\(‖x0‖2\+‖v‖𝒱^\)\.\\left\\\|x\_\{t\}\\right\\\|\_\{2\}\\leq\\left\(\\left\\\|x\_\{0\}\\right\\\|\_\{2\}\+\\int\_\{0\}^\{t\}\\left\\\|v\_\{s\}\(0\)\\right\\\|\_\{2\}\\,\\mathrm\{d\}s\\right\)\\exp\\left\(\\int\_\{0\}^\{t\}\\left\\\|\\nabla v\_\{s\}\\right\\\|\_\{\\infty\}\\,\\mathrm\{d\}s\\right\)\\leq e^\{\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}\}\(\\left\\\|x\_\{0\}\\right\\\|\_\{2\}\+\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}\)\.Therefore, the regularity condition‖v‖𝒱^<∞\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}<\\inftyimplies that the ODE admits a unique nonexplosive solution\. ∎

#### A\.1\.2Ambient Density Class

Here, we identify a broad class of terminal densities compatible with the ambient velocity space\. The first result shows that the flow of everyv∈𝒱^v\\in\\widehat\{\\mathscr\{V\}\}transports the Gaussian source to a positive continuous density with finite second moment\.

###### Lemma 54\.

Consider the source densityp0=𝒩⁡\(0,I\)p\_\{0\}=\\mathcal\{N\}\(0,I\)and the probability path\(ℙtv\)t∈\[0,1\]\(\\mathbb\{P\}\_\{t\}^\{v\}\)\_\{t\\in\[0,1\]\}generated by the pushforward

ℙtv:=\(Φ0→tv\)\#​\(p0​d​x\)\.\\mathbb\{P\}\_\{t\}^\{v\}:=\(\\Phi\_\{0\\to t\}^\{v\}\)\_\{\\\#\}\(p\_\{0\}\\,\\mathrm\{d\}x\)\.Then,ℙtv=ptv​d​x\\mathbb\{P\}\_\{t\}^\{v\}=p\_\{t\}^\{v\}\\,\\mathrm\{d\}xhas finite second moment, whereptvp\_\{t\}^\{v\}is a positive, normalized, continuous density\. The densityptvp\_\{t\}^\{v\}also satisfies the continuity equation

∂tptv​\(xt\)\+∇xt⋅\(ptv​\(xt\)​vt​\(xt\)\)=0,p0v=p0\\partial\_\{t\}p\_\{t\}^\{v\}\(x\_\{t\}\)\+\\nabla\_\{x\_\{t\}\}\\cdot\\left\(p\_\{t\}^\{v\}\(x\_\{t\}\)v\_\{t\}\(x\_\{t\}\)\\right\)=0,\\qquad p\_\{0\}^\{v\}=p\_\{0\}in the weak sense\.

###### Proof\.

According to\[[29](https://arxiv.org/html/2609.05727#bib.bib47), Chapter 6\],Φt→0v\\Phi\_\{t\\to 0\}^\{v\}is aC1C^\{1\}diffeomorphism\. Then, we have the explicit expression

ptv​\(y\)=p0​\(Φt→0v​\(y\)\)​\|det∇Φt→0v​\(y\)\|\.p\_\{t\}^\{v\}\(y\)=p\_\{0\}\(\\Phi\_\{t\\to 0\}^\{v\}\(y\)\)\\left\|\\det\\nabla\\Phi\_\{t\\to 0\}^\{v\}\(y\)\\right\|\.Hence,ptvp\_\{t\}^\{v\}is a positive, normalized, continuous density, rather than merely a probability measure\. Furthermore, \([152](https://arxiv.org/html/2609.05727#A1.E152)\) implies that

𝔼Xt∼ptv​\[‖Xt‖22\]≤𝔼X0∼𝒩⁡\(0,I\)​\[e2​‖v‖𝒱^​\(‖X0‖2\+‖v‖𝒱^\)2\]<∞\.\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{v\}\}\\left\[\\left\\\|X\_\{t\}\\right\\\|\_\{2\}^\{2\}\\right\]\\leq\\mathbb\{E\}\_\{X\_\{0\}\\sim\\mathcal\{N\}\(0,I\)\}\\left\[e^\{2\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}\}\(\\left\\\|X\_\{0\}\\right\\\|\_\{2\}\+\\left\\\|v\\right\\\|\_\{\\widehat\{\\mathscr\{V\}\}\}\)^\{2\}\\right\]<\\infty\.In other words,ptvp\_\{t\}^\{v\}has finite second moment\. The continuity equation is a classical result\[[3](https://arxiv.org/html/2609.05727#bib.bib33), Section 8\.1\]\. ∎

Throughout this paper, we identify a densityppwith its associated probability measurep​d​xp\\,\\mathrm\{d\}x\. Therefore, we can writeptv:=\(Φ0→tv\)\#​p0p\_\{t\}^\{v\}:=\(\\Phi\_\{0\\to t\}^\{v\}\)\_\{\\\#\}p\_\{0\}\.

Lemma[54](https://arxiv.org/html/2609.05727#Thmtheorem54)shows that every ambient velocity field generates a density of the following form\. This motivates a density class that is broad enough to contain all ambient terminal densities while retaining the moment and positivity assumptions needed later\.

###### Definition 55\.

The ambient density class is defined by

𝒫^:=\{ρ∈C\(ℝd;\(0,\+∞\)\):∫ℝdρ\(x\)dx=1,𝔼X∼ρ\[‖X‖22\]<∞\}\.\\widehat\{\\mathscr\{P\}\}:=\\left\\\{\\rho\\in C\\left\(\\mathbb\{R\}^\{d\};\(0,\+\\infty\)\\right\):\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(x\)\\,\\mathrm\{d\}x=1,\\,\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\left\\\|X\\right\\\|\_\{2\}^\{2\}\\right\]<\\infty\\right\\\}\.
Define the ambient terminal\-density map by

𝒯^:𝒱^→𝒫^,v↦ρv:=\(Φ0→1v\)\#​p0,\\widehat\{\\mathcal\{T\}\}:\\widehat\{\\mathscr\{V\}\}\\to\\widehat\{\\mathscr\{P\}\},\\qquad v\\mapsto\\rho^\{v\}:=\(\\Phi\_\{0\\to 1\}^\{v\}\)\_\{\\\#\}p\_\{0\},whereΦs→tv:ℝd→ℝd\\Phi\_\{s\\to t\}^\{v\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}is the Carathéodory ODE flow generated byvv\.

The finite\-second\-moment condition𝔼X1∼ρ​\[‖X1‖22\]<∞\\mathbb\{E\}\_\{X\_\{1\}\\sim\\rho\}\\left\[\\left\\\|X\_\{1\}\\right\\\|\_\{2\}^\{2\}\\right\]<\\infty, together with the boundedness ofα˙\\dot\{\\alpha\}andβ˙\\dot\{\\beta\}implied by their continuity on the compact interval\[0,1\]\[0,1\], is precisely what is needed for square integrability of the CFM target\. For everyρ∈𝒫^\\rho\\in\\widehat\{\\mathscr\{P\}\}, we have

𝔼t∼U⁡\(0,1\),X0∼p0,X1∼ρ​\[‖α˙t​X1\+β˙t​X0‖22\]<∞\.\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{1\}\\sim\\rho\}\\left\[\\left\\\|\\dot\{\\alpha\}\_\{t\}X\_\{1\}\+\\dot\{\\beta\}\_\{t\}X\_\{0\}\\right\\\|\_\{2\}^\{2\}\\right\]<\\infty\.Consequently, the CFM loss \([3](https://arxiv.org/html/2609.05727#S2.E3)\) and the conditional\-expectation velocity are integrable whenever

𝔼t∼U⁡\(0,1\),X0∼p0,X1∼ρ​\[‖vtθ​\(Xt\)‖22\]<∞\.\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{1\}\\sim\\rho\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]<\\infty\.Hereθ\\thetais the parameter of the CFM neural network\. Strict positivityρ⁡\(x\)\>0\\rho\(x\)\>0is imposed to avoid singularities in the log\-density ratios, the KL divergence, and the Fisher–Rao metric used later\.

#### A\.1\.3Ambient Canonical\-Velocity Map

Fix the interpolant schedule\(αt,βt\)t∈\(0,1\)\(\\alpha\_\{t\},\\beta\_\{t\}\)\_\{t\\in\(0,1\)\}\. For each terminal density, the population minimizer of the CFM loss \([3](https://arxiv.org/html/2609.05727#S2.E3)\) defines a pointwise velocity field through conditional expectation \([4](https://arxiv.org/html/2609.05727#S2.E4)\)\. We first establish that this assignment and its marginal probability path are well\-defined\. The resulting velocity need not belong to𝒱^\\widehat\{\\mathscr\{V\}\}for everyρ∈𝒫^\\rho\\in\\widehat\{\\mathscr\{P\}\}; terminal correctness is asserted whenever the required ambient velocity regularity holds\.

###### Lemma 56\.

Consider any terminal densityρ∈𝒫^\\rho\\in\\widehat\{\\mathscr\{P\}\}\. For allt∈\[0,1\)t\\in\[0,1\)andxt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}, the computations

ptρ​\(xt\):=∫ℝdpt\|1​\(xt\|x1\)​ρ​\(x1\)​d​x1\>0,vtρ​\(xt\):=∫ℝdvt\|1​\(xt\|x1\)​pt\|1​\(xt\|x1\)​ρ​\(x1\)​d​x1ptρ​\(xt\)p\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\int\_\{\\mathbb\{R\}^\{d\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\rho\(x\_\{1\}\)\\,\\mathrm\{d\}x\_\{1\}\>0,\\qquad v\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\frac\{\\int\_\{\\mathbb\{R\}^\{d\}\}v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\rho\(x\_\{1\}\)\\,\\mathrm\{d\}x\_\{1\}\}\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}are well\-defined\. Fort=1t=1, we define

p1ρ​\(x1\):=ρ⁡\(x1\),v1ρ​\(x1\):=α˙1​x1\.p\_\{1\}^\{\\rho\}\(x\_\{1\}\):=\\rho\(x\_\{1\}\),\\qquad v\_\{1\}^\{\\rho\}\(x\_\{1\}\):=\\dot\{\\alpha\}\_\{1\}x\_\{1\}\.In particular,\(vtρ,ptρ\)\(v\_\{t\}^\{\\rho\},p\_\{t\}^\{\\rho\}\)satisfies the continuity equation weakly:

∂tptρ​\(xt\)\+∇xt⋅\(ptρ​\(xt\)​vtρ​\(xt\)\)=0,p0ρ=p0,p1ρ=ρ\.\\partial\_\{t\}p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\nabla\_\{x\_\{t\}\}\\cdot\\left\(p\_\{t\}^\{\\rho\}\(x\_\{t\}\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)=0,\\qquad p\_\{0\}^\{\\rho\}=p\_\{0\},\\qquad p\_\{1\}^\{\\rho\}=\\rho\.
Moreover, ifvρ∈𝒱^v^\{\\rho\}\\in\\widehat\{\\mathscr\{V\}\}, then

ρ=𝒯^​\(vρ\)\.\\rho=\\widehat\{\\mathcal\{T\}\}\(v^\{\\rho\}\)\.

###### Proof\.

According to \([8](https://arxiv.org/html/2609.05727#S3.E8)\),pt\|1\(xt\|⋅\)p\_\{t\|1\}\(x\_\{t\}\|\\cdot\)is bounded and positive ift∈\[0,1\)t\\in\[0,1\); hence, the integralptρ​\(xt\)p\_\{t\}^\{\\rho\}\(x\_\{t\}\)is finite and positive\. Ift=1t=1, thenp1ρ​\(x1\)=ρ⁡\(x1\)p\_\{1\}^\{\\rho\}\(x\_\{1\}\)=\\rho\(x\_\{1\}\)is also positive\. By \([9](https://arxiv.org/html/2609.05727#S3.E9)\),vt\|1\(xt\|⋅\)pt\|1\(xt\|⋅\)v\_\{t\|1\}\(x\_\{t\}\|\\cdot\)p\_\{t\|1\}\(x\_\{t\}\|\\cdot\)has at most linear growth ift∈\[0,1\)t\\in\[0,1\); hence, the numerator ofvtρ​\(xt\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)is absolutely convergent\.

According to\[[33](https://arxiv.org/html/2609.05727#bib.bib1), Theorem 1\],vtρv\_\{t\}^\{\\rho\}andptρp\_\{t\}^\{\\rho\}satisfy the continuity equation\.

By Jensen’s inequality, we have

∫01𝔼Xt∼ptρ​\[‖vtρ​\(Xt\)‖22\]​dt=\\displaystyle\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t=∫01𝔼Xt∼ptρ\[‖𝔼X1∼p1\|tρ\(⋅\|Xt\)\[vt\|1\(Xt\|X1\)\]‖22\]dt\\displaystyle\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\}\}\\left\[\\left\\\|\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\]\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t≤\\displaystyle\\leq∫01𝔼Xt∼ptρ,X1∼p1\|tρ\(⋅\|Xt\)\[‖vt\|1\(Xt\|X1\)‖22\]dt\\displaystyle\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\},\\,X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\}\\left\[\\left\\\|v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t=\\displaystyle=∫01𝔼X0∼p0,X1∼ρ​\[‖α˙t​X1\+β˙t​X0‖22\]​𝑑t\\displaystyle\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{X\_\{0\}\\sim p\_\{0\},\\,X\_\{1\}\\sim\\rho\}\\left\[\\left\\\|\\dot\{\\alpha\}\_\{t\}X\_\{1\}\+\\dot\{\\beta\}\_\{t\}X\_\{0\}\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t=\\displaystyle=𝔼X0∼p0​\[‖X0‖22\]​∫01β˙t2​𝑑t\+𝔼X1∼ρ​\[‖X1‖22\]​∫01α˙t2​𝑑t\\displaystyle\\mathbb\{E\}\_\{X\_\{0\}\\sim p\_\{0\}\}\\left\[\\left\\\|X\_\{0\}\\right\\\|\_\{2\}^\{2\}\\right\]\\int\_\{0\}^\{1\}\\dot\{\\beta\}\_\{t\}^\{2\}\\mathrm\{d\}t\+\\mathbb\{E\}\_\{X\_\{1\}\\sim\\rho\}\\left\[\\left\\\|X\_\{1\}\\right\\\|\_\{2\}^\{2\}\\right\]\\int\_\{0\}^\{1\}\\dot\{\\alpha\}\_\{t\}^\{2\}\\mathrm\{d\}t<\\displaystyle<∞,\\displaystyle\\infty,which implies the regularity condition\[[3](https://arxiv.org/html/2609.05727#bib.bib33), Equation \(8\.1\.2\)\]\. Ifvρ∈𝒱^v^\{\\rho\}\\in\\widehat\{\\mathscr\{V\}\}, then the integrability condition\[[3](https://arxiv.org/html/2609.05727#bib.bib33), Equation \(8\.1\.7\)\]holds, and the ODE flowΦvρ\\Phi^\{v^\{\\rho\}\}is well\-defined\. By the representation formula for the continuity equation\[[3](https://arxiv.org/html/2609.05727#bib.bib33), Proposition 8\.1\.8\],ρ\\rhois the terminal density ofvv\. In this case, we haveρ=𝒯^​\(vρ\)\\rho=\\widehat\{\\mathcal\{T\}\}\(v^\{\\rho\}\)\. ∎

###### Definition 57\.

For any terminal densityρ∈𝒫^\\rho\\in\\widehat\{\\mathscr\{P\}\}, its canonical velocity fieldvρ=\(vtρ\)t∈\[0,1\]v^\{\\rho\}=\(v\_\{t\}^\{\\rho\}\)\_\{t\\in\[0,1\]\}is defined as

vtρ​\(xt\)=\{𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|X1\)\],t∈\[0,1\),α˙1​x1,t=1,v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\begin\{cases\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\],&t\\in\[0,1\),\\\\ \\dot\{\\alpha\}\_\{1\}x\_\{1\},&t=1,\\end\{cases\}which is pointwise well\-defined and serves as the population minimizer of the CFM loss \([3](https://arxiv.org/html/2609.05727#S2.E3)\)\. We define the ambient canonical\-velocity map as

𝒞^:𝒫^→\{v:\[0,1\]×ℝd→ℝd\},ρ↦vρ=\(vtρ\)t∈\[0,1\]\.\\widehat\{\\mathcal\{C\}\}:\\widehat\{\\mathscr\{P\}\}\\to\\left\\\{v:\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}\\right\\\},\\qquad\\rho\\mapsto v^\{\\rho\}=\(v\_\{t\}^\{\\rho\}\)\_\{t\\in\[0,1\]\}\.

###### Corollary 58\.

On the set of densities whose canonical velocity belongs to𝒱^\\widehat\{\\mathscr\{V\}\}, the ambient canonical\-velocity map is the inverse of the restricted ambient terminal\-density map:

𝒯^\|𝒞^​\(𝒫^\)∩𝒱^=\(𝒞^\|𝒞^−1​\(𝒱^\)\)−1\.\\left\.\\widehat\{\\mathcal\{T\}\}\\right\|\_\{\\widehat\{\\mathcal\{C\}\}\\left\(\\widehat\{\\mathscr\{P\}\}\\right\)\\cap\\widehat\{\\mathscr\{V\}\}\}=\\left\(\\left\.\\widehat\{\\mathcal\{C\}\}\\right\|\_\{\\widehat\{\\mathcal\{C\}\}^\{\-1\}\\left\(\\widehat\{\\mathscr\{V\}\}\\right\)\}\\right\)^\{\-1\}\.

###### Proof\.

For everyρ∈𝒞^−1​\(𝒱^\)\\rho\\in\\widehat\{\\mathcal\{C\}\}^\{\-1\}\(\\widehat\{\\mathscr\{V\}\}\), Lemma[56](https://arxiv.org/html/2609.05727#Thmtheorem56)gives𝒯^​\(𝒞^​\(ρ\)\)=ρ\\widehat\{\\mathcal\{T\}\}\(\\widehat\{\\mathcal\{C\}\}\(\\rho\)\)=\\rho, which finishes the proof\. ∎

The relations among these ambient classes and their maps are summarized in Figure[8](https://arxiv.org/html/2609.05727#A1.F8)\.

### A\.2Set\-Theoretic Definitions

The ambient classes in Appendix[A\.1](https://arxiv.org/html/2609.05727#A1.SS1)are deliberately broad and need not be closed under the operations required by Newton Matching\. We therefore select a density class and a velocity class on which both the terminal\-density map and the canonical\-velocity map take values in the desired classes\. At this stage, the density and velocity classes are treated simply as sets, and the associated maps as set maps; no topology or smooth manifold structure is introduced\.

#### A\.2\.1Admissible Subclasses and Restricted Maps

𝒱^\\widehat\{\\mathscr\{V\}\}ambient velocity space𝒞^​\(𝒫\)⊂𝒱⊂𝒯^−1​\(𝒫\)\\widehat\{\\mathcal\{C\}\}\(\\mathscr\{P\}\)\\subset\\mathscr\{V\}\\subset\\widehat\{\\mathcal\{T\}\}^\{\-1\}\(\\mathscr\{P\}\)velocity space𝒫^\\widehat\{\\mathscr\{P\}\}ambient density class𝒫⊂𝒞^−1​\(𝒱^\)\\mathscr\{P\}\\subset\\widehat\{\\mathcal\{C\}\}^\{\-1\}\(\\widehat\{\\mathscr\{V\}\}\)density manifoldℰ\\mathscr\{E\}energy space𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}open energy domain𝒯^\\widehat\{\\mathcal\{T\}\}𝒞^\\widehat\{\\mathcal\{C\}\}𝒯\\mathcal\{T\}𝒞\\mathcal\{C\}Log\\mathrm\{Log\}Exp\\mathrm\{Exp\}Figure 8:Construction of ambient classes, admissible subclasses, and smooth structures\.We first select terminal densities whose canonical velocity fields satisfy the ambient velocity regularity\. This condition guarantees that the canonical assignment is compatible with the ambient terminal\-density map\.

###### Definition 60\.

A density class𝒫\\mathscr\{P\}is a non\-empty subset of the ambient density class𝒫^\\widehat\{\\mathscr\{P\}\}satisfying

𝒫⊂𝒞^−1​\(𝒱^\)⊂𝒫^\.\\mathscr\{P\}\\subset\\widehat\{\\mathcal\{C\}\}^\{\-1\}\\left\(\\widehat\{\\mathscr\{V\}\}\\right\)\\subset\\widehat\{\\mathscr\{P\}\}\.

Definition[60](https://arxiv.org/html/2609.05727#Thmtheorem60)ensures that every selected density has a canonical velocity in𝒱^\\widehat\{\\mathscr\{V\}\}\. Ambient terminal correctness then places that canonical velocity in the corresponding terminal\-density preimage\.

###### Corollary 61\.

𝒞^​\(𝒫\)⊂𝒯^−1​\(𝒫\)\\widehat\{\\mathcal\{C\}\}\(\\mathscr\{P\}\)\\subset\\widehat\{\\mathcal\{T\}\}^\{\-1\}\(\\mathscr\{P\}\)\.

###### Proof\.

By Definition[60](https://arxiv.org/html/2609.05727#Thmtheorem60), we have𝒫⊂𝒞^−1​\(𝒱^\)\\mathscr\{P\}\\subset\\widehat\{\\mathcal\{C\}\}^\{\-1\}\\left\(\\widehat\{\\mathscr\{V\}\}\\right\)\. Then,

𝒞^​\(𝒫\)=𝒞^\|𝒞^−1​\(𝒱^\)​\(𝒫\)=\(𝒯^\|𝒞^​\(𝒫^\)∩𝒱^\)−1​\(𝒫\)⊂𝒯^−1​\(𝒫\)\\widehat\{\\mathcal\{C\}\}\(\\mathscr\{P\}\)=\\left\.\\widehat\{\\mathcal\{C\}\}\\right\|\_\{\\widehat\{\\mathcal\{C\}\}^\{\-1\}\\left\(\\widehat\{\\mathscr\{V\}\}\\right\)\}\(\\mathscr\{P\}\)=\\left\(\\left\.\\widehat\{\\mathcal\{T\}\}\\right\|\_\{\\widehat\{\\mathcal\{C\}\}\\left\(\\widehat\{\\mathscr\{P\}\}\\right\)\\cap\\widehat\{\\mathscr\{V\}\}\}\\right\)^\{\-1\}\(\\mathscr\{P\}\)\\subset\\widehat\{\\mathcal\{T\}\}^\{\-1\}\(\\mathscr\{P\}\)is given by Corollary[58](https://arxiv.org/html/2609.05727#Thmtheorem58)\. ∎

Based on Corollary[61](https://arxiv.org/html/2609.05727#Thmtheorem61), we can define the admissible velocity subclasses as follows\. The selected velocity class may be strictly smaller than the full terminal\-density preimage, which allows additional regularity to be imposed without excluding the canonical velocities\. We also define the maps between these admissible subclasses as the restricted maps in Appendix[A\.1](https://arxiv.org/html/2609.05727#A1.SS1)\.

###### Definition 62\.

A velocity class𝒱\\mathscr\{V\}compatible with𝒫\\mathscr\{P\}is defined as a subset of the ambient velocity space𝒱^\\widehat\{\\mathscr\{V\}\}satisfying

𝒞^​\(𝒫\)⊂𝒱⊂𝒯^−1​\(𝒫\)⊂𝒱^\.\\widehat\{\\mathcal\{C\}\}\(\\mathscr\{P\}\)\\subset\\mathscr\{V\}\\subset\\widehat\{\\mathcal\{T\}\}^\{\-1\}\(\\mathscr\{P\}\)\\subset\\widehat\{\\mathscr\{V\}\}\.The terminal\-density map is defined as

𝒯:=𝒯^\|𝒱:𝒱→𝒫,v↦ρv:=\(Φ0→1v\)\#​p0\.\\mathcal\{T\}:=\\left\.\\widehat\{\\mathcal\{T\}\}\\right\|\_\{\\mathscr\{V\}\}:\\mathscr\{V\}\\to\\mathscr\{P\},\\qquad v\\mapsto\\rho^\{v\}:=\(\\Phi\_\{0\\to 1\}^\{v\}\)\_\{\\\#\}p\_\{0\}\.The canonical\-velocity map is defined as

𝒞:=𝒞^\|𝒫:𝒫→𝒞\(𝒫\)⊂𝒱,ρ↦vρ,vtρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt\|1\(xt\|X1\)\]\.\\mathcal\{C\}:=\\left\.\\widehat\{\\mathcal\{C\}\}\\right\|\_\{\\mathscr\{P\}\}:\\mathscr\{P\}\\to\\mathcal\{C\}\(\\mathscr\{P\}\)\\subset\\mathscr\{V\},\\qquad\\rho\\mapsto v^\{\\rho\},\\qquad v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]\.

The defining restrictions preserve the ambient terminal\-correctness identity\.

###### Corollary 63\.

The canonical\-velocity map𝒞\\mathcal\{C\}is a right inverse of the terminal\-density map𝒯\\mathcal\{T\}:

𝒯∘𝒞=Id𝒫\.\\mathcal\{T\}\\circ\\mathcal\{C\}=\\mathrm\{Id\}\_\{\\mathscr\{P\}\}\.

###### Proof\.

This corollary is the restriction of Corollary[58](https://arxiv.org/html/2609.05727#Thmtheorem58)to𝒫\\mathscr\{P\}\. ∎

The relations among the two admissible subclasses and their maps are summarized in Figure[8](https://arxiv.org/html/2609.05727#A1.F8)\.

#### A\.2\.2Canonical Subclass

The following definition singles out the canonical velocity fieldvρv^\{\\rho\}for each densityρ\\rho\. Its image forms the canonical subclass\.

###### Definition 64\.

The canonical subclass𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}consists of the canonical velocity field of densities in𝒫\\mathscr\{P\}, i\.e\.,

𝒱can:=𝒞⁡\(𝒫\)⊂𝒱\.\\mathscr\{V\}\_\{\\mathrm\{can\}\}:=\\mathcal\{C\}\(\\mathscr\{P\}\)\\subset\\mathscr\{V\}\.The canonical projection is defined as

ℛ:=𝒞∘𝒯:𝒱→𝒱can\.\\mathcal\{R\}:=\\mathcal\{C\}\\circ\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

###### Theorem 65\.

The canonical\-velocity map𝒞\\mathcal\{C\}is the inverse of the restricted terminal\-density map, i\.e\.,

𝒯\|𝒱can=𝒞−1\.\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathcal\{C\}^\{\-1\}\.In other words, ifvρ=𝒞⁡\(ρ\)∈𝒱v^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}is a canonical velocity field of a densityρ∈𝒫\\rho\\in\\mathscr\{P\}, thenρ\\rhois also the terminal density𝒯⁡\(vρ\)\\mathcal\{T\}\(v^\{\\rho\}\)ofvρv^\{\\rho\}\. Therefore, the canonical\-velocity map𝒞\\mathcal\{C\}induces a set\-theoretic identification, i\.e\.,

𝒫≅set𝒱can\.\\mathscr\{P\}\\cong\_\{\\mathrm\{set\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.
Consequently, the canonical projectionℛ\\mathcal\{R\}preserves the terminal density, i\.e\.,

𝒯∘ℛ=𝒯\.\\mathcal\{T\}\\circ\\mathcal\{R\}=\\mathcal\{T\}\.In other words, for anyv∈𝒱v\\in\\mathscr\{V\}, its canonical velocityℛ⁡\(v\)∈𝒱\\mathcal\{R\}\(v\)\\in\\mathscr\{V\}has the same terminal density asvv, i\.e\.,𝒯⁡\(v\)=𝒯⁡\(ℛ⁡\(v\)\)∈𝒫\\mathcal\{T\}\(v\)=\\mathcal\{T\}\(\\mathcal\{R\}\(v\)\)\\in\\mathscr\{P\}\.

Furthermore, the canonical projectionℛ\\mathcal\{R\}is idempotent, i\.e\.,

ℛ2=ℛ,ℛ\|𝒱can=Id𝒱can\.\\mathcal\{R\}^\{2\}=\\mathcal\{R\},\\qquad\\left\.\\mathcal\{R\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathrm\{Id\}\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\.In other words, the projection of a canonical velocity fieldvρ∈𝒱canv^\{\\rho\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}is itself, i\.e\.,ℛ⁡\(vρ\)=vρ\\mathcal\{R\}\(v^\{\\rho\}\)=v^\{\\rho\}\.

###### Proof\.

According to Corollary[63](https://arxiv.org/html/2609.05727#Thmtheorem63), we have𝒯\|𝒱can=𝒞−1\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathcal\{C\}^\{\-1\}\. Consequently, we have

𝒯∘ℛ=𝒯∘𝒞∘𝒯=𝒯\|𝒱can∘𝒞∘𝒯=𝒞−1∘𝒞∘𝒯=𝒯\.\\mathcal\{T\}\\circ\\mathcal\{R\}=\\mathcal\{T\}\\circ\\mathcal\{C\}\\circ\\mathcal\{T\}=\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\\circ\\mathcal\{C\}\\circ\\mathcal\{T\}=\\mathcal\{C\}^\{\-1\}\\circ\\mathcal\{C\}\\circ\\mathcal\{T\}=\\mathcal\{T\}\.Furthermore,

ℛ2=𝒞∘𝒯∘ℛ=𝒞∘𝒯=ℛ,ℛ\|𝒱can=𝒞∘𝒯\|𝒱can=𝒞∘𝒞−1=Id𝒱can,\\mathcal\{R\}^\{2\}=\\mathcal\{C\}\\circ\\mathcal\{T\}\\circ\\mathcal\{R\}=\\mathcal\{C\}\\circ\\mathcal\{T\}=\\mathcal\{R\},\\qquad\\left\.\\mathcal\{R\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathcal\{C\}\\circ\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathcal\{C\}\\circ\\mathcal\{C\}^\{\-1\}=\\mathrm\{Id\}\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\},which finishes the proof\. ∎

The construction of the canonical subclass is shown in Figure[3](https://arxiv.org/html/2609.05727#S4.F3)\.

### A\.3Smooth Geometric Realization

Appendix[A\.2](https://arxiv.org/html/2609.05727#A1.SS2)establishes the canonical correspondence only at the level of sets\. Those identities identify a unique canonical velocity representative for each terminal density, but they do not yet support the differential operations used in Sections[4\.2](https://arxiv.org/html/2609.05727#S4.SS2)and[4\.3](https://arxiv.org/html/2609.05727#S4.SS3)\. Tangent and cotangent spaces, metric duality, covariant differentiation, and smooth retractions require compatible Banach\-manifold structures\. In this subsection, we therefore equip𝒫\\mathscr\{P\}with a global centered\-energy chart and𝒱\\mathscr\{V\}with the Banach structure inherited from its realization as a closed linear subspace of𝒱^\\widehat\{\\mathscr\{V\}\}\. Assuming that𝒯\\mathcal\{T\}and𝒞\\mathcal\{C\}are smooth, the set\-theoretic identities from Appendix[A\.2](https://arxiv.org/html/2609.05727#A1.SS2)then become smooth geometric statements:𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a split embedded Banach submanifold of𝒱\\mathscr\{V\},𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a diffeomorphism, andℛ\\mathcal\{R\}is a smooth terminal\-density\-preserving projection\.

Appendix[A\.3\.1](https://arxiv.org/html/2609.05727#A1.SS3.SSS1)specifies this compatible smooth realization and establishes the global manifold structure\. Appendix[A\.3\.2](https://arxiv.org/html/2609.05727#A1.SS3.SSS2)then identifies the tangent and cotangent spaces in centered\-energy coordinates and states the additional smoothness condition required by the Fisher–Rao metric\. Finally, Appendix[A\.3\.3](https://arxiv.org/html/2609.05727#A1.SS3.SSS3)pulls the Fisher–Rao metric back to the global energy chart, constructs the flat energy connection and the mixture connection, and proves the fixed\-observable characterization used in Section[4\.3](https://arxiv.org/html/2609.05727#S4.SS3)\.

#### A\.3\.1Compatible Smooth Structures

We use a global centered log\-density coordinate on𝒫\\mathscr\{P\}\. This choice is in the spirit of exponential\-coordinate constructions in information geometry, while the assumptions below are stated directly for the selected density class and need not coincide with a maximal nonparametric statistical manifold\[[4](https://arxiv.org/html/2609.05727#bib.bib46)\]\.

###### Definition 66\.

A compatible smooth realization of\(𝒫,𝒱\)\(\\mathscr\{P\},\\mathscr\{V\}\)consists of the following structures\.

1. 1\.There exists a Banach function space𝔹⊂C⁡\(ℝd,ℝ\)\\mathbb\{B\}\\subset C\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}\)which contains the constant functions and induces an energy space ℰ:=𝔹/ℝ=\{\[f\]:=\{f\+c:c∈ℝ\}:f∈𝔹\},\\mathscr\{E\}:=\{\\left\.\\mathbb\{B\}\\middle/\\mathbb\{R\}\\right\.\}=\\left\\\{\[f\]:=\\left\\\{f\+c:c\\in\\mathbb\{R\}\\right\\\}:f\\in\\mathbb\{B\}\\right\\\},such that the energy spaceℰ\\mathscr\{E\}contains a non\-empty open subset 𝒪ℰ⊂\{\[f\]∈ℰ:∫ℝdef⁡\(x\)​𝑑x<∞\},\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\subset\\left\\\{\[f\]\\in\\mathscr\{E\}:\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{f\(x\)\}\\,\\mathrm\{d\}x<\\infty\\right\\\},and the centered\-energy map Log:𝒫→𝒪ℰ,ρ↦\[log⁡ρ\]\\mathrm\{Log\}:\\mathscr\{P\}\\to\\mathcal\{O\}\_\{\\mathscr\{E\}\},\\qquad\\rho\\mapsto\[\\log\\rho\]is bijective\.
2. 2\.The density class𝒫\\mathscr\{P\}is equipped with the global Banach chart\(𝒫,Log\)\(\\mathscr\{P\},\\mathrm\{Log\}\)\. Equivalently, this is the unique smooth Banach\-manifold structure on𝒫\\mathscr\{P\}for whichLog:𝒫→∼𝒪ℰ\\mathrm\{Log\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}is a diffeomorphism\.
3. 3\.The velocity class𝒱\\mathscr\{V\}is a closed linear subspace of𝒱^\\widehat\{\\mathscr\{V\}\}; hence,𝒱\\mathscr\{V\}is a Banach space under the restricted norm\. We equip𝒱\\mathscr\{V\}with the induced smooth Banach\-manifold structure\.
4. 4\.The terminal\-density map𝒯:𝒱→𝒫\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{P\}and the canonical\-velocity map𝒞:𝒫→𝒱\\mathcal\{C\}:\\mathscr\{P\}\\to\\mathscr\{V\}are smooth\.

In Definition[66](https://arxiv.org/html/2609.05727#Thmtheorem66),ℰ=𝔹/ℝ\\mathscr\{E\}=\{\\left\.\\mathbb\{B\}\\middle/\\mathbb\{R\}\\right\.\}is also a Banach space\. Let‖⋅‖𝔹\\left\\\|\\cdot\\right\\\|\_\{\\mathbb\{B\}\}denote the norm on𝔹\\mathbb\{B\}\. Then, the norm ofℰ\\mathscr\{E\}is

‖\[f\]‖ℰ:=infc∈ℝ‖f\+c‖𝔹\.\\left\\\|\[f\]\\right\\\|\_\{\\mathscr\{E\}\}:=\\inf\_\{c\\in\\mathbb\{R\}\}\\left\\\|f\+c\\right\\\|\_\{\\mathbb\{B\}\}\.Since\{x↦c:c∈ℝ\}\\left\\\{x\\mapsto c:c\\in\\mathbb\{R\}\\right\\\}is a closed one\-dimensional subspace of𝔹\\mathbb\{B\}, the energy spaceℰ\\mathscr\{E\}is a Banach space under the norm‖⋅‖ℰ\\left\\\|\\cdot\\right\\\|\_\{\\mathscr\{E\}\}\.

Note that for anyc∈ℝc\\in\\mathbb\{R\},

∫ℝdef⁡\(x\)dx<∞⇔∫ℝdef⁡\(x\)\+cdx=ec∫ℝdef⁡\(x\)dx<∞\.\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{f\(x\)\}\\,\\mathrm\{d\}x<\\infty\\quad\\Leftrightarrow\\quad\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{f\(x\)\+c\}\\,\\mathrm\{d\}x=e^\{c\}\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{f\(x\)\}\\,\\mathrm\{d\}x<\\infty\.Therefore, the open subset𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}is well\-defined\.

We assume the existence of compatible smooth structures as follows and fix them throughout this paper\.

###### Assumption 67\.

The density class𝒫\\mathscr\{P\}and the velocity class𝒱\\mathscr\{V\}are chosen so that they admit at least one pair of compatible smooth structures in Definition[66](https://arxiv.org/html/2609.05727#Thmtheorem66)\.

###### Proposition 68\.

The canonical projectionℛ\\mathcal\{R\}is smooth\. The canonical\-velocity map embeds𝒫\\mathscr\{P\}as a split submanifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}of𝒱\\mathscr\{V\}, i\.e\.,

𝒞:𝒫→∼𝒱can⊂𝒱\.\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\subset\\mathscr\{V\}\.In particular,𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a split embedded Banach submanifold of𝒱\\mathscr\{V\}\. The inverse ofLog\\mathrm\{Log\}is the normalized exponential map

Exp:=Log−1:𝒪ℰ→∼𝒫,\[f\]↦ef∫ℝdef⁡\(x\)​𝑑x\.\\mathrm\{Exp\}:=\\mathrm\{Log\}^\{\-1\}:\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\xrightarrow\{\\sim\}\\mathscr\{P\},\\qquad\[f\]\\mapsto\\frac\{e^\{f\}\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{f\(x\)\}\\,\\mathrm\{d\}x\}\.

###### Proof\.

Since𝒞\\mathcal\{C\}and𝒯\\mathcal\{T\}are both smooth, the canonical projectionℛ=𝒞∘𝒯:𝒱→𝒱\\mathcal\{R\}=\\mathcal\{C\}\\circ\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{V\}is smooth under the topology of𝒱\\mathscr\{V\}\. Sinceℛ2=ℛ\\mathcal\{R\}^\{2\}=\\mathcal\{R\},\[[36](https://arxiv.org/html/2609.05727#bib.bib51), Lemma 9\.46\]implies that the image ofℛ\\mathcal\{R\}, i\.e\., the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}, is a split embedded Banach submanifold of𝒱\\mathscr\{V\}\.

Sinceℛ∘𝒞=𝒞\\mathcal\{R\}\\circ\\mathcal\{C\}=\\mathcal\{C\}, the corestriction𝒞:𝒫→𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\to\\mathscr\{V\}\_\{\\mathrm\{can\}\}is smooth\. Since𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is an embedded submanifold of𝒱\\mathscr\{V\}and𝒯:𝒱→𝒫\\mathcal\{T\}:\\mathscr\{V\}\\to\\mathscr\{P\}is smooth, the restriction𝒯\|𝒱can:𝒱can→𝒫\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}:\\mathscr\{V\}\_\{\\mathrm\{can\}\}\\to\\mathscr\{P\}is also smooth\. By Theorem[65](https://arxiv.org/html/2609.05727#Thmtheorem65),𝒞−1=𝒯\|𝒱can\.\\mathcal\{C\}^\{\-1\}=\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\.Therefore,𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a diffeomorphism\.

For every\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}, denoteρ=Log−1​\(\[f\]\)∈𝒫\\rho=\\mathrm\{Log\}^\{\-1\}\(\[f\]\)\\in\\mathscr\{P\}\. Then,f=log⁡ρ\+constf=\\log\\rho\+\\mathrm\{const\}\. Sinceρ\\rhois a normalized positive density, we haveρ=ef∫ℝdef⁡\(x\)​𝑑x\\rho=\\frac\{e^\{f\}\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{f\(x\)\}\\,\\mathrm\{d\}x\}, which gives the explicit expression ofExp=Log−1\\mathrm\{Exp\}=\\mathrm\{Log\}^\{\-1\}\. ∎

In Proposition[68](https://arxiv.org/html/2609.05727#Thmtheorem68),Exp=Log−1\\mathrm\{Exp\}=\\mathrm\{Log\}^\{\-1\}denotes the normalized exponential parametrization associated with the global energy chart\. It should not be conflated with the Riemannian exponential map of the density manifold\.

From now on,𝒫\\mathscr\{P\}and𝒱\\mathscr\{V\}carry the smooth structures fixed above, and𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}carries its split embedded submanifold structure\. We refer to𝒫\\mathscr\{P\}as the density manifold, to𝒱\\mathscr\{V\}as the velocity space, and to𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}as the canonical manifold\. The open set𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}is the global energy\-coordinate domain, whereasℰ\\mathscr\{E\}is its Banach model space\. We suppress the chosen smooth structures from the notation\.

For convenience of computation, we apply the following assumption in this paper\.

###### Assumption 69\.

Unless stated otherwise, all functions, curves, and vector fields considered throughout this paper are assumed sufficiently integrable and smooth for the displayed quantities and derivatives to be well defined\. Differentiation may be interchanged with integration, and repeated integrals may be reordered\.

#### A\.3\.2First\-Order Geometry

The global chartLog:𝒫→∼𝒪ℰ\\mathrm\{Log\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}reduces the first\-order geometry of𝒫\\mathscr\{P\}to differentiation in the Banach energy spaceℰ\\mathscr\{E\}\. We first differentiate the inverse chartExp:𝒪ℰ→∼𝒫\\mathrm\{Exp\}:\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\xrightarrow\{\\sim\}\\mathscr\{P\}, which converts additive energy perturbations into centered density perturbations and thereby identifies the tangent and cotangent spaces of𝒫\\mathscr\{P\}\. We then specialize this identification to the reverse\-KL log\-density residual and transport the resulting first\-order structures to𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}through𝒞\\mathcal\{C\}\. Finally, we impose the additional smoothness condition required by the Fisher–Rao metric in Section[4\.2\.3](https://arxiv.org/html/2609.05727#S4.SS2.SSS3)\.

The differential ofExp\\mathrm\{Exp\}involves expectations of representatives in𝔹\\mathbb\{B\}\. We therefore impose the following integrability condition\.

###### Assumption 70\.

Assume that for everyf∈𝔹f\\in\\mathbb\{B\}, andρ∈𝒫\\rho\\in\\mathscr\{P\}, we have𝔼ρ​\[\|f\|\]<∞\\mathbb\{E\}\_\{\\rho\}\[\\left\|f\\right\|\]<\\infty\.

###### Proposition 71\.

Since𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}is an open subset of the Banach spaceℰ\\mathscr\{E\}, we use the canonical identification

T\[f\]​𝒪ℰ≅ℰ,\[f\]∈𝒪ℰ\.T\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\cong\\mathscr\{E\},\\qquad\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}\.Consequently, every smooth tangent vector fieldE∈𝔛⁡\(𝒪ℰ\)E\\in\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\)can be identified with a smooth map

E:𝒪ℰ→ℰ\.E:\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\to\\mathscr\{E\}\.
Letρ=Exp⁡\(\[f\]\)∈𝒫\\rho=\\mathrm\{Exp\}\(\[f\]\)\\in\\mathscr\{P\}\. Then, for every\[h\]∈T\[f\]​ℰ≅ℰ\[h\]\\in T\_\{\[f\]\}\\mathscr\{E\}\\cong\\mathscr\{E\}, the differential ofExp\\mathrm\{Exp\}has the density\-function representation

d\[f\]​Exp​\[\[h\]\]=ρ⁡\(h−𝔼ρ​\[h\]\)\.\\mathrm\{d\}\_\{\[f\]\}\\mathrm\{Exp\}\[\[h\]\]=\\rho\\left\(h\-\\mathbb\{E\}\_\{\\rho\}\[h\]\\right\)\.

###### Proof\.

Since𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}is an open set of the Banach spaceℰ\\mathscr\{E\}, we haveT\[f\]​𝒪ℰ≅ℰT\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\cong\\mathscr\{E\}, and every smooth tangent vector field can be identified with a smooth map\.

Consider a smooth curve\(\[fε\]\)ε∈\(−δ,δ\)\(\[f^\{\\varepsilon\}\]\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}passing through\[f\]\[f\]atε=0\\varepsilon=0\. Letρε=Exp⁡\(\[fε\]\)\\rho^\{\\varepsilon\}=\\mathrm\{Exp\}\(\[f^\{\\varepsilon\}\]\)andh=dd​ε​fε\|ε=0h=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}f^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}, with\[h\]∈T\[f\]​𝒪ℰ\[h\]\\in T\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\. By Assumption[69](https://arxiv.org/html/2609.05727#Thmtheorem69), differentiation may be interchanged with integration, and hence

d\[f\]​Exp​\[h\]=dd​ε​ρε\|ε=0=dd​ε​efε∫ℝdefε​\(x\)​𝑑x\|ε=0=ρ⁡\(h−𝔼ρ​\[h\]\),\\mathrm\{d\}\_\{\[f\]\}\\mathrm\{Exp\}\[h\]=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\rho^\{\\varepsilon\}\\right\|\_\{\\varepsilon=0\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\frac\{e^\{f^\{\\varepsilon\}\}\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{f^\{\\varepsilon\}\(x\)\}\\,\\mathrm\{d\}x\}\\right\|\_\{\\varepsilon=0\}=\\rho\\left\(h\-\\mathbb\{E\}\_\{\\rho\}\[h\]\\right\),which finishes the proof\. ∎

The differential formula above converts an additive energy perturbation into its density\-function representation\. Consequently, the global energy chartLog\\mathrm\{Log\}identifies every density tangent space with the common Banach model spaceℰ\\mathscr\{E\}, and every density cotangent space with its continuous dualℰ∗\\mathscr\{E\}^\{\*\}\.

###### Theorem 72\.

For any densityρ∈𝒫\\rho\\in\\mathscr\{P\}, the tangent space of the density manifold𝒫\\mathscr\{P\}atρ\\rhois

Tρ​𝒫=\{ρ⋅\(f−𝔼ρ​\[f\]\):\[f\]∈ℰ\}≅ℰ\.T\_\{\\rho\}\\mathscr\{P\}=\\left\\\{\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\):\[f\]\\in\\mathscr\{E\}\\right\\\}\\cong\\mathscr\{E\}\.We can also relax the quotient and write

Tρ𝒫=\{ρ⋅\(f−𝔼ρ\[f\]\):f∈𝔹\}≅𝔹/ℝ≅ℰ\.T\_\{\\rho\}\\mathscr\{P\}=\\left\\\{\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\):f\\in\\mathbb\{B\}\\right\\\}\\cong\{\\left\.\\mathbb\{B\}\\middle/\\mathbb\{R\}\\right\.\}\\cong\\mathscr\{E\}\.The cotangent space of𝒫\\mathscr\{P\}atρ\\rhois

Tρ∗​𝒫=\{ℓ∘dρ​Log:ℓ∈ℰ∗\}≅ℰ∗\.T\_\{\\rho\}^\{\*\}\\mathscr\{P\}=\\left\\\{\\ell\\circ\\mathrm\{d\}\_\{\\rho\}\\mathrm\{Log\}:\\ell\\in\\mathscr\{E\}^\{\*\}\\right\\\}\\cong\\mathscr\{E\}^\{\*\}\.

###### Proof\.

By Proposition[71](https://arxiv.org/html/2609.05727#Thmtheorem71), we have

Tρ​𝒫=dLog⁡\(ρ\)​Exp​\(TLog⁡\(ρ\)​ℰ\)=\{ρ⋅\(f−𝔼ρ​\[f\]\):\[f\]∈ℰ\}\.T\_\{\\rho\}\\mathscr\{P\}=\\mathrm\{d\}\_\{\\mathrm\{Log\}\(\\rho\)\}\\mathrm\{Exp\}\(T\_\{\\mathrm\{Log\}\(\\rho\)\}\\mathscr\{E\}\)=\\left\\\{\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\):\[f\]\\in\\mathscr\{E\}\\right\\\}\.Evidently, we have

\{ρ⋅\(f−𝔼ρ​\[f\]\):\[f\]∈ℰ\}=\{ρ⋅\(f−𝔼ρ​\[f\]\):f∈𝔹\}\.\\left\\\{\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\):\[f\]\\in\\mathscr\{E\}\\right\\\}=\\left\\\{\\rho\\cdot\\left\(f\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\):f\\in\\mathbb\{B\}\\right\\\}\.The expression forTρ∗​𝒫T\_\{\\rho\}^\{\*\}\\mathscr\{P\}is straightforward\. ∎

Theorem[72](https://arxiv.org/html/2609.05727#Thmtheorem72)provides a common Banach model for all density tangent and cotangent spaces\. We next record the two consequences used directly by Newton Matching\.

###### Corollary 73\.

For anyρ,π∈𝒫\\rho,\\pi\\in\\mathscr\{P\}, we have

\[log⁡ρπ\]∈ℰ,ρ⋅\(log⁡ρπ−KL⁡\(ρ∥π\)\)∈Tρ​𝒫\.\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\in\\mathscr\{E\},\\qquad\\rho\\cdot\\left\(\\log\\frac\{\\rho\}\{\\pi\}\-\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\right\)\\in T\_\{\\rho\}\\mathscr\{P\}\.

###### Proof\.

Note that\[log⁡ρπ\]=Log⁡\(ρ\)−Log⁡\(π\)∈ℰ\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]=\\mathrm\{Log\}\(\\rho\)\-\\mathrm\{Log\}\(\\pi\)\\in\\mathscr\{E\}\. According to Theorem[72](https://arxiv.org/html/2609.05727#Thmtheorem72), we haveρ⋅\(log⁡ρπ−KL⁡\(ρ∥π\)\)∈Tρ​𝒫\\rho\\cdot\\left\(\\log\\frac\{\\rho\}\{\\pi\}\-\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\right\)\\in T\_\{\\rho\}\\mathscr\{P\}\. ∎

Corollary[73](https://arxiv.org/html/2609.05727#Thmtheorem73)shows that the centered reverse\-KL log\-density residual defines an admissible tangent vector on𝒫\\mathscr\{P\}\. Newton Matching, however, performs its updates in velocity\-field coordinates\. We therefore differentiate the canonical diffeomorphism to transport the tangent and cotangent structures from𝒫\\mathscr\{P\}to the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

###### Corollary 74\.

Letvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. Then

dρ​𝒞:Tρ​𝒫→∼Tvρ​𝒱can\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}:T\_\{\\rho\}\\mathscr\{P\}\\xrightarrow\{\\sim\}T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a Banach\-space isomorphism, and

Tvρ​𝒱can=dρ​𝒞​\(Tρ​𝒫\)\.T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}=\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\(T\_\{\\rho\}\\mathscr\{P\}\)\.Dually,

\(\(dρ​𝒞\)−1\)∗:Tρ∗​𝒫→∼Tvρ∗​𝒱can\\left\(\(\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\)^\{\-1\}\\right\)^\{\*\}:T\_\{\\rho\}^\{\*\}\\mathscr\{P\}\\xrightarrow\{\\sim\}T\_\{v^\{\\rho\}\}^\{\*\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}is a Banach\-space isomorphism\.

###### Proof\.

This follows by differentiating the diffeomorphism𝒞:𝒫→∼𝒱can\\mathcal\{C\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}and taking continuous duals\. ∎

The preceding results determine the admissible first\-order perturbations on both𝒫\\mathscr\{P\}and𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. To represent the relevant objective differentials by tangent update directions whenever their metric duals exist, we additionally equip𝒫\\mathscr\{P\}with the Fisher–Rao metric introduced in Section[4\.2\.3](https://arxiv.org/html/2609.05727#S4.SS2.SSS3)\. The Banach\-manifold structure alone does not guarantee that this bilinear form varies smoothly with the base point and tangent vectors, so we impose the following condition\.

###### Assumption 75\.

For everyρ∈𝒫\\rho\\in\\mathscr\{P\},

gρFR​\(ξ1,ξ2\)=∫ℝdξ1​\(x\)​ξ2​\(x\)ρ⁡\(x\)​𝑑xg\_\{\\rho\}^\{\\mathrm\{FR\}\}\(\\xi\_\{1\},\\xi\_\{2\}\)=\\int\_\{\\mathbb\{R\}^\{d\}\}\\frac\{\\xi\_\{1\}\(x\)\\xi\_\{2\}\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}xdefines a finite, continuous, symmetric, and positive\-definite bilinear form onTρ​𝒫T\_\{\\rho\}\\mathscr\{P\}\. Moreover, the map

gFR:\(ρ,ξ1,ξ2\)↦gρFR​\(ξ1,ξ2\),g^\{\\mathrm\{FR\}\}:\(\\rho,\\xi\_\{1\},\\xi\_\{2\}\)\\mapsto g\_\{\\rho\}^\{\\mathrm\{FR\}\}\(\\xi\_\{1\},\\xi\_\{2\}\),is smooth\.

Under Assumption[75](https://arxiv.org/html/2609.05727#Thmtheorem75), the Fisher–Rao metric defines the smooth, possibly weak, Riemannian structure used on𝒫\\mathscr\{P\}, and its transport through𝒞\\mathcal\{C\}defines the corresponding smooth metric on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. This completes the first\-order smooth realization of𝒫\\mathscr\{P\}and𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

To proceed to the second\-order geometry, we pull back the Fisher–Rao metric to𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}by defining at every\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}:

g\[f\]ℰ:ℰ×ℰ→ℝ,\(\[h1\],\[h2\]\)↦g\[f\]ℰ​\(\[h1\],\[h2\]\):=gExp⁡\(\[f\]\)FR​\(d\[f\]​Exp​\[h1\],d\[f\]​Exp​\[h2\]\)\.g^\{\\mathscr\{E\}\}\_\{\[f\]\}:\\mathscr\{E\}\\times\\mathscr\{E\}\\to\\mathbb\{R\},\\qquad\(\[h\_\{1\}\],\[h\_\{2\}\]\)\\mapsto g^\{\\mathscr\{E\}\}\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\]\):=g^\{\\mathrm\{FR\}\}\_\{\\mathrm\{Exp\}\(\[f\]\)\}\\left\(\\mathrm\{d\}\_\{\[f\]\}\\mathrm\{Exp\}\[h\_\{1\}\],\\mathrm\{d\}\_\{\[f\]\}\\mathrm\{Exp\}\[h\_\{2\}\]\\right\)\.In particular, denoteρ=Exp⁡\(\[f\]\)\\rho=\\mathrm\{Exp\}\(\[f\]\), and we have

g\[f\]ℰ​\(\[h1\],\[h2\]\)=gρFR​\(ξρ,h1,ξρ,h2\)=𝔼ρ​\[\(h1−𝔼ρ​\[h1\]\)​\(h2−𝔼ρ​\[h2\]\)\]\.g^\{\\mathscr\{E\}\}\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\]\)=g^\{\\mathrm\{FR\}\}\_\{\\rho\}\\left\(\\xi^\{\\rho,h\_\{1\}\},\\xi^\{\\rho,h\_\{2\}\}\\right\)=\\mathbb\{E\}\_\{\\rho\}\\left\[\(h\_\{1\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{1\}\\right\]\)\(h\_\{2\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{2\}\\right\]\)\\right\]\.
The Fisher–Rao flat map on𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}at\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}is defined as

G\[f\]♭:ℰ→ℰ∗,G\[f\]♭​\(\[h1\]\)​\(\[h2\]\):=g\[f\]ℰ​\(\[h1\],\[h2\]\)\.G\_\{\[f\]\}^\{\\flat\}:\\mathscr\{E\}\\to\\mathscr\{E\}^\{\*\},\\qquad G\_\{\[f\]\}^\{\\flat\}\(\[h\_\{1\}\]\)\(\[h\_\{2\}\]\):=g^\{\\mathscr\{E\}\}\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\]\)\.Due to the positive definiteness ofg\[f\]ℰg^\{\\mathscr\{E\}\}\_\{\[f\]\}, the Fisher–Rao flat mapG\[f\]♭G\_\{\[f\]\}^\{\\flat\}is injective\. Sincegℰg^\{\\mathscr\{E\}\}is possibly weak,G\[f\]♭G\_\{\[f\]\}^\{\\flat\}need not be surjective\.

#### A\.3\.3Second\-Order Geometry

The first\-order construction identifies tangent and cotangent vectors at each fixed density and equips the density manifold with the Fisher–Rao metric\. To define covariant derivatives and Hessians, we need to additionally compare tangent vector fields as the base point varies, which requires an affine connection\.

We first define the energy flat connection and then construct its dual connection with respect to the Fisher–Rao metric, which we call the energy mixture connection\. We subsequently transport the latter to𝒫\\mathscr\{P\}\. This construction is objective\-agnostic; the coincidence between the Newton direction and the negative Fisher–Rao gradient for the reverse\-KL objective is derived later as a consequence in Section[6\.2](https://arxiv.org/html/2609.05727#S6.SS2)\.

###### Proposition 76\.

For two smooth tangent vector fieldsE,H∈𝔛⁡\(𝒪ℰ\)E,H\\in\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\), define

\(∇Eℰ,flatH\)\[f\]:=D\[f\]​H​\[E\[f\]\]∈ℰ≅T\[f\]​𝒪ℰ\.\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}H\\right\)\_\{\[f\]\}:=\\mathrm\{D\}\_\{\[f\]\}H\\left\[E\_\{\[f\]\}\\right\]\\in\\mathscr\{E\}\\cong T\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\.Equivalently, consider a smooth curve\(\[fε\]\)ε∈\(−δ,δ\)⊂𝒪ℰ\(\[f^\{\\varepsilon\}\]\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathcal\{O\}\_\{\\mathscr\{E\}\}which passes through\[f\]\[f\]atε=0\\varepsilon=0and satisfiesE\[f\]=dd​ε​\[fε\]\|ε=0E\_\{\[f\]\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\[f^\{\\varepsilon\}\]\\right\|\_\{\\varepsilon=0\}\. Then, we have

D\[f\]​H​\[E\[f\]\]=dd​ε​H\[fε\]\|ε=0,\\mathrm\{D\}\_\{\[f\]\}H\\left\[E\_\{\[f\]\}\\right\]=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}H\_\{\[f^\{\\varepsilon\}\]\}\\right\|\_\{\\varepsilon=0\},where the derivative is taken under the canonical identificationT\[fε\]​𝒪ℰ≅ℰT\_\{\[f^\{\\varepsilon\}\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\cong\\mathscr\{E\}\.

Then, the map

∇ℰ,flat:𝔛⁡\(𝒪ℰ\)×𝔛⁡\(𝒪ℰ\)→𝔛⁡\(𝒪ℰ\),\(E,H\)↦∇Eℰ,flatH,\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}:\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\)\\times\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\)\\to\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\),\\qquad\(E,H\)\\mapsto\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}H,is an affine connection on𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}, which we call the energy flat connection\.

###### Proof\.

The map∇ℰ,flat\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}is well\-defined and smooth\. We now verify the definition of a connection \([6](https://arxiv.org/html/2609.05727#S2.E6)\)\. Consider any tangent vector fieldsE,F,H∈𝔛⁡\(𝒪ℰ\)E,F,H\\in\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\), smooth scalar functionsh1,h2:𝒪ℰ→ℝ\{h\_\{1\}\},\{h\_\{2\}\}:\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\to\\mathbb\{R\}, and constantsa,b∈ℝa,b\\in\\mathbb\{R\}\. First,

\(∇h1​E\+h2​Fℰ,flatH\)\[f\]=h1​\(\[f\]\)​D\[f\]​H​\[E\[f\]\]\+h2​\(\[f\]\)​D\[f\]​H​\[F\[f\]\]=\(h1​∇Eℰ,flatH\+h2​∇Fℰ,flatH\)\[f\]\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{\{h\_\{1\}\}E\+\{h\_\{2\}\}F\}H\\right\)\_\{\[f\]\}=\{h\_\{1\}\}\(\{\[f\]\}\)\\mathrm\{D\}\_\{\[f\]\}H\[E\_\{\[f\]\}\]\+\{h\_\{2\}\}\(\{\[f\]\}\)\\mathrm\{D\}\_\{\[f\]\}H\[F\_\{\[f\]\}\]=\\left\(\{h\_\{1\}\}\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}H\+\{h\_\{2\}\}\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{F\}H\\right\)\_\{\[f\]\}gives \([6a](https://arxiv.org/html/2609.05727#S2.E6.1)\)\. Then,

\(∇Eℰ,flat\(a​F\+b​H\)\)\[f\]=a​D\[f\]​F​\[E\[f\]\]\+b​D\[f\]​H​\[E\[f\]\]=\(a​∇Eℰ,flatF\+b​∇Eℰ,flatH\)\[f\]\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}\(aF\+bH\)\\right\)\_\{\[f\]\}=a\\mathrm\{D\}\_\{\[f\]\}F\[E\_\{\[f\]\}\]\+b\\mathrm\{D\}\_\{\[f\]\}H\[E\_\{\[f\]\}\]=\\left\(a\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}F\+b\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}H\\right\)\_\{\[f\]\}gives \([6b](https://arxiv.org/html/2609.05727#S2.E6.2)\)\. Finally, \([6c](https://arxiv.org/html/2609.05727#S2.E6.3)\) is implied by

\(∇Eℰ,flat\(h1​F\)\)\[f\]=dd​ε​h1​\(\[fε\]\)​F\[fε\]\|ε=0=h1​\(\[f\]\)​\(∇Eℰ,flat\(F\)\)\[f\]\+E⁡\[h1\]​\(\[f\]\)​F\[f\]\.\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}\(\{h\_\{1\}\}F\)\\right\)\_\{\[f\]\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}h\_\{1\}\(\{\[f^\{\\varepsilon\}\]\}\)F\_\{\{\[f^\{\\varepsilon\}\]\}\}\\right\|\_\{\\varepsilon=0\}=\{h\_\{1\}\}\(\{\[f\]\}\)\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}\(F\)\\right\)\_\{\[f\]\}\+E\[h\_\{1\}\]\(\{\[f\]\}\)F\_\{\[f\]\}\.Therefore,∇ℰ,flat\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}is an affine connection on𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}\. ∎

The energy flat connection accounts for the variation of tangent vector fields in the fixed Banach model spaceℰ\\mathscr\{E\}\. To construct its dual connection with respect to the Fisher–Rao metric, we additionally account for the variation of the metric itself with the base point\. The next proposition computes this variation as a trilinear form\.

###### Proposition 77\.

For every\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\},ρ:=Exp⁡\(\[f\]\)\\rho:=\\mathrm\{Exp\}\(\[f\]\), and\[h1\],\[h2\],\[h3\]∈T\[f\]​𝒪ℰ≅ℰ\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\\in T\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\cong\\mathscr\{E\}, consider the tensor

C\[f\]​\(\[h1\],\[h2\],\[h3\]\):=D\[f\]​\(gℰ​\(\[h1\],\[h2\]\)\)​\[\[h3\]\]\.C\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\):=\\mathrm\{D\}\_\{\[f\]\}\\left\(g^\{\\mathscr\{E\}\}\\left\(\[h\_\{1\}\],\[h\_\{2\}\]\\right\)\\right\)\[\[h\_\{3\}\]\]\.Here

gℰ​\(\[h1\],\[h2\]\):\[f\]↦g\[f\]ℰ​\(\[h1\],\[h2\]\)g^\{\\mathscr\{E\}\}\\left\(\[h\_\{1\}\],\[h\_\{2\}\]\\right\):\[f\]\\mapsto g\_\{\[f\]\}^\{\\mathscr\{E\}\}\\left\(\[h\_\{1\}\],\[h\_\{2\}\]\\right\)is a real\-valued map\. Then, the map

\(\[f\],\[h1\],\[h2\],\[h3\]\)↦C\[f\]​\(\[h1\],\[h2\],\[h3\]\)\(\[f\],\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\)\\mapsto C\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\)is smooth\. Furthermore, we have

C\[f\]​\(\[h1\],\[h2\],\[h3\]\)=𝔼ρ​\[\(h1−𝔼ρ​\[h1\]\)​\(h2−𝔼ρ​\[h2\]\)​\(h3−𝔼ρ​\[h3\]\)\]\.C\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\)=\\mathbb\{E\}\_\{\\rho\}\\left\[\\left\(h\_\{1\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{1\}\\right\]\\right\)\\left\(h\_\{2\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{2\}\\right\]\\right\)\\left\(h\_\{3\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{3\}\\right\]\\right\)\\right\]\.

###### Proof\.

By Assumption[75](https://arxiv.org/html/2609.05727#Thmtheorem75), the map\(\[f\],\[h1\],\[h2\]\)↦g\[f\]ℰ​\(\[h1\],\[h2\]\)\(\[f\],\[h\_\{1\}\],\[h\_\{2\}\]\)\\mapsto g\_\{\[f\]\}^\{\\mathscr\{E\}\}\\left\(\[h\_\{1\}\],\[h\_\{2\}\]\\right\)is smooth; hence,\(\[f\],\[h1\],\[h2\],\[h3\]\)↦C\[f\]​\(\[h1\],\[h2\],\[h3\]\)=D\[f\]​\(gℰ​\(\[h1\],\[h2\]\)\)​\[\[h3\]\]\(\[f\],\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\)\\mapsto C\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\)=\\mathrm\{D\}\_\{\[f\]\}\\left\(g^\{\\mathscr\{E\}\}\\left\(\[h\_\{1\}\],\[h\_\{2\}\]\\right\)\\right\)\[\[h\_\{3\}\]\]is smooth\. Consider a smooth curve\(\[fε\]\)ε∈\(−δ,δ\)⊂𝒪ℰ\(\[f^\{\\varepsilon\}\]\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathcal\{O\}\_\{\\mathscr\{E\}\}which passes through\[f\]\[f\]atε=0\\varepsilon=0and satisfies\[h3\]=dd​ε​\[fε\]\|ε=0\[h\_\{3\}\]=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\[f^\{\\varepsilon\}\]\\right\|\_\{\\varepsilon=0\}\. Letρε=Exp⁡\(\[fε\]\)\\rho^\{\\varepsilon\}=\\mathrm\{Exp\}\(\[f^\{\\varepsilon\}\]\)\. Then, we have

D\[f\]​\(gℰ​\(\[h1\],\[h2\]\)\)​\[\[h3\]\]\\displaystyle\\mathrm\{D\}\_\{\[f\]\}\\left\(g^\{\\mathscr\{E\}\}\\left\(\[h\_\{1\}\],\[h\_\{2\}\]\\right\)\\right\)\[\[h\_\{3\}\]\]=\\displaystyle=dd​ε​𝔼ρε​\[\(h1−𝔼ρε​\[h1\]\)​\(h2−𝔼ρε​\[h2\]\)\]\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\rho^\{\\varepsilon\}\}\\left\[\\left\(h\_\{1\}\-\\mathbb\{E\}\_\{\\rho^\{\\varepsilon\}\}\\left\[h\_\{1\}\\right\]\\right\)\\left\(h\_\{2\}\-\\mathbb\{E\}\_\{\\rho^\{\\varepsilon\}\}\\left\[h\_\{2\}\\right\]\\right\)\\right\]\\right\|\_\{\\varepsilon=0\}=\\displaystyle=dd​ε​𝔼ρ​\[\(h1−𝔼ρε​\[h1\]\)​\(h2−𝔼ρε​\[h2\]\)\]\|ε=0\+dd​ε​𝔼ρε​\[\(h1−𝔼ρ​\[h1\]\)​\(h2−𝔼ρ​\[h2\]\)\]\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\rho\}\\left\[\\left\(h\_\{1\}\-\\mathbb\{E\}\_\{\\rho^\{\\varepsilon\}\}\\left\[h\_\{1\}\\right\]\\right\)\\left\(h\_\{2\}\-\\mathbb\{E\}\_\{\\rho^\{\\varepsilon\}\}\\left\[h\_\{2\}\\right\]\\right\)\\right\]\\right\|\_\{\\varepsilon=0\}\+\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\\mathbb\{E\}\_\{\\rho^\{\\varepsilon\}\}\\left\[\\left\(h\_\{1\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{1\}\\right\]\\right\)\\left\(h\_\{2\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{2\}\\right\]\\right\)\\right\]\\right\|\_\{\\varepsilon=0\}=\\displaystyle=0\+𝔼ρ​\[\(h1−𝔼ρ​\[h1\]\)​\(h2−𝔼ρ​\[h2\]\)​\(h3−𝔼ρ​\[h3\]\)\],\\displaystyle 0\+\\mathbb\{E\}\_\{\\rho\}\\left\[\\left\(h\_\{1\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{1\}\\right\]\\right\)\\left\(h\_\{2\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{2\}\\right\]\\right\)\\left\(h\_\{3\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[h\_\{3\}\\right\]\\right\)\\right\],which finishes the proof\. ∎

The trilinear formC\[f\]C\_\{\[f\]\}measures the base\-point variation of the Fisher–Rao metric when its two tangent arguments are held fixed\. For every\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}and\[h1\],\[h2\]∈ℰ\[h\_\{1\}\],\[h\_\{2\}\]\\in\\mathscr\{E\}, it determines the continuous linear functionalC\[f\]\(\[h1\],\[h2\],⋅\)∈ℰ∗C\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\],\\cdot\)\\in\\mathscr\{E\}^\{\*\}\. To incorporate this metric variation into an affine connection, this functional must admit a tangent\-vector representation under the Fisher–Rao metric\. Since the metric may be weak, the flat mapG\[f\]♭G\_\{\[f\]\}^\{\\flat\}need not be surjective, and the existence of such a representation is therefore an additional second\-order requirement\.

###### Assumption 78\.

For every\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}and\[h1\],\[h2\]∈T\[f\]​𝒪ℰ≅ℰ\[h\_\{1\}\],\[h\_\{2\}\]\\in T\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\cong\\mathscr\{E\}, assume that

C\[f\]\(\[h1\],\[h2\],⋅\)∈Im\(G\[f\]♭\)\.C\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\],\\cdot\)\\in\\mathrm\{Im\}\\left\(G\_\{\[f\]\}^\{\\flat\}\\right\)\.SinceG\[f\]♭G\_\{\[f\]\}^\{\\flat\}is injective, there exists a uniqueR\[f\]​\(\[h1\],\[h2\]\)∈T\[f\]​𝒪ℰ≅ℰR\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\]\)\\in T\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\cong\\mathscr\{E\}such that for every\[h3\]∈T\[f\]​𝒪ℰ≅ℰ\[h\_\{3\}\]\\in T\_\{\[f\]\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\cong\\mathscr\{E\},

C\[f\]​\(\[h1\],\[h2\],\[h3\]\)=g\[f\]ℰ​\(R\[f\]​\(\[h1\],\[h2\]\),\[h3\]\)\.C\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\],\[h\_\{3\}\]\)=g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(R\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\]\),\[h\_\{3\}\]\\right\)\.We additionally assume that the map

\(\[f\],\[h1\],\[h2\]\)↦R\[f\]​\(\[h1\],\[h2\]\)\(\[f\],\[h\_\{1\}\],\[h\_\{2\}\]\)\\mapsto R\_\{\[f\]\}\(\[h\_\{1\}\],\[h\_\{2\}\]\)is smooth\.

Assumption[78](https://arxiv.org/html/2609.05727#Thmtheorem78)supplies the tangent correction that represents the base\-point variation of the Fisher–Rao metric\. Adding this correction to the energy flat connection gives the candidate connection dual to∇ℰ,flat\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\. The next proposition verifies that this construction indeed defines an affine connection and then transports it to the density manifold\.

###### Proposition 79\.

The map

∇ℰ,mix:𝔛⁡\(𝒪ℰ\)×𝔛⁡\(𝒪ℰ\)→𝔛⁡\(𝒪ℰ\),\(∇Eℰ,mixH\)\[f\]:=\(∇Eℰ,flatH\)\[f\]\+R\[f\]​\(E\[f\],H\[f\]\)\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}:\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\)\\times\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\)\\to\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\),\\qquad\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}H\\right\)\_\{\[f\]\}:=\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}H\\right\)\_\{\[f\]\}\+R\_\{\[f\]\}\\left\(E\_\{\[f\]\},H\_\{\[f\]\}\\right\)is an affine connection on𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}, which we call the energy mixture connection\. Then, we transport it to the density manifold𝒫\\mathscr\{P\}through the diffeomorphismLog:𝒫→∼𝒪ℰ\\mathrm\{Log\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}\. Specifically, forP,Q∈𝔛⁡\(𝒫\)P,Q\\in\\mathfrak\{X\}\(\\mathscr\{P\}\), defineE,H∈𝔛⁡\(𝒪ℰ\)E,H\\in\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\)by

E\[h\]:=dExp⁡\(\[h\]\)​Log​\[PExp⁡\(\[h\]\)\],H\[h\]:=dExp⁡\(\[h\]\)​Log​\[QExp⁡\(\[h\]\)\]\.E\_\{\[h\]\}:=\\mathrm\{d\}\_\{\\mathrm\{Exp\}\(\[h\]\)\}\\mathrm\{Log\}\\left\[P\_\{\\mathrm\{Exp\}\(\[h\]\)\}\\right\],\\qquad H\_\{\[h\]\}:=\\mathrm\{d\}\_\{\\mathrm\{Exp\}\(\[h\]\)\}\\mathrm\{Log\}\\left\[Q\_\{\\mathrm\{Exp\}\(\[h\]\)\}\\right\]\.Then, the transported mixture connection on𝒫\\mathscr\{P\}is defined by

∇mix:𝔛⁡\(𝒫\)×𝔛⁡\(𝒫\)→𝔛⁡\(𝒫\),\(∇PmixQ\)ρ:=dLog⁡\(ρ\)​Exp​\[\(∇Eℰ,mixH\)Log⁡\(ρ\)\]\.\\nabla^\{\\mathrm\{mix\}\}:\\mathfrak\{X\}\(\\mathscr\{P\}\)\\times\\mathfrak\{X\}\(\\mathscr\{P\}\)\\to\\mathfrak\{X\}\(\\mathscr\{P\}\),\\qquad\\left\(\\nabla^\{\\mathrm\{mix\}\}\_\{P\}Q\\right\)\_\{\\rho\}:=\\mathrm\{d\}\_\{\\mathrm\{Log\}\(\\rho\)\}\\mathrm\{Exp\}\\left\[\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}H\\right\)\_\{\\mathrm\{Log\}\(\\rho\)\}\\right\]\.

###### Proof\.

By Assumption[78](https://arxiv.org/html/2609.05727#Thmtheorem78),∇ℰ,mix\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}is well\-defined and smooth\. We now verify the definition of a connection \([6](https://arxiv.org/html/2609.05727#S2.E6)\)\. Consider any energy\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}, tangent vector fieldsE,F,H∈𝔛⁡\(𝒪ℰ\)E,F,H\\in\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\), smooth scalar functionsh1,h2:𝒪ℰ→ℝ\{h\_\{1\}\},\{h\_\{2\}\}:\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\to\\mathbb\{R\}, and constantsa,b∈ℝa,b\\in\\mathbb\{R\}\.

First, for every\[h3\]∈ℰ\[h\_\{3\}\]\\in\\mathscr\{E\},

g\[f\]ℰ​\(R\[f\]​\(h1​\(\[f\]\)​E\[f\]\+h2​\(\[f\]\)​F\[f\],H\[f\]\),\[h3\]\)\\displaystyle g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(R\_\{\[f\]\}\\left\(h\_\{1\}\(\[f\]\)E\_\{\[f\]\}\+h\_\{2\}\(\[f\]\)F\_\{\[f\]\},H\_\{\[f\]\}\\right\),\[h\_\{3\}\]\\right\)=\\displaystyle=C\[f\]​\(h1​\(\[f\]\)​E\[f\]\+h2​\(\[f\]\)​F\[f\],H\[f\],\[h3\]\)\\displaystyle C\_\{\[f\]\}\(h\_\{1\}\(\[f\]\)E\_\{\[f\]\}\+h\_\{2\}\(\[f\]\)F\_\{\[f\]\},H\_\{\[f\]\},\[h\_\{3\}\]\)=\\displaystyle=g\[f\]ℰ​\(h1​\(\[f\]\)​R\[f\]​\(E\[f\],H\[f\]\)\+h2​\(\[f\]\)​R\[f\]​\(F\[f\],H\[f\]\),\[h3\]\)\.\\displaystyle g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(h\_\{1\}\(\[f\]\)R\_\{\[f\]\}\\left\(E\_\{\[f\]\},H\_\{\[f\]\}\\right\)\+h\_\{2\}\(\[f\]\)R\_\{\[f\]\}\\left\(F\_\{\[f\]\},H\_\{\[f\]\}\\right\),\[h\_\{3\}\]\\right\)\.By the positive definiteness ofg\[f\]ℰg^\{\\mathscr\{E\}\}\_\{\[f\]\}, we have

R⁡\(h1​E\+h2​F,H\)=h1​R​\(E,H\)\+h2​R​\(F,H\)\.R\\left\(h\_\{1\}E\+h\_\{2\}F,H\\right\)=h\_\{1\}R\\left\(E,H\\right\)\+h\_\{2\}R\\left\(F,H\\right\)\.Then, \([6a](https://arxiv.org/html/2609.05727#S2.E6.1)\) is given by

∇h1​E\+h2​Fℰ,mixH=\\displaystyle\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{\{h\_\{1\}\}E\+\{h\_\{2\}\}F\}H=∇h1​E\+h2​Fℰ,flatH\+R⁡\(h1​E\+h2​F,H\)\\displaystyle\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{\{h\_\{1\}\}E\+\{h\_\{2\}\}F\}H\+R\\left\(h\_\{1\}E\+h\_\{2\}F,H\\right\)=\\displaystyle=h1​∇Eℰ,flatH\+h2​∇Fℰ,flatH\+h1​R​\(E,H\)\+h2​R​\(F,H\)\\displaystyle\{h\_\{1\}\}\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}H\+\{h\_\{2\}\}\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{F\}H\+h\_\{1\}R\\left\(E,H\\right\)\+h\_\{2\}R\\left\(F,H\\right\)=\\displaystyle=h1​∇Eℰ,mixH\+h2​∇Fℰ,mixH\.\\displaystyle\{h\_\{1\}\}\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}H\+\{h\_\{2\}\}\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{F\}H\.
Second, for every\[h3\]∈ℰ\[h\_\{3\}\]\\in\\mathscr\{E\},

g\[f\]ℰ​\(R\[f\]​\(E\[f\],a​F\[f\]\+b​H\[f\]\),\[h3\]\)=\\displaystyle g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(R\_\{\[f\]\}\\left\(E\_\{\[f\]\},aF\_\{\[f\]\}\+bH\_\{\[f\]\}\\right\),\[h\_\{3\}\]\\right\)=C\[f\]​\(E\[f\],a​F\[f\]\+b​H\[f\],\[h3\]\)\\displaystyle C\_\{\[f\]\}\(E\_\{\[f\]\},aF\_\{\[f\]\}\+bH\_\{\[f\]\},\[h\_\{3\}\]\)=\\displaystyle=g\[f\]ℰ​\(a​R\[f\]​\(E\[f\],F\[f\]\)\+b​R\[f\]​\(E\[f\],H\[f\]\),\[h3\]\)\.\\displaystyle g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(aR\_\{\[f\]\}\\left\(E\_\{\[f\]\},F\_\{\[f\]\}\\right\)\+bR\_\{\[f\]\}\\left\(E\_\{\[f\]\},H\_\{\[f\]\}\\right\),\[h\_\{3\}\]\\right\)\.By the positive definiteness ofg\[f\]ℰg^\{\\mathscr\{E\}\}\_\{\[f\]\}, we have

R⁡\(E,a​F\+b​H\)=a​R​\(E,F\)\+b​R​\(E,H\)\.R\\left\(E,aF\+bH\\right\)=aR\\left\(E,F\\right\)\+bR\\left\(E,H\\right\)\.Then, \([6b](https://arxiv.org/html/2609.05727#S2.E6.2)\) is given by

∇Eℰ,mix\(a​F\+b​H\)=\\displaystyle\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}\(aF\+bH\)=∇Eℰ,flat\(a​F\+b​H\)\+R⁡\(E,a​F\+b​H\)\\displaystyle\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}\(aF\+bH\)\+R\\left\(E,aF\+bH\\right\)=\\displaystyle=a​∇Eℰ,flatF\+b​∇Eℰ,flatH\+a​R​\(E,F\)\+b​R​\(E,H\)\\displaystyle a\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}F\+b\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}H\+aR\\left\(E,F\\right\)\+bR\\left\(E,H\\right\)=\\displaystyle=a​∇Eℰ,mixF\+b​∇Eℰ,mixH\.\\displaystyle a\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}F\+b\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}H\.
Finally, applying a similar analysis gives

R⁡\(E,h1​F\)=h1​R​\(E,F\)\.R\(E,h\_\{1\}F\)=h\_\{1\}R\(E,F\)\.Then, \([6c](https://arxiv.org/html/2609.05727#S2.E6.3)\) is given by

∇Eℰ,mix\(h1​F\)=\\displaystyle\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}\(\{h\_\{1\}\}F\)=∇Eℰ,flat\(h1​F\)\+R⁡\(E,h1​F\)\\displaystyle\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}\(\{h\_\{1\}\}F\)\+R\(E,h\_\{1\}F\)=\\displaystyle=h1​\(\[f\]\)​\(∇Eℰ,flat\(F\)\)\[f\]\+E⁡\[h1\]​\(\[f\]\)​F\[f\]\+h1​R​\(E,F\)\\displaystyle\{h\_\{1\}\}\(\{\[f\]\}\)\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}\(F\)\\right\)\_\{\[f\]\}\+E\[h\_\{1\}\]\(\{\[f\]\}\)F\_\{\[f\]\}\+h\_\{1\}R\(E,F\)=\\displaystyle=h1​\(\[f\]\)​\(∇Eℰ,mix\(F\)\)\[f\]\+E⁡\[h1\]​\(\[f\]\)​F\[f\]\.\\displaystyle\{h\_\{1\}\}\(\{\[f\]\}\)\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}\(F\)\\right\)\_\{\[f\]\}\+E\[h\_\{1\}\]\(\{\[f\]\}\)F\_\{\[f\]\}\.Therefore,∇ℰ,mix\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}is an affine connection on𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}\. Since affine connections are preserved under transport through a diffeomorphism,∇mix\\nabla^\{\\mathrm\{mix\}\}is an affine connection on𝒫\\mathscr\{P\}\. ∎

The above proposition establishes the connection axioms, but it does not yet identify the metric relation that motivates the correctionRR\. We next prove that the energy mixture connection is dual to the energy flat connection with respect to the Fisher–Rao metric\. Transporting this duality identity to𝒫\\mathscr\{P\}then yields the fixed\-observable characterization used in Section[4\.3](https://arxiv.org/html/2609.05727#S4.SS3)\.

###### Theorem 80\.

For everyE,F,H∈𝔛⁡\(𝒪ℰ\)E,F,H\\in\\mathfrak\{X\}\(\\mathcal\{O\}\_\{\\mathscr\{E\}\}\), the energy flat connection and the energy mixture connection satisfy the following duality identity:

E⁡\[gℰ​\(F,H\)\]=gℰ​\(∇Eℰ,flatF,H\)\+gℰ​\(F,∇Eℰ,mixH\)\.E\\left\[g^\{\\mathscr\{E\}\}\(F,H\)\\right\]=g^\{\\mathscr\{E\}\}\\left\(\\nabla^\{\\mathscr\{E\},\\mathrm\{flat\}\}\_\{E\}F,H\\right\)\+g^\{\\mathscr\{E\}\}\\left\(F,\\nabla^\{\\mathscr\{E\},\\mathrm\{mix\}\}\_\{E\}H\\right\)\.Consequently, for everyP,Q∈𝔛⁡\(𝒫\)P,Q\\in\\mathfrak\{X\}\(\\mathscr\{P\}\)andρ∈𝒫\\rho\\in\\mathscr\{P\},\(∇PmixQ\)ρ\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\}is the unique tangent vector inTρ​𝒫T\_\{\\rho\}\\mathscr\{P\}which satisfies

gρFR​\(\(∇PmixQ\)ρ,ξρ,h\)=P⁡\[gFR​\(Q,ξ⋅,h\)\]​\(ρ\)g\_\{\\rho\}^\{\\mathrm\{FR\}\}\\left\(\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\},\\xi^\{\\rho,h\}\\right\)=P\\left\[g^\{\\mathrm\{FR\}\}\\left\(Q,\\xi^\{\\cdot,h\}\\right\)\\right\]\(\\rho\)\(153\)for every\[h\]∈ℰ\[h\]\\in\\mathscr\{E\}\. Here we consider

gFR​\(Q,ξ⋅,h\):𝒫→ℝ,q↦gqFR​\(Qq,ξq,h\)g^\{\\mathrm\{FR\}\}\\left\(Q,\\xi^\{\\cdot,h\}\\right\):\\mathscr\{P\}\\to\\mathbb\{R\},\\qquad q\\mapsto g^\{\\mathrm\{FR\}\}\_\{q\}\\left\(Q\_\{q\},\\xi^\{q,h\}\\right\)as a real\-valued function on𝒫\\mathscr\{P\}, whereasξ⋅,h∈𝔛⁡\(𝒫\)\\xi^\{\\cdot,h\}\\in\\mathfrak\{X\}\(\\mathscr\{P\}\)is a tangent vector field whose terminal observable is alwayshh\.

###### Proof\.

For every\[f\]∈𝒪ℰ\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}, consider a smooth curve\(\[fε\]\)ε∈\(−δ,δ\)⊂𝒪ℰ\(\[f^\{\\varepsilon\}\]\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathcal\{O\}\_\{\\mathscr\{E\}\}which passes through\[f\]\[f\]atε=0\\varepsilon=0and satisfiesE\[f\]=dd​ε​\[fε\]\|ε=0E\_\{\[f\]\}=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}\[f^\{\\varepsilon\}\]\\right\|\_\{\\varepsilon=0\}\. Then, the duality identity is given by

E⁡\[gℰ​\(F,H\)\]​\(\[f\]\)\\displaystyle E\\left\[g^\{\\mathscr\{E\}\}\(F,H\)\\right\]\(\[f\]\)=\\displaystyle=dd​ε​g\[fε\]ℰ​\(F\[fε\],H\[fε\]\)\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\[f^\{\\varepsilon\}\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f^\{\\varepsilon\}\]\},H\_\{\[f^\{\\varepsilon\}\]\}\\right\)\\right\|\_\{\\varepsilon=0\}=\\displaystyle=dd​ε​g\[fε\]ℰ​\(F\[f\],H\[f\]\)\|ε=0\+dd​ε​g\[f\]ℰ​\(F\[fε\],H\[f\]\)\|ε=0\+dd​ε​g\[f\]ℰ​\(F\[f\],H\[fε\]\)\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\[f^\{\\varepsilon\}\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f\]\},H\_\{\[f\]\}\\right\)\\right\|\_\{\\varepsilon=0\}\+\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\[f\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f^\{\\varepsilon\}\]\},H\_\{\[f\]\}\\right\)\\right\|\_\{\\varepsilon=0\}\+\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\[f\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f\]\},H\_\{\[f^\{\\varepsilon\}\]\}\\right\)\\right\|\_\{\\varepsilon=0\}=\\displaystyle=C\[f\]​\(F\[f\],H\[f\],E\[f\]\)\+g\[f\]ℰ​\(\(∇Eℰ,flatF\)\[f\],H\[f\]\)\+g\[f\]ℰ​\(F\[f\],\(∇Eℰ,flatH\)\[f\]\)\\displaystyle C\_\{\[f\]\}\\left\(F\_\{\[f\]\},H\_\{\[f\]\},E\_\{\[f\]\}\\right\)\+g\_\{\[f\]\}^\{\\mathscr\{E\}\}\\left\(\\left\(\\nabla\_\{E\}^\{\\mathscr\{E\},\\mathrm\{flat\}\}F\\right\)\_\{\[f\]\},H\_\{\[f\]\}\\right\)\+g\_\{\[f\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f\]\},\\left\(\\nabla\_\{E\}^\{\\mathscr\{E\},\\mathrm\{flat\}\}H\\right\)\_\{\[f\]\}\\right\)=\\displaystyle=g\[f\]ℰ​\(\(∇Eℰ,flatF\)\[f\],H\[f\]\)\+g\[f\]ℰ​\(F\[f\],\(∇Eℰ,mixH\)\[f\]\)\.\\displaystyle g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(\\left\(\\nabla\_\{E\}^\{\\mathscr\{E\},\\mathrm\{flat\}\}F\\right\)\_\{\[f\]\},H\_\{\[f\]\}\\right\)\+g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(F\_\{\[f\]\},\\left\(\\nabla\_\{E\}^\{\\mathscr\{E\},\\mathrm\{mix\}\}H\\right\)\_\{\[f\]\}\\right\)\.Consequently, we have

g\[f\]ℰ​\(F\[f\],\(∇Eℰ,mixH\)\[f\]\)=\\displaystyle g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(F\_\{\[f\]\},\\left\(\\nabla\_\{E\}^\{\\mathscr\{E\},\\mathrm\{mix\}\}H\\right\)\_\{\[f\]\}\\right\)=E⁡\[gℰ​\(F,H\)\]​\(\[f\]\)−g\[f\]ℰ​\(\(∇Eℰ,flatF\)\[f\],H\[f\]\)\\displaystyle E\\left\[g^\{\\mathscr\{E\}\}\(F,H\)\\right\]\(\[f\]\)\-g^\{\\mathscr\{E\}\}\_\{\[f\]\}\\left\(\\left\(\\nabla\_\{E\}^\{\\mathscr\{E\},\\mathrm\{flat\}\}F\\right\)\_\{\[f\]\},H\_\{\[f\]\}\\right\)=\\displaystyle=dd​ε​g\[fε\]ℰ​\(F\[f\],H\[f\]\)\|ε=0\+dd​ε​g\[f\]ℰ​\(F\[f\],H\[fε\]\)\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\[f^\{\\varepsilon\}\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f\]\},H\_\{\[f\]\}\\right\)\\right\|\_\{\\varepsilon=0\}\+\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\[f\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f\]\},H\_\{\[f^\{\\varepsilon\}\]\}\\right\)\\right\|\_\{\\varepsilon=0\}=\\displaystyle=dd​ε​g\[fε\]ℰ​\(F\[f\],H\[fε\]\)\|ε=0\\displaystyle\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\[f^\{\\varepsilon\}\]\}^\{\\mathscr\{E\}\}\\left\(F\_\{\[f\]\},H\_\{\[f^\{\\varepsilon\}\]\}\\right\)\\right\|\_\{\\varepsilon=0\}=\\displaystyle=D\[f\]​\(gℰ​\(F\[f\],H\)\)​\[E\[f\]\]\.\\displaystyle\\mathrm\{D\}\_\{\[f\]\}\\left\(g^\{\\mathscr\{E\}\}\\left\(F\_\{\[f\]\},H\\right\)\\right\)\\left\[E\_\{\[f\]\}\\right\]\.Transporting the above equation to𝒫\\mathscr\{P\}implies \([153](https://arxiv.org/html/2609.05727#A1.E153)\)\. ∎

Equation \([153](https://arxiv.org/html/2609.05727#A1.E153)\) also admits a geometric interpretation in density coordinates\. For every fixed terminal observableffand tangent vector fieldQ∈𝔛⁡\(𝒫\)Q\\in\\mathfrak\{X\}\(\\mathscr\{P\}\), sinceQqQ\_\{q\}has zero total mass for everyq∈𝒫q\\in\\mathscr\{P\}, we have

gqFR​\(Qq,ξq,f\)=∫ℝdf⁡\(x\)​Qq​\(x\)​𝑑x\.g\_\{q\}^\{\\mathrm\{FR\}\}\\left\(Q\_\{q\},\\xi^\{q,f\}\\right\)=\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\)Q\_\{q\}\(x\)\\,\\mathrm\{d\}x\.Therefore, along a smooth curve\(ρε\)ε∈\(−δ,δ\)⊂𝒫\(\\rho^\{\\varepsilon\}\)\_\{\\varepsilon\\in\(\-\\delta,\\delta\)\}\\subset\\mathscr\{P\}which passes throughρ\\rhoatε=0\\varepsilon=0,

dd​ε​gρεFR​\(Qρε,ξρε,f\)\|ε=0=∫ℝdf⁡\(x\)​dd​ε​Qρε​\(x\)\|ε=0​𝑑x\.\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}g\_\{\\rho^\{\\varepsilon\}\}^\{\\mathrm\{FR\}\}\\left\(Q\_\{\\rho^\{\\varepsilon\}\},\\xi^\{\\rho^\{\\varepsilon\},f\}\\right\)\\right\|\_\{\\varepsilon=0\}=\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\)\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\varepsilon\}Q\_\{\\rho^\{\\varepsilon\}\}\(x\)\\right\|\_\{\\varepsilon=0\}\\mathrm\{d\}x\.In this variation, the terminal observableffis held fixed, whereas the base pointρε\\rho^\{\\varepsilon\}, the tangent vectorQρεQ\_\{\\rho^\{\\varepsilon\}\}, the Fisher–Rao metricgρεFRg\_\{\\rho^\{\\varepsilon\}\}^\{\\mathrm\{FR\}\}, and the corresponding tangent vectorξρε,f\\xi^\{\\rho^\{\\varepsilon\},f\}all vary withε\\varepsilon\. Thus, the pairing compares the tangent vector fieldQQat neighboring densities against the same terminal observable\. The first identity shows that the explicit density dependence of the Fisher–Rao metric and the representationξq,f\\xi^\{q,f\}cancels in this pairing; the remaining dependence onqqis entirely throughQqQ\_\{q\}\. Consequently, \([153](https://arxiv.org/html/2609.05727#A1.E153)\) characterizes\(∇PmixQ\)ρ\\left\(\\nabla\_\{P\}^\{\\mathrm\{mix\}\}Q\\right\)\_\{\\rho\}as the unique tangent vector whose Fisher–Rao pairing with everyξρ,f\\xi^\{\\rho,f\}reproduces this first\-order variation along the tangent vectorPρP\_\{\\rho\}\. In this sense, the mixture connection differentiatesQQwhile keeping the terminal observablefffixed\.

### A\.4Non\-Affineness of the Canonical Manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}

In this subsection, we show that the canonical manifold, despite being a split embedded Banach submanifold of𝒱\\mathscr\{V\}, need not be affine, which explains why the canonical retraction must applyℛ\\mathcal\{R\}after ambient addition\. Assume that the density class𝒫\\mathscr\{P\}contains the centered Gaussian family and that the function space𝔹\\mathbb\{B\}contains all quadratic forms\. In this setting, we construct a canonical velocity field and a canonical tangent direction such that every nonzero finite ambient step along that direction leaves𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

Throughout this subsection, we arbitrarily select and fix a nonzero symmetric matrixA∈𝕊dA\\in\\mathbb\{S\}^\{d\}\. Fix the densityρ=𝒩⁡\(0,I\)\\rho=\\mathcal\{N\}\(0,I\)and its canonical velocity fieldvρ=𝒞⁡\(ρ\)∈𝒱canv^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. Letf⁡\(x\):=12​x⊤​A​xf\(x\):=\\frac\{1\}\{2\}x^\{\\top\}Ax\. According to the assumption,f∈𝔹f\\in\\mathbb\{B\}\. Based on Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5), the fieldΓρ,f:=\(κt∇Vtρ\[f\]\)t∈\[0,1\]∈Tvρ𝒱can\\Gamma^\{\\rho,f\}:=\(\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\)\_\{t\\in\[0,1\]\}\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}should be a tangent vector atvρv^\{\\rho\}\. We then prove that for all scalar stepsizesη≠0\\eta\\neq 0,

v¯η≠ℛ⁡\(v¯η\),v¯η:=vρ\+η​Γρ,f\.\\bar\{v\}^\{\\eta\}\\neq\\mathcal\{R\}\(\\bar\{v\}^\{\\eta\}\),\\qquad\\bar\{v\}^\{\\eta\}:=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\.\(154\)According to Theorem[3](https://arxiv.org/html/2609.05727#Thmtheorem3), we haveℛ\|𝒱can=Id𝒱can\\left\.\\mathcal\{R\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}=\\mathrm\{Id\}\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}\. Therefore, Equation \([154](https://arxiv.org/html/2609.05727#A1.E154)\) impliesv¯η∉𝒱can\\bar\{v\}^\{\\eta\}\\notin\\mathscr\{V\}\_\{\\mathrm\{can\}\}for allη≠0\\eta\\neq 0\.

For the first step, we compute the canonical velocity fieldvρ∈𝒱canv^\{\\rho\}\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. Sinceρ=𝒩⁡\(0,I\)\\rho=\\mathcal\{N\}\(0,I\), we have

𝔼X1∼p1\|tρ\(⋅\|xt\)\[X1\]=αtαt2\+βt2xt,𝔼X0∼p0\|tρ\(⋅\|xt\)\[X0\]=βtαt2\+βt2xt\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[X\_\{1\}\\right\]=\\frac\{\\alpha\_\{t\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}x\_\{t\},\\qquad\\mathbb\{E\}\_\{X\_\{0\}\\sim p\_\{0\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[X\_\{0\}\\right\]=\\frac\{\\beta\_\{t\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}x\_\{t\}\.Therefore,

vtρ\(xt\)=α˙t𝔼X1∼p1\|tρ\(⋅\|xt\)\[X1\]\+β˙t𝔼X0∼p0\|tρ\(⋅\|xt\)\[X0\]=αt​α˙t\+βt​β˙tαt2\+βt2xt\.v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\dot\{\\alpha\}\_\{t\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[X\_\{1\}\\right\]\+\\dot\{\\beta\}\_\{t\}\\mathbb\{E\}\_\{X\_\{0\}\\sim p\_\{0\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[X\_\{0\}\\right\]=\\frac\{\\alpha\_\{t\}\\dot\{\\alpha\}\_\{t\}\+\\beta\_\{t\}\\dot\{\\beta\}\_\{t\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}x\_\{t\}\.
For the second step, we compute the tangent vectorΓtρ,f=κt∇Vtρ\[f\]\\Gamma^\{\\rho,f\}\_\{t\}=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\], wheref⁡\(x\)=12​x⊤​A​xf\(x\)=\\frac\{1\}\{2\}x^\{\\top\}Ax\. Sincep1\|tρ\(⋅\|xt\)=𝒩\(αtαt2\+βt2xt,βt2αt2\+βt2I\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)=\\mathcal\{N\}\\left\(\\frac\{\\alpha\_\{t\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}x\_\{t\},\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}I\\right\), we have

Vtρ\[f\]\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[12X1⊤AX1\]=αt22​\(αt2\+βt2\)2xt⊤Axt\+βt22​\(αt2\+βt2\)tr\(A\)\.V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\frac\{1\}\{2\}X\_\{1\}^\{\\top\}AX\_\{1\}\\right\]=\\frac\{\\alpha\_\{t\}^\{2\}\}\{2\(\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\)^\{2\}\}x\_\{t\}^\{\\top\}Ax\_\{t\}\+\\frac\{\\beta\_\{t\}^\{2\}\}\{2\(\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\)\}\\mathrm\{tr\}\(A\)\.SinceA∈𝕊dA\\in\\mathbb\{S\}^\{d\}, we have

Γtρ,f\(xt\)=κt∇Vtρ\[f\]\(xt\)=κt​αt2\(αt2\+βt2\)2Axt\.\\Gamma^\{\\rho,f\}\_\{t\}\(x\_\{t\}\)=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\frac\{\\kappa\_\{t\}\\alpha\_\{t\}^\{2\}\}\{\(\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\)^\{2\}\}Ax\_\{t\}\.
For the third step, we compute the updated velocity fieldv¯η=vρ\+η​Γρ,f∈𝒱\\bar\{v\}^\{\\eta\}=v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,f\}\\in\\mathscr\{V\}and its terminal densityqη:=𝒯⁡\(v¯η\)∈𝒫q^\{\\eta\}:=\\mathcal\{T\}\(\\bar\{v\}^\{\\eta\}\)\\in\\mathscr\{P\}\. Note that

v¯tη​\(xt\)=vtρ​\(xt\)\+η​Γtρ,f​\(xt\)=12​dd​t​\(log⁡\(αt2\+βt2\)​I\+η​αt2αt2\+βt2​A\)​xt\.\\bar\{v\}\_\{t\}^\{\\eta\}\(x\_\{t\}\)=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\eta\\Gamma^\{\\rho,f\}\_\{t\}\(x\_\{t\}\)=\\frac\{1\}\{2\}\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(\\log\\left\(\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\\right\)I\+\\frac\{\\eta\\alpha\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}A\\right\)x\_\{t\}\.Denote the ODE flow ofv¯η\\bar\{v\}^\{\\eta\}byΦ0→tv¯η\\Phi\_\{0\\to t\}^\{\\bar\{v\}^\{\\eta\}\}\. Solving the ODE gives

Φ0→tv¯η​\(x0\)=exp⁡\(12​\(log⁡\(αt2\+βt2\)​I\+η​αt2αt2\+βt2​A\)\)​x0=αt2\+βt2​exp⁡\(η​αt22​\(αt2\+βt2\)​A\)​x0\.\\Phi\_\{0\\to t\}^\{\\bar\{v\}^\{\\eta\}\}\(x\_\{0\}\)=\\exp\\left\(\\frac\{1\}\{2\}\\left\(\\log\\left\(\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\\right\)I\+\\frac\{\\eta\\alpha\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}A\\right\)\\right\)x\_\{0\}=\\sqrt\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}\\exp\\left\(\\frac\{\\eta\\alpha\_\{t\}^\{2\}\}\{2\(\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\)\}A\\right\)x\_\{0\}\.Then, the time\-ttmarginal density induced by the ODE flowΦ0→tv¯η\\Phi\_\{0\\to t\}^\{\\bar\{v\}^\{\\eta\}\}is

\(Φ0→tv¯η\)\#​p0=𝒩⁡\(0,\(αt2\+βt2\)​exp⁡\(η​αt2αt2\+βt2​A\)\)\.\(\\Phi\_\{0\\to t\}^\{\\bar\{v\}^\{\\eta\}\}\)\_\{\\\#\}p\_\{0\}=\\mathcal\{N\}\\left\(0,\\left\(\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\\right\)\\exp\\left\(\\frac\{\\eta\\alpha\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}A\\right\)\\right\)\.In particular, the terminal density is

qη=\(Φ0→1v¯η\)\#​p0=𝒩⁡\(0,exp⁡\(η​A\)\)\.q^\{\\eta\}=\(\\Phi\_\{0\\to 1\}^\{\\bar\{v\}^\{\\eta\}\}\)\_\{\\\#\}p\_\{0\}=\\mathcal\{N\}\\left\(0,\\exp\\left\(\\eta A\\right\)\\right\)\.
For the last step, we compare the canonical velocityvqη:=𝒞⁡\(qη\)∈𝒱canv^\{q^\{\\eta\}\}:=\\mathcal\{C\}\(q^\{\\eta\}\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}withv¯η\\bar\{v\}^\{\\eta\}\. For simplicity, we compare their time\-ttmarginal densities\. Since

ptqη=𝒩⁡\(0,αt2​eη​A\+βt2​I\)≠\(Φ0→tv¯η\)\#​p0,p\_\{t\}^\{q^\{\\eta\}\}=\\mathcal\{N\}\\left\(0,\\alpha\_\{t\}^\{2\}e^\{\\eta A\}\+\\beta\_\{t\}^\{2\}I\\right\)\\neq\(\\Phi\_\{0\\to t\}^\{\\bar\{v\}^\{\\eta\}\}\)\_\{\\\#\}p\_\{0\},we have

αt2​eη​A\+βt2​Iαt2\+βt2−eη​αt2αt2\+βt2​A∈𝕊\+d\\\{0\}\\frac\{\\alpha\_\{t\}^\{2\}e^\{\\eta A\}\+\\beta\_\{t\}^\{2\}I\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}\-e^\{\\frac\{\\eta\\alpha\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\+\\beta\_\{t\}^\{2\}\}A\}\\in\{\\left\.\\mathbb\{S\}\_\{\+\}^\{d\}\\middle\\backslash\\\{0\\\}\\right\.\}for allη≠0\\eta\\neq 0,A≠0A\\neq 0\. That meansptqη≠\(Φ0→tv¯η\)\#​p0p\_\{t\}^\{q^\{\\eta\}\}\\neq\(\\Phi\_\{0\\to t\}^\{\\bar\{v\}^\{\\eta\}\}\)\_\{\\\#\}p\_\{0\}fort∈\(0,1\)t\\in\(0,1\); hence,

v¯η≠vqη=𝒞⁡\(qη\)=𝒞⁡\(𝒯⁡\(v¯η\)\)=ℛ⁡\(v¯η\)\.\\bar\{v\}^\{\\eta\}\\neq v^\{q^\{\\eta\}\}=\\mathcal\{C\}\(q^\{\\eta\}\)=\\mathcal\{C\}\(\\mathcal\{T\}\(\\bar\{v\}^\{\\eta\}\)\)=\\mathcal\{R\}\(\\bar\{v\}^\{\\eta\}\)\.Therefore,v¯η∉𝒱can\\bar\{v\}^\{\\eta\}\\notin\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

In summary, once the density class𝒫\\mathscr\{P\}contains the centered Gaussian family\{𝒩⁡\(0,Σ\):Σ∈𝕊\+⁣\+d\}\\\{\\mathcal\{N\}\(0,\\Sigma\):\\Sigma\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}\\\}, the canonical manifold𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}is non\-affine\. This establishes the geometric necessity of canonicalization in the canonical retraction of Section[5\.1](https://arxiv.org/html/2609.05727#S5.SS1)\.

## Appendix BConvergence Proofs for Newton Matching

This appendix contains the proofs of the convergence results stated in Section[7](https://arxiv.org/html/2609.05727#S7)\. Appendix[B\.1](https://arxiv.org/html/2609.05727#A2.SS1)proves self\-calibration and global KL convergence from the exact terminal\-density recursion, whereas Appendix[B\.2](https://arxiv.org/html/2609.05727#A2.SS2)uses centered\-energy coordinates to prove local quadratic convergence and the continuation result\.

### B\.1Proofs of Global Convergence

This subsection proves the global convergence theorems in Sections[7\.1\.2](https://arxiv.org/html/2609.05727#S7.SS1.SSS2)and[7\.1\.3](https://arxiv.org/html/2609.05727#S7.SS1.SSS3)\. We first record the required Toeplitz lemma\[[43](https://arxiv.org/html/2609.05727#bib.bib55)\]\.

###### Lemma 81\(Toeplitz averaging\)\.

Let\{ai\}i=0∞⊂ℝ\\left\\\{a\_\{i\}\\right\\\}\_\{i=0\}^\{\\infty\}\\subset\\mathbb\{R\}satisfy

limi→∞ai=a\.\\lim\_\{i\\to\\infty\}a\_\{i\}=a\.Let\{wk,i\}0≤i<k⊂ℝ\\left\\\{w\_\{k,i\}\\right\\\}\_\{0\\leq i<k\}\\subset\\mathbb\{R\}satisfy

supk≥1∑i=0k−1\|wk,i\|<∞,limk→∞∑i=0k−1wk,i=1,\\sup\_\{k\\geq 1\}\\sum\_\{i=0\}^\{k\-1\}\\left\|w\_\{k,i\}\\right\|<\\infty,\\qquad\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}=1,and for everyii,

limk→∞wk,i=0\.\\lim\_\{k\\to\\infty\}w\_\{k,i\}=0\.Then, we have

limk→∞∑i=0k−1wk,i​ai=a\.\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}a\_\{i\}=a\.

###### Proof\.

Denote

M:=supk≥1∑i=0k−1\|wk,i\|<∞\.M:=\\sup\_\{k\\geq 1\}\\sum\_\{i=0\}^\{k\-1\}\\left\|w\_\{k,i\}\\right\|<\\infty\.Then, we have

∑i=0k−1wk,i​ai−a=∑i=0k−1wk,i​\(ai−a\)\+a⁡\(∑i=0k−1wk,i−1\)\.\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}a\_\{i\}\-a=\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}\(a\_\{i\}\-a\)\+a\\left\(\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}\-1\\right\)\.It remains to show that both terms on the right\-hand side converge to00ask→∞k\\to\\infty\.

For anyε\>0\\varepsilon\>0, there exists a sufficiently largeIεI\_\{\\varepsilon\}such that for everyi≥Iεi\\geq I\_\{\\varepsilon\}, we have\|ai−a\|<ε\\left\|a\_\{i\}\-a\\right\|<\\varepsilon\. Therefore, fork\>Iεk\>I\_\{\\varepsilon\},

\|∑i=0k−1wk,i​\(ai−a\)\|≤∑i=0k−1\|wk,i\|​\|ai−a\|≤∑i=0Iε−1\|wk,i\|​\|ai−a\|\+ε​∑i=Iεk−1\|wk,i\|\.\\displaystyle\\left\|\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}\(a\_\{i\}\-a\)\\right\|\\leq\\sum\_\{i=0\}^\{k\-1\}\\left\|w\_\{k,i\}\\right\|\\left\|a\_\{i\}\-a\\right\|\\leq\\sum\_\{i=0\}^\{I\_\{\\varepsilon\}\-1\}\\left\|w\_\{k,i\}\\right\|\\left\|a\_\{i\}\-a\\right\|\+\\varepsilon\\sum\_\{i=I\_\{\\varepsilon\}\}^\{k\-1\}\\left\|w\_\{k,i\}\\right\|\.Takingk→∞k\\to\\inftygives

∀ε\>0,lim supk→∞\|∑i=0k−1wk,i​\(ai−a\)\|≤M​ε\.\\forall\\varepsilon\>0,\\,\\limsup\_\{k\\to\\infty\}\\left\|\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}\(a\_\{i\}\-a\)\\right\|\\leq M\\varepsilon\.Takingε→0\\varepsilon\\to 0gives

limk→∞\|∑i=0k−1wk,i​\(ai−a\)\|=0\.\\lim\_\{k\\to\\infty\}\\left\|\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}\(a\_\{i\}\-a\)\\right\|=0\.Sincelimk→∞∑i=0k−1wk,i=1\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}=1, we havelimk→∞∑i=0k−1wk,i​ai=a\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}w\_\{k,i\}a\_\{i\}=a\. ∎

See[26](https://arxiv.org/html/2609.05727#Thmtheorem26)

###### Proof\.

By Theorem[24](https://arxiv.org/html/2609.05727#Thmtheorem24), the sequence\{KL⁡\(ρk∥π\)\}k=0∞\\left\\\{\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\\right\\\}\_\{k=0\}^\{\\infty\}is nonincreasing and nonnegative\. Therefore, it converges to someK∞∈\[0,KL⁡\(ρ0∥π\)\]K\_\{\\infty\}\\in\[0,\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\]\.

Denoteλk:=ηkτ∈\(0,1\]\\lambda\_\{k\}:=\\frac\{\\eta\_\{k\}\}\{\\tau\}\\in\(0,1\]\. By Proposition[19](https://arxiv.org/html/2609.05727#Thmtheorem19), it holds pointwise that

log⁡ρkπ=B0,k​log⁡ρ0π\+∑i=0k−1Bi\+1,k​\(λi​KL⁡\(ρi∥π\)−𝒟ρi,ηi,r~ρi\),\\log\\frac\{\\rho\_\{k\}\}\{\\pi\}=B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\}\{\\pi\}\+\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\left\(\\lambda\_\{i\}\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\-\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},\\tilde\{r\}^\{\\rho\_\{i\}\}\}\\right\),\(155\)where the coefficientBi,kB\_\{i,k\}is defined as

Bi,k:=\{∏j=ik−1\(1−λj\),i<k,1,i=k\.B\_\{i,k\}:=\\begin\{cases\}\\prod\_\{j=i\}^\{k\-1\}\(1\-\\lambda\_\{j\}\),&i<k,\\\\ 1,&i=k\.\\end\{cases\}Note that𝒟ρk,ηk,r~ρk\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}is nonnegative for allkk\. Therefore, we have

𝔼X∼ρk​\[\|log⁡ρk​\(X\)π⁡\(X\)−K∞\|\]≤\\displaystyle\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[\\left\|\\log\\frac\{\\rho\_\{k\}\(X\)\}\{\\pi\(X\)\}\-K\_\{\\infty\}\\right\|\\right\]\\leq\|∑i=0k−1Bi\+1,k​λi​KL⁡\(ρi∥π\)−K∞\|\+𝔼X∼ρk​\[B0,k​\|log⁡ρ0​\(X\)π⁡\(X\)\|\]\\displaystyle\\left\|\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\-K\_\{\\infty\}\\right\|\+\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[B\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\\right\|\\right\]\+𝔼X∼ρk​\[∑i=0k−1Bi\+1,k​𝒟ρi,ηi,r~ρi​\(X\)\]\.\\displaystyle\+\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},\\tilde\{r\}^\{\\rho\_\{i\}\}\}\(X\)\\right\]\.The remaining task is to prove that the three terms on the right\-hand side converge to zero ask→∞k\\to\\infty\.

For the first step, we verify the conditions of the Toeplitz averaging lemma in Lemma[81](https://arxiv.org/html/2609.05727#Thmtheorem81)for\{KL⁡\(ρi∥π\)\}i=0∞\\left\\\{\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\\right\\\}\_\{i=0\}^\{\\infty\}and\{Bi\+1,k​λi\}0≤i<k\\left\\\{B\_\{i\+1,k\}\\lambda\_\{i\}\\right\\\}\_\{0\\leq i<k\}\. Since∑i=0∞λi=∞\\sum\_\{i=0\}^\{\\infty\}\\lambda\_\{i\}=\\infty, we have

limk→∞Bi,k=0\\lim\_\{k\\to\\infty\}B\_\{i,k\}=0\(156a\)for everyii\. Note thatBi\+1,k​λi=Bi\+1,k−Bi,kB\_\{i\+1,k\}\\lambda\_\{i\}=B\_\{i\+1,k\}\-B\_\{i,k\}; hence,limk→∞∑i=0k−1Bi\+1,k​λi=limk→∞1−B0,k=1\.\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}=\\lim\_\{k\\to\\infty\}1\-B\_\{0,k\}=1\.\(156b\)Sinceλi\>0\\lambda\_\{i\}\>0,Bi\+1,k≥0B\_\{i\+1,k\}\\geq 0, andlimk→∞∑i=0k−1Bi\+1,k​λi=1\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}=1, we havesupk≥1∑i=0k−1\|Bi\+1,k​λi\|=supk≥1∑i=0k−1Bi\+1,k​λi<∞\.\\sup\_\{k\\geq 1\}\\sum\_\{i=0\}^\{k\-1\}\\left\|B\_\{i\+1,k\}\\lambda\_\{i\}\\right\|=\\sup\_\{k\\geq 1\}\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}<\\infty\.\(156c\)Sincelimi→∞KL⁡\(ρi∥π\)=K∞\\lim\_\{i\\to\\infty\}\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)=K\_\{\\infty\}, the Toeplitz averaging lemma in Lemma[81](https://arxiv.org/html/2609.05727#Thmtheorem81)gives

limk→∞∑i=0k−1Bi\+1,k​λi​KL⁡\(ρi∥π\)=K∞\.\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)=K\_\{\\infty\}\.
For the second step, \([155](https://arxiv.org/html/2609.05727#A2.E155)\),KL⁡\(ρi∥π\)≤KL⁡\(ρ0∥π\)\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\), and𝒟ρi,ηi,r~ρi​\(x\)≥0\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},\\tilde\{r\}^\{\\rho\_\{i\}\}\}\(x\)\\geq 0imply

log⁡ρkπ≤B0,k​log⁡ρ0π\+KL⁡\(ρ0∥π\)​∑i=0k−1Bi\+1,k​λi≤B0,k​log⁡ρ0π\+KL⁡\(ρ0∥π\)\.\\log\\frac\{\\rho\_\{k\}\}\{\\pi\}\\leq B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\}\{\\pi\}\+\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}\\leq B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\}\{\\pi\}\+\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\.Denote the setA=\{x∈ℝd:ρ0​\(x\)≥π⁡\(x\)\}A=\\left\\\{x\\in\\mathbb\{R\}^\{d\}:\\rho\_\{0\}\(x\)\\geq\\pi\(x\)\\right\\\}\. Then, we have

𝔼X∼ρk​\[B0,k​\|log⁡ρ0​\(X\)π⁡\(X\)\|\]=\\displaystyle\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[B\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\\right\|\\right\]=𝔼X∼π​\[ρk​\(X\)π⁡\(X\)​B0,k​\|log⁡ρ0​\(X\)π⁡\(X\)\|\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\frac\{\\rho\_\{k\}\(X\)\}\{\\pi\(X\)\}B\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\\right\|\\right\]≤\\displaystyle\\leqeKL⁡\(ρ0∥π\)​𝔼X∼π​\[B0,k​\|log⁡ρ0​\(X\)π⁡\(X\)\|​exp⁡\(B0,k​log⁡ρ0​\(X\)π⁡\(X\)\)\]\\displaystyle e^\{\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\}\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[B\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\\right\|\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\\right\)\\right\]=\\displaystyle=eKL⁡\(ρ0∥π\)​∫Aπ⁡\(x\)​B0,k​log⁡ρ0​\(x\)π⁡\(x\)​exp⁡\(B0,k​log⁡ρ0​\(x\)π⁡\(x\)\)​𝑑x\\displaystyle e^\{\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\}\\int\_\{A\}\\pi\(x\)B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\,\\mathrm\{d\}x−eKL⁡\(ρ0∥π\)∫Acπ\(x\)B0,klogρ0​\(x\)π⁡\(x\)exp\(B0,klogρ0​\(x\)π⁡\(x\)\)dx\\displaystyle\-e^\{\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\}\\int\_\{A^\{\\mathrm\{c\}\}\}\\pi\(x\)B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\,\\mathrm\{d\}xBy the elementary inequalityx​e−x≤1exe^\{\-x\}\\leq\\frac\{1\}\{e\}forx≥0x\\geq 0, we bound the integral overAAby

∫Aπ⁡\(x\)​B0,k​log⁡ρ0​\(x\)π⁡\(x\)​exp⁡\(B0,k​log⁡ρ0​\(x\)π⁡\(x\)\)​𝑑x\\displaystyle\\int\_\{A\}\\pi\(x\)B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\,\\mathrm\{d\}x=\\displaystyle=B0,k1−B0,k​∫Aπ⁡\(x\)​exp⁡\(log⁡ρ0​\(x\)π⁡\(x\)\)⋅\(1−B0,k\)​log​ρ0​\(x\)π⁡\(x\)​exp⁡\(−\(1−B0,k\)​log​ρ0​\(x\)π⁡\(x\)\)​𝑑x\\displaystyle\\frac\{B\_\{0,k\}\}\{1\-B\_\{0,k\}\}\\int\_\{A\}\\pi\(x\)\\exp\\left\(\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\cdot\(1\-B\_\{0,k\}\)\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\exp\\left\(\-\(1\-B\_\{0,k\}\)\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\,\\mathrm\{d\}x≤\\displaystyle\\leqB0,k1−B0,k​∫Aρ0​\(x\)​1e​𝑑x\\displaystyle\\frac\{B\_\{0,k\}\}\{1\-B\_\{0,k\}\}\\int\_\{A\}\\rho\_\{0\}\(x\)\\frac\{1\}\{e\}\\,\\mathrm\{d\}x≤\\displaystyle\\leqB0,ke⁡\(1−B0,k\)\.\\displaystyle\\frac\{B\_\{0,k\}\}\{e\(1\-B\_\{0,k\}\)\}\.Lettingk→∞k\\to\\inftygives

∫Aπ⁡\(x\)​B0,k​log⁡ρ0​\(x\)π⁡\(x\)​exp⁡\(B0,k​log⁡ρ0​\(x\)π⁡\(x\)\)​𝑑x→0\.\\int\_\{A\}\\pi\(x\)B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\,\\mathrm\{d\}x\\to 0\.Next, we bound the integral overAc=\{x∈ℝd:ρ0​\(x\)<π⁡\(x\)\}A^\{\\mathrm\{c\}\}=\\left\\\{x\\in\\mathbb\{R\}^\{d\}:\\rho\_\{0\}\(x\)<\\pi\(x\)\\right\\\}\. By the bound

B0,k​\|log⁡ρ0​\(x\)π⁡\(x\)\|​exp⁡\(B0,k​log⁡ρ0​\(x\)π⁡\(x\)\)≤1eB\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\|\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\leq\\frac\{1\}\{e\}and the pointwise convergence

B0,k​\|log⁡ρ0​\(x\)π⁡\(x\)\|​exp⁡\(B0,k​log⁡ρ0​\(x\)π⁡\(x\)\)→0,∀x∈Ac,B\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\|\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\to 0,\\,\\forall x\\in A^\{\\mathrm\{c\}\},the dominated convergence theorem gives

∫Acπ⁡\(x\)​B0,k​\|log⁡ρ0​\(x\)π⁡\(x\)\|​exp⁡\(B0,k​log⁡ρ0​\(x\)π⁡\(x\)\)​𝑑x→0\.\\int\_\{A^\{\\mathrm\{c\}\}\}\\pi\(x\)B\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\|\\exp\\left\(B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(x\)\}\{\\pi\(x\)\}\\right\)\\,\\mathrm\{d\}x\\to 0\.Therefore, we have

limk→∞𝔼X∼ρk​\[B0,k​\|log⁡ρ0​\(X\)π⁡\(X\)\|\]=0\.\\lim\_\{k\\to\\infty\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[B\_\{0,k\}\\left\|\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\\right\|\\right\]=0\.
For the third step, we have

𝔼X∼ρk​\[∑i=0k−1Bi\+1,k​𝒟ρi,ηi,r~ρi​\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},\\tilde\{r\}^\{\\rho\_\{i\}\}\}\(X\)\\right\]=\\displaystyle=𝔼X∼ρk​\[B0,k​log⁡ρ0​\(X\)π⁡\(X\)−log⁡ρk​\(X\)π⁡\(X\)\+∑i=0k−1Bi\+1,k​λi​KL⁡\(ρi∥π\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\-\\log\\frac\{\\rho\_\{k\}\(X\)\}\{\\pi\(X\)\}\+\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\\right\]=\\displaystyle=𝔼X∼ρk​\[B0,k​log⁡ρ0​\(X\)π⁡\(X\)\]−KL⁡\(ρk∥π\)\+∑i=0k−1Bi\+1,k​λi​KL⁡\(ρi∥π\)\.\\displaystyle\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[B\_\{0,k\}\\log\\frac\{\\rho\_\{0\}\(X\)\}\{\\pi\(X\)\}\\right\]\-\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\+\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\.Based on the conclusion in the first two steps, we have

lim supk→∞𝔼X∼ρk​\[∑i=0k−1Bi\+1,k​𝒟ρi,ηi,r~ρi​\(X\)\]≤0−K∞\+K∞=0\.\\limsup\_\{k\\to\\infty\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},\\tilde\{r\}^\{\\rho\_\{i\}\}\}\(X\)\\right\]\\leq 0\-K\_\{\\infty\}\+K\_\{\\infty\}=0\.
Together, these three steps prove \([38](https://arxiv.org/html/2609.05727#S7.E38)\)\. ∎

We next prove the two global\-convergence theorems\. We first record the following lemma\.

###### Lemma 82\.

Consider two positive normalized densitiesρ,π\\rho,\\pionℝd\\mathbb\{R\}^\{d\}\. LetA⊂ℝdA\\subset\\mathbb\{R\}^\{d\}be measurable\. BothAAandAcA^\{\\mathrm\{c\}\}are of positive measure\. Then, we have

KL⁡\(π∥ρ\)≥π⁡\(Ac\)​log​π⁡\(Ac\)ρ⁡\(Ac\)\+π⁡\(A\)​log​π⁡\(A\)ρ⁡\(A\)\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\\geq\\pi\(A^\{\\mathrm\{c\}\}\)\\log\\frac\{\\pi\(A^\{\\mathrm\{c\}\}\)\}\{\\rho\(A^\{\\mathrm\{c\}\}\)\}\+\\pi\(A\)\\log\\frac\{\\pi\(A\)\}\{\\rho\(A\)\}\.

###### Proof\.

LetρA:=ρρ⁡\(A\)\\rho\_\{A\}:=\\frac\{\\rho\}\{\\rho\(A\)\}andπA:=ππ⁡\(A\)\\pi\_\{A\}:=\\frac\{\\pi\}\{\\pi\(A\)\}be the conditional densities onAA\. Then, we have

∫Aπ⁡\(x\)​log⁡π⁡\(x\)ρ⁡\(x\)​𝑑x=\\displaystyle\\int\_\{A\}\\pi\(x\)\\log\\frac\{\\pi\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}x=π⁡\(A\)​∫AπA​\(x\)​log⁡πA​\(x\)​π​\(A\)ρA​\(x\)​ρ​\(A\)​𝑑x=π⁡\(A\)​\(log⁡π⁡\(A\)ρ⁡\(A\)\+KL⁡\(πA∥ρA\)\)\.\\displaystyle\\pi\(A\)\\int\_\{A\}\\pi\_\{A\}\(x\)\\log\\frac\{\\pi\_\{A\}\(x\)\\pi\(A\)\}\{\\rho\_\{A\}\(x\)\\rho\(A\)\}\\,\\mathrm\{d\}x=\\pi\(A\)\\left\(\\log\\frac\{\\pi\(A\)\}\{\\rho\(A\)\}\+\\operatorname\{KL\}\\left\(\\pi\_\{A\}\\middle\\\|\\rho\_\{A\}\\right\)\\right\)\.Thus,

∫Aπ⁡\(x\)​log​π⁡\(x\)ρ⁡\(x\)​𝑑x≥π⁡\(A\)​log​π⁡\(A\)ρ⁡\(A\)\.\\int\_\{A\}\\pi\(x\)\\log\\frac\{\\pi\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}x\\geq\\pi\(A\)\\log\\frac\{\\pi\(A\)\}\{\\rho\(A\)\}\.For the same reason,

∫Acπ⁡\(x\)​log​π⁡\(x\)ρ⁡\(x\)​𝑑x≥π⁡\(Ac\)​log​π⁡\(Ac\)ρ⁡\(Ac\)\.\\int\_\{A^\{\\mathrm\{c\}\}\}\\pi\(x\)\\log\\frac\{\\pi\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}x\\geq\\pi\(A^\{\\mathrm\{c\}\}\)\\log\\frac\{\\pi\(A^\{\\mathrm\{c\}\}\)\}\{\\rho\(A^\{\\mathrm\{c\}\}\)\}\.Therefore,

KL⁡\(π∥ρ\)=∫Aπ⁡\(x\)​log​π⁡\(x\)ρ⁡\(x\)​𝑑x\+∫Acπ⁡\(x\)​log​π⁡\(x\)ρ⁡\(x\)​𝑑x≥π⁡\(A\)​log​π⁡\(A\)ρ⁡\(A\)\+π⁡\(Ac\)​log​π⁡\(Ac\)ρ⁡\(Ac\),\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)=\\int\_\{A\}\\pi\(x\)\\log\\frac\{\\pi\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}x\+\\int\_\{A^\{\\mathrm\{c\}\}\}\\pi\(x\)\\log\\frac\{\\pi\(x\)\}\{\\rho\(x\)\}\\,\\mathrm\{d\}x\\geq\\pi\(A\)\\log\\frac\{\\pi\(A\)\}\{\\rho\(A\)\}\+\\pi\(A^\{\\mathrm\{c\}\}\)\\log\\frac\{\\pi\(A^\{\\mathrm\{c\}\}\)\}\{\\rho\(A^\{\\mathrm\{c\}\}\)\},which finishes the proof\. ∎

See[28](https://arxiv.org/html/2609.05727#Thmtheorem28)

###### Proof\.

Assume that \([39](https://arxiv.org/html/2609.05727#S7.E39)\) holds butK∞\>0K\_\{\\infty\}\>0, whereK∞K\_\{\\infty\}is defined in \([37](https://arxiv.org/html/2609.05727#S7.E37)\)\. We consider the setAk:=\{x∈ℝd:\|log⁡ρk​\(x\)π⁡\(x\)−K∞\|≤K∞2\}A\_\{k\}:=\\left\\\{x\\in\\mathbb\{R\}^\{d\}:\\left\|\\log\\frac\{\\rho\_\{k\}\(x\)\}\{\\pi\(x\)\}\-K\_\{\\infty\}\\right\|\\leq\\frac\{K\_\{\\infty\}\}\{2\}\\right\\\}\. According to Lemma[82](https://arxiv.org/html/2609.05727#Thmtheorem82), we have

KL⁡\(π∥ρk\)≥π⁡\(Akc\)​log​π⁡\(Akc\)ρk​\(Akc\)\+π⁡\(Ak\)​log​π⁡\(Ak\)ρk​\(Ak\)\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\\geq\\pi\(A\_\{k\}^\{\\mathrm\{c\}\}\)\\log\\frac\{\\pi\(A\_\{k\}^\{\\mathrm\{c\}\}\)\}\{\\rho\_\{k\}\(A\_\{k\}^\{\\mathrm\{c\}\}\)\}\+\\pi\(A\_\{k\}\)\\log\\frac\{\\pi\(A\_\{k\}\)\}\{\\rho\_\{k\}\(A\_\{k\}\)\}\.Applying the inequalityx​log⁡x≥−1ex\\log x\\geq\-\\frac\{1\}\{e\}forx\>0x\>0gives

KL⁡\(π∥ρk\)≥−π⁡\(Akc\)​log​ρk​\(Akc\)−π⁡\(Ak\)​log​ρk​\(Ak\)−2e\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\\geq\-\\pi\(A\_\{k\}^\{\\mathrm\{c\}\}\)\\log\\rho\_\{k\}\(A\_\{k\}^\{\\mathrm\{c\}\}\)\-\\pi\(A\_\{k\}\)\\log\\rho\_\{k\}\(A\_\{k\}\)\-\\frac\{2\}\{e\}\.We can boundπ⁡\(Ak\)\\pi\(A\_\{k\}\)by

π\(Ak\)=∫Akπ\(x\)dx≤∫Akρk\(x\)e−K∞/2dx≤e−K∞/2<1\.\\pi\(A\_\{k\}\)=\\int\_\{A\_\{k\}\}\\pi\(x\)\\,\\mathrm\{d\}x\\leq\\int\_\{A\_\{k\}\}\\rho\_\{k\}\(x\)e^\{\-K\_\{\\infty\}/2\}\\,\\mathrm\{d\}x\\leq e^\{\-K\_\{\\infty\}/2\}<1\.Therefore,

KL\(π∥ρk\)≥−\(1−e−K∞/2\)logρk\(Akc\)−2e\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\\geq\-\(1\-e^\{\-K\_\{\\infty\}/2\}\)\\log\\rho\_\{k\}\(A\_\{k\}^\{\\mathrm\{c\}\}\)\-\\frac\{2\}\{e\}\.Note that

∫Akcρk​\(x\)​𝑑x≤∫Akcρk​\(x\)​2K∞​\|log⁡ρk​\(x\)π⁡\(x\)−K∞\|​𝑑x≤2K∞​𝔼X∼ρk​\[\|log⁡ρk​\(X\)π⁡\(X\)−K∞\|\]\.\\int\_\{A\_\{k\}^\{\\mathrm\{c\}\}\}\\rho\_\{k\}\(x\)\\,\\mathrm\{d\}x\\leq\\int\_\{A\_\{k\}^\{\\mathrm\{c\}\}\}\\rho\_\{k\}\(x\)\\frac\{2\}\{K\_\{\\infty\}\}\\left\|\\log\\frac\{\\rho\_\{k\}\(x\)\}\{\\pi\(x\)\}\-K\_\{\\infty\}\\right\|\\,\\mathrm\{d\}x\\leq\\frac\{2\}\{K\_\{\\infty\}\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{k\}\}\\left\[\\left\|\\log\\frac\{\\rho\_\{k\}\(X\)\}\{\\pi\(X\)\}\-K\_\{\\infty\}\\right\|\\right\]\.By Theorem[26](https://arxiv.org/html/2609.05727#Thmtheorem26), we have

limk→∞ρk​\(Akc\)=0\.\\lim\_\{k\\to\\infty\}\\rho\_\{k\}\(A\_\{k\}^\{\\mathrm\{c\}\}\)=0\.Therefore, ask→∞k\\to\\infty,

KL\(π∥ρk\)≥−\(1−e−K∞/2\)logρk\(Akc\)−2e→\+∞,\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\\geq\-\(1\-e^\{\-K\_\{\\infty\}/2\}\)\\log\\rho\_\{k\}\(A\_\{k\}^\{\\mathrm\{c\}\}\)\-\\frac\{2\}\{e\}\\to\+\\infty,which leads to a contradiction with \([39](https://arxiv.org/html/2609.05727#S7.E39)\)\. ∎

See[30](https://arxiv.org/html/2609.05727#Thmtheorem30)

###### Proof\.

Denote𝒟i:=1ηi2​𝒟ρi,ηi,r~ρi\\mathcal\{D\}\_\{i\}:=\\frac\{1\}\{\\eta\_\{i\}^\{2\}\}\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},\\tilde\{r\}^\{\\rho\_\{i\}\}\}\. Taking the expectation of \([155](https://arxiv.org/html/2609.05727#A2.E155)\) overπ\\pigives

KL⁡\(π∥ρk\)=B0,k​KL⁡\(π∥ρ0\)−∑i=0k−1Bi\+1,k​λi​\(KL⁡\(ρi∥π\)−τ​ηi​𝔼X∼π​\[𝒟i​\(X\)\]\)\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)=B\_\{0,k\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{0\}\\right\)\-\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}\\left\(\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\-\\tau\\eta\_\{i\}\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{i\}\(X\)\\right\]\\right\)\.By \([37](https://arxiv.org/html/2609.05727#S7.E37)\) and condition \([40](https://arxiv.org/html/2609.05727#S7.E40)\), we have

limi→∞KL⁡\(ρi∥π\)−τ​ηi​𝔼X∼π​\[𝒟i​\(X\)\]=K∞\.\\lim\_\{i\\to\\infty\}\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\-\\tau\\eta\_\{i\}\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{i\}\(X\)\\right\]=K\_\{\\infty\}\.By \([156](https://arxiv.org/html/2609.05727#A2.E156)\), the Toeplitz averaging lemma in Lemma[81](https://arxiv.org/html/2609.05727#Thmtheorem81)gives

limk→∞∑i=0k−1Bi\+1,k​λi​\(KL⁡\(ρi∥π\)−τ​ηi​𝔼X∼π​\[𝒟i​\(X\)\]\)=K∞\.\\lim\_\{k\\to\\infty\}\\sum\_\{i=0\}^\{k\-1\}B\_\{i\+1,k\}\\lambda\_\{i\}\\left\(\\operatorname\{KL\}\\left\(\\rho\_\{i\}\\middle\\\|\\pi\\right\)\-\\tau\\eta\_\{i\}\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{i\}\(X\)\\right\]\\right\)=K\_\{\\infty\}\.Therefore, we have

limk→∞KL⁡\(π∥ρk\)=−K∞≥0;\\lim\_\{k\\to\\infty\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)=\-K\_\{\\infty\}\\geq 0;hence,K∞=0K\_\{\\infty\}=0\. ∎

### B\.2Proofs of Local Convergence

The global\-convergence proofs above use the exact terminal\-density recursion and asymptotic averaging\. The local analysis instead represents the full\-step update in centered\-energy coordinates, where Newton cancellation appears as a vanishing first derivative at the target\. Appendix[B\.2\.1](https://arxiv.org/html/2609.05727#A2.SS2.SSS1)proves the resulting quadratic estimate, and Appendix[B\.2\.2](https://arxiv.org/html/2609.05727#A2.SS2.SSS2)makes this estimate uniform along the inverse\-temperature path used by the continuation method\.

#### B\.2\.1Local Quadratic Convergence

We adopt the notation in Section[7\.2\.1](https://arxiv.org/html/2609.05727#S7.SS2.SSS1)\. Taking the full stepsizeηk=τ\\eta\_\{k\}=\\tauin \([34](https://arxiv.org/html/2609.05727#S6.E34)\) gives

log⁡ρk\+1π=KL⁡\(ρk∥π\)−𝒟ρk,1,log⁡\(π/ρk\),\\log\\frac\{\\rho\_\{k\+1\}\}\{\\pi\}=\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\-\\mathcal\{D\}\_\{\\rho\_\{k\},1,\\log\(\\pi/\\rho\_\{k\}\)\},\(157\)where

𝒟ρ,1,log⁡\(π/ρ\)​\(x\)=𝒟ρ,τ,r~ρ​\(x\)=∫01κt​‖∇Vtρ​\[log⁡πρ\]​\(Φ1→tρ,1,log⁡\(π/ρ\)​\(x\)\)‖22​𝑑t\\mathcal\{D\}\_\{\\rho,1,\\log\(\\pi/\\rho\)\}\(x\)=\\mathcal\{D\}\_\{\\rho,\\tau,\\tilde\{r\}^\{\\rho\}\}\(x\)=\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\\left\[\\log\\frac\{\\pi\}\{\\rho\}\\right\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,1,\\log\(\\pi/\\rho\)\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}tis independent ofτ\\taufor a fixedπ\\pi\.

Recall the centered\-energy mapLog:𝒫→𝒪ℰ⊂ℰ\\mathrm\{Log\}:\\mathscr\{P\}\\to\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\subset\\mathscr\{E\}and the equivalence class in Definition[66](https://arxiv.org/html/2609.05727#Thmtheorem66)\. We have

\[log⁡ρπ\]=Log⁡\(ρ\)−Log⁡\(π\)∈ℰ\.\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]=\\mathrm\{Log\}\(\\rho\)\-\\mathrm\{Log\}\(\\pi\)\\in\\mathscr\{E\}\.Taking equivalence classes in \([157](https://arxiv.org/html/2609.05727#A2.E157)\) removes the constant termKL⁡\(ρk∥π\)\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)and gives

\[log⁡ρk\+1π\]=−\[𝒟ρk,1,log⁡\(π/ρk\)\]\.\\left\[\\log\\frac\{\\rho\_\{k\+1\}\}\{\\pi\}\\right\]=\-\\left\[\\mathcal\{D\}\_\{\\rho\_\{k\},1,\\log\(\\pi/\\rho\_\{k\}\)\}\\right\]\.We therefore represent the full\-step iteration by the centered\-energy update map

\[log⁡ρk\+1π\]=ℐπ​\(\[log⁡ρkπ\]\):=−\[𝒟ρk,1,log⁡\(π/ρk\)\]\.\\left\[\\log\\frac\{\\rho\_\{k\+1\}\}\{\\pi\}\\right\]=\\mathcal\{I\}\_\{\\pi\}\\left\(\\left\[\\log\\frac\{\\rho\_\{k\}\}\{\\pi\}\\right\]\\right\):=\-\\left\[\\mathcal\{D\}\_\{\\rho\_\{k\},1,\\log\(\\pi/\\rho\_\{k\}\)\}\\right\]\.The following proposition provides its smooth local realization and the full\-step Newton cancellation, which contributes to the local quadratic convergence\.

###### Proposition 83\.

We define

χπ:𝒫→ℰ,ρ↦\[log⁡ρπ\]=Log⁡\(ρ\)−Log⁡\(π\)\.\\chi\_\{\\pi\}:\\mathscr\{P\}\\to\\mathscr\{E\},\\qquad\\rho\\mapsto\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]=\\mathrm\{Log\}\(\\rho\)\-\\mathrm\{Log\}\(\\pi\)\.Then,χπ\\chi\_\{\\pi\}is a diffeomorphism onto the open set

𝒪π:=Im⁡\(χπ\)=𝒪ℰ−Log⁡\(π\)⊂ℰ,\\mathcal\{O\}\_\{\\pi\}:=\\mathrm\{Im\}\(\\chi\_\{\\pi\}\)=\\mathcal\{O\}\_\{\\mathscr\{E\}\}\-\\mathrm\{Log\}\(\\pi\)\\subset\\mathscr\{E\},and its inverse

χπ−1:𝒪π→𝒫,\(χπ−1​\(\[f\]\)\)​\(x\):=π⁡\(x\)​ef⁡\(x\)𝔼X∼π​\[ef⁡\(X\)\]\\chi\_\{\\pi\}^\{\-1\}:\\mathcal\{O\}\_\{\\pi\}\\to\\mathscr\{P\},\\qquad\\left\(\\chi\_\{\\pi\}^\{\-1\}\(\[f\]\)\\right\)\(x\):=\\frac\{\\pi\(x\)e^\{f\(x\)\}\}\{\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[e^\{f\(X\)\}\\right\]\}is well\-defined\. The map

ℐ^π:𝒫→𝒫,ρ↦𝒯⁡\(𝒞⁡\(ρ\)\+Γρ,log⁡\(π/ρ\)\)\\widehat\{\\mathcal\{I\}\}\_\{\\pi\}:\\mathscr\{P\}\\to\\mathscr\{P\},\\qquad\\rho\\mapsto\\mathcal\{T\}\\left\(\\mathcal\{C\}\(\\rho\)\+\\Gamma^\{\\rho,\\log\(\\pi/\\rho\)\}\\right\)is well\-defined and smooth\. Moreover,

ℐπ:=χπ∘ℐ^π∘χπ−1:𝒪π→𝒪π\\mathcal\{I\}\_\{\\pi\}:=\\chi\_\{\\pi\}\\circ\\widehat\{\\mathcal\{I\}\}\_\{\\pi\}\\circ\\chi\_\{\\pi\}^\{\-1\}:\\mathcal\{O\}\_\{\\pi\}\\to\\mathcal\{O\}\_\{\\pi\}is smooth and satisfies

ℐπ​\(0\)=0,D0​ℐπ=0\.\\mathcal\{I\}\_\{\\pi\}\(0\)=0,\\qquad\\mathrm\{D\}\_\{0\}\\mathcal\{I\}\_\{\\pi\}=0\.\(158\)

###### Proof\.

SinceLog:𝒫→∼𝒪ℰ\\mathrm\{Log\}:\\mathscr\{P\}\\xrightarrow\{\\sim\}\\mathcal\{O\}\_\{\\mathscr\{E\}\}is a diffeomorphism,χπ​\(ρ\)=Log⁡\(ρ\)−Log⁡\(π\)\\chi\_\{\\pi\}\(\\rho\)=\\mathrm\{Log\}\(\\rho\)\-\\mathrm\{Log\}\(\\pi\)is a diffeomorphism from𝒫\\mathscr\{P\}onto the open set𝒪ℰ−Log⁡\(π\)\\mathcal\{O\}\_\{\\mathscr\{E\}\}\-\\mathrm\{Log\}\(\\pi\)\. Its inverse is

χπ−1​\(\[f\]\)=Exp⁡\(Log⁡\(π\)\+\[f\]\),\(χπ−1​\(\[f\]\)\)​\(x\)=π⁡\(x\)​ef⁡\(x\)𝔼X∼π​\[ef⁡\(X\)\],\\chi\_\{\\pi\}^\{\-1\}\(\[f\]\)=\\mathrm\{Exp\}\\left\(\\mathrm\{Log\}\(\\pi\)\+\[f\]\\right\),\\qquad\\left\(\\chi\_\{\\pi\}^\{\-1\}\(\[f\]\)\\right\)\(x\)=\\frac\{\\pi\(x\)e^\{f\(x\)\}\}\{\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[e^\{f\(X\)\}\\right\]\},which is independent of the selected representative of\[f\]\[f\]\.

Consider\[f\]∈𝒪ℰ−Log⁡\(π\)\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}\-\\mathrm\{Log\}\(\\pi\)and denoteρ=χπ−1​\(\[f\]\)\\rho=\\chi\_\{\\pi\}^\{\-1\}\(\[f\]\)\. Since\[f\]=−\[log⁡πρ\]\[f\]=\-\\left\[\\log\\frac\{\\pi\}\{\\rho\}\\right\], Proposition[71](https://arxiv.org/html/2609.05727#Thmtheorem71)gives

dLog⁡\(π\)\+\[f\]​Exp​\[−\[f\]\]=ρ⁡\(log⁡πρ−𝔼ρ​\[log⁡πρ\]\)\.\\mathrm\{d\}\_\{\\mathrm\{Log\}\(\\pi\)\+\[f\]\}\\mathrm\{Exp\}\[\-\[f\]\]=\\rho\\left\(\\log\\frac\{\\pi\}\{\\rho\}\-\\mathbb\{E\}\_\{\\rho\}\\left\[\\log\\frac\{\\pi\}\{\\rho\}\\right\]\\right\)\.Therefore, by Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5),

DLog⁡\(π\)\+\[f\]​\(𝒞∘Exp\)​\[−\[f\]\]=dρ​𝒞​\[dLog⁡\(π\)\+\[f\]​Exp​\[−\[f\]\]\]=Γρ,log⁡\(π/ρ\)\.\\mathrm\{D\}\_\{\\mathrm\{Log\}\(\\pi\)\+\[f\]\}\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\[\-\[f\]\]=\\mathrm\{d\}\_\{\\rho\}\\mathcal\{C\}\\left\[\\mathrm\{d\}\_\{\\mathrm\{Log\}\(\\pi\)\+\[f\]\}\\mathrm\{Exp\}\[\-\[f\]\]\\right\]=\\Gamma^\{\\rho,\\log\(\\pi/\\rho\)\}\.
Hence, the ambient full\-step update in the energy coordinates is

\(𝒞∘Exp\)​\(Log⁡\(π\)\+\[f\]\)\+DLog⁡\(π\)\+\[f\]​\(𝒞∘Exp\)​\[−\[f\]\]\.\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\\left\(\\mathrm\{Log\}\(\\pi\)\+\[f\]\\right\)\+\\mathrm\{D\}\_\{\\mathrm\{Log\}\(\\pi\)\+\[f\]\}\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\[\-\[f\]\]\.This expression depends smoothly on\[f\]\[f\]\. At\[f\]=0\[f\]=0, it is equal to𝒞⁡\(π\)∈𝒱\\mathcal\{C\}\(\\pi\)\\in\\mathscr\{V\}\. Since𝒱\\mathscr\{V\}is a Banach space,ℐ^π\\widehat\{\\mathcal\{I\}\}\_\{\\pi\}is well\-defined for every\[f\]∈𝒪π\[f\]\\in\\mathcal\{O\}\_\{\\pi\}\. Moreover,

ℐπ​\(\[f\]\)=Log∘𝒯⁡\(\(𝒞∘Exp\)​\(Log⁡\(π\)\+\[f\]\)\+DLog⁡\(π\)\+\[f\]​\(𝒞∘Exp\)​\[−\[f\]\]\)−Log⁡\(π\)\.\\mathcal\{I\}\_\{\\pi\}\(\[f\]\)=\\mathrm\{Log\}\\circ\\mathcal\{T\}\\left\(\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\\left\(\\mathrm\{Log\}\(\\pi\)\+\[f\]\\right\)\+\\mathrm\{D\}\_\{\\mathrm\{Log\}\(\\pi\)\+\[f\]\}\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\[\-\[f\]\]\\right\)\-\\mathrm\{Log\}\(\\pi\)\.Therefore,ℐ^π\\widehat\{\\mathcal\{I\}\}\_\{\\pi\}andℐπ\\mathcal\{I\}\_\{\\pi\}are smooth\.

At\[f\]=0\[f\]=0, we have

ℐπ​\(0\)=Log∘𝒯⁡\(𝒞⁡\(π\)\)−Log⁡\(π\)=0\.\\mathcal\{I\}\_\{\\pi\}\(0\)=\\mathrm\{Log\}\\circ\\mathcal\{T\}\\left\(\\mathcal\{C\}\(\\pi\)\\right\)\-\\mathrm\{Log\}\(\\pi\)=0\.For anyh∈ℰh\\in\\mathscr\{E\}, differentiating the ambient update at\[f\]=0\[f\]=0gives

DLog⁡\(π\)​\(𝒞∘Exp\)​\[h\]\+DLog⁡\(π\)2​\(𝒞∘Exp\)​\[h,0\]−DLog⁡\(π\)​\(𝒞∘Exp\)​\[h\]=0\.\\displaystyle\\mathrm\{D\}\_\{\\mathrm\{Log\}\(\\pi\)\}\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\[h\]\+\\mathrm\{D\}^\{2\}\_\{\\mathrm\{Log\}\(\\pi\)\}\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\[h,0\]\-\\mathrm\{D\}\_\{\\mathrm\{Log\}\(\\pi\)\}\\left\(\\mathcal\{C\}\\circ\\mathrm\{Exp\}\\right\)\[h\]=0\.Applying the chain rule to the coordinate expression ofℐπ\\mathcal\{I\}\_\{\\pi\}givesD0​ℐπ​\[h\]=0\\mathrm\{D\}\_\{0\}\\mathcal\{I\}\_\{\\pi\}\[h\]=0\. Therefore, we haveD0​ℐπ=0\\mathrm\{D\}\_\{0\}\\mathcal\{I\}\_\{\\pi\}=0\. ∎

To express the centered\-energy estimate intrinsically on the density and canonical manifolds, define

d𝒫​\(ρ1,ρ2\):=‖Log⁡\(ρ1\)−Log⁡\(ρ2\)‖ℰ,d\_\{\\mathscr\{P\}\}\(\\rho\_\{1\},\\rho\_\{2\}\):=\\left\\\|\\mathrm\{Log\}\(\\rho\_\{1\}\)\-\\mathrm\{Log\}\(\\rho\_\{2\}\)\\right\\\|\_\{\\mathscr\{E\}\},which is the pullback under the global energy chartLog:𝒫→𝒪ℰ\\mathrm\{Log\}:\\mathscr\{P\}\\to\\mathcal\{O\}\_\{\\mathscr\{E\}\}of the metric on𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}induced by the norm‖⋅‖ℰ\\left\\\|\\cdot\\right\\\|\_\{\\mathscr\{E\}\}\. The following proposition verifies thatd𝒫d\_\{\\mathscr\{P\}\}is a metric on𝒫\\mathscr\{P\}and that the transported metricd𝒱cand\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}in \([41](https://arxiv.org/html/2609.05727#S7.E41)\) is a metric on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.

See[31](https://arxiv.org/html/2609.05727#Thmtheorem31)

###### Proof\.

The map

ℰ×ℰ→\[0,∞\),\(\[f1\],\[f2\]\)↦‖\[f1−f2\]‖ℰ\\mathscr\{E\}\\times\\mathscr\{E\}\\to\[0,\\infty\),\\qquad\(\[f\_\{1\}\],\[f\_\{2\}\]\)\\mapsto\\left\\\|\[f\_\{1\}\-f\_\{2\}\]\\right\\\|\_\{\\mathscr\{E\}\}is a metric onℰ\\mathscr\{E\}\. SinceLog\\mathrm\{Log\}is a diffeomorphism between𝒫\\mathscr\{P\}and the open subset𝒪ℰ⊂ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\subset\\mathscr\{E\},d𝒫d\_\{\\mathscr\{P\}\}is a metric on𝒫\\mathscr\{P\}\. As the pullback of the norm metric on𝒪ℰ\\mathcal\{O\}\_\{\\mathscr\{E\}\},d𝒫d\_\{\\mathscr\{P\}\}induces exactly the Banach\-manifold topology on𝒫\\mathscr\{P\}\. Consequently, since𝒯\|𝒱can\\left\.\\mathcal\{T\}\\right\|\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}is a diffeomorphism between𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}and𝒫\\mathscr\{P\},d𝒱cand\_\{\\mathscr\{V\}\_\{\\mathrm\{can\}\}\}is a metric on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}and induces exactly the Banach\-manifold topology on𝒱can\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. ∎

With the centered\-energy update and the induced metrics in place, local quadratic convergence follows from a second\-order Taylor expansion ofℐπ\\mathcal\{I\}\_\{\\pi\}at the fixed point0∈ℰ0\\in\\mathscr\{E\}\.

See[32](https://arxiv.org/html/2609.05727#Thmtheorem32)

###### Proof\.

Sinceℐπ\\mathcal\{I\}\_\{\\pi\}isC2C^\{2\}on the open neighborhood𝒪π\\mathcal\{O\}\_\{\\pi\}of0∈ℰ0\\in\\mathscr\{E\}, there exist a constantCπ\>0C\_\{\\pi\}\>0and a sufficiently small radiusRπ\>0R\_\{\\pi\}\>0such that, for anyρ^∈𝒫\\hat\{\\rho\}\\in\\mathscr\{P\}satisfyingd𝒫​\(ρ^,π\)=‖\[log⁡ρ^π\]‖ℰ<Rπd\_\{\\mathscr\{P\}\}\(\\hat\{\\rho\},\\pi\)=\\left\\\|\\left\[\\log\\frac\{\\hat\{\\rho\}\}\{\\pi\}\\right\]\\right\\\|\_\{\\mathscr\{E\}\}<R\_\{\\pi\}, we have

\[log⁡ρ^π\]∈𝒪π,‖D\[log⁡ρ^π\]2​ℐπ‖op≤2​Cπ\.\\left\[\\log\\frac\{\\hat\{\\rho\}\}\{\\pi\}\\right\]\\in\\mathcal\{O\}\_\{\\pi\},\\qquad\\left\\\|\\mathrm\{D\}\_\{\\left\[\\log\\frac\{\\hat\{\\rho\}\}\{\\pi\}\\right\]\}^\{2\}\\mathcal\{I\}\_\{\\pi\}\\right\\\|\_\{\\mathrm\{op\}\}\\leq 2C\_\{\\pi\}\.Here, the norm‖⋅‖op\\left\\\|\\cdot\\right\\\|\_\{\\mathrm\{op\}\}is the operator norm with respect to‖⋅‖ℰ\\left\\\|\\cdot\\right\\\|\_\{\\mathscr\{E\}\}\. According to Proposition[83](https://arxiv.org/html/2609.05727#Thmtheorem83), taking the Taylor expansion at00gives

ℐπ​\(\[log⁡ρπ\]\)=∫01\(1−ζ\)​D\[ζ​log⁡ρπ\]2​ℐπ​\[\[log⁡ρπ\],\[log⁡ρπ\]\]​𝑑ζ\.\\mathcal\{I\}\_\{\\pi\}\\left\(\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\right\)=\\int\_\{0\}^\{1\}\(1\-\\zeta\)\\mathrm\{D\}\_\{\\left\[\\zeta\\log\\frac\{\\rho\}\{\\pi\}\\right\]\}^\{2\}\\mathcal\{I\}\_\{\\pi\}\\left\[\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\],\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\right\]\\mathrm\{d\}\\zeta\.If‖\[log⁡ρπ\]‖ℰ<Rπ\\left\\\|\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\right\\\|\_\{\\mathscr\{E\}\}<R\_\{\\pi\}, then we have‖\[ζ​log⁡ρπ\]‖ℰ<Rπ\\left\\\|\\left\[\\zeta\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\right\\\|\_\{\\mathscr\{E\}\}<R\_\{\\pi\}for allζ∈\[0,1\]\\zeta\\in\[0,1\]\. By \([158](https://arxiv.org/html/2609.05727#A2.E158)\),

‖ℐπ​\(\[log⁡ρπ\]\)‖ℰ≤∫01\(1−ζ\)​‖D\[ζ​log⁡ρπ\]2​ℐπ‖op​‖\[log⁡ρπ\]‖ℰ2​𝑑ζ≤Cπ​‖\[log⁡ρπ\]‖ℰ2,\\left\\\|\\mathcal\{I\}\_\{\\pi\}\\left\(\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\right\)\\right\\\|\_\{\\mathscr\{E\}\}\\leq\\int\_\{0\}^\{1\}\(1\-\\zeta\)\\left\\\|\\mathrm\{D\}\_\{\\left\[\\zeta\\log\\frac\{\\rho\}\{\\pi\}\\right\]\}^\{2\}\\mathcal\{I\}\_\{\\pi\}\\right\\\|\_\{\\mathrm\{op\}\}\\left\\\|\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\right\\\|\_\{\\mathscr\{E\}\}^\{2\}\\mathrm\{d\}\\zeta\\leq C\_\{\\pi\}\\left\\\|\\left\[\\log\\frac\{\\rho\}\{\\pi\}\\right\]\\right\\\|\_\{\\mathscr\{E\}\}^\{2\},which gives \([42](https://arxiv.org/html/2609.05727#S7.E42)\)\.

Consequently, suppose that the initial pointρ0\\rho\_\{0\}satisfiesd𝒫​\(ρ0,π\)<min⁡\{Rπ,1Cπ\}d\_\{\\mathscr\{P\}\}\(\\rho\_\{0\},\\pi\)<\\min\\left\\\{R\_\{\\pi\},\\frac\{1\}\{C\_\{\\pi\}\}\\right\\\}\. By \([42](https://arxiv.org/html/2609.05727#S7.E42)\), we have

d𝒫​\(ρ1,π\)≤Cπ​d𝒫​\(ρ0,π\)2≤d𝒫​\(ρ0,π\)<min⁡\{Rπ,1Cπ\}\.d\_\{\\mathscr\{P\}\}\(\\rho\_\{1\},\\pi\)\\leq C\_\{\\pi\}d\_\{\\mathscr\{P\}\}\(\\rho\_\{0\},\\pi\)^\{2\}\\leq d\_\{\\mathscr\{P\}\}\(\\rho\_\{0\},\\pi\)<\\min\\left\\\{R\_\{\\pi\},\\frac\{1\}\{C\_\{\\pi\}\}\\right\\\}\.By induction, \([43](https://arxiv.org/html/2609.05727#S7.E43)\) holds for allkk\. Therefore,\{ρk\}k=0∞\\\{\\rho\_\{k\}\\\}\_\{k=0\}^\{\\infty\}converges quadratically toπ\\piunder the metricd𝒫d\_\{\\mathscr\{P\}\}, which also gives the local quadratic convergence of the canonical velocity fields\. ∎

#### B\.2\.2Proof of Continuation Method

We adopt the notation in Section[7\.2\.2](https://arxiv.org/html/2609.05727#S7.SS2.SSS2)\. The local estimate above has constants that may depend on the fixed target\. To obtain uniform constants over the compact family\{πs:s∈\[τ0,τ\]\}\\\{\\pi\_\{s\}:s\\in\[\\tau\_\{0\},\\tau\]\\\}, we first establish joint smoothness of the centered\-energy update in\(s,\[f\]\)\(s,\[f\]\)\.

###### Proposition 84\.

Ifπs:=πμ,s,r∈𝒫\\pi\_\{s\}:=\\pi\_\{\\mu,s,r\}\\in\\mathscr\{P\}for alls∈\[τ0,τ\]s\\in\[\\tau\_\{0\},\\tau\], then the map

\(s,\[f\]\)↦ℐπs​\(\[f\]\)\(s,\[f\]\)\\mapsto\\mathcal\{I\}\_\{\\pi\_\{s\}\}\(\[f\]\)is smooth on an open neighborhood of\[τ0,τ\]×\{0\}\[\\tau\_\{0\},\\tau\]\\times\\\{0\\\}inℝ×ℰ\\mathbb\{R\}\\times\\mathscr\{E\}\.

###### Proof\.

SinceLog⁡\(πs\)−Log⁡\(πτ0\)=\(s−τ0\)​\[r\]∈ℰ\\mathrm\{Log\}\(\\pi\_\{s\}\)\-\\mathrm\{Log\}\(\\pi\_\{\\tau\_\{0\}\}\)=\(s\-\\tau\_\{0\}\)\[r\]\\in\\mathscr\{E\}, we have\[r\]∈ℰ\[r\]\\in\\mathscr\{E\}\. Therefore,\(s,\[f\]\)↦ℐπs​\(\[f\]\)\(s,\[f\]\)\\mapsto\\mathcal\{I\}\_\{\\pi\_\{s\}\}\(\[f\]\)is smooth wherever it is defined\. Note that

\{\(s,\[f\]\):Log⁡\(πs\)\+\[f\]∈𝒪ℰ\}⊃\[τ0,τ\]×\{0\}\\left\\\{\(s,\[f\]\):\\mathrm\{Log\}\(\\pi\_\{s\}\)\+\[f\]\\in\\mathcal\{O\}\_\{\\mathscr\{E\}\}\\right\\\}\\supset\[\\tau\_\{0\},\\tau\]\\times\\\{0\\\}is open; hence, the map is smooth on an open neighborhood of\[τ0,τ\]×\{0\}\[\\tau\_\{0\},\\tau\]\\times\\\{0\\\}\. ∎

Joint smoothness and compactness now yield a common coordinate neighborhood and a uniform second\-derivative bound over the entire inverse\-temperature interval\[τ0,τ\]\[\\tau\_\{0\},\\tau\]\.

See[33](https://arxiv.org/html/2609.05727#Thmtheorem33)

###### Proof\.

Let

C:=sups∈\[τ0,τ\]‖D02​ℐπs‖op\+1<∞,M:=‖\[r\]‖ℰ<∞\.C:=\\sup\_\{s\\in\[\\tau\_\{0\},\\tau\]\}\\left\\\|\\mathrm\{D\}\_\{0\}^\{2\}\\mathcal\{I\}\_\{\\pi\_\{s\}\}\\right\\\|\_\{\\mathrm\{op\}\}\+1<\\infty,\\qquad M:=\\left\\\|\\left\[r\\right\]\\right\\\|\_\{\\mathscr\{E\}\}<\\infty\.Since the domain of\(s,\[f\]\)↦ℐπs​\(\[f\]\)\(s,\[f\]\)\\mapsto\\mathcal\{I\}\_\{\\pi\_\{s\}\}\(\[f\]\)is an open neighborhood of the compact set\[τ0,τ\]×\{0\}\[\\tau\_\{0\},\\tau\]\\times\\\{0\\\}, there existsR\>0R\>0such that

s∈\[τ0,τ\],‖\[f\]‖ℰ<R⇒\[f\]∈𝒪πs\.s\\in\[\\tau\_\{0\},\\tau\],\\,\\left\\\|\[f\]\\right\\\|\_\{\\mathscr\{E\}\}<R\\,\\Rightarrow\\,\[f\]\\in\\mathcal\{O\}\_\{\\pi\_\{s\}\}\.
For the first step, we prove that, after decreasingRRif necessary, for anys∈\[τ0,τ\]s\\in\[\\tau\_\{0\},\\tau\]andρ∈𝒫\\rho\\in\\mathscr\{P\}which satisfiesd𝒫​\(ρ,πs\)<Rd\_\{\\mathscr\{P\}\}\(\\rho,\\pi\_\{s\}\)<R, we have

‖D\[log⁡ρπs\]2​ℐπs‖op<C\.\\left\\\|\\mathrm\{D\}\_\{\\left\[\\log\\frac\{\\rho\}\{\\pi\_\{s\}\}\\right\]\}^\{2\}\\mathcal\{I\}\_\{\\pi\_\{s\}\}\\right\\\|\_\{\\mathrm\{op\}\}<C\.Otherwise, for everyj\>0j\>0, there existsj∈\[τ0,τ\]s\_\{j\}\\in\[\\tau\_\{0\},\\tau\]andρj∈𝒫\\rho\_\{j\}\\in\\mathscr\{P\}satisfyingd𝒫​\(ρj,πsj\)<min⁡\{R,1j\+1\}d\_\{\\mathscr\{P\}\}\(\\rho\_\{j\},\\pi\_\{s\_\{j\}\}\)<\\min\\left\\\{R,\\frac\{1\}\{j\+1\}\\right\\\}such that

‖D\[log⁡ρjπsj\]2​ℐπsj‖op≥C\.\\left\\\|\\mathrm\{D\}\_\{\\left\[\\log\\frac\{\\rho\_\{j\}\}\{\\pi\_\{s\_\{j\}\}\}\\right\]\}^\{2\}\\mathcal\{I\}\_\{\\pi\_\{s\_\{j\}\}\}\\right\\\|\_\{\\mathrm\{op\}\}\\geq C\.Note that\{sj\}j=0∞\\left\\\{s\_\{j\}\\right\\\}\_\{j=0\}^\{\\infty\}has a convergent subsequence\. Without loss of generality, we assume that the sequence\{sj\}j=0∞\\left\\\{s\_\{j\}\\right\\\}\_\{j=0\}^\{\\infty\}converges tos⋆s^\{\\star\}asj→∞j\\to\\infty\. Then, we have

d𝒫​\(πsj,πs⋆\)=\|sj−s⋆\|​‖\[r\]‖ℰ≤\|sj−s⋆\|​M→0,d\_\{\\mathscr\{P\}\}\(\\pi\_\{s\_\{j\}\},\\pi\_\{s^\{\\star\}\}\)=\\left\|s\_\{j\}\-s^\{\\star\}\\right\|\\left\\\|\\left\[r\\right\]\\right\\\|\_\{\\mathscr\{E\}\}\\leq\\left\|s\_\{j\}\-s^\{\\star\}\\right\|M\\to 0,and

d𝒫​\(ρj,πs⋆\)≤d𝒫​\(ρj,πsj\)\+d𝒫​\(πs⋆,πsj\)→0\.d\_\{\\mathscr\{P\}\}\(\\rho\_\{j\},\\pi\_\{s^\{\\star\}\}\)\\leq d\_\{\\mathscr\{P\}\}\(\\rho\_\{j\},\\pi\_\{s\_\{j\}\}\)\+d\_\{\\mathscr\{P\}\}\(\\pi\_\{s^\{\\star\}\},\\pi\_\{s\_\{j\}\}\)\\to 0\.Moreover,

‖\[log⁡ρjπsj\]‖ℰ=d𝒫​\(ρj,πsj\)→0\.\\left\\\|\\left\[\\log\\frac\{\\rho\_\{j\}\}\{\\pi\_\{s\_\{j\}\}\}\\right\]\\right\\\|\_\{\\mathscr\{E\}\}=d\_\{\\mathscr\{P\}\}\(\\rho\_\{j\},\\pi\_\{s\_\{j\}\}\)\\to 0\.Therefore, Proposition[84](https://arxiv.org/html/2609.05727#Thmtheorem84)gives

‖D\[log⁡ρjπsj\]2​ℐπsj‖op→‖D02​ℐπs⋆‖op,\\left\\\|\\mathrm\{D\}\_\{\\left\[\\log\\frac\{\\rho\_\{j\}\}\{\\pi\_\{s\_\{j\}\}\}\\right\]\}^\{2\}\\mathcal\{I\}\_\{\\pi\_\{s\_\{j\}\}\}\\right\\\|\_\{\\mathrm\{op\}\}\\to\\left\\\|\\mathrm\{D\}\_\{0\}^\{2\}\\mathcal\{I\}\_\{\\pi\_\{s^\{\\star\}\}\}\\right\\\|\_\{\\mathrm\{op\}\},which leads to a contradiction since‖D02​ℐπs⋆‖op≤C−1\\left\\\|\\mathrm\{D\}\_\{0\}^\{2\}\\mathcal\{I\}\_\{\\pi\_\{s^\{\\star\}\}\}\\right\\\|\_\{\\mathrm\{op\}\}\\leq C\-1\. This completes the first step\.

For the second step, we defineΔmax=1M\+1​min⁡\{R,1C\}\>0\\Delta\_\{\\max\}=\\frac\{1\}\{M\+1\}\\min\\left\\\{R,\\frac\{1\}\{C\}\\right\\\}\>0\. Then, for anys∈\[τ0,τ\]s\\in\[\\tau\_\{0\},\\tau\]andΔ∈\[0,min⁡\{s−τ0,Δmax\}\]\\Delta\\in\[0,\\min\\\{s\-\\tau\_\{0\},\\Delta\_\{\\max\}\\\}\], we have

d𝒫​\(πs−Δ,πs\)=Δ​‖\[r\]‖ℰ≤Δmax​M<min⁡\{R,1C\}\.d\_\{\\mathscr\{P\}\}\(\\pi\_\{s\-\\Delta\},\\pi\_\{s\}\)=\\Delta\\left\\\|\\left\[r\\right\]\\right\\\|\_\{\\mathscr\{E\}\}\\leq\\Delta\_\{\\max\}M<\\min\\left\\\{R,\\frac\{1\}\{C\}\\right\\\}\.
For the last step, consider any finite gridτ0<τ1<⋯<τn=τ\\tau\_\{0\}<\\tau\_\{1\}<\\cdots<\\tau\_\{n\}=\\tausatisfyingΔi:=τi−τi−1≤Δmax\\Delta\_\{i\}:=\\tau\_\{i\}\-\\tau\_\{i\-1\}\\leq\\Delta\_\{\\max\}for allii\. The same Taylor expansion as in the proof of Theorem[32](https://arxiv.org/html/2609.05727#Thmtheorem32), together with the uniform bound in the first step, gives

d𝒫​\(ρk\+1\(i\),π\(i\)\)≤C​d𝒫​\(ρk\(i\),π\(i\)\)2d\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{k\+1\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)\\leq Cd\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{k\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)^\{2\}wheneverd𝒫​\(ρk\(i\),π\(i\)\)<Rd\_\{\\mathscr\{P\}\}\(\\rho\_\{k\}^\{\(i\)\},\\pi^\{\(i\)\}\)<R\. The second step gives

d𝒫​\(ρ0\(i\),π\(i\)\)<min⁡\{R,1C\}\.d\_\{\\mathscr\{P\}\}\\left\(\\rho\_\{0\}^\{\(i\)\},\\pi^\{\(i\)\}\\right\)<\\min\\left\\\{R,\\frac\{1\}\{C\}\\right\\\}\.By induction, the quadratic estimate holds for allkk, andρk\(i\)\\rho\_\{k\}^\{\(i\)\}converges quadratically toπ\(i\)\\pi^\{\(i\)\}for everyii\. The local quadratic convergence of the canonical velocity fields follows immediately\. ∎

## Appendix CNewton Matching for the Isotropic Gaussian Family

This appendix presents an analytically tractable instance of Newton Matching on the isotropic Gaussian family

\{𝒩\(m,σ2I\):m∈ℝd,σ\>0\}\.\\left\\\{\\mathcal\{N\}\(m,\\sigma^\{2\}I\):m\\in\\mathbb\{R\}^\{d\},\\ \\sigma\>0\\right\\\}\.Appendix[C\.1](https://arxiv.org/html/2609.05727#A3.SS1)shows that this family is closed under the ideal Newton Matching update and derives the exact one\-step parameter update\. Appendix[C\.2](https://arxiv.org/html/2609.05727#A3.SS2)then establishes global convergence of both forward and reverse KL divergences, together with local quadratic rates\.

In this appendix, it suffices to analyze the normalized caseτ=1\\tau=1, as we can equivalently applyr^:=τ​r\\hat\{r\}:=\\tau randη^:=ητ\\hat\{\\eta\}:=\\frac\{\\eta\}\{\\tau\}for generalτ\>0\\tau\>0\.

### C\.1Gaussian Closure

This subsection derives the closure property of Newton Matching in the isotropic Gaussian family\. In other words, if the current densityρ\\rhoand the target densityπ\\piare both isotropic Gaussian, then the updated densityq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\)is also isotropic Gaussian\.

We first derive the canonical velocity field of a Gaussian distribution\.

###### Proposition 85\.

Consider a Gaussian distributionρ=𝒩⁡\(m,Σ\)\\rho=\\mathcal\{N\}\(m,\\Sigma\),m∈ℝdm\\in\\mathbb\{R\}^\{d\},Σ∈𝕊\+⁣\+d\\Sigma\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}\. Its canonical velocity field is

vtρ​\(xt\)=α˙t​m\+\(αt​α˙t​Σ\+βt​β˙t​I\)​\(αt2​Σ\+βt2​I\)−1​\(xt−αt​m\)\.v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\dot\{\\alpha\}\_\{t\}m\+\\left\(\\alpha\_\{t\}\\dot\{\\alpha\}\_\{t\}\\Sigma\+\\beta\_\{t\}\\dot\{\\beta\}\_\{t\}I\\right\)\\left\(\\alpha\_\{t\}^\{2\}\\Sigma\+\\beta\_\{t\}^\{2\}I\\right\)^\{\-1\}\(x\_\{t\}\-\\alpha\_\{t\}m\)\.\(159\)In particular, ifρ=𝒩⁡\(m,σ2​I\)\\rho=\\mathcal\{N\}\(m,\\sigma^\{2\}I\)is isotropic, then its canonical velocity field is

vtρ​\(xt\)=α˙t​m\+β˙t​βt\+α˙t​αt​σ2βt2\+αt2​σ2​\(xt−αt​m\)\.v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\dot\{\\alpha\}\_\{t\}m\+\\frac\{\\dot\{\\beta\}\_\{t\}\\beta\_\{t\}\+\\dot\{\\alpha\}\_\{t\}\\alpha\_\{t\}\\sigma^\{2\}\}\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma^\{2\}\}\(x\_\{t\}\-\\alpha\_\{t\}m\)\.\(160\)

###### Proof\.

Consider a Gaussian distributionρ=𝒩⁡\(m,Σ\)\\rho=\\mathcal\{N\}\(m,\\Sigma\)\. Under independent coupling, the interpolantXt=αt​X1\+βt​X0X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}satisfies

Xt∼𝒩⁡\(αt​m,αt2​Σ\+βt2​I\),\[X1Xt\]∼𝒩⁡\(\[mαt​m\],\[Σαt​Σαt​Σαt2​Σ\+βt2​I\]\),X\_\{t\}\\sim\\mathcal\{N\}\\left\(\\alpha\_\{t\}m,\\alpha\_\{t\}^\{2\}\\Sigma\+\\beta\_\{t\}^\{2\}I\\right\),\\qquad\\left\[\\begin\{array\}\[\]\{c\}X\_\{1\}\\\\ X\_\{t\}\\end\{array\}\\right\]\\sim\\mathcal\{N\}\\left\(\\left\[\\begin\{array\}\[\]\{c\}m\\\\ \\alpha\_\{t\}m\\end\{array\}\\right\],\\left\[\\begin\{array\}\[\]\{cc\}\\Sigma&\\alpha\_\{t\}\\Sigma\\\\ \\alpha\_\{t\}\\Sigma&\\alpha\_\{t\}^\{2\}\\Sigma\+\\beta\_\{t\}^\{2\}I\\end\{array\}\\right\]\\right\),which gives

p1\|tρ\(⋅\|xt\)=𝒩\(m\+αtΣ\(αt2Σ\+βt2I\)−1\(xt−αtm\),βt2Σ\(αt2Σ\+βt2I\)−1\)\.p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)=\\mathcal\{N\}\\left\(m\+\\alpha\_\{t\}\\Sigma\\left\(\\alpha\_\{t\}^\{2\}\\Sigma\+\\beta\_\{t\}^\{2\}I\\right\)^\{\-1\}\(x\_\{t\}\-\\alpha\_\{t\}m\),\\beta\_\{t\}^\{2\}\\Sigma\\left\(\\alpha\_\{t\}^\{2\}\\Sigma\+\\beta\_\{t\}^\{2\}I\\right\)^\{\-1\}\\right\)\.Then,

vtρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[β˙tβtxt\+αt​κtβt2X1\]v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\+\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}X\_\{1\}\\right\]gives \([159](https://arxiv.org/html/2609.05727#A3.E159)\)\. In particular, substitutingΣ=σ2​I\\Sigma=\\sigma^\{2\}Iinto \([159](https://arxiv.org/html/2609.05727#A3.E159)\) implies \([160](https://arxiv.org/html/2609.05727#A3.E160)\)\. ∎

The following proposition provides the one\-step updated density for the isotropic Gaussian, which is a special case of Proposition[19](https://arxiv.org/html/2609.05727#Thmtheorem19)\. The full Gaussian family

\{𝒩\(m,Σ\):m∈ℝd,Σ∈𝕊\+⁣\+d\}\\left\\\{\\mathcal\{N\}\(m,\\Sigma\):m\\in\\mathbb\{R\}^\{d\},\\Sigma\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}\\right\\\}is also closed under the ideal Newton Matching update; however, the resulting matrix\-valued parameter recursion is considerably more involved\. For simplicity, we restrict our analysis to the isotropic case in this appendix\.

###### Proposition 86\.

Consider the target densityπ=𝒩⁡\(mπ,σπ2​I\)\\pi=\\mathcal\{N\}\(m\_\{\\pi\},\\sigma\_\{\\pi\}^\{2\}I\)and the current densityρ=𝒩⁡\(mρ,σρ2​I\)\\rho=\\mathcal\{N\}\(m\_\{\\rho\},\\sigma\_\{\\rho\}^\{2\}I\)\. The regularized reward satisfiesr~ρ​\(x\)=log⁡π⁡\(x\)ρ⁡\(x\)\+const\\tilde\{r\}^\{\\rho\}\(x\)=\\log\\frac\{\\pi\(x\)\}\{\\rho\(x\)\}\+\\mathrm\{const\}\. Consider a stepsizeη∈ℝ\\eta\\in\\mathbb\{R\}\. Letq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\)\. Then, the updated density is also Gaussian:

q=𝒩⁡\(mq,σq2​I\),mq:=mρ\+\(mπ−mρ\)⋅wη​\(σρ2σπ2\),σq:=σρ​exp⁡\(η2​\(1−σρ2σπ2\)\),q=\\mathcal\{N\}\(m\_\{q\},\\sigma\_\{q\}^\{2\}I\),\\quad m\_\{q\}:=m\_\{\\rho\}\+\(m\_\{\\pi\}\-m\_\{\\rho\}\)\\cdot w\_\{\\eta\}\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\),\\quad\\sigma\_\{q\}:=\\sigma\_\{\\rho\}\\exp\\left\(\\frac\{\\eta\}\{2\}\\left\(1\-\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\right\),where

wη​\(z\):=η​z​∫01exp⁡\(η2​\(1−z\)​\(1−s2\)\)​𝑑s\.w\_\{\\eta\}\(z\):=\\eta z\\int\_\{0\}^\{1\}\\exp\\left\(\\frac\{\\eta\}\{2\}\(1\-z\)\(1\-s^\{2\}\)\\right\)\\,\\mathrm\{d\}s\.

###### Proof\.

Since

r~ρ​\(x1\)=log⁡π⁡\(x1\)ρ⁡\(x1\)\+const=12​σρ2​‖x1−mρ‖22−12​σπ2​‖x1−mπ‖22\+const,\\tilde\{r\}^\{\\rho\}\(x\_\{1\}\)=\\log\\frac\{\\pi\(x\_\{1\}\)\}\{\\rho\(x\_\{1\}\)\}\+\\mathrm\{const\}=\\frac\{1\}\{2\\sigma\_\{\\rho\}^\{2\}\}\\left\\\|x\_\{1\}\-m\_\{\\rho\}\\right\\\|\_\{2\}^\{2\}\-\\frac\{1\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\left\\\|x\_\{1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\+\\mathrm\{const\},we have

∇r~ρ​\(x1\)=\(1σρ2−1σπ2\)​x1\+mπσπ2−mρσρ2\.\\nabla\\tilde\{r\}^\{\\rho\}\(x\_\{1\}\)=\\left\(\\frac\{1\}\{\\sigma\_\{\\rho\}^\{2\}\}\-\\frac\{1\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)x\_\{1\}\+\\frac\{m\_\{\\pi\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\-\\frac\{m\_\{\\rho\}\}\{\\sigma\_\{\\rho\}^\{2\}\}\.
According to Appendix[E\.5](https://arxiv.org/html/2609.05727#A5.SS5), the covariance matrix ofp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)is a posterior Stein kernel\. Therefore,

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,r~ρ\(X1\)\)\\displaystyle\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\)=\\displaystyle=βt2​σρ2βt2\+αt2​σρ2𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r~ρ\(X1\)\]\\displaystyle\\frac\{\\beta\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\]=\\displaystyle=βt2​σρ2βt2\+αt2​σρ2​\(\(1σρ2−1σπ2\)​\(mρ\+αt​σρ2βt2\+αt2​σρ2​\(xt−αt​mρ\)\)\+mπσπ2−mρσρ2\)\\displaystyle\\frac\{\\beta\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\\left\(\\left\(\\frac\{1\}\{\\sigma\_\{\\rho\}^\{2\}\}\-\\frac\{1\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\left\(m\_\{\\rho\}\+\\frac\{\\alpha\_\{t\}\\sigma\_\{\\rho\}^\{2\}\}\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\(x\_\{t\}\-\\alpha\_\{t\}m\_\{\\rho\}\)\\right\)\+\\frac\{m\_\{\\pi\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\-\\frac\{m\_\{\\rho\}\}\{\\sigma\_\{\\rho\}^\{2\}\}\\right\)=\\displaystyle=βt2​σρ2βt2\+αt2​σρ2​\(mπ−mρσπ2\+αt​\(σπ2−σρ2\)σπ2​\(βt2\+αt2​σρ2\)​\(xt−αt​mρ\)\)\.\\displaystyle\\frac\{\\beta\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\\left\(\\frac\{m\_\{\\pi\}\-m\_\{\\rho\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\+\\frac\{\\alpha\_\{t\}\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\}\{\\sigma\_\{\\pi\}^\{2\}\\left\(\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\\right\)\}\(x\_\{t\}\-\\alpha\_\{t\}m\_\{\\rho\}\)\\right\)\.Consequently, we have

Γtρ,r~ρ​\(xt\)=\\displaystyle\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(x\_\{t\}\)=αt​κtβt2CovX1∼p1\|tρ\(⋅\|xt\)\(X1,r~ρ\(X1\)\)\\displaystyle\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\\right\)=\\displaystyle=αt​κt​σρ2\(βt2\+αt2​σρ2\)​σπ2​\(mπ−mρ\+αt​\(σπ2−σρ2\)βt2\+αt2​σρ2​\(xt−αt​mρ\)\)\.\\displaystyle\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\\sigma\_\{\\rho\}^\{2\}\}\{\(\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\)\\sigma\_\{\\pi\}^\{2\}\}\\left\(m\_\{\\pi\}\-m\_\{\\rho\}\+\\frac\{\\alpha\_\{t\}\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\}\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\(x\_\{t\}\-\\alpha\_\{t\}m\_\{\\rho\}\)\\right\)\.
LetYtY\_\{t\}solve

d​Ytd​t=vtρ​\(Yt\)\+η​Γtρ,r~ρ​\(Yt\)\.\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=v\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\+\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(Y\_\{t\}\)\.Then,Y0∼p0Y\_\{0\}\\sim p\_\{0\}impliesY1∼qY\_\{1\}\\sim q\. The remaining task is to solve the above ODE\. By Proposition[85](https://arxiv.org/html/2609.05727#Thmtheorem85), a direct calculation gives

dd​t​\(Yt−αt​mρβt2\+αt2​σρ2\)=η​Γtρ,r~ρ​\(Yt\)βt2\+αt2​σρ2,dd​t​\(σρ​αtβt2\+αt2​σρ2\)=σρ​αt​κt\(βt2\+αt2​σρ2\)3/2\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(\\frac\{Y\_\{t\}\-\\alpha\_\{t\}m\_\{\\rho\}\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}\\right\)=\\frac\{\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\(Y\_\{t\}\)\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\},\\qquad\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(\\frac\{\\sigma\_\{\\rho\}\\alpha\_\{t\}\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}\\right\)=\\frac\{\\sigma\_\{\\rho\}\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\left\(\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\\right\)^\{3/2\}\}\.Then, we have

dd​t​\(Yt−αt​mρβt2\+αt2​σρ2\)=η​σρσπ2​\(mπ−mρ\+αt​\(σπ2−σρ2\)βt2\+αt2​σρ2​Yt−αt​mρβt2\+αt2​σρ2\)​dd​t​\(σρ​αtβt2\+αt2​σρ2\)\.\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(\\frac\{Y\_\{t\}\-\\alpha\_\{t\}m\_\{\\rho\}\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}\\right\)=\\frac\{\\eta\\sigma\_\{\\rho\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\left\(m\_\{\\pi\}\-m\_\{\\rho\}\+\\frac\{\\alpha\_\{t\}\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}\\frac\{Y\_\{t\}\-\\alpha\_\{t\}m\_\{\\rho\}\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}\\right\)\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(\\frac\{\\sigma\_\{\\rho\}\\alpha\_\{t\}\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}\\right\)\.LetZt=Yt−αt​mρβt2\+αt2​σρ2Z\_\{t\}=\\frac\{Y\_\{t\}\-\\alpha\_\{t\}m\_\{\\rho\}\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}andγt=σρ​αtβt2\+αt2​σρ2\\gamma\_\{t\}=\\frac\{\\sigma\_\{\\rho\}\\alpha\_\{t\}\}\{\\sqrt\{\\beta\_\{t\}^\{2\}\+\\alpha\_\{t\}^\{2\}\\sigma\_\{\\rho\}^\{2\}\}\}\. Equivalently,

Z˙t=ησπ2​\(σρ​\(mπ−mρ\)\+\(σπ2−σρ2\)​γt​Zt\)​γ˙t\.\\dot\{Z\}\_\{t\}=\\frac\{\\eta\}\{\\sigma\_\{\\pi\}^\{2\}\}\\left\(\\sigma\_\{\\rho\}\(m\_\{\\pi\}\-m\_\{\\rho\}\)\+\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\\gamma\_\{t\}Z\_\{t\}\\right\)\\dot\{\\gamma\}\_\{t\}\.In other words,

dd​t​\(Zt​exp⁡\(−η⁡\(σπ2−σρ2\)2​σπ2​γt2\)\)=\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(Z\_\{t\}\\exp\\left\(\-\\frac\{\\eta\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\gamma\_\{t\}^\{2\}\\right\)\\right\)=η​σρ​\(mπ−mρ\)σπ2​exp⁡\(−η⁡\(σπ2−σρ2\)2​σπ2​γt2\)​γ˙t\.\\displaystyle\\frac\{\\eta\\sigma\_\{\\rho\}\(m\_\{\\pi\}\-m\_\{\\rho\}\)\}\{\\sigma\_\{\\pi\}^\{2\}\}\\exp\\left\(\-\\frac\{\\eta\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\gamma\_\{t\}^\{2\}\\right\)\\dot\{\\gamma\}\_\{t\}\.Integrating fromt=0t=0tot=1t=1gives

Y1−mρσρ​exp⁡\(−η⁡\(σπ2−σρ2\)2​σπ2\)−X0=η​σρ​\(mπ−mρ\)σπ2​∫01exp⁡\(−η⁡\(σπ2−σρ2\)2​σπ2​s2\)​ds,\\displaystyle\\frac\{Y\_\{1\}\-m\_\{\\rho\}\}\{\\sigma\_\{\\rho\}\}\\exp\\left\(\-\\frac\{\\eta\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\right\)\-X\_\{0\}=\\frac\{\\eta\\sigma\_\{\\rho\}\(m\_\{\\pi\}\-m\_\{\\rho\}\)\}\{\\sigma\_\{\\pi\}^\{2\}\}\\int\_\{0\}^\{1\}\\exp\\left\(\-\\frac\{\\eta\(\\sigma\_\{\\pi\}^\{2\}\-\\sigma\_\{\\rho\}^\{2\}\)\}\{2\\sigma\_\{\\pi\}^\{2\}\}s^\{2\}\\right\)\\,\\mathrm\{d\}s,which impliesq=𝒩⁡\(mq,σq2​I\)q=\\mathcal\{N\}\(m\_\{q\},\\sigma\_\{q\}^\{2\}I\)\. ∎

### C\.2Convergence Guarantee

This subsection investigates the convergence properties of Newton Matching in the isotropic Gaussian family\. In Appendix[C\.2\.1](https://arxiv.org/html/2609.05727#A3.SS2.SSS1), we prove that the reverse KL decreases whenever the stepsizeη∈\(0,1\]\\eta\\in\(0,1\]\. The forward and reverse KL divergences both converge to zero\. In Appendix[C\.2\.2](https://arxiv.org/html/2609.05727#A3.SS2.SSS2), we further prove the local quadratic convergence of both forward and reverse KL divergences\.

#### C\.2\.1KL Descent and Global Convergence

Here, we investigate the KL descent and global convergence of Newton Matching\. The following lemma estimateswηw\_\{\\eta\}when the stepsize is not larger than 1\.

###### Lemma 87\.

Use the notation of Proposition[86](https://arxiv.org/html/2609.05727#Thmtheorem86)\. Ifη∈\(0,1\]\\eta\\in\(0,1\], then for anyz\>0z\>0, we havewη​\(z\)∈\(0,2\)w\_\{\\eta\}\(z\)\\in\(0,2\)\.

###### Proof\.

Evidently,wη​\(z\)\>0w\_\{\\eta\}\(z\)\>0holds for allz\>0z\>0\. For thosez≤1z\\leq 1, we havewη​\(z\)≤z​exp⁡\(1−z2\)≤1w\_\{\\eta\}\(z\)\\leq z\\exp\\left\(\\frac\{1\-z\}\{2\}\\right\)\\leq 1\. For thosez\>1z\>1, denoteζ=η2​\(z−1\)\>0\\zeta=\\frac\{\\eta\}\{2\}\(z\-1\)\>0, and we have

wη​\(z\)=\(2​ζ\+η\)​∫01e−ζ⁡\(1−s2\)​𝑑s≤\(2​ζ\+1\)​∫01e−ζ⁡\(1−s2\)​𝑑s\.w\_\{\\eta\}\(z\)=\(2\\zeta\+\\eta\)\\int\_\{0\}^\{1\}e^\{\-\\zeta\(1\-s^\{2\}\)\}\\,\\mathrm\{d\}s\\leq\(2\\zeta\+1\)\\int\_\{0\}^\{1\}e^\{\-\\zeta\(1\-s^\{2\}\)\}\\,\\mathrm\{d\}s\.Integrating by parts gives

∫01e−ζ⁡\(1−s2\)​𝑑s=1−2​ζ​∫01s2​e−ζ⁡\(1−s2\)​𝑑s\.\\int\_\{0\}^\{1\}e^\{\-\\zeta\(1\-s^\{2\}\)\}\\,\\mathrm\{d\}s=1\-2\\zeta\\int\_\{0\}^\{1\}s^\{2\}e^\{\-\\zeta\(1\-s^\{2\}\)\}\\,\\mathrm\{d\}s\.Hence,

wη​\(z\)≤1\+2​∫01ζ⁡\(1−s2\)​e−ζ⁡\(1−s2\)​𝑑s≤1\+2​e−1<2\.w\_\{\\eta\}\(z\)\\leq 1\+2\\int\_\{0\}^\{1\}\\zeta\(1\-s^\{2\}\)e^\{\-\\zeta\(1\-s^\{2\}\)\}\\,\\mathrm\{d\}s\\leq 1\+2e^\{\-1\}<2\.Therefore, for allz\>0z\>0, we havewη​\(z\)∈\(0,2\)w\_\{\\eta\}\(z\)\\in\(0,2\)\. ∎

We next give the KL divergence between two isotropic Gaussian distributions\.

###### Lemma 88\.

Consider two Gaussian distributionsρ=𝒩⁡\(mρ,σρ2​I\)\\rho=\\mathcal\{N\}\(m\_\{\\rho\},\\sigma\_\{\\rho\}^\{2\}I\)andπ=𝒩⁡\(mπ,σπ2​I\)\\pi=\\mathcal\{N\}\(m\_\{\\pi\},\\sigma\_\{\\pi\}^\{2\}I\)\. Then, we have

KL⁡\(ρ∥π\)=‖mρ−mπ‖222​σπ2\+d2​\(σρ2σπ2−1−log⁡σρ2σπ2\)\.\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)=\\frac\{\\left\\\|m\_\{\\rho\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\+\\frac\{d\}\{2\}\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\-1\-\\log\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\.\(161\)

###### Proof\.

Taking the expectation of

log⁡ρ⁡\(x\)π⁡\(x\)=d2​log⁡σπ2σρ2−‖x−mρ‖222​σρ2\+‖x−mπ‖222​σπ2\\log\\frac\{\\rho\(x\)\}\{\\pi\(x\)\}=\\frac\{d\}\{2\}\\log\\frac\{\\sigma\_\{\\pi\}^\{2\}\}\{\\sigma\_\{\\rho\}^\{2\}\}\-\\frac\{\\left\\\|x\-m\_\{\\rho\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\rho\}^\{2\}\}\+\\frac\{\\left\\\|x\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}underρ\\rhofinishes the proof\. ∎

Based on Lemmas[87](https://arxiv.org/html/2609.05727#Thmtheorem87)and[88](https://arxiv.org/html/2609.05727#Thmtheorem88), we prove the descent property of the reverse KL\. The following result serves as a specialization of Theorem[24](https://arxiv.org/html/2609.05727#Thmtheorem24)\.

###### Proposition 89\.

Use the notation of Proposition[86](https://arxiv.org/html/2609.05727#Thmtheorem86)\. Denote

g:ℝ\+⁣\+→ℝ,z↦z−1−log⁡z\.g:\\mathbb\{R\}\_\{\+\+\}\\to\\mathbb\{R\},\\qquad z\\mapsto z\-1\-\\log z\.Then, for anyη∈ℝ\\eta\\in\\mathbb\{R\}, the reverse KL ofq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\)is

KL⁡\(q∥π\)=‖mρ−mπ‖222​σπ2​\(1−wη​\(σρ2σπ2\)\)2\+d2​g​\(σρ2σπ2​exp⁡\(η⁡\(1−σρ2σπ2\)\)\)\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\frac\{\\left\\\|m\_\{\\rho\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\left\(1\-w\_\{\\eta\}\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\right\)^\{2\}\+\\frac\{d\}\{2\}g\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\exp\\left\(\\eta\\left\(1\-\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\right\)\\right\)\.\(162\)Ifη∈\(0,1\]\\eta\\in\(0,1\], the reverse KL decreases:

KL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.Furthermore,

KL\(q∥π\)=KL\(ρ∥π\)⇔q=ρ=π\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\quad\\Leftrightarrow\\quad q=\\rho=\\pi\.

###### Proof\.

By Lemma[88](https://arxiv.org/html/2609.05727#Thmtheorem88)and Proposition[86](https://arxiv.org/html/2609.05727#Thmtheorem86), we obtain the expression forKL⁡\(q∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)in \([162](https://arxiv.org/html/2609.05727#A3.E162)\)\. Lemma[87](https://arxiv.org/html/2609.05727#Thmtheorem87)implies that, forη∈\(0,1\]\\eta\\in\(0,1\],wη​\(σρ2σπ2\)∈\(0,2\)w\_\{\\eta\}\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\in\(0,2\); hence,

‖mq−mπ‖222​σπ2=‖mρ−mπ‖222​σπ2​\(1−wη​\(σρ2σπ2\)\)2≤‖mρ−mπ‖222​σπ2\.\\frac\{\\left\\\|m\_\{q\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}=\\frac\{\\left\\\|m\_\{\\rho\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\left\(1\-w\_\{\\eta\}\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\right\)^\{2\}\\leq\\frac\{\\left\\\|m\_\{\\rho\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\.Lethz​\(η\):=g⁡\(z​eη⁡\(1−z\)\)−g⁡\(z\)=z​eη⁡\(1−z\)−η⁡\(1−z\)−zh\_\{z\}\(\\eta\):=g\(ze^\{\\eta\(1\-z\)\}\)\-g\(z\)=ze^\{\\eta\(1\-z\)\}\-\\eta\(1\-z\)\-z,z\>0z\>0,η∈\(0,1\]\\eta\\in\(0,1\]\. We aim to provehz​\(η\)≤0h\_\{z\}\(\\eta\)\\leq 0\. Note thathz′′​\(η\)=z​\(1−z\)2​eη⁡\(1−z\)≥0h\_\{z\}^\{\\prime\\prime\}\(\\eta\)=z\(1\-z\)^\{2\}e^\{\\eta\(1\-z\)\}\\geq 0; hence,hzh\_\{z\}is convex w\.r\.t\.η\\eta\. Therefore, we havehz​\(η\)≤\(1−η\)​hz​\(0\)\+η​hz​\(1\)=η⁡\(z​e1−z−1\)≤0h\_\{z\}\(\\eta\)\\leq\(1\-\\eta\)h\_\{z\}\(0\)\+\\eta h\_\{z\}\(1\)=\\eta\(ze^\{1\-z\}\-1\)\\leq 0\. Then,hσρ2/σπ2​\(η\)≤0h\_\{\\sigma\_\{\\rho\}^\{2\}/\\sigma\_\{\\pi\}^\{2\}\}\(\\eta\)\\leq 0implies

g⁡\(σq2σπ2\)≤g⁡\(σρ2σπ2\),g\\left\(\\frac\{\\sigma\_\{q\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\leq g\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\),and

g\(σq2σπ2\)=g\(σρ2σπ2\)⇔σρ=σq=σπ\.g\\left\(\\frac\{\\sigma\_\{q\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)=g\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\quad\\Leftrightarrow\\quad\\sigma\_\{\\rho\}=\\sigma\_\{q\}=\\sigma\_\{\\pi\}\.Therefore,KL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.

Assume thatKL⁡\(q∥π\)=KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\. Then, we have

\|1−wη​\(σρ2σπ2\)\|​‖mρ−mπ‖2=‖mρ−mπ‖2,g⁡\(σq2σπ2\)=g⁡\(σρ2σπ2\)\.\\left\|1\-w\_\{\\eta\}\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\\right\|\\left\\\|m\_\{\\rho\}\-m\_\{\\pi\}\\right\\\|\_\{2\}=\\left\\\|m\_\{\\rho\}\-m\_\{\\pi\}\\right\\\|\_\{2\},\\qquad g\\left\(\\frac\{\\sigma\_\{q\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)=g\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)\.Hence,σρ=σq=σπ\\sigma\_\{\\rho\}=\\sigma\_\{q\}=\\sigma\_\{\\pi\}\. Then,wη​\(σρ2σπ2\)=η\>0w\_\{\\eta\}\\left\(\\frac\{\\sigma\_\{\\rho\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\\right\)=\\eta\>0, which impliesq=ρ=πq=\\rho=\\pi\.

Assume thatq=ρ=πq=\\rho=\\pi\. Then,KL⁡\(q∥π\)=0=KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=0=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)evidently holds\. ∎

In contrast to the case in Section[7\.1](https://arxiv.org/html/2609.05727#S7.SS1), any isotropic Gaussian distribution𝒩⁡\(m,σ2​I\)\\mathcal\{N\}\(m,\\sigma^\{2\}I\)can be represented by\(d\+1\)\(d\+1\)\-dimensional parametersm,σm,\\sigma\. Therefore, global convergence can be proved directly from the explicit parameter recursions, without the additional anti\-collapse or dissipation\-vanishing conditions in \([39](https://arxiv.org/html/2609.05727#S7.E39)\) and \([40](https://arxiv.org/html/2609.05727#S7.E40)\)\.

###### Proposition 90\.

Use the notation of Proposition[86](https://arxiv.org/html/2609.05727#Thmtheorem86)and apply Newton Matching iteratively\. For thekkth step, the density is denoted byρk=𝒩⁡\(mk,σk2​I\)\\rho\_\{k\}=\\mathcal\{N\}\(m\_\{k\},\\sigma\_\{k\}^\{2\}I\)\. Letρk\+1=𝒯⁡\(vρk\+ηk​Γρk,r~ρk\)\\rho\_\{k\+1\}=\\mathcal\{T\}\(v^\{\\rho\_\{k\}\}\+\\eta\_\{k\}\\Gamma^\{\\rho\_\{k\},\\tilde\{r\}^\{\\rho\_\{k\}\}\}\)where the stepsizeηk∈\(0,1\]\\eta\_\{k\}\\in\(0,1\]\. If∑k=0∞ηk=∞\\sum\_\{k=0\}^\{\\infty\}\\eta\_\{k\}=\\infty, then ask→∞k\\to\\infty,

mk→mπ,σk→σπ,KL⁡\(ρk∥π\)→0,KL⁡\(π∥ρk\)→0\.m\_\{k\}\\to m\_\{\\pi\},\\qquad\\sigma\_\{k\}\\to\\sigma\_\{\\pi\},\\qquad\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\\to 0,\\qquad\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\\to 0\.Consequently, the velocity field also converges:

∫011κt​𝔼Xt∼ptρk​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​𝑑t→0,∫011κt​𝔼Xt∼ptπ​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​𝑑t→0\.\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\_\{k\}\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\\to 0,\\,\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\pi\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\\to 0\.

###### Proof\.

For the first step, we provelimk→∞σk=σπ\\lim\_\{k\\to\\infty\}\\sigma\_\{k\}=\\sigma\_\{\\pi\}\. Denotezk:=σk2σπ2\>0z\_\{k\}:=\\frac\{\\sigma\_\{k\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\>0\. Consider the case where there exists ak0k\_\{0\}, such thatzk0≤1z\_\{k\_\{0\}\}\\leq 1\. Then, for allk≥k0k\\geq k\_\{0\}, we havezk≤zk\+1=zk​eηk​\(1−zk\)≤1z\_\{k\}\\leq z\_\{k\+1\}=z\_\{k\}e^\{\\eta\_\{k\}\(1\-z\_\{k\}\)\}\\leq 1\. Denotez⋆:=limk→∞zk∈\(0,1\]z\_\{\\star\}:=\\lim\_\{k\\to\\infty\}z\_\{k\}\\in\(0,1\]\. Then, for allk≥k0k\\geq k\_\{0\},zk≤1\+z⋆2z\_\{k\}\\leq\\frac\{1\+z\_\{\\star\}\}\{2\}; hence,log⁡zk\+1−log⁡zk=ηk​\(1−zk\)≥1−z⋆2​ηk\\log z\_\{k\+1\}\-\\log z\_\{k\}=\\eta\_\{k\}\(1\-z\_\{k\}\)\\geq\\frac\{1\-z\_\{\\star\}\}\{2\}\\eta\_\{k\}\. Then, we havelimk→∞log⁡zk≥log⁡zk0\+1−z⋆2​∑k=k0∞ηk=∞\\lim\_\{k\\to\\infty\}\\log z\_\{k\}\\geq\\log z\_\{k\_\{0\}\}\+\\frac\{1\-z\_\{\\star\}\}\{2\}\\sum\_\{k=k\_\{0\}\}^\{\\infty\}\\eta\_\{k\}=\\infty, which leads to a contradiction\. Therefore, we must havez⋆=1z\_\{\\star\}=1\. In other words,

∃k0\>0​, s\.t\.​σk0≤σπ⇒limk→∞σk=σπ\.\\exists k\_\{0\}\>0\\text\{, s\.t\. \}\\sigma\_\{k\_\{0\}\}\\leq\\sigma\_\{\\pi\}\\,\\Rightarrow\\,\\lim\_\{k\\to\\infty\}\\sigma\_\{k\}=\\sigma\_\{\\pi\}\.Consider another case where for allkk,zk\>1z\_\{k\}\>1\. Then,1<zk\+1=zk​eηk​\(1−zk\)<zk1<z\_\{k\+1\}=z\_\{k\}e^\{\\eta\_\{k\}\(1\-z\_\{k\}\)\}<z\_\{k\}\. Denotez⋆:=limk→∞zk∈\[1,∞\)z\_\{\\star\}:=\\lim\_\{k\\to\\infty\}z\_\{k\}\\in\[1,\\infty\)\. Assume thatz⋆\>1z\_\{\\star\}\>1\. Then,zk\>1\+z⋆2z\_\{k\}\>\\frac\{1\+z\_\{\\star\}\}\{2\}; hence,log⁡zk\+1−log⁡zk=ηk​\(1−zk\)≤1−z⋆2​ηk<0\\log z\_\{k\+1\}\-\\log z\_\{k\}=\\eta\_\{k\}\(1\-z\_\{k\}\)\\leq\\frac\{1\-z\_\{\\star\}\}\{2\}\\eta\_\{k\}<0\. Then, we havelimk→∞log⁡zk≤log⁡z0\+1−z⋆2​∑k=0∞ηk=−∞\\lim\_\{k\\to\\infty\}\\log z\_\{k\}\\leq\\log z\_\{0\}\+\\frac\{1\-z\_\{\\star\}\}\{2\}\\sum\_\{k=0\}^\{\\infty\}\\eta\_\{k\}=\-\\infty, which leads to a contradiction\. Therefore, we must havez⋆=1z\_\{\\star\}=1\. In other words,

∀k,σk\>σπ⇒limk→∞σk=σπ\.\\forall k,\\,\\sigma\_\{k\}\>\\sigma\_\{\\pi\}\\,\\Rightarrow\\,\\lim\_\{k\\to\\infty\}\\sigma\_\{k\}=\\sigma\_\{\\pi\}\.In summary, we have proved unconditionally that

limk→∞σk=σπ\.\\lim\_\{k\\to\\infty\}\\sigma\_\{k\}=\\sigma\_\{\\pi\}\.
For the second step, we provelimk→∞mk=mπ\\lim\_\{k\\to\\infty\}m\_\{k\}=m\_\{\\pi\}\. Denotezk:=σk2σπ2z\_\{k\}:=\\frac\{\\sigma\_\{k\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}andek=‖mk−mπ‖2e\_\{k\}=\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}\. We have proved in the first step thatlimk→∞zk=1\\lim\_\{k\\to\\infty\}z\_\{k\}=1\. Hence, there exists a sufficiently largek0k\_\{0\}, such that for allk≥k0k\\geq k\_\{0\},zk∈\[12,32\]z\_\{k\}\\in\\left\[\\frac\{1\}\{2\},\\frac\{3\}\{2\}\\right\]\. Then, we have

ηk2​e1/4≤wk=ηk​zk​∫01eηk2​\(1−zk\)​\(1−s2\)​𝑑s≤32\.\\frac\{\\eta\_\{k\}\}\{2e^\{1/4\}\}\\leq w\_\{k\}=\\eta\_\{k\}z\_\{k\}\\int\_\{0\}^\{1\}e^\{\\frac\{\\eta\_\{k\}\}\{2\}\(1\-z\_\{k\}\)\(1\-s^\{2\}\)\}\\,\\mathrm\{d\}s\\leq\\frac\{3\}\{2\}\.Hence,

‖mk\+1−mπ‖22=\(1−wk​\(2−wk\)\)​‖mk−mπ‖22≤\(1−ηk4​e1/4\)​‖mk−mπ‖22\.\\left\\\|m\_\{k\+1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}=\(1\-w\_\{k\}\(2\-w\_\{k\}\)\)\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\\leq\\left\(1\-\\frac\{\\eta\_\{k\}\}\{4e^\{1/4\}\}\\right\)\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\.By∑k=k0∞ηk=∞\\sum\_\{k=k\_\{0\}\}^\{\\infty\}\\eta\_\{k\}=\\infty, we have

limk→∞‖mk−mπ‖2=0\.\\lim\_\{k\\to\\infty\}\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}=0\.
By \([161](https://arxiv.org/html/2609.05727#A3.E161)\), we haveKL⁡\(ρk∥π\)→0\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\\to 0andKL⁡\(π∥ρk\)→0\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\\to 0\. The convergence of the canonical velocity field is further given by Corollary[107](https://arxiv.org/html/2609.05727#Thmtheorem107)\. ∎

#### C\.2\.2Local Quadratic Convergence

Proposition[90](https://arxiv.org/html/2609.05727#Thmtheorem90)establishes global convergence of Newton Matching in the isotropic Gaussian family\. Here, we establish the local quadratic convergence rate\. The following proposition serves as a specialization of Theorem[32](https://arxiv.org/html/2609.05727#Thmtheorem32)\.

###### Proposition 91\.

Use the notation of Proposition[89](https://arxiv.org/html/2609.05727#Thmtheorem89)and letηk=1\\eta\_\{k\}=1for allkk\. If the reverse KL satisfies

KL⁡\(ρ0∥π\)<d2​\(12−log⁡32\),\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)<\\frac\{d\}\{2\}\\left\(\\frac\{1\}\{2\}\-\\log\\frac\{3\}\{2\}\\right\),then we have

KL⁡\(ρk\+1∥π\)≤392​d​KL⁡\(ρk∥π\)2\.\\operatorname\{KL\}\\left\(\\rho\_\{k\+1\}\\middle\\\|\\pi\\right\)\\leq\\frac\{39\}\{2d\}\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)^\{2\}\.Therefore, the reverse KL and the canonical velocity field exhibit local quadratic convergence:

392​d​KL⁡\(ρk∥π\)=392​d​∫011κt​𝔼Xt∼ptρk​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​𝑑t≤\(392​d​KL⁡\(ρ0∥π\)\)2k→0\.\\displaystyle\\frac\{39\}\{2d\}\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)=\\frac\{39\}\{2d\}\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\rho\_\{k\}\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\\leq\\left\(\\frac\{39\}\{2d\}\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)\\right\)^\{2^\{k\}\}\\to 0\.
If the forward KL satisfies

KL⁡\(π∥ρ0\)<d2​\(12−log⁡32\),\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{0\}\\right\)<\\frac\{d\}\{2\}\\left\(\\frac\{1\}\{2\}\-\\log\\frac\{3\}\{2\}\\right\),then we have

KL⁡\(π∥ρk\+1\)≤87548​d​KL⁡\(π∥ρk\)2\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\+1\}\\right\)\\leq\\frac\{875\}\{48d\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)^\{2\}\.Therefore, the forward KL and the canonical velocity field exhibit local quadratic convergence:

87548​d​KL⁡\(π∥ρk\)=87548​d​∫011κt​𝔼Xt∼ptπ​\[‖vtρk​\(Xt\)−vtπ​\(Xt\)‖22\]​𝑑t≤\(87548​d​KL⁡\(π∥ρ0\)\)2k→0\.\\displaystyle\\frac\{875\}\{48d\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)=\\frac\{875\}\{48d\}\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\mathbb\{E\}\_\{X\_\{t\}\\sim p\_\{t\}^\{\\pi\}\}\\left\[\\left\\\|v\_\{t\}^\{\\rho\_\{k\}\}\(X\_\{t\}\)\-v\_\{t\}^\{\\pi\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\\leq\\left\(\\frac\{875\}\{48d\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{0\}\\right\)\\right\)^\{2^\{k\}\}\\to 0\.

###### Proof\.

Denotezk=σk2σπ2z\_\{k\}=\\frac\{\\sigma\_\{k\}^\{2\}\}\{\\sigma\_\{\\pi\}^\{2\}\}\. By Proposition[86](https://arxiv.org/html/2609.05727#Thmtheorem86), we havezk\+1=zk​exp⁡\(1−zk\)z\_\{k\+1\}=z\_\{k\}\\exp\(1\-z\_\{k\}\)\. Then, we have

KL⁡\(ρk∥π\)=‖mk−mπ‖222​σπ2\+d2​g​\(zk\),KL⁡\(π∥ρk\)=‖mk−mπ‖222​σk2\+d2​g​\(zk−1\)\.\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)=\\frac\{\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\+\\frac\{d\}\{2\}g\(z\_\{k\}\),\\qquad\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)=\\frac\{\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{k\}^\{2\}\}\+\\frac\{d\}\{2\}g\(z\_\{k\}^\{\-1\}\)\.Denote

w⁡\(z\)=z​exp⁡\(1−z2\)​∫01exp⁡\(−1−z2​s2\)​𝑑s=z​∫01exp⁡\(1−z2​\(1−s2\)\)​𝑑s\.w\(z\)=z\\exp\\left\(\\frac\{1\-z\}\{2\}\\right\)\\int\_\{0\}^\{1\}\\exp\\left\(\-\\frac\{1\-z\}\{2\}s^\{2\}\\right\)\\,\\mathrm\{d\}s=z\\int\_\{0\}^\{1\}\\exp\\left\(\\frac\{1\-z\}\{2\}\\left\(1\-s^\{2\}\\right\)\\right\)\\,\\mathrm\{d\}s\.Then, we have

mk\+1−mπ=\(1−w⁡\(zk\)\)​\(mk−mπ\)\.m\_\{k\+1\}\-m\_\{\\pi\}=\(1\-w\(z\_\{k\}\)\)\(m\_\{k\}\-m\_\{\\pi\}\)\.
We first estimateKL⁡\(ρk∥π\)\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\. ByKL⁡\(ρk∥π\)≤KL⁡\(ρ0∥π\)<d2​\(12−log⁡32\)\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\_\{0\}\\middle\\\|\\pi\\right\)<\\frac\{d\}\{2\}\\left\(\\frac\{1\}\{2\}\-\\log\\frac\{3\}\{2\}\\right\), we havezk∈\(12,32\)z\_\{k\}\\in\\left\(\\frac\{1\}\{2\},\\frac\{3\}\{2\}\\right\)\. We havew⁡\(1\)=1w\(1\)=1and0≤w′​\(zk\)≤10\\leq w^\{\\prime\}\(z\_\{k\}\)\\leq 1; hence,

\|w⁡\(zk\)−1\|≤\|zk−1\|\.\\left\|w\(z\_\{k\}\)\-1\\right\|\\leq\\left\|z\_\{k\}\-1\\right\|\.Elementary calculus onz∈\(12,32\)z\\in\\left\(\\frac\{1\}\{2\},\\frac\{3\}\{2\}\\right\)also gives

g⁡\(zk\)≥13​\(zk−1\)2,g⁡\(zk\+1\)≤\(zk−1\)4\.g\(z\_\{k\}\)\\geq\\frac\{1\}\{3\}\(z\_\{k\}\-1\)^\{2\},\\qquad g\(z\_\{k\+1\}\)\\leq\(z\_\{k\}\-1\)^\{4\}\.Since

‖mk\+1−mπ‖222​σπ2=\(w⁡\(zk\)−1\)2​‖mk−mπ‖222​σπ2,\(w⁡\(zk\)−1\)2≤\(zk−1\)2≤3​g​\(zk\),\\frac\{\\left\\\|m\_\{k\+1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}=\(w\(z\_\{k\}\)\-1\)^\{2\}\\frac\{\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\},\\qquad\(w\(z\_\{k\}\)\-1\)^\{2\}\\leq\(z\_\{k\}\-1\)^\{2\}\\leq 3g\(z\_\{k\}\),we have

‖mk\+1−mπ‖222​σπ2≤6d⋅‖mk−mπ‖222​σπ2⋅d2​g​\(zk\)≤32​d​KL⁡\(ρk∥π\)2\.\\frac\{\\left\\\|m\_\{k\+1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\leq\\frac\{6\}\{d\}\\cdot\\frac\{\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\\cdot\\frac\{d\}\{2\}g\(z\_\{k\}\)\\leq\\frac\{3\}\{2d\}\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)^\{2\}\.Since

d2​g​\(zk\+1\)≤d2​\(zk−1\)4≤9​d2​g​\(zk\)2≤18d​KL⁡\(ρk∥π\)2,\\frac\{d\}\{2\}g\(z\_\{k\+1\}\)\\leq\\frac\{d\}\{2\}\(z\_\{k\}\-1\)^\{4\}\\leq\\frac\{9d\}\{2\}g\(z\_\{k\}\)^\{2\}\\leq\\frac\{18\}\{d\}\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)^\{2\},we have

KL⁡\(ρk\+1∥π\)=‖mk\+1−mπ‖222​σπ2\+d2​g​\(zk\+1\)≤392​d​KL⁡\(ρk∥π\)2\.\\operatorname\{KL\}\\left\(\\rho\_\{k\+1\}\\middle\\\|\\pi\\right\)=\\frac\{\\left\\\|m\_\{k\+1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\pi\}^\{2\}\}\+\\frac\{d\}\{2\}g\(z\_\{k\+1\}\)\\leq\\frac\{39\}\{2d\}\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)^\{2\}\.Therefore,392​d⋅d2​\(12−log⁡32\)=394​\(12−log⁡32\)<0\.93<1\\frac\{39\}\{2d\}\\cdot\\frac\{d\}\{2\}\\left\(\\frac\{1\}\{2\}\-\\log\\frac\{3\}\{2\}\\right\)=\\frac\{39\}\{4\}\\left\(\\frac\{1\}\{2\}\-\\log\\frac\{3\}\{2\}\\right\)<0\.93<1proves the local quadratic convergence ofKL⁡\(ρk∥π\)\\operatorname\{KL\}\\left\(\\rho\_\{k\}\\middle\\\|\\pi\\right\)\.

Next, we estimateKL⁡\(π∥ρk\)\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\. SinceKL⁡\(π∥ρ0\)<d2​g​\(32\)\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{0\}\\right\)<\\frac\{d\}\{2\}g\\left\(\\frac\{3\}\{2\}\\right\), we havez0∈\(23,53\)z\_\{0\}\\in\\left\(\\frac\{2\}\{3\},\\frac\{5\}\{3\}\\right\)\. A direct calculation shows thatz1=z0​e1−z0∈\(56,1\]⊂\(23,53\)z\_\{1\}=z\_\{0\}e^\{1\-z\_\{0\}\}\\in\\left\(\\frac\{5\}\{6\},1\\right\]\\subset\\left\(\\frac\{2\}\{3\},\\frac\{5\}\{3\}\\right\)\. Consequently, for allkk, we havezk∈\(23,53\)z\_\{k\}\\in\\left\(\\frac\{2\}\{3\},\\frac\{5\}\{3\}\\right\)\. Onz∈\(23,53\)z\\in\\left\(\\frac\{2\}\{3\},\\frac\{5\}\{3\}\\right\), elementary calculus gives

\|w⁡\(zk\)−1\|≤\|zk−1\|,g⁡\(zk−1\)≥625​\(zk−1\)2,g⁡\(zk\+1−1\)≤81200​\(zk−1\)4\.\\left\|w\(z\_\{k\}\)\-1\\right\|\\leq\\left\|z\_\{k\}\-1\\right\|,\\qquad g\(z\_\{k\}^\{\-1\}\)\\geq\\frac\{6\}\{25\}\(z\_\{k\}\-1\)^\{2\},\\qquad g\(z\_\{k\+1\}^\{\-1\}\)\\leq\\frac\{81\}\{200\}\(z\_\{k\}\-1\)^\{4\}\.Since

‖mk\+1−mπ‖222​σk\+12≤zkzk\+1​\(w⁡\(zk\)−1\)2​‖mk−mπ‖222​σk2≤253​g​\(zk−1\)​‖mk−mπ‖222​σk2,\\frac\{\\left\\\|m\_\{k\+1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{k\+1\}^\{2\}\}\\leq\\frac\{z\_\{k\}\}\{z\_\{k\+1\}\}\(w\(z\_\{k\}\)\-1\)^\{2\}\\frac\{\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{k\}^\{2\}\}\\leq\\frac\{25\}\{3\}g\(z\_\{k\}^\{\-1\}\)\\frac\{\\left\\\|m\_\{k\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{k\}^\{2\}\},we obtain

‖mk\+1−mπ‖222​σk\+12≤256​d​KL⁡\(π∥ρk\)2\.\\frac\{\\left\\\|m\_\{k\+1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{k\+1\}^\{2\}\}\\leq\\frac\{25\}\{6d\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)^\{2\}\.We also have

d2​g​\(zk\+1−1\)≤81​d400​\(zk−1\)4≤225​d64​g​\(zk−1\)2≤22516​d​KL⁡\(π∥ρk\)2\.\\frac\{d\}\{2\}g\(z\_\{k\+1\}^\{\-1\}\)\\leq\\frac\{81d\}\{400\}\(z\_\{k\}\-1\)^\{4\}\\leq\\frac\{225d\}\{64\}g\(z\_\{k\}^\{\-1\}\)^\{2\}\\leq\\frac\{225\}\{16d\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)^\{2\}\.Thus,

KL⁡\(π∥ρk\+1\)=‖mk\+1−mπ‖222​σk\+12\+d2​g​\(zk\+1−1\)≤87548​d​KL⁡\(π∥ρk\)2\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\+1\}\\right\)=\\frac\{\\left\\\|m\_\{k\+1\}\-m\_\{\\pi\}\\right\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{k\+1\}^\{2\}\}\+\\frac\{d\}\{2\}g\(z\_\{k\+1\}^\{\-1\}\)\\leq\\frac\{875\}\{48d\}\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)^\{2\}\.Since87548​d⋅d2​\(12−log⁡32\)=87596​\(12−log⁡32\)<0\.87<1\\frac\{875\}\{48d\}\\cdot\\frac\{d\}\{2\}\\left\(\\frac\{1\}\{2\}\-\\log\\frac\{3\}\{2\}\\right\)=\\frac\{875\}\{96\}\\left\(\\frac\{1\}\{2\}\-\\log\\frac\{3\}\{2\}\\right\)<0\.87<1, the recurrence establishes local quadratic convergence ofKL⁡\(π∥ρk\)\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{k\}\\right\)\. ∎

By the explicit Gaussian KL formula, both the forward and reverse KL divergences are locally comparable to the squared Euclidean error in the parameters\(m,σ\)\(m,\\sigma\)near the target\(mπ,σπ\)\(m\_\{\\pi\},\\sigma\_\{\\pi\}\)\. Since all norms on this finite\-dimensional parameter space are equivalent, the one\-step KL bounds above imply quadratic convergence of\(mk,σk\)\(m\_\{k\},\\sigma\_\{k\}\)under any fixed norm\. We therefore do not state the corresponding norm\-based bounds separately\.

## Appendix DStochastic Foundations and Computational Tools

This appendix develops stochastic foundations and computational tools used primarily in Sections[8](https://arxiv.org/html/2609.05727#S8)and[9](https://arxiv.org/html/2609.05727#S9); some results are also used in the convergence analysis of Section[7\.1\.3](https://arxiv.org/html/2609.05727#S7.SS1.SSS3)\. All results depend on the terminal condition

α1=1,β1=0,\\alpha\_\{1\}=1,\\qquad\\beta\_\{1\}=0,and are independent of the initial valuesα0\\alpha\_\{0\},β0\\beta\_\{0\}\. Therefore, conclusions in this appendix apply both to the standard flow matching construction, where

α0=0,β0=1,α1=1,β1=0,\\alpha\_\{0\}=0,\\qquad\\beta\_\{0\}=1,\\qquad\\alpha\_\{1\}=1,\\qquad\\beta\_\{1\}=0,and to the one\-sided interpolant considered in Appendix[H\.2](https://arxiv.org/html/2609.05727#A8.SS2), where

α~0=0,β~0=0,α~1=1,β~1=0\.\\widetilde\{\\alpha\}\_\{0\}=0,\\qquad\\widetilde\{\\beta\}\_\{0\}=0,\\qquad\\widetilde\{\\alpha\}\_\{1\}=1,\\qquad\\widetilde\{\\beta\}\_\{1\}=0\.Here,α~t\\widetilde\{\\alpha\}\_\{t\}andβ~t\\widetilde\{\\beta\}\_\{t\}denote the schedule in one\-sided interpolants\. The extension of Newton Matching to one\-sided interpolants is discussed in Appendix[H\.2](https://arxiv.org/html/2609.05727#A8.SS2)\.

Appendix[D\.1](https://arxiv.org/html/2609.05727#A4.SS1)presents the SDE setup and assumptions used throughout the appendix\. Appendix[D\.2](https://arxiv.org/html/2609.05727#A4.SS2)then derives the backward equation satisfied by the posterior valueVtρ​\[f\]V\_\{t\}^\{\\rho\}\[f\]defined in \([14](https://arxiv.org/html/2609.05727#S4.E14)\)\. This equation contributes to the value\-ascent certificate in Section[5\.2](https://arxiv.org/html/2609.05727#S5.SS2)\.

The remaining subsections of this appendix provide stochastic computational tools for Newton Matching\. Appendix[D\.3](https://arxiv.org/html/2609.05727#A4.SS3)constructs the posterior\-preserving SDE, whose terminal transition kernel is exactlyp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)in \([10](https://arxiv.org/html/2609.05727#S3.E10)\)\. This SDE supplies the exact posterior samplerX1∼p1\|tρ\(⋅\|xt\)X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)used by the reverse construction in Section[8\.2](https://arxiv.org/html/2609.05727#S8.SS2)\. Its path distribution also supports the SDE\-based estimation in Section[8\.3](https://arxiv.org/html/2609.05727#S8.SS3), the gradient form in Section[8\.4](https://arxiv.org/html/2609.05727#S8.SS4), and the reference\-adjoint approximation in Section[9\.2\.1](https://arxiv.org/html/2609.05727#S9.SS2.SSS1)\.

Appendix[D\.4](https://arxiv.org/html/2609.05727#A4.SS4)establishes the bridge universality of the posterior\-preserving SDE: the law of the intervening path conditioning on its two endpoints is independent of the terminal densityρ\\rhoand admits an explicit Gaussian representation\. This property enables path\-dependent computations under the forward construction by separating the generation of the endpoint pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)from the generation of a path with the required posterior\-compatible law\. Once the endpoint pair has been obtained, a universal bridge supplies the path needed for the SDE\-based covariance calculation in Section[8\.3](https://arxiv.org/html/2609.05727#S8.SS3)and the pathwise adjoint calculation in Section[8\.4](https://arxiv.org/html/2609.05727#S8.SS4)\.

Appendix[D\.5](https://arxiv.org/html/2609.05727#A4.SS5)develops SDE\-based representations of the log densitylog⁡ρ⁡\(x\)\\log\\rho\(x\)and the log\-density ratiolog⁡ρ2​\(x\)ρ1​\(x\)\\log\\frac\{\\rho\_\{2\}\(x\)\}\{\\rho\_\{1\}\(x\)\}\. In Section[8\.3](https://arxiv.org/html/2609.05727#S8.SS3), the pathwise identities provide exact evaluations of the regularized reward appearing in the covariance\-form targets\. In Section[8\.4](https://arxiv.org/html/2609.05727#S8.SS4), the conditional\-expectation identities move the density\-ratio correction from the terminal objective to a running objective, while retaining an explicit marginal correction\. This decomposition yields the Bolza realization of the gradient form\. Section[9\.3](https://arxiv.org/html/2609.05727#S9.SS3)uses a related decomposition to isolate the posterior\-specific correction that may be approximated\.

Appendix[D\.6](https://arxiv.org/html/2609.05727#A4.SS6)develops initial\-state sensitivity and adjoint calculus for generic SDEs\. The pathwise and endpoint\-conditioned identities underlie the posterior sensitivity kernel in Appendix[E\.4](https://arxiv.org/html/2609.05727#A5.SS4), while the conditional\-mean adjoint equation is used to prove critical\-point consistency in Appendix[F\.2](https://arxiv.org/html/2609.05727#A6.SS2)\.

### D\.1Common SDE Setup and Basic Identities

This subsection provides the common SDE setup, standing assumptions, and stochastic\-calculus tools used throughout the appendix\. Appendix[D\.1\.1](https://arxiv.org/html/2609.05727#A4.SS1.SSS1)introduces a generic SDE with a scalar state\-independent diffusion coefficient, together with its solution, path laws, transition densities, and generator\. Appendix[D\.1\.2](https://arxiv.org/html/2609.05727#A4.SS1.SSS2)states the required regularity assumptions, while Appendix[D\.1\.3](https://arxiv.org/html/2609.05727#A4.SS1.SSS3)records some lemmas, including the backward Kolmogorov equation, Itô’s formula, Dynkin’s formula, and the Doobhh\-transform\.

Throughout this appendix, identities involving realized SDE paths are understood to hold almost surely \(a\.s\.\) under the indicated path distribution, unless stated otherwise\.

#### D\.1\.1SDE with a State\-Independent Diffusion Coefficient

Fix an initial timet∈\[0,1\)t\\in\[0,1\)and an initial statext∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}\. We consider the SDE

d​Ysb,σ​\(t,xt\)=bs​\(Ysb,σ​\(t,xt\)\)​d​s\+σs​d​Ws,s∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)=b\_\{s\}\\left\(Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\\mathrm\{d\}s\+\\sigma\_\{s\}\\mathrm\{d\}W\_\{s\},\\qquad s\\in\[t,1\],\(163a\)Ytb,σ​\(t,xt\)=xt\.\\displaystyle Y\_\{t\}^\{b,\\sigma\}\(t,x\_\{t\}\)=x\_\{t\}\.\(163b\)Here,bs:ℝd→ℝdb\_\{s\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}is the drift,σs∈ℝ\\sigma\_\{s\}\\in\\mathbb\{R\}is a scalar state\-independent diffusion coefficient, andWWis a standarddd\-dimensional Brownian motion\. The integral form of SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\) is

Ysb,σ​\(t,xt\)=xt\+∫tsbu​\(Yub,σ​\(t,xt\)\)​𝑑u\+∫tsσu​d​Wu\.Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)=x\_\{t\}\+\\int\_\{t\}^\{s\}b\_\{u\}\\left\(Y\_\{u\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\\mathrm\{d\}u\+\\int\_\{t\}^\{s\}\\sigma\_\{u\}\\mathrm\{d\}W\_\{u\}\.\(164\)Whenever derivatives with respect to the initial statextx\_\{t\}are considered, the same Brownian motion\(Ws\)s∈\[t,1\]\(W\_\{s\}\)\_\{s\\in\[t,1\]\}is used for all initial states\.

Fort≤s≤1t\\leq s\\leq 1, we denote the random path by

𝒀\[t,s\]b,σ​\(t,xt\):=\(Yub,σ​\(t,xt\)\)u∈\[t,s\]\.\\bm\{Y\}\_\{\[t,s\]\}^\{b,\\sigma\}\(t,x\_\{t\}\):=\\left\(Y\_\{u\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\_\{u\\in\[t,s\]\}\.We denote the path distribution of𝒀\[t,s\]b,σ​\(t,xt\)\\bm\{Y\}\_\{\[t,s\]\}^\{b,\\sigma\}\(t,x\_\{t\}\)by

ℙ\[t,s\]\|tb,σ\(⋅\|xt\)\.\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\.The transition density is denoted by

qs\|tb,σ\(⋅\|xt\)\.q\_\{s\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\.
For every functionf∈C2​\(ℝd\)f\\in C^\{2\}\(\\mathbb\{R\}^\{d\}\)ands∈\[t,1\]s\\in\[t,1\], define the generator of SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\) by

𝒜sb,σ​f​\(x\):=bs​\(x\)⋅∇f​\(x\)\+σs22​Δ​f​\(x\)\.\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}f\(x\):=b\_\{s\}\(x\)\\cdot\\nabla f\(x\)\+\\frac\{\\sigma\_\{s\}^\{2\}\}\{2\}\\Delta f\(x\)\.\(165\)

#### D\.1\.2Regularity Assumptions

To avoid repeating technical qualifications in every result, we impose the following standing conditions throughout this appendix\.

For everyt∈\(0,1\)t\\in\(0,1\), assume that the map\(s,x\)↦bs​\(x\)\(s,x\)\\mapsto b\_\{s\}\(x\)is jointly Borel measurable on\[t,1\]×ℝd\[t,1\]\\times\\mathbb\{R\}^\{d\}, thatbsb\_\{s\}isC1C^\{1\}for a\.e\.s∈\[t,1\]s\\in\[t,1\], and thatσ\\sigmais a deterministic Borel\-measurable scalar function on\[t,1\]\[t,1\]\. We assume

∫t1\(‖bs​\(0\)‖2\+‖∇bs‖∞\+σs2\)​𝑑s<∞\.\\int\_\{t\}^\{1\}\\left\(\\left\\\|b\_\{s\}\(0\)\\right\\\|\_\{2\}\+\\left\\\|\\nabla b\_\{s\}\\right\\\|\_\{\\infty\}\+\\sigma\_\{s\}^\{2\}\\right\)\\mathrm\{d\}s<\\infty\.Consequently, the SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\) admits a unique strong solution on\[t,1\]\[t,1\]\. Whenever initial\-state sensitivities are invoked, solutions from different initial states are driven by the same Brownian motion, and the resulting solution mapxt↦Ysb,σ​\(t,xt\)x\_\{t\}\\mapsto Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)is assumed to be continuously differentiable\. Since some coefficients used below—notably those of the posterior\-preserving SDE—may be singular att=0t=0, the preceding well\-posedness assumptions are imposed only on intervals\[t,1\]\[t,1\]witht\>0t\>0\. Whenever an SDE is written with an initial law att=0t=0, it is understood as a solution on\(0,1\]\(0,1\]with the stated initial distribution as its zero\-time entrance law; existence and uniqueness in law are assumed\.

Unless stated otherwise, the densities, conditional densities, test functions, and path functionals appearing below are assumed to have sufficient regularity and integrability for all displayed expressions, derivatives, conditional expectations, and PDE identities to be well defined\. We assume that all required interchanges of differentiation, integration, and conditional expectation, together with all applications of Tonelli’s theorem and spatial integration by parts, are justified\. Whenever a martingale property is invoked, the relevant stochastic integral or stochastic exponential is assumed to be a true martingale\. We further assume that the required transition densities and endpoint\-conditioned path distributions exist and that the backward equations invoked below admit unique classical solutions in the relevant function classes\.

#### D\.1\.3Basic Identities

We next record several standard results\[[59](https://arxiv.org/html/2609.05727#bib.bib56)\]\. The backward Kolmogorov equation is used in Appendix[D\.3](https://arxiv.org/html/2609.05727#A4.SS3)to identify the terminal transition kernel of the posterior\-preserving SDE\. The Doobhh\-transform is used in Appendix[D\.4](https://arxiv.org/html/2609.05727#A4.SS4)to compare posterior path distributions and establish bridge universality\. Itô’s formula and Dynkin’s formula are used in Appendix[D\.5](https://arxiv.org/html/2609.05727#A4.SS5)to derive pathwise and conditional\-expectation representations of terminal log densities and log\-density ratios\. Dynkin’s formula is also used in Appendix[D\.6](https://arxiv.org/html/2609.05727#A4.SS6)to derive the generator\-form sensitivity identity and the conditional\-mean adjoint equation\.

In the following lemmas,Ysb,σY\_\{s\}^\{b,\\sigma\}denotesYsb,σ​\(t,xt\)Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)generated by SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\) for simplicity\.

###### Lemma 92\(Backward Kolmogorov equation for terminal observables\)\.

Consider SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\) and an integrable terminal observableφ:ℝd→ℝ\\varphi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}satisfying∀s∈\[t,1\]\\forall s\\in\[t,1\],x∈ℝdx\\in\\mathbb\{R\}^\{d\},

𝔼Y1∼q1\|sb,σ\(⋅\|x\)\[\|φ\(Y1\)\|\]<∞\.\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|s\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[\\left\|\\varphi\(Y\_\{1\}\)\\right\|\\right\]<\\infty\.For everys∈\[t,1\]s\\in\[t,1\], define

φs\(x\):=𝔼Y1∼q1\|sb,σ\(⋅\|x\)\[φ\(Y1\)\]\.\\varphi\_\{s\}\(x\):=\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|s\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[\\varphi\(Y\_\{1\}\)\\right\]\.Then,∀s∈\[t,1\]\\forall s\\in\[t,1\],x∈ℝdx\\in\\mathbb\{R\}^\{d\}, we have the backward Kolmogorov equation:

\(∂s\+𝒜sb,σ\)φs\(x\)=0,φ1\(x\)=φ\(x\)\.\(\\partial\_\{s\}\+\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}\)\\varphi\_\{s\}\(x\)=0,\\qquad\\varphi\_\{1\}\(x\)=\\varphi\(x\)\.

###### Proof\.

Fors∈\[t,1\)s\\in\[t,1\)and sufficiently smallε\>0\\varepsilon\>0, the Markov property gives

φs​\(x\)=\\displaystyle\\varphi\_\{s\}\(x\)=𝔼Y1∼q1\|sb,σ\(⋅\|x\)\[φ\(Y1\)\]\\displaystyle\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|s\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[\\varphi\(Y\_\{1\}\)\\right\]=\\displaystyle=𝔼Ys\+ε∼qs\+ε\|sb,σ\(⋅\|x\)\[𝔼Y1∼q1\|s\+εb,σ\(⋅\|Ys\+ε\)\[φ\(Y1\)\]\]\\displaystyle\\mathbb\{E\}\_\{Y\_\{s\+\\varepsilon\}\\sim q\_\{s\+\\varepsilon\|s\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|s\+\\varepsilon\}^\{b,\\sigma\}\(\\cdot\|Y\_\{s\+\\varepsilon\}\)\}\\left\[\\varphi\(Y\_\{1\}\)\\right\]\\right\]=\\displaystyle=𝔼Ys\+ε∼qs\+ε\|sb,σ\(⋅\|x\)\[φs\+ε\(Ys\+ε\)\]\.\\displaystyle\\mathbb\{E\}\_\{Y\_\{s\+\\varepsilon\}\\sim q\_\{s\+\\varepsilon\|s\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[\\varphi\_\{s\+\\varepsilon\}\(Y\_\{s\+\\varepsilon\}\)\\right\]\.The short\-time expansion associated with the backward operator∂s\+𝒜sb,σ\\partial\_\{s\}\+\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}gives

0=𝔼Ys\+ε∼qs\+ε\|sb,σ\(⋅\|x\)\[φs\+ε\(Ys\+ε\)−φs\(x\)\]=ε\(∂s\+𝒜sb,σ\)φs\(x\)\+o\(ε\)\.0=\\mathbb\{E\}\_\{Y\_\{s\+\\varepsilon\}\\sim q\_\{s\+\\varepsilon\|s\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[\\varphi\_\{s\+\\varepsilon\}\(Y\_\{s\+\\varepsilon\}\)\-\\varphi\_\{s\}\(x\)\\right\]=\\varepsilon\(\\partial\_\{s\}\+\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}\)\\varphi\_\{s\}\(x\)\+o\(\\varepsilon\)\.Lettingε↓0\\varepsilon\\downarrow 0proves the backward equation\. The terminal condition follows directly from the definition\. ∎

###### Lemma 93\(Itô’s formula\)\.

LetFs​\(x\)F\_\{s\}\(x\)be a regular function that isC1C^\{1\}inssandC2C^\{2\}inxx\. For the SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\),∀t≤s≤1\\forall t\\leq s\\leq 1, we have Itô’s formula

Fs\(Ysb,σ\)=Ft\(Ytb,σ\)\+∫ts\(∂u\+𝒜ub,σ\)Fu\(Yub,σ\)du\+∫tsσu∇Fu\(Yub,σ\)⊤dWu\.\\displaystyle F\_\{s\}\(Y\_\{s\}^\{b,\\sigma\}\)=F\_\{t\}\(Y\_\{t\}^\{b,\\sigma\}\)\+\\int\_\{t\}^\{s\}\(\\partial\_\{u\}\+\\mathcal\{A\}\_\{u\}^\{b,\\sigma\}\)F\_\{u\}\(Y\_\{u\}^\{b,\\sigma\}\)\\mathrm\{d\}u\+\\int\_\{t\}^\{s\}\\sigma\_\{u\}\\nabla F\_\{u\}\(Y\_\{u\}^\{b,\\sigma\}\)^\{\\top\}\\mathrm\{d\}W\_\{u\}\.\(166\)

###### Proof\.

This is the standard form of Itô’s formula applied to \([163](https://arxiv.org/html/2609.05727#A4.E163)\)\. ∎

###### Lemma 94\(Dynkin’s formula\)\.

LetFs​\(x\)F\_\{s\}\(x\)beC1,2C^\{1,2\}and satisfy the usual integrability assumptions\. For the SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\),t≤s≤1t\\leq s\\leq 1, we have Dynkin’s formula

𝔼Ys∼qs\|tb,σ\(⋅\|x\)\[Fs\(Ys\)\]=Ft\(x\)\+𝔼𝒀\[t,s\]∼ℙ\[t,s\]\|tb,σ\(⋅\|x\)\[∫ts\(∂u\+𝒜ub,σ\)Fu\(Yu\)du\]\.\\mathbb\{E\}\_\{Y\_\{s\}\\sim q\_\{s\|t\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[F\_\{s\}\(Y\_\{s\}\)\\right\]=F\_\{t\}\(x\)\+\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,s\]\}\\sim\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[\\int\_\{t\}^\{s\}\(\\partial\_\{u\}\+\\mathcal\{A\}\_\{u\}^\{b,\\sigma\}\)F\_\{u\}\(Y\_\{u\}\)\\mathrm\{d\}u\\right\]\.

###### Proof\.

Taking conditional expectation in \([166](https://arxiv.org/html/2609.05727#A4.E166)\) and using that the stochastic integral has conditional\-mean zero gives the result\. ∎

###### Lemma 95\(Doobhh\-transform\)\.

Consider SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\)\. Assumehs​\(x\)\>0h\_\{s\}\(x\)\>0isC1C^\{1\}inssandC2C^\{2\}inxx, satisfying

\(∂s\+𝒜sb,σ\)hs\(x\)=0\.\(\\partial\_\{s\}\+\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}\)h\_\{s\}\(x\)=0\.\(167\)Assume that, for the solution started fromYtb,σ=xtY\_\{t\}^\{b,\\sigma\}=x\_\{t\}, the process\(hu​\(Yub,σ\)ht​\(xt\)\)u∈\[t,1\]\\left\(\\frac\{h\_\{u\}\(Y\_\{u\}^\{b,\\sigma\}\)\}\{h\_\{t\}\(x\_\{t\}\)\}\\right\)\_\{u\\in\[t,1\]\}is a true martingale\. Consider another SDE

dZs=\[bs\(Zs\)\+σs2∇loghs\(Zs\)\]ds\+σsdWs\\mathrm\{d\}Z\_\{s\}=\\left\[b\_\{s\}\(Z\_\{s\}\)\+\\sigma\_\{s\}^\{2\}\\nabla\\log h\_\{s\}\(Z\_\{s\}\)\\right\]\\mathrm\{d\}s\+\\sigma\_\{s\}\\mathrm\{d\}W\_\{s\}\(168\)which also starts fromZt=xtZ\_\{t\}=x\_\{t\}\. Denote the conditional path distribution of𝐙\[t,s\]=\(Zu\)u∈\[t,s\]\\bm\{Z\}\_\{\[t,s\]\}=\(Z\_\{u\}\)\_\{u\\in\[t,s\]\}byℙ\[t,s\]\|tZ\(⋅\|xt\)\\mathbb\{P\}^\{Z\}\_\{\[t,s\]\|t\}\(\\cdot\|x\_\{t\}\)\. Then,

ℙ\[t,s\]\|tZ\(⋅\|xt\)≪ℙ\[t,s\]\|tb,σ\(⋅\|xt\),d​ℙ\[t,s\]\|tZd​ℙ\[t,s\]\|tb,σ=hs​\(Ysb,σ\)ht​\(xt\)\.\\mathbb\{P\}^\{Z\}\_\{\[t,s\]\|t\}\(\\cdot\|x\_\{t\}\)\\ll\\mathbb\{P\}^\{b,\\sigma\}\_\{\[t,s\]\|t\}\(\\cdot\|x\_\{t\}\),\\qquad\\frac\{\\mathrm\{d\}\\mathbb\{P\}^\{Z\}\_\{\[t,s\]\|t\}\}\{\\mathrm\{d\}\\mathbb\{P\}^\{b,\\sigma\}\_\{\[t,s\]\|t\}\}=\\frac\{h\_\{s\}\(Y\_\{s\}^\{b,\\sigma\}\)\}\{h\_\{t\}\(x\_\{t\}\)\}\.\(169\)Letqr\|uZq^\{Z\}\_\{r\|u\}denote the transition densities fromuutorrof SDE \([168](https://arxiv.org/html/2609.05727#A4.E168)\)\. Then, we have

qr\|uZ​\(xr\|xu\)=qr\|ub,σ​\(xr\|xu\)​hr​\(xr\)hu​\(xu\),t≤u≤r≤s\.q^\{Z\}\_\{r\|u\}\(x\_\{r\}\|x\_\{u\}\)=q^\{b,\\sigma\}\_\{r\|u\}\(x\_\{r\}\|x\_\{u\}\)\\frac\{h\_\{r\}\(x\_\{r\}\)\}\{h\_\{u\}\(x\_\{u\}\)\},\\qquad t\\leq u\\leq r\\leq s\.\(170\)

###### Proof\.

By \([167](https://arxiv.org/html/2609.05727#A4.E167)\), Itô’s formula in Lemma[93](https://arxiv.org/html/2609.05727#Thmtheorem93)implies

dhs\(Ysb,σ\)=σs∇hs\(Ysb,σ\)⊤dWs\.\\mathrm\{d\}h\_\{s\}\(Y\_\{s\}^\{b,\\sigma\}\)=\\sigma\_\{s\}\\nabla h\_\{s\}\(Y\_\{s\}^\{b,\\sigma\}\)^\{\\top\}\\mathrm\{d\}W\_\{s\}\.Sohs​\(Ysb,σ\)ht​\(xt\)\\frac\{h\_\{s\}\(Y\_\{s\}^\{b,\\sigma\}\)\}\{h\_\{t\}\(x\_\{t\}\)\}is a positive local martingale and, by assumption, a true martingale\. Therefore, we have

𝔼Ys∼qs\|tb,σ\(⋅\|xt\)\[hs​\(Ys\)ht​\(xt\)\]=𝔼Yt∼qt\|tb,σ\(⋅\|xt\)\[ht​\(Yt\)ht​\(xt\)\]=ht​\(xt\)ht​\(xt\)=1\.\\mathbb\{E\}\_\{Y\_\{s\}\\sim q\_\{s\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\frac\{h\_\{s\}\(Y\_\{s\}\)\}\{h\_\{t\}\(x\_\{t\}\)\}\\right\]=\\mathbb\{E\}\_\{Y\_\{t\}\\sim q\_\{t\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\frac\{h\_\{t\}\(Y\_\{t\}\)\}\{h\_\{t\}\(x\_\{t\}\)\}\\right\]=\\frac\{h\_\{t\}\(x\_\{t\}\)\}\{h\_\{t\}\(x\_\{t\}\)\}=1\.Therefore,ℙ\[t,s\]\|tZ\\mathbb\{P\}^\{Z\}\_\{\[t,s\]\|t\}in \([169](https://arxiv.org/html/2609.05727#A4.E169)\) is a normalized path distribution\. Then, by Bayes’ rule, \([170](https://arxiv.org/html/2609.05727#A4.E170)\) holds\. In the following, we prove thatℙ\[t,s\]\|tZ\\mathbb\{P\}^\{Z\}\_\{\[t,s\]\|t\}andqr\|uZq^\{Z\}\_\{r\|u\}are the conditional path distribution and the transition density of SDE \([168](https://arxiv.org/html/2609.05727#A4.E168)\), respectively\.

Fixu∈\[t,s\)u\\in\[t,s\), and letε\>0\\varepsilon\>0be sufficiently small thatu\+ε≤su\+\\varepsilon\\leq s\. Then

𝔼Zu\+ε∼qZu\+ε\|u\(⋅\|x\)\[φ\(Zu\+ε\)\]\\displaystyle\\mathbb\{E\}\_\{Z\_\{u\+\\varepsilon\}\\sim q^\{Z\}\_\{u\+\\varepsilon\|u\}\(\\cdot\|x\)\}\\left\[\\varphi\(Z\_\{u\+\\varepsilon\}\)\\right\]=1hu​\(x\)𝔼Yu\+ε∼qu\+ε\|ub,σ\(⋅\|x\)\[hu\+ε\(Yu\+ε\)φ\(Yu\+ε\)\]\.\\displaystyle=\\frac\{1\}\{h\_\{u\}\(x\)\}\\mathbb\{E\}\_\{Y\_\{u\+\\varepsilon\}\\sim q\_\{u\+\\varepsilon\|u\}^\{b,\\sigma\}\(\\cdot\|x\)\}\\left\[h\_\{u\+\\varepsilon\}\(Y\_\{u\+\\varepsilon\}\)\\varphi\(Y\_\{u\+\\varepsilon\}\)\\right\]\.Denotebsh\(x\):=bs\(x\)\+σs2∇loghs\(x\)b\_\{s\}^\{h\}\(x\):=b\_\{s\}\(x\)\+\\sigma\_\{s\}^\{2\}\\nabla\\log h\_\{s\}\(x\)\. Consequently,

𝒜ubh,σφ\(x\)=1hu​\(x\)\(∂u\+𝒜ub,σ\)\(huφ\)\(x\)=𝒜ub,σφ\(x\)\+σu2∇loghu\(x\)⋅∇φ\(x\)\.\\mathcal\{A\}\_\{u\}^\{b^\{h\},\\sigma\}\\varphi\(x\)=\\frac\{1\}\{h\_\{u\}\(x\)\}\(\\partial\_\{u\}\+\\mathcal\{A\}\_\{u\}^\{b,\\sigma\}\)\(h\_\{u\}\\varphi\)\(x\)=\\mathcal\{A\}\_\{u\}^\{b,\\sigma\}\\varphi\(x\)\+\\sigma\_\{u\}^\{2\}\\nabla\\log h\_\{u\}\(x\)\\cdot\\nabla\\varphi\(x\)\.By weak uniqueness for \([168](https://arxiv.org/html/2609.05727#A4.E168)\), the path measureℙ\[t,s\]\|tZ\(⋅\|xt\)\\mathbb\{P\}^\{Z\}\_\{\[t,s\]\|t\}\(\\cdot\|x\_\{t\}\)coincides with the path distribution of𝒁\[t,s\]\\bm\{Z\}\_\{\[t,s\]\}; hence,ℙ\[t,s\]\|tZ\\mathbb\{P\}^\{Z\}\_\{\[t,s\]\|t\}is the conditional path distribution of SDE \([168](https://arxiv.org/html/2609.05727#A4.E168)\)\. ∎

### D\.2Backward Equation for Posterior Value

The posterior value is defined as a conditional expectation underp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\. In this appendix, we call this conditional law the*interpolant posterior*to distinguish its interpolant\-based definition from the terminal transition kernel of an SDE\. To obtain a dynamic characterization, we derive the backward equation for the posterior value, which is the key analytic link for the constructions that follow\. In Appendix[D\.3](https://arxiv.org/html/2609.05727#A4.SS3), we construct an SDE whose conditional expectations of terminal observables solve the same backward terminal\-value problem; uniqueness then implies that its terminal transition kernel coincides with the interpolant posterior\. In Appendix[D\.4](https://arxiv.org/html/2609.05727#A4.SS4), taking the terminal observable to beρ2/ρ1\\rho\_\{2\}/\\rho\_\{1\}shows that the marginal density ratioptρ2/ptρ1p\_\{t\}^\{\\rho\_\{2\}\}/p\_\{t\}^\{\\rho\_\{1\}\}satisfies the backward equation associated with theρ1\\rho\_\{1\}posterior\-preserving SDE\. This relation enables the Doobhh\-transform between posterior\-preserving SDEs and leads to the universal\-bridge result\.

For a terminal observablef:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}, recall the posterior value defined in \([14](https://arxiv.org/html/2609.05727#S4.E14)\):

Vtρ\[f\]\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[f\(X1\)\]\.V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[f\(X\_\{1\}\)\]\.Define the weighted marginal

Utρ​\[f\]​\(xt\):=ptρ​\(xt\)​Vtρ​\[f\]​\(xt\)=∫ℝdf⁡\(x1\)​ρ​\(x1\)​pt\|1​\(xt\|x1\)​d​x1\.U\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\):=p\_\{t\}^\{\\rho\}\(x\_\{t\}\)V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)=\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\_\{1\}\)\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\.At terminal time, we have

V1ρ​\[f\]​\(x1\)=f⁡\(x1\),U1ρ​\[f\]​\(x1\)=f⁡\(x1\)​ρ​\(x1\)\.V\_\{1\}^\{\\rho\}\[f\]\(x\_\{1\}\)=f\(x\_\{1\}\),\\qquad U\_\{1\}^\{\\rho\}\[f\]\(x\_\{1\}\)=f\(x\_\{1\}\)\\rho\(x\_\{1\}\)\.At initial time, the independent coupling gives

V0ρ​\[f\]​\(x0\)=𝔼X1∼ρ​\[f⁡\(X1\)\],V\_\{0\}^\{\\rho\}\[f\]\(x\_\{0\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim\\rho\}\[f\(X\_\{1\}\)\],which is independent ofx0x\_\{0\}\.

###### Proposition 96\.

For every sufficiently regular terminal observableffand everyt∈\(0,1\)t\\in\(0,1\),

∂tUtρ\[f\]\(x\)\+∇⋅\(α˙tαtxUtρ\[f\]\(x\)\+κt∇Utρ\[f\]\(x\)\)=0\.\\partial\_\{t\}U\_\{t\}^\{\\rho\}\[f\]\(x\)\+\\nabla\\cdot\\left\(\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}xU\_\{t\}^\{\\rho\}\[f\]\(x\)\+\\kappa\_\{t\}\\nabla U\_\{t\}^\{\\rho\}\[f\]\(x\)\\right\)=0\.\(171\)Consequently, we have the backward equation for the posterior value:

∂tVtρ\[f\]\(x\)\+\(α˙tαtx\+2κt∇logptρ\(x\)\)⋅∇Vtρ\[f\]\(x\)\+κtΔVtρ\[f\]\(x\)=0\.\\partial\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(x\)\+\\left\(\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\+2\\kappa\_\{t\}\\nabla\\log p\_\{t\}^\{\\rho\}\(x\)\\right\)\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\)\+\\kappa\_\{t\}\\Delta V\_\{t\}^\{\\rho\}\[f\]\(x\)=0\.\(172\)

###### Proof\.

For fixedx1x\_\{1\}, the conditional density satisfies

∂tpt\|1​\(xt\|x1\)\+∇xt⋅\(pt\|1​\(xt\|x1\)​vt\|1​\(xt\|x1\)\)=0\.\\partial\_\{t\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\+\\nabla\_\{x\_\{t\}\}\\cdot\\left\(p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\right\)=0\.Multiplying byf⁡\(x1\)​ρ​\(x1\)f\(x\_\{1\}\)\\rho\(x\_\{1\}\)and integrating gives

∂tUtρ​\[f\]​\(xt\)\+∇xt⋅\(∫ℝdf⁡\(x1\)​ρ​\(x1\)​pt\|1​\(xt\|x1\)​vt\|1​\(xt\|x1\)​d​x1\)=0\.\\partial\_\{t\}U\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\+\\nabla\_\{x\_\{t\}\}\\cdot\\left\(\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\_\{1\}\)\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\\right\)=0\.\(173\)Since

∇xtpt\|1​\(xt\|x1\)=−xt−αt​x1βt2​pt\|1​\(xt\|x1\),\\nabla\_\{x\_\{t\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\-\\frac\{x\_\{t\}\-\\alpha\_\{t\}x\_\{1\}\}\{\\beta\_\{t\}^\{2\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\),we have

∫ℝdx1pt\|1\(xt\|x1\)f\(x1\)ρ\(x1\)dx1=βt2αt∇Utρ\[f\]\(xt\)\+xtαtUtρ\[f\]\(xt\)\.\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}x\_\{1\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)f\(x\_\{1\}\)\\rho\(x\_\{1\}\)\\mathrm\{d\}x\_\{1\}=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}\\nabla U\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\+\\frac\{x\_\{t\}\}\{\\alpha\_\{t\}\}U\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)\.Substituting \([9](https://arxiv.org/html/2609.05727#S3.E9)\) and the above equation into \([173](https://arxiv.org/html/2609.05727#A4.E173)\) proves \([171](https://arxiv.org/html/2609.05727#A4.E171)\)\. Takingf≡1f\\equiv 1gives

∂tptρ\(x\)\+∇⋅\(α˙tαtxptρ\(x\)\+κt∇ptρ\(x\)\)=0\.\\partial\_\{t\}p\_\{t\}^\{\\rho\}\(x\)\+\\nabla\\cdot\\left\(\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}xp\_\{t\}^\{\\rho\}\(x\)\+\\kappa\_\{t\}\\nabla p\_\{t\}^\{\\rho\}\(x\)\\right\)=0\.\(174\)For generalff, substituteUtρ​\[f\]=ptρ​Vtρ​\[f\]U\_\{t\}^\{\\rho\}\[f\]=p\_\{t\}^\{\\rho\}V\_\{t\}^\{\\rho\}\[f\]into \([171](https://arxiv.org/html/2609.05727#A4.E171)\)\. Expanding the derivatives and dividing byptρ​\(x\)p\_\{t\}^\{\\rho\}\(x\)gives

∂tVtρ\[f\]\(x\)\+\(α˙tαtx\+2κt∇logptρ\(x\)\)⋅∇Vtρ\[f\]\(x\)\+κtΔVtρ\[f\]\(x\)\\displaystyle\\partial\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(x\)\+\\left\(\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\+2\\kappa\_\{t\}\\nabla\\log p\_\{t\}^\{\\rho\}\(x\)\\right\)\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\)\+\\kappa\_\{t\}\\Delta V\_\{t\}^\{\\rho\}\[f\]\(x\)=\\displaystyle=−Vtρ​\[f\]​\(x\)ptρ​\(x\)\[∂tptρ\(x\)\+∇⋅\(α˙tαtxptρ\(x\)\+κt∇ptρ\(x\)\)\]=0,\\displaystyle\-\\frac\{V\_\{t\}^\{\\rho\}\[f\]\(x\)\}\{p\_\{t\}^\{\\rho\}\(x\)\}\\left\[\\partial\_\{t\}p\_\{t\}^\{\\rho\}\(x\)\+\\nabla\\cdot\\left\(\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}xp\_\{t\}^\{\\rho\}\(x\)\+\\kappa\_\{t\}\\nabla p\_\{t\}^\{\\rho\}\(x\)\\right\)\\right\]=0,which proves \([172](https://arxiv.org/html/2609.05727#A4.E172)\)\. ∎

### D\.3Posterior\-Preserving SDE

The backward equation derived in Appendix[D\.2](https://arxiv.org/html/2609.05727#A4.SS2)suggests an SDE realization of the interpolant posterior\. We define an SDE whose generator is the spatial differential operator appearing in \([172](https://arxiv.org/html/2609.05727#A4.E172)\)\. The main result of this subsection shows that, when initialized atYt=xtY\_\{t\}=x\_\{t\}, its terminal stateY1Y\_\{1\}has densityp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\. Thus, the SDE provides an exact sampler from the interpolant posterior\.

###### Definition 97\.

For a terminal densityρ\\rhoandt∈\(0,1\]t\\in\(0,1\], consider the drift

btρ\(x\):=vtρ\(x\)\+κt∇logptρ\(x\)=α˙tαtx\+2κt∇logptρ\(x\)=2vtρ\(x\)−α˙tαtx\.b\_\{t\}^\{\\rho\}\(x\):=v\_\{t\}^\{\\rho\}\(x\)\+\\kappa\_\{t\}\\nabla\\log p\_\{t\}^\{\\rho\}\(x\)=\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\+2\\kappa\_\{t\}\\nabla\\log p\_\{t\}^\{\\rho\}\(x\)=2v\_\{t\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\.\(175\)The posterior\-preserving SDE is defined as

d​Ytρ=btρ​\(Ytρ\)​d​t\+2​κt​d​Wt\.\\mathrm\{d\}Y\_\{t\}^\{\\rho\}=b\_\{t\}^\{\\rho\}\(Y\_\{t\}^\{\\rho\}\)\\mathrm\{d\}t\+\\sqrt\{2\\kappa\_\{t\}\}\\mathrm\{d\}W\_\{t\}\.\(176\)By definition \([165](https://arxiv.org/html/2609.05727#A4.E165)\), the generator of the posterior\-preserving SDE \([176](https://arxiv.org/html/2609.05727#A4.E176)\) is

ℒtρ​f​\(x\)=btρ​\(x\)⋅∇f​\(x\)\+κt​Δ​f​\(x\)\.\\mathcal\{L\}\_\{t\}^\{\\rho\}f\(x\)=b\_\{t\}^\{\\rho\}\(x\)\\cdot\\nabla f\(x\)\+\\kappa\_\{t\}\\Delta f\(x\)\.For0<t<s≤10<t<s\\leq 1, denote its transition density byqs\|tρ​\(xs\|xt\)q\_\{s\|t\}^\{\\rho\}\(x\_\{s\}\|x\_\{t\}\), and denote its conditional path distribution byℙ\[t,s\]\|tρ\(⋅\|xt\)\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)for simplicity\.

###### Theorem 98\.

For every sufficiently regular terminal observableff,

\(∂t\+ℒtρ\)Vtρ\[f\]\(x\)=0,t∈\(0,1\),\\displaystyle\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\}\)V\_\{t\}^\{\\rho\}\[f\]\(x\)=0,\\qquad t\\in\(0,1\),\(177a\)V1ρ​\[f\]​\(x\)=f⁡\(x\)\.\\displaystyle V\_\{1\}^\{\\rho\}\[f\]\(x\)=f\(x\)\.\(177b\)Consequently, the terminal transition density of the posterior\-preserving SDE, denoted byq1\|tρ\(⋅\|xt\)q\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\), satisfies

q1\|tρ\(⋅\|xt\)=p1\|tρ\(⋅\|xt\)\.q\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)=p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\.\(178\)
Moreover, consider the marginal\-preserving SDE family

dYtρ,σ=\[vtρ\(Ytρ,σ\)\+σt22∇logptρ\(Ytρ,σ\)\]dt\+σtdWt,\\mathrm\{d\}Y\_\{t\}^\{\\rho,\\sigma\}=\\left\[v\_\{t\}^\{\\rho\}\(Y\_\{t\}^\{\\rho,\\sigma\}\)\+\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\\nabla\\log p\_\{t\}^\{\\rho\}\(Y\_\{t\}^\{\\rho,\\sigma\}\)\\right\]\\mathrm\{d\}t\+\\sigma\_\{t\}\\mathrm\{d\}W\_\{t\},\(179\)whose transition density fromtttossis denoted byqs\|tρ,σ\(⋅\|xt\)q\_\{s\|t\}^\{\\rho,\\sigma\}\(\\cdot\|x\_\{t\}\)\. Assume that, for everyt∈\(0,1\)t\\in\(0,1\),xt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}, and sufficiently regular terminal observableff,

𝔼Y1∼q1\|tρ,σ\(⋅\|xt\)\[f\(Y1\)\]=Vtρ\[f\]\(xt\)\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{\\rho,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\\left\(Y\_\{1\}\\right\)\\right\]=V\_\{t\}^\{\\rho\}\[f\]\(x\_\{t\}\)Assume additionally that, for everyt∈\(0,1\)t\\in\(0,1\), there exists a sufficiently regular terminal observableffsuch that

Δ​Vtρ​\[f\]\+∇log⁡ptρ⋅∇Vtρ​\[f\]≢0\.\\Delta V\_\{t\}^\{\\rho\}\[f\]\+\\nabla\\log p\_\{t\}^\{\\rho\}\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\\not\\equiv 0\.\(180\)Then, for a\.e\.t∈\(0,1\)t\\in\(0,1\), we have

σt2=2​κt\.\\sigma\_\{t\}^\{2\}=2\\kappa\_\{t\}\.In other words, the scheduleσt2=2​κt\\sigma\_\{t\}^\{2\}=2\\kappa\_\{t\}is the unique one whose conditional values agree with the interpolant posteriorp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)for every sufficiently regular terminal observable\.

###### Proof\.

Equation \([172](https://arxiv.org/html/2609.05727#A4.E172)\) is exactly \([177](https://arxiv.org/html/2609.05727#A4.E177)\)\. Under the posterior\-preserving SDE \([176](https://arxiv.org/html/2609.05727#A4.E176)\), the conditional value

Ft\(xt\):=𝔼Y1∼q1\|tρ\(⋅\|xt\)\[f\(Y1\)\]F\_\{t\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(Y\_\{1\}\)\\right\]solves the same backward equation and terminal condition by Lemma[92](https://arxiv.org/html/2609.05727#Thmtheorem92)\. Uniqueness of the classical backward problem givesFt=Vtρ​\[f\]F\_\{t\}=V\_\{t\}^\{\\rho\}\[f\]\. Since this holds for every sufficiently regular terminal observable, the corresponding terminal transition kernels agree, proving \([178](https://arxiv.org/html/2609.05727#A4.E178)\)\.

Every SDE in \([179](https://arxiv.org/html/2609.05727#A4.E179)\) has marginal densityptρp\_\{t\}^\{\\rho\}, since its Fokker–Planck equation reduces to the continuity equation\. Its generator is

𝒜tbρ,σg=\(vtρ\+σt22∇logptρ\)⋅∇g\+σt22Δg\.\\mathcal\{A\}\_\{t\}^\{b^\{\\rho\},\\sigma\}g=\\left\(v\_\{t\}^\{\\rho\}\+\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\\nabla\\log p\_\{t\}^\{\\rho\}\\right\)\\cdot\\nabla g\+\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\\Delta g\.Forσt2=2​κt\\sigma\_\{t\}^\{2\}=2\\kappa\_\{t\}, we have𝒜tbρ,σ=ℒtρ\\mathcal\{A\}\_\{t\}^\{b^\{\\rho\},\\sigma\}=\\mathcal\{L\}\_\{t\}^\{\\rho\}, so the preceding posterior\-preservation argument applies\.

Conversely, suppose a noise scheduleσt\\sigma\_\{t\}in \([179](https://arxiv.org/html/2609.05727#A4.E179)\) has conditional values equal toVtρ​\[f\]V\_\{t\}^\{\\rho\}\[f\]for every sufficiently regular terminal observableff\. ThenVtρ​\[f\]V\_\{t\}^\{\\rho\}\[f\]solves the backward equations associated with both𝒜tbρ,σ\\mathcal\{A\}\_\{t\}^\{b^\{\\rho\},\\sigma\}andℒtρ\\mathcal\{L\}\_\{t\}^\{\\rho\}\. Subtracting them gives

0=\(σt22−κt\)​\(Δ​Vtρ​\[f\]\+∇log⁡ptρ⋅∇Vtρ​\[f\]\)0=\\left\(\\frac\{\\sigma\_\{t\}^\{2\}\}\{2\}\-\\kappa\_\{t\}\\right\)\\left\(\\Delta V\_\{t\}^\{\\rho\}\[f\]\+\\nabla\\log p\_\{t\}^\{\\rho\}\\cdot\\nabla V\_\{t\}^\{\\rho\}\[f\]\\right\)for every suchff\. Under the condition \([180](https://arxiv.org/html/2609.05727#A4.E180)\), the second factor is nonzero for at least one pair\(f,x\)\(f,x\)at a\.e\.t∈\(0,1\)t\\in\(0,1\)\. Therefore,σt2=2​κt\\sigma\_\{t\}^\{2\}=2\\kappa\_\{t\}for a\.e\.t∈\(0,1\)t\\in\(0,1\), proving the uniqueness claim\. ∎

### D\.4Universal Bridges

This subsection establishes the universal bridge property of the posterior\-preserving SDE \([176](https://arxiv.org/html/2609.05727#A4.E176)\): when conditioned on two endpoints, the bridge distribution depends only on the interpolant schedule\(αt,βt\)t∈\[0,1\]\(\\alpha\_\{t\},\\beta\_\{t\}\)\_\{t\\in\[0,1\]\}and not on the terminal densityρ\\rho\. Moreover, the bridge can be sampled on any chosen time grid from an explicit Gaussian bridge distribution\.

Letρ1,ρ2∈𝒫\\rho\_\{1\},\\rho\_\{2\}\\in\\mathscr\{P\}be positive terminal densities\. Define the marginal ratio as

htρ1,ρ2​\(x\)=ptρ2​\(x\)ptρ1​\(x\)\.h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\)=\\frac\{p\_\{t\}^\{\\rho\_\{2\}\}\(x\)\}\{p\_\{t\}^\{\\rho\_\{1\}\}\(x\)\}\.
###### Proposition 99\.

For everyt∈\(0,1\)t\\in\(0,1\),

htρ1,ρ2​\(xt\)=Vtρ1​\[ρ2ρ1\]​\(xt\)\.h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)=V\_\{t\}^\{\\rho\_\{1\}\}\\left\[\\frac\{\\rho\_\{2\}\}\{\\rho\_\{1\}\}\\right\]\(x\_\{t\}\)\.Consequently,

\(∂t\+ℒtρ1\)htρ1,ρ2=0\.\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\_\{1\}\}\)h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}=0\.\(181\)

###### Proof\.

Direct expansion gives

Vtρ1​\[ρ2ρ1\]​\(xt\)=∫ℝdρ2​\(x1\)ρ1​\(x1\)​ρ1​\(x1\)​pt\|1​\(xt\|x1\)ptρ1​\(xt\)​d​x1=ptρ2​\(xt\)ptρ1​\(xt\)=htρ1,ρ2​\(xt\)\.V\_\{t\}^\{\\rho\_\{1\}\}\\left\[\\frac\{\\rho\_\{2\}\}\{\\rho\_\{1\}\}\\right\]\(x\_\{t\}\)=\\int\_\{\\mathbb\{R\}^\{d\}\}\\frac\{\\rho\_\{2\}\(x\_\{1\}\)\}\{\\rho\_\{1\}\(x\_\{1\}\)\}\\frac\{\\rho\_\{1\}\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{p\_\{t\}^\{\\rho\_\{1\}\}\(x\_\{t\}\)\}\\mathrm\{d\}x\_\{1\}=\\frac\{p\_\{t\}^\{\\rho\_\{2\}\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\rho\_\{1\}\}\(x\_\{t\}\)\}=h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)\.Equation \([181](https://arxiv.org/html/2609.05727#A4.E181)\) follows from Theorem[98](https://arxiv.org/html/2609.05727#Thmtheorem98)\. ∎

###### Proposition 100\.

Fort∈\(0,1\)t\\in\(0,1\),

vtρ2\(x\)−vtρ1\(x\)=κt∇loghtρ1,ρ2\(x\),\\displaystyle v\_\{t\}^\{\\rho\_\{2\}\}\(x\)\-v\_\{t\}^\{\\rho\_\{1\}\}\(x\)=\\kappa\_\{t\}\\nabla\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\),\(182a\)btρ2\(x\)−btρ1\(x\)=2κt∇loghtρ1,ρ2\(x\)\.\\displaystyle b\_\{t\}^\{\\rho\_\{2\}\}\(x\)\-b\_\{t\}^\{\\rho\_\{1\}\}\(x\)=2\\kappa\_\{t\}\\nabla\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\)\.\(182b\)

###### Proof\.

Substitutingvtρi\(x\)=α˙tαtx\+κt∇logptρi\(x\)v\_\{t\}^\{\\rho\_\{i\}\}\(x\)=\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\+\\kappa\_\{t\}\\nabla\\log p\_\{t\}^\{\\rho\_\{i\}\}\(x\)into \([175](https://arxiv.org/html/2609.05727#A4.E175)\) proves the result\. ∎

The following proposition gives the Radon–Nikodym derivative between the path distributions of two posterior\-preserving SDEs\.

###### Proposition 101\.

For0<t<s≤10<t<s\\leq 1, letℙ\[t,s\]\|tρi\(⋅\|xt\)\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\rho\_\{i\}\}\(\\cdot\|x\_\{t\}\)be the path distribution of the posterior\-preserving SDE with terminal densityρi\\rho\_\{i\}, started fromYt=xtY\_\{t\}=x\_\{t\}\. Then

d​ℙ\[t,s\]\|tρ2d​ℙ\[t,s\]\|tρ1​\(𝒀\[t,s\]\)=hsρ1,ρ2​\(Ys\)htρ1,ρ2​\(xt\)\.\\frac\{\\mathrm\{d\}\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\rho\_\{2\}\}\}\{\\mathrm\{d\}\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\rho\_\{1\}\}\}\(\\bm\{Y\}\_\{\[t,s\]\}\)=\\frac\{h\_\{s\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(Y\_\{s\}\)\}\{h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)\}\.\(183\)

###### Proof\.

Apply the Doobhh\-transform in Lemma[95](https://arxiv.org/html/2609.05727#Thmtheorem95)to the posterior\-preserving SDE \([176](https://arxiv.org/html/2609.05727#A4.E176)\) ofρ1\\rho\_\{1\}with the positive functionhuρ1,ρ2h\_\{u\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\. By Proposition[99](https://arxiv.org/html/2609.05727#Thmtheorem99),huρ1,ρ2h\_\{u\}^\{\\rho\_\{1\},\\rho\_\{2\}\}satisfies the required backward equation \([167](https://arxiv.org/html/2609.05727#A4.E167)\)\. Moreover, posterior preservation and the Markov property imply that\(huρ1,ρ2​\(Yu\)htρ1,ρ2​\(xt\)\)u∈\[t,s\]\\left\(\\frac\{h\_\{u\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(Y\_\{u\}\)\}\{h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)\}\\right\)\_\{u\\in\[t,s\]\}is a true martingale\. By \([182b](https://arxiv.org/html/2609.05727#A4.E182.2)\), the drift by Doobhh\-transform isbuρ1\+2κu∇loghuρ1,ρ2=buρ2b\_\{u\}^\{\\rho\_\{1\}\}\+2\\kappa\_\{u\}\\nabla\\log h\_\{u\}^\{\\rho\_\{1\},\\rho\_\{2\}\}=b\_\{u\}^\{\\rho\_\{2\}\}\. Hence, the transformed path distribution isℙ\[t,s\]\|tρ2\(⋅\|xt\)\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\rho\_\{2\}\}\(\\cdot\|x\_\{t\}\), and the path\-likelihood ratio \([169](https://arxiv.org/html/2609.05727#A4.E169)\) gives \([183](https://arxiv.org/html/2609.05727#A4.E183)\)\. ∎

The Radon–Nikodym derivative between path distributions in \([183](https://arxiv.org/html/2609.05727#A4.E183)\) depends on a realized path only through its two endpoints\. This observation yields the following bridge universality\.

###### Theorem 102\.

For0<t<s≤10<t<s\\leq 1, letℙ\[t,s\]\|t,sρi\(⋅\|xt,xs\)\\mathbb\{P\}\_\{\[t,s\]\|t,s\}^\{\\rho\_\{i\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\)denote the conditional path distribution of the posterior\-preserving SDE ofρi\\rho\_\{i\}, givenYt=xtY\_\{t\}=x\_\{t\}andYs=xsY\_\{s\}=x\_\{s\}\. Then, we have

ℙ\[t,s\]\|t,sρ2\(⋅\|xt,xs\)=ℙ\[t,s\]\|t,sρ1\(⋅\|xt,xs\)\.\\mathbb\{P\}\_\{\[t,s\]\|t,s\}^\{\\rho\_\{2\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\)=\\mathbb\{P\}\_\{\[t,s\]\|t,s\}^\{\\rho\_\{1\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\)\.\(184\)We denote the above universal conditional path distribution by

ℙ\[t,s\]uni\(⋅\|xt,xs\),\\mathbb\{P\}\_\{\[t,s\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\),which is independent of the terminal densityρ\\rho\.

Assume thatγt:=βt2αt2\\gamma\_\{t\}:=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}is strictly decreasing on the interval under consideration\. For0<t<u<s≤10<t<u<s\\leq 1, the universal bridge conditioned onYt=xtY\_\{t\}=x\_\{t\}andYs=xsY\_\{s\}=x\_\{s\}satisfies

Yu\|\(Yt=xt,Ys=xs\)∼𝒩⁡\(mu\|t,s​\(xt,xs\),Σu\|t,s\),Y\_\{u\}\|\(Y\_\{t\}=x\_\{t\},Y\_\{s\}=x\_\{s\}\)\\sim\\mathcal\{N\}\\left\(m\_\{u\|t,s\}\(x\_\{t\},x\_\{s\}\),\\Sigma\_\{u\|t,s\}\\right\),where

mu\|t,s​\(xt,xs\)\\displaystyle m\_\{u\|t,s\}\(x\_\{t\},x\_\{s\}\):=αu​\[γu−γsγt−γs​xtαt\+γt−γuγt−γs​xsαs\],Σu\|t,s:=αu2​\(γt−γu\)​\(γu−γs\)γt−γs​I\.\\displaystyle:=\\alpha\_\{u\}\\left\[\\frac\{\\gamma\_\{u\}\-\\gamma\_\{s\}\}\{\\gamma\_\{t\}\-\\gamma\_\{s\}\}\\frac\{x\_\{t\}\}\{\\alpha\_\{t\}\}\+\\frac\{\\gamma\_\{t\}\-\\gamma\_\{u\}\}\{\\gamma\_\{t\}\-\\gamma\_\{s\}\}\\frac\{x\_\{s\}\}\{\\alpha\_\{s\}\}\\right\],\\qquad\\Sigma\_\{u\|t,s\}:=\\alpha\_\{u\}^\{2\}\\frac\{\(\\gamma\_\{t\}\-\\gamma\_\{u\}\)\(\\gamma\_\{u\}\-\\gamma\_\{s\}\)\}\{\\gamma\_\{t\}\-\\gamma\_\{s\}\}I\.

###### Proof\.

The Radon–Nikodym derivative in \([183](https://arxiv.org/html/2609.05727#A4.E183)\) is constant after conditioning on both endpoints\. It therefore cancels in Bayes’ rule for every bounded path functional, which proves \([184](https://arxiv.org/html/2609.05727#A4.E184)\)\.

Consider the bridge under the following linear reference process:

d​Yr=α˙rαr​Yr​d​r\+2​κr​d​Wr\.\\mathrm\{d\}Y\_\{r\}=\\frac\{\\dot\{\\alpha\}\_\{r\}\}\{\\alpha\_\{r\}\}Y\_\{r\}\\mathrm\{d\}r\+\\sqrt\{2\\kappa\_\{r\}\}\\mathrm\{d\}W\_\{r\}\.SetZr=YrαrZ\_\{r\}=\\frac\{Y\_\{r\}\}\{\\alpha\_\{r\}\}\. Then

d​Zr=2​κrαr​d​Wr,γ˙r=−2​κrαr2\.\\mathrm\{d\}Z\_\{r\}=\\frac\{\\sqrt\{2\\kappa\_\{r\}\}\}\{\\alpha\_\{r\}\}\\mathrm\{d\}W\_\{r\},\\qquad\\dot\{\\gamma\}\_\{r\}=\-\\frac\{2\\kappa\_\{r\}\}\{\\alpha\_\{r\}^\{2\}\}\.ThusZrZ\_\{r\}is a Brownian motion in the decreasing clockγr\\gamma\_\{r\}\. The result is the standard Brownian\-bridge conditional distribution in that clock, multiplied byαu\\alpha\_\{u\}\. ∎

According to Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102), once an endpoint pair\(xt,xs\)\(x\_\{t\},x\_\{s\}\)has been obtained—either from a rollout of the posterior\-preserving SDE or from the forward construction—the same endpoints can be reused to sample arbitrarily many conditionally independent bridge trajectories𝒀\[t,s\]=\(Yu\)u∈\[t,s\]\\bm\{Y\}\_\{\[t,s\]\}=\(Y\_\{u\}\)\_\{u\\in\[t,s\]\}without rerunning the model\-dependent SDE\.

The universal bridge provides the state path, whereas the stochastic\-integral terms in the SDE\-based estimation of the covariance form are expressed in terms of the Brownian motion driving the posterior\-preserving SDE before endpoint conditioning\. Since the diffusion coefficient is state\-independent, the required noise increments can be reconstructed from a bridge𝒀\[t,s\]=\(Yu\)u∈\[t,s\]\\bm\{Y\}\_\{\[t,s\]\}=\(Y\_\{u\}\)\_\{u\\in\[t,s\]\}relative to the driftbρb^\{\\rho\}:

d​W~u:=d​Yu−buρ​\(Yu\)​d​u2​κu,u∈\[t,s\]\.\\mathrm\{d\}\\widetilde\{W\}\_\{u\}:=\\frac\{\\mathrm\{d\}Y\_\{u\}\-b\_\{u\}^\{\\rho\}\(Y\_\{u\}\)\\mathrm\{d\}u\}\{\\sqrt\{2\\kappa\_\{u\}\}\},\\qquad u\\in\[t,s\]\.Under the universal bridge lawℙ\[t,s\]uni\(⋅\|xt,xs\)\\mathbb\{P\}\_\{\[t,s\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{s\}\), the reconstructed processW~\\widetilde\{W\}has the conditional law of the original Brownian driver given the two endpoints\. In general, it is not a Brownian motion under the bridge measure\.

### D\.5Log Density and Log\-Density Ratio

This subsection develops the SDE\-based computational approaches for the log densitylog⁡ρ\\log\\rhoand the log\-density ratiolog⁡ρ2ρ1\\log\\frac\{\\rho\_\{2\}\}\{\\rho\_\{1\}\}, which are used in the exact Newton Matching\.

###### Proposition 103\.

The marginal log\-density ratio satisfies

\(∂t\+ℒtρ1\)loghtρ1,ρ2\(x\)=−1κt‖vtρ2\(x\)−vtρ1\(x\)‖22,\\displaystyle\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\_\{1\}\}\)\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\)=\-\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\rho\_\{2\}\}\(x\)\-v\_\{t\}^\{\\rho\_\{1\}\}\(x\)\\right\\\|\_\{2\}^\{2\},\(185a\)\(∂t\+ℒtρ2\)loghtρ1,ρ2\(x\)=1κt‖vtρ2\(x\)−vtρ1\(x\)‖22\.\\displaystyle\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\_\{2\}\}\)\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\)=\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\rho\_\{2\}\}\(x\)\-v\_\{t\}^\{\\rho\_\{1\}\}\(x\)\\right\\\|\_\{2\}^\{2\}\.\(185b\)

###### Proof\.

The logarithmic chain rule forℒtρ1=btρ1⋅∇\+κtΔ\\mathcal\{L\}\_\{t\}^\{\\rho\_\{1\}\}=b\_\{t\}^\{\\rho\_\{1\}\}\\cdot\\nabla\+\\kappa\_\{t\}\\Delta, together with \([181](https://arxiv.org/html/2609.05727#A4.E181)\), gives

\(∂t\+ℒtρ1\)loghtρ1,ρ2\\displaystyle\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\_\{1\}\}\)\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}=1htρ1,ρ2\(∂t\+ℒtρ1\)htρ1,ρ2−κt‖∇loghtρ1,ρ2‖22=−κt‖∇loghtρ1,ρ2‖22\.\\displaystyle=\\frac\{1\}\{h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\}\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\_\{1\}\}\)h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\-\\kappa\_\{t\}\\left\\\|\\nabla\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\\right\\\|\_\{2\}^\{2\}=\-\\kappa\_\{t\}\\left\\\|\\nabla\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\\right\\\|\_\{2\}^\{2\}\.Then, \([182a](https://arxiv.org/html/2609.05727#A4.E182.1)\) implies \([185a](https://arxiv.org/html/2609.05727#A4.E185.1)\)\. Moreover, substitutingf=log⁡htρ1,ρ2f=\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}into

ℒtρ2f=ℒtρ1f\+2κt∇loghtρ1,ρ2⋅∇f\\mathcal\{L\}\_\{t\}^\{\\rho\_\{2\}\}f=\\mathcal\{L\}\_\{t\}^\{\\rho\_\{1\}\}f\+2\\kappa\_\{t\}\\nabla\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\\cdot\\nabla fand using \([185a](https://arxiv.org/html/2609.05727#A4.E185.1)\) give \([185b](https://arxiv.org/html/2609.05727#A4.E185.2)\)\. ∎

###### Proposition 104\.

Let\(Ysρi\)s∈\[t,1\]\(Y\_\{s\}^\{\\rho\_\{i\}\}\)\_\{s\\in\[t,1\]\}follow theρi\\rho\_\{i\}posterior\-preserving SDE, started fromYtρi=xtY\_\{t\}^\{\\rho\_\{i\}\}=x\_\{t\}and driven byWρiW^\{\\rho\_\{i\}\}, fori∈\{1,2\}i\\in\\\{1,2\\\}\. Along theρ2\\rho\_\{2\}path,

log⁡ρ2​\(Y1ρ2\)ρ1​\(Y1ρ2\)=\\displaystyle\\log\\frac\{\\rho\_\{2\}\(Y\_\{1\}^\{\\rho\_\{2\}\}\)\}\{\\rho\_\{1\}\(Y\_\{1\}^\{\\rho\_\{2\}\}\)\}=log⁡htρ1,ρ2​\(xt\)\+∫t1‖vsρ2​\(Ysρ2\)−vsρ1​\(Ysρ2\)‖22κs​𝑑s\\displaystyle\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)\+\\int\_\{t\}^\{1\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\+∫t12κs\(vsρ2\(Ysρ2\)−vsρ1\(Ysρ2\)\)⊤dWsρ2\.\\displaystyle\+\\int\_\{t\}^\{1\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}^\{\\rho\_\{2\}\}\.\(186\)Along theρ1\\rho\_\{1\}path,

log⁡ρ2​\(Y1ρ1\)ρ1​\(Y1ρ1\)=\\displaystyle\\log\\frac\{\\rho\_\{2\}\(Y\_\{1\}^\{\\rho\_\{1\}\}\)\}\{\\rho\_\{1\}\(Y\_\{1\}^\{\\rho\_\{1\}\}\)\}=log⁡htρ1,ρ2​\(xt\)−∫t1‖vsρ2​\(Ysρ1\)−vsρ1​\(Ysρ1\)‖22κs​𝑑s\\displaystyle\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)\-\\int\_\{t\}^\{1\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\+∫t12κs\(vsρ2\(Ysρ1\)−vsρ1\(Ysρ1\)\)⊤dWsρ1\.\\displaystyle\+\\int\_\{t\}^\{1\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}^\{\\rho\_\{1\}\}\.\(187\)

###### Proof\.

Apply Itô’s formula in Lemma[93](https://arxiv.org/html/2609.05727#Thmtheorem93)toFs​\(x\)=log⁡hsρ1,ρ2​\(x\)F\_\{s\}\(x\)=\\log h\_\{s\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\)\. Along the posterior\-preserving SDE ofρ2\\rho\_\{2\}, \([185b](https://arxiv.org/html/2609.05727#A4.E185.2)\) gives

dloghsρ1,ρ2\(Ysρ2\)=1κs‖vsρ2\(Ysρ2\)−vsρ1\(Ysρ2\)‖22ds\+2​κs∇loghsρ1,ρ2\(Ysρ2\)⊤dWsρ2\.\\displaystyle\\mathrm\{d\}\\log h\_\{s\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)=\\frac\{1\}\{\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}s\+\\sqrt\{2\\kappa\_\{s\}\}\\nabla\\log h\_\{s\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)^\{\\top\}\\mathrm\{d\}W\_\{s\}^\{\\rho\_\{2\}\}\.By \([182a](https://arxiv.org/html/2609.05727#A4.E182.1)\), the integrand in the stochastic term can be written as

2​κs∇loghsρ1,ρ2\(Ysρ2\)=2κs\(vsρ2\(Ysρ2\)−vsρ1\(Ysρ2\)\)\.\\sqrt\{2\\kappa\_\{s\}\}\\nabla\\log h\_\{s\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)=\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{2\}\}\)\\right\)\.Substituting this identity, integrating fromttto11, and usingh1ρ1,ρ2=ρ2ρ1h\_\{1\}^\{\\rho\_\{1\},\\rho\_\{2\}\}=\\frac\{\\rho\_\{2\}\}\{\\rho\_\{1\}\}proves \([186](https://arxiv.org/html/2609.05727#A4.E186)\)\.

Along the posterior\-preserving SDE ofρ1\\rho\_\{1\}, the same calculation uses \([185a](https://arxiv.org/html/2609.05727#A4.E185.1)\) and gives

d​log⁡hsρ1,ρ2​\(Ysρ1\)=\\displaystyle\\mathrm\{d\}\\log h\_\{s\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)=−1κs​‖vsρ2​\(Ysρ1\)−vsρ1​\(Ysρ1\)‖22​d​s\+2κs​\(vsρ2​\(Ysρ1\)−vsρ1​\(Ysρ1\)\)⊤​d​Wsρ1\.\\displaystyle\-\\frac\{1\}\{\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}s\+\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}^\{\\rho\_\{1\}\}\)\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}^\{\\rho\_\{1\}\}\.Integration proves \([187](https://arxiv.org/html/2609.05727#A4.E187)\)\. ∎

###### Proposition 105\.

In the setting of Proposition[104](https://arxiv.org/html/2609.05727#Thmtheorem104), we have

Vtρ2​\[log⁡ρ2ρ1\]​\(xt\)=\\displaystyle V\_\{t\}^\{\\rho\_\{2\}\}\\left\[\\log\\frac\{\\rho\_\{2\}\}\{\\rho\_\{1\}\}\\right\]\(x\_\{t\}\)=loghtρ1,ρ2\(xt\)\+𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ2\(⋅\|xt\)\[∫t1‖vsρ2​\(Ys\)−vsρ1​\(Ys\)‖22κsds\],\\displaystyle\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)\+\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\{\\rho\_\{2\}\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\int\_\{t\}^\{1\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\\right\],\(188\)Vtρ1​\[log⁡ρ2ρ1\]​\(xt\)=\\displaystyle V\_\{t\}^\{\\rho\_\{1\}\}\\left\[\\log\\frac\{\\rho\_\{2\}\}\{\\rho\_\{1\}\}\\right\]\(x\_\{t\}\)=loghtρ1,ρ2\(xt\)−𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ1\(⋅\|xt\)\[∫t1‖vsρ2​\(Ys\)−vsρ1​\(Ys\)‖22κsds\]\.\\displaystyle\\log h\_\{t\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(x\_\{t\}\)\-\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\{\\rho\_\{1\}\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\int\_\{t\}^\{1\}\\frac\{\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\}\{\\kappa\_\{s\}\}\\mathrm\{d\}s\\right\]\.

###### Proof\.

Take conditional expectations in \([186](https://arxiv.org/html/2609.05727#A4.E186)\) and \([187](https://arxiv.org/html/2609.05727#A4.E187)\)\. By the martingale assumptions, the stochastic integrals have conditional\-mean zero\. Theorem[98](https://arxiv.org/html/2609.05727#Thmtheorem98)identifies the endpoint expectations with the corresponding posterior values, which finishes the proof\. ∎

Recall from Definitions[55](https://arxiv.org/html/2609.05727#Thmtheorem55)and[60](https://arxiv.org/html/2609.05727#Thmtheorem60)that every densityρ∈𝒫\\rho\\in\\mathscr\{P\}has finite second moment\. In this class, we can represent the KL divergence between marginal densities by the difference of canonical velocity fields\. Lemma[106](https://arxiv.org/html/2609.05727#Thmtheorem106)and Corollary[107](https://arxiv.org/html/2609.05727#Thmtheorem107)below are stated for the standard flow\-matching setting; in the one\-sided interpolant setting, the conditionlimε↓0KL⁡\(pερ2∥pερ1\)=0\\lim\_\{\\varepsilon\\downarrow 0\}\\operatorname\{KL\}\\left\(p\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\\middle\\\|p\_\{\\varepsilon\}^\{\\rho\_\{1\}\}\\right\)=0in Corollary[107](https://arxiv.org/html/2609.05727#Thmtheorem107)holds automatically\.

###### Lemma 106\.

For everyρ1,ρ2∈𝒫\\rho\_\{1\},\\rho\_\{2\}\\in\\mathscr\{P\}, theW2W\_\{2\}distance is finite

W2​\(ρ1,ρ2\):=infγ∈Π⁡\(ρ1,ρ2\)𝔼\(X,Y\)∼γ​\[‖X−Y‖22\]<∞,W\_\{2\}\(\\rho\_\{1\},\\rho\_\{2\}\):=\\sqrt\{\\inf\_\{\\gamma\\in\\Pi\(\\rho\_\{1\},\\rho\_\{2\}\)\}\\mathbb\{E\}\_\{\(X,Y\)\\sim\\gamma\}\\left\[\\left\\\|X\-Y\\right\\\|\_\{2\}^\{2\}\\right\]\}<\\infty,whereΠ⁡\(ρ1,ρ2\)\\Pi\(\\rho\_\{1\},\\rho\_\{2\}\)is the set of couplings ofρ1\\rho\_\{1\}andρ2\\rho\_\{2\}\.

Moreover, for everyt∈\[0,1\)t\\in\[0,1\), we have

KL⁡\(ptρ1∥ptρ2\)≤αt22​βt2​W2​\(ρ1,ρ2\)2\.\\operatorname\{KL\}\\left\(p\_\{t\}^\{\\rho\_\{1\}\}\\middle\\\|p\_\{t\}^\{\\rho\_\{2\}\}\\right\)\\leq\\frac\{\\alpha\_\{t\}^\{2\}\}\{2\\beta\_\{t\}^\{2\}\}W\_\{2\}\(\\rho\_\{1\},\\rho\_\{2\}\)^\{2\}\.\(189\)

###### Proof\.

Applying the independent couplingρ1​\(x\)​ρ2​\(y\)​d​x​d​y\\rho\_\{1\}\(x\)\\rho\_\{2\}\(y\)\\,\\mathrm\{d\}x\\,\\mathrm\{d\}ygives

W2​\(ρ1,ρ2\)2≤𝔼X∼ρ1,Y∼ρ2​\[‖X−Y‖22\]≤2​𝔼X∼ρ1,Y∼ρ2​\[‖X‖22\+‖Y‖22\]<∞\.\\displaystyle W\_\{2\}\(\\rho\_\{1\},\\rho\_\{2\}\)^\{2\}\\leq\\mathbb\{E\}\_\{X\\sim\\rho\_\{1\},Y\\sim\\rho\_\{2\}\}\\left\[\\left\\\|X\-Y\\right\\\|\_\{2\}^\{2\}\\right\]\\leq 2\\mathbb\{E\}\_\{X\\sim\\rho\_\{1\},Y\\sim\\rho\_\{2\}\}\\left\[\\left\\\|X\\right\\\|\_\{2\}^\{2\}\+\\left\\\|Y\\right\\\|\_\{2\}^\{2\}\\right\]<\\infty\.
For everyγ∈Π⁡\(ρ1,ρ2\)\\gamma\\in\\Pi\(\\rho\_\{1\},\\rho\_\{2\}\)andt∈\[0,1\)t\\in\[0,1\), the joint convexity of relative entropy gives

KL⁡\(ptρ1∥ptρ2\)=\\displaystyle\\operatorname\{KL\}\\left\(p\_\{t\}^\{\\rho\_\{1\}\}\\middle\\\|p\_\{t\}^\{\\rho\_\{2\}\}\\right\)=KL\(𝔼\(X,Y\)∼γ\[pt\|1\(⋅\|X\)\]∥𝔼\(X,Y\)∼γ\[pt\|1\(⋅\|Y\)\]\)\\displaystyle\\operatorname\{KL\}\\left\(\\mathbb\{E\}\_\{\(X,Y\)\\sim\\gamma\}\\left\[p\_\{t\|1\}\(\\cdot\|X\)\\right\]\\middle\\\|\\mathbb\{E\}\_\{\(X,Y\)\\sim\\gamma\}\\left\[p\_\{t\|1\}\(\\cdot\|Y\)\\right\]\\right\)≤\\displaystyle\\leq𝔼\(X,Y\)∼γ\[KL\(pt\|1\(⋅\|X\)∥pt\|1\(⋅\|Y\)\)\]\\displaystyle\\mathbb\{E\}\_\{\(X,Y\)\\sim\\gamma\}\\left\[\\operatorname\{KL\}\\left\(p\_\{t\|1\}\(\\cdot\|X\)\\middle\\\|p\_\{t\|1\}\(\\cdot\|Y\)\\right\)\\right\]=\\displaystyle=𝔼\(X,Y\)∼γ​\[KL⁡\(𝒩⁡\(αt​X,βt2​I\)∥𝒩⁡\(αt​Y,βt2​I\)\)\]\\displaystyle\\mathbb\{E\}\_\{\(X,Y\)\\sim\\gamma\}\\left\[\\operatorname\{KL\}\\left\(\\mathcal\{N\}\(\\alpha\_\{t\}X,\\beta\_\{t\}^\{2\}I\)\\middle\\\|\\mathcal\{N\}\(\\alpha\_\{t\}Y,\\beta\_\{t\}^\{2\}I\)\\right\)\\right\]=\\displaystyle=αt22​βt2​𝔼\(X,Y\)∼γ​\[‖X−Y‖22\]\.\\displaystyle\\frac\{\\alpha\_\{t\}^\{2\}\}\{2\\beta\_\{t\}^\{2\}\}\\mathbb\{E\}\_\{\(X,Y\)\\sim\\gamma\}\\left\[\\left\\\|X\-Y\\right\\\|\_\{2\}^\{2\}\\right\]\.Therefore, \([189](https://arxiv.org/html/2609.05727#A4.E189)\) holds\. ∎

###### Corollary 107\.

For everyρ1,ρ2∈𝒫\\rho\_\{1\},\\rho\_\{2\}\\in\\mathscr\{P\}, we have

KL⁡\(ρ2∥ρ1\)=∫011κs​𝔼Ys∼psρ2​\[‖vsρ2​\(Ys\)−vsρ1​\(Ys\)‖22\]​𝑑s\.\\operatorname\{KL\}\\left\(\\rho\_\{2\}\\middle\\\|\\rho\_\{1\}\\right\)=\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{s\}\}\\mathbb\{E\}\_\{Y\_\{s\}\\sim p\_\{s\}^\{\\rho\_\{2\}\}\}\\left\[\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}s\.\(190\)

###### Proof\.

To derive the endpoint KL identity under a potentially singular schedule, fixε∈\(0,1\)\\varepsilon\\in\(0,1\), apply \([188](https://arxiv.org/html/2609.05727#A4.E188)\) att=εt=\\varepsilon, and average overXε∼pερ2X\_\{\\varepsilon\}\\sim p\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\. Posterior preservation gives

𝔼Xε∼pερ2​\[Vερ2​\[log⁡ρ2ρ1\]​\(Xε\)\]=KL⁡\(ρ2∥ρ1\),\\mathbb\{E\}\_\{X\_\{\\varepsilon\}\\sim p\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\}\\left\[V\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\\left\[\\log\\frac\{\\rho\_\{2\}\}\{\\rho\_\{1\}\}\\right\]\(X\_\{\\varepsilon\}\)\\right\]=\\operatorname\{KL\}\\left\(\\rho\_\{2\}\\middle\\\|\\rho\_\{1\}\\right\),whereas

𝔼Xε∼pερ2​\[log⁡hερ1,ρ2​\(Xε\)\]=KL⁡\(pερ2∥pερ1\)\.\\mathbb\{E\}\_\{X\_\{\\varepsilon\}\\sim p\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\}\\left\[\\log h\_\{\\varepsilon\}^\{\\rho\_\{1\},\\rho\_\{2\}\}\(X\_\{\\varepsilon\}\)\\right\]=\\operatorname\{KL\}\\left\(p\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\\middle\\\|p\_\{\\varepsilon\}^\{\\rho\_\{1\}\}\\right\)\.Using Tonelli’s theorem and the fact that the posterior\-preserving SDE ofρ2\\rho\_\{2\}has marginalpsρ2p\_\{s\}^\{\\rho\_\{2\}\}, we obtain

KL⁡\(ρ2∥ρ1\)=\\displaystyle\\operatorname\{KL\}\\left\(\\rho\_\{2\}\\middle\\\|\\rho\_\{1\}\\right\)=KL⁡\(pερ2∥pερ1\)\+∫ε11κs​𝔼Ys∼psρ2​\[‖vsρ2​\(Ys\)−vsρ1​\(Ys\)‖22\]​ds\.\\displaystyle\\operatorname\{KL\}\\left\(p\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\\middle\\\|p\_\{\\varepsilon\}^\{\\rho\_\{1\}\}\\right\)\+\\int\_\{\\varepsilon\}^\{1\}\\frac\{1\}\{\\kappa\_\{s\}\}\\mathbb\{E\}\_\{Y\_\{s\}\\sim p\_\{s\}^\{\\rho\_\{2\}\}\}\\left\[\\left\\\|v\_\{s\}^\{\\rho\_\{2\}\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\rho\_\{1\}\}\(Y\_\{s\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}s\.By Lemma[106](https://arxiv.org/html/2609.05727#Thmtheorem106),KL⁡\(pερ2∥pερ1\)→0\\operatorname\{KL\}\\left\(p\_\{\\varepsilon\}^\{\\rho\_\{2\}\}\\middle\\\|p\_\{\\varepsilon\}^\{\\rho\_\{1\}\}\\right\)\\to 0asε↓0\\varepsilon\\downarrow 0\. Since the integrand in the energy term is nonnegative, the monotone convergence theorem allows the lower integration limit to pass to zero\. Takingε↓0\\varepsilon\\downarrow 0in the preceding identity therefore proves \([190](https://arxiv.org/html/2609.05727#A4.E190)\)\. If the coefficients extend regularly tot=0t=0, the same argument can be written directly by settingt=0t=0and averaging overX0∼p0X\_\{0\}\\sim p\_\{0\}\. ∎

###### Proposition 108\.

Let\(Ysρ\)s∈\[t,1\]\(Y\_\{s\}^\{\\rho\}\)\_\{s\\in\[t,1\]\}follow the posterior\-preserving SDE with terminal densityρ\\rho, started fromYtρ=xtY\_\{t\}^\{\\rho\}=x\_\{t\}\. Then

log⁡ρ⁡\(Y1ρ\)=\\displaystyle\\log\\rho\(Y\_\{1\}^\{\\rho\}\)=log⁡\(αtd​ptρ​\(xt\)\)\+∫t11κs​‖vsρ​\(Ysρ\)−α˙sαs​Ysρ‖22​𝑑s\\displaystyle\\log\\left\(\\alpha\_\{t\}^\{d\}p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\+\\int\_\{t\}^\{1\}\\frac\{1\}\{\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\}\(Y\_\{s\}^\{\\rho\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}^\{\\rho\}\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}s\+∫t12κs\(vsρ\(Ysρ\)−α˙sαsYsρ\)⊤dWsρ\.\\displaystyle\+\\int\_\{t\}^\{1\}\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\rho\}\(Y\_\{s\}^\{\\rho\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}^\{\\rho\}\\right\)^\{\\top\}\\mathrm\{d\}W\_\{s\}^\{\\rho\}\.\(191\)

###### Proof\.

Apply Itô’s formula toFs​\(x\)=log⁡psρ​\(x\)F\_\{s\}\(x\)=\\log p\_\{s\}^\{\\rho\}\(x\)under the posterior\-preserving SDE ofρ\\rho\. Its generator is

ℒsρf=\(α˙sαsx\+2κs∇logpsρ\)⋅∇f\+κsΔf\.\\mathcal\{L\}\_\{s\}^\{\\rho\}f=\\left\(\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x\+2\\kappa\_\{s\}\\nabla\\log p\_\{s\}^\{\\rho\}\\right\)\\cdot\\nabla f\+\\kappa\_\{s\}\\Delta f\.By \([174](https://arxiv.org/html/2609.05727#A4.E174)\),

∂spsρ\+∇⋅\(psρ​α˙sαs​x\)\+κs​Δ​psρ=0\.\\partial\_\{s\}p\_\{s\}^\{\\rho\}\+\\nabla\\cdot\\left\(p\_\{s\}^\{\\rho\}\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x\\right\)\+\\kappa\_\{s\}\\Delta p\_\{s\}^\{\\rho\}=0\.Dividing bypsρp\_\{s\}^\{\\rho\}and using

Δ​psρpsρ=Δ​log⁡psρ\+‖∇log⁡psρ‖22\\frac\{\\Delta p\_\{s\}^\{\\rho\}\}\{p\_\{s\}^\{\\rho\}\}=\\Delta\\log p\_\{s\}^\{\\rho\}\+\\left\\\|\\nabla\\log p\_\{s\}^\{\\rho\}\\right\\\|\_\{2\}^\{2\}gives

\(∂s\+ℒsρ\)logpsρ\(x\)=−dα˙sαs\+κs‖∇logpsρ\(x\)‖22\.\(\\partial\_\{s\}\+\\mathcal\{L\}\_\{s\}^\{\\rho\}\)\\log p\_\{s\}^\{\\rho\}\(x\)=\-d\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}\+\\kappa\_\{s\}\\left\\\|\\nabla\\log p\_\{s\}^\{\\rho\}\(x\)\\right\\\|\_\{2\}^\{2\}\.Sincep1ρ=ρp\_\{1\}^\{\\rho\}=\\rho, Itô’s formula fromttto11yields the score\-form identity

log⁡ρ⁡\(Y1ρ\)=\\displaystyle\\log\\rho\(Y\_\{1\}^\{\\rho\}\)=logptρ\(xt\)\+∫t1\(κs‖∇logpsρ\(Ysρ\)‖22−dα˙sαs\)ds\+∫t12​κs∇logpsρ\(Ysρ\)⊤dWsρ\.\\displaystyle\\log p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\int\_\{t\}^\{1\}\\left\(\\kappa\_\{s\}\\left\\\|\\nabla\\log p\_\{s\}^\{\\rho\}\(Y\_\{s\}^\{\\rho\}\)\\right\\\|\_\{2\}^\{2\}\-d\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}\\right\)\\mathrm\{d\}s\+\\int\_\{t\}^\{1\}\\sqrt\{2\\kappa\_\{s\}\}\\nabla\\log p\_\{s\}^\{\\rho\}\(Y\_\{s\}^\{\\rho\}\)^\{\\top\}\\mathrm\{d\}W\_\{s\}^\{\\rho\}\.Finally,vsρ\(x\)−α˙sαsx=κs∇logpsρ\(x\)v\_\{s\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x=\\kappa\_\{s\}\\nabla\\log p\_\{s\}^\{\\rho\}\(x\)and∫t1−dα˙sαsds=dlogαt\\int\_\{t\}^\{1\}\-d\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}\\mathrm\{d\}s=d\\log\\alpha\_\{t\}give \([191](https://arxiv.org/html/2609.05727#A4.E191)\)\. ∎

###### Proposition 109\.

In the setting of Proposition[108](https://arxiv.org/html/2609.05727#Thmtheorem108),

Vtρ​\[log⁡ρ\]​\(xt\)=\\displaystyle V\_\{t\}^\{\\rho\}\[\\log\\rho\]\(x\_\{t\}\)=log\(αtdptρ\(xt\)\)\+𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[∫t11κs‖vsρ\(Ys\)−α˙sαsYs‖22ds\]\.\\displaystyle\\log\\left\(\\alpha\_\{t\}^\{d\}p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\+\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\int\_\{t\}^\{1\}\\frac\{1\}\{\\kappa\_\{s\}\}\\left\\\|v\_\{s\}^\{\\rho\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}s\\right\]\.\(192\)

###### Proof\.

Take conditional expectation in \([191](https://arxiv.org/html/2609.05727#A4.E191)\)\. The stochastic integral has conditional\-mean zero by the standing assumptions, and Theorem[98](https://arxiv.org/html/2609.05727#Thmtheorem98)identifies

𝔼Y1∼q1\|tρ\(⋅\|xt\)\[logρ\(Y1\)\]=𝔼Y1∼p1\|tρ\(⋅\|xt\)\[logρ\(Y1\)\]=Vtρ\[logρ\]\(xt\),\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\log\\rho\(Y\_\{1\}\)\\right\]=\\mathbb\{E\}\_\{Y\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\log\\rho\(Y\_\{1\}\)\\right\]=V\_\{t\}^\{\\rho\}\[\\log\\rho\]\(x\_\{t\}\),which gives \([192](https://arxiv.org/html/2609.05727#A4.E192)\)\. ∎

### D\.6Initial\-State Sensitivity and Adjoint Calculus

This subsection develops pathwise and endpoint\-conditioned sensitivity for generic SDEs, together with pathwise and conditional\-mean adjoint identities\.

#### D\.6\.1Pathwise Adjoint

We first develop the sensitivity calculus for a generic SDE\[[52](https://arxiv.org/html/2609.05727#bib.bib57)\]\. The pathwise state\-transition matrix propagates infinitesimal perturbations of the initial state along a realized path\. The transposed state\-transition matrices pull the state derivatives of the terminal and running objectives back to the initial time and aggregate their contributions; the same quantity can equivalently be computed by a backward adjoint\. These pathwise constructions are well\-defined for any sufficiently regular path and drift; they coincide with actual initial\-state derivatives when the path is generated by the same drift used in the sensitivity equation\.

Fixt∈\[0,1\)t\\in\[0,1\)and an initial statext∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}\. Recall the SDE in \([163](https://arxiv.org/html/2609.05727#A4.E163)\):

d​Ysb,σ​\(t,xt\)=bs​\(Ysb,σ​\(t,xt\)\)​d​s\+σs​d​Ws,s∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)=b\_\{s\}\\left\(Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\\mathrm\{d\}s\+\\sigma\_\{s\}\\mathrm\{d\}W\_\{s\},\\qquad s\\in\[t,1\],Ytb,σ​\(t,xt\)=xt\.\\displaystyle Y\_\{t\}^\{b,\\sigma\}\(t,x\_\{t\}\)=x\_\{t\}\.
###### Definition 110\.

For a continuous path𝐘\[t,1\]=\(Ys\)s∈\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}=\(Y\_\{s\}\)\_\{s\\in\[t,1\]\}, define the pathwise state\-transition matrixGt→sb​\(𝐘\[t,1\]\)∈ℝd×dG\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\in\\mathbb\{R\}^\{d\\times d\}by the forward variational ODE

dd​s​Gt→sb​\(𝒀\[t,1\]\)=∇bs​\(Ys\)​Gt→sb​\(𝒀\[t,1\]\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla b\_\{s\}\(Y\_\{s\}\)G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\),\\qquad s\\in\[t,1\],\(193a\)Gt→tb​\(𝒀\[t,1\]\)=I\.\\displaystyle G\_\{t\\to t\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=I\.\(193b\)

Definition[110](https://arxiv.org/html/2609.05727#Thmtheorem110)does not require that the path be generated by \([163](https://arxiv.org/html/2609.05727#A4.E163)\)\. The matrixGt→sb​\(𝒀\[t,1\]\)G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)is defined for any sufficiently regular path𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}and driftbb\. The following lemma shows that, when the path is generated by \([163](https://arxiv.org/html/2609.05727#A4.E163)\) with the same drift, the pathwise state\-transition matrix is the actual derivative of the SDE path with respect to its initial state\.

###### Lemma 111\.

The following identity holds a\.s\. simultaneously for everys∈\[t,1\]s\\in\[t,1\]:

∇xtYsb,σ​\(t,xt\)=Gt→sb​\(𝒀\[t,1\]b,σ​\(t,xt\)\)\.\\nabla\_\{x\_\{t\}\}Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)=G\_\{t\\to s\}^\{b\}\\left\(\\bm\{Y\}\_\{\[t,1\]\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\.\(194\)

###### Proof\.

The integral form of \([163](https://arxiv.org/html/2609.05727#A4.E163)\) has been given in \([164](https://arxiv.org/html/2609.05727#A4.E164)\), i\.e\.,

Ysb,σ​\(t,xt\)=xt\+∫tsbu​\(Yub,σ​\(t,xt\)\)​𝑑u\+∫tsσu​d​Wu\.Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)=x\_\{t\}\+\\int\_\{t\}^\{s\}b\_\{u\}\\left\(Y\_\{u\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\\mathrm\{d\}u\+\\int\_\{t\}^\{s\}\\sigma\_\{u\}\\mathrm\{d\}W\_\{u\}\.Recall that, when considering initial\-state sensitivities, the same Brownian motion is used\. The stochastic integral∫tsσu​d​Wu\\int\_\{t\}^\{s\}\\sigma\_\{u\}\\mathrm\{d\}W\_\{u\}is independent ofxtx\_\{t\}\. Differentiating \([164](https://arxiv.org/html/2609.05727#A4.E164)\) with respect toxtx\_\{t\}gives

∇xtYsb,σ​\(t,xt\)=I\+∫ts∇bu​\(Yub,σ​\(t,xt\)\)​∇xtYub,σ​\(t,xt\)​𝑑ua\.s\.\\nabla\_\{x\_\{t\}\}Y\_\{s\}^\{b,\\sigma\}\(t,x\_\{t\}\)=I\+\\int\_\{t\}^\{s\}\\nabla b\_\{u\}\\left\(Y\_\{u\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\\nabla\_\{x\_\{t\}\}Y\_\{u\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\mathrm\{d\}u\\quad\\mathrm\{a\.s\.\}This is exactly the integral form of \([193](https://arxiv.org/html/2609.05727#A4.E193)\) along the path𝒀\[t,1\]b,σ​\(t,xt\)\\bm\{Y\}\_\{\[t,1\]\}^\{b,\\sigma\}\(t,x\_\{t\}\), which proves \([194](https://arxiv.org/html/2609.05727#A4.E194)\)\. ∎

Consider the pathwise objective

Jtφ,l​\(𝒀\[t,1\]\):=φ⁡\(Y1\)\+∫t1ls​\(Ys\)​𝑑s,J\_\{t\}^\{\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\):=\\varphi\(Y\_\{1\}\)\+\\int\_\{t\}^\{1\}l\_\{s\}\(Y\_\{s\}\)\\mathrm\{d\}s,\(195\)where the terminal objectiveφ:ℝd→ℝ\\varphi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}and the running objectivels:ℝd→ℝl\_\{s\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}areC1C^\{1\}and satisfy the regularity conditions in Appendix[D\.1\.2](https://arxiv.org/html/2609.05727#A4.SS1.SSS2)\.

When applying the objective \([195](https://arxiv.org/html/2609.05727#A4.E195)\) to SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\), the initial\-state sensitivity of the objectiveJtφ,lJ\_\{t\}^\{\\varphi,l\}can be evaluated by the adjoint\.

###### Lemma 112\.

For a continuous path𝐘\[t,1\]=\(Ys\)s∈\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}=\(Y\_\{s\}\)\_\{s\\in\[t,1\]\}, define the pathwise adjoint by

dd​sλsb,φ,l\(𝒀\[t,1\]\)=−∇bs\(Ys\)⊤λsb,φ,l\(𝒀\[t,1\]\)−∇ls\(Ys\),s∈\[t,1\],\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla b\_\{s\}\(Y\_\{s\}\)^\{\\top\}\\lambda\_\{s\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\-\\nabla l\_\{s\}\(Y\_\{s\}\),\\qquad s\\in\[t,1\],\(196a\)λ1b,φ,l​\(𝒀\[t,1\]\)=∇φ​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla\\varphi\(Y\_\{1\}\)\.\(196b\)Then, we have

λtb,φ,l​\(𝒀\[t,1\]\)=\\displaystyle\\lambda\_\{t\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=Gt→1b\(𝒀\[t,1\]\)⊤∇φ\(Y1\)\+∫t1Gt→sb\(𝒀\[t,1\]\)⊤∇ls\(Ys\)ds\.\\displaystyle G\_\{t\\to 1\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla\\varphi\(Y\_\{1\}\)\+\\int\_\{t\}^\{1\}G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla l\_\{s\}\(Y\_\{s\}\)\\mathrm\{d\}s\.\(197\)If the path is generated by \([163](https://arxiv.org/html/2609.05727#A4.E163)\), i\.e\.,𝐘\[t,1\]=𝐘\[t,1\]b,σ​\(t,xt\)\\bm\{Y\}\_\{\[t,1\]\}=\\bm\{Y\}\_\{\[t,1\]\}^\{b,\\sigma\}\(t,x\_\{t\}\), then the pathwise adjoint is the initial\-state sensitivity of the objectiveJtφ,lJ\_\{t\}^\{\\varphi,l\}:

λtb,φ,l​\(𝒀\[t,1\]b,σ​\(t,xt\)\)\\displaystyle\\lambda\_\{t\}^\{b,\\varphi,l\}\\left\(\\bm\{Y\}\_\{\[t,1\]\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)=∇xtJtφ,l​\(𝒀\[t,1\]b,σ​\(t,xt\)\)\.\\displaystyle=\\nabla\_\{x\_\{t\}\}J\_\{t\}^\{\\varphi,l\}\\left\(\\bm\{Y\}\_\{\[t,1\]\}^\{b,\\sigma\}\(t,x\_\{t\}\)\\right\)\.\(198\)

###### Proof\.

By \([193](https://arxiv.org/html/2609.05727#A4.E193)\) and \([196](https://arxiv.org/html/2609.05727#A4.E196)\), we have

dd​s​\(Gt→sb​\(𝒀\[t,1\]\)⊤​λsb,φ,l​\(𝒀\[t,1\]\)\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\left\(G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\lambda\_\{s\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)=\\displaystyle=\(∇bs\(Ys\)Gt→sb\(𝒀\[t,1\]\)\)⊤λsb,φ,l\(𝒀\[t,1\]\)−Gt→sb\(𝒀\[t,1\]\)⊤\(∇bs\(Ys\)⊤λsb,φ,l\(𝒀\[t,1\]\)\+∇ls\(Ys\)\)\\displaystyle\\left\(\\nabla b\_\{s\}\(Y\_\{s\}\)G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)^\{\\top\}\\lambda\_\{s\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\-G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\left\(\\nabla b\_\{s\}\(Y\_\{s\}\)^\{\\top\}\\lambda\_\{s\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+\\nabla l\_\{s\}\(Y\_\{s\}\)\\right\)=\\displaystyle=−Gt→sb\(𝒀\[t,1\]\)⊤∇ls\(Ys\)\.\\displaystyle\-G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla l\_\{s\}\(Y\_\{s\}\)\.Integrating fromttto11gives

Gt→1b\(𝒀\[t,1\]\)⊤λ1b,φ,l\(𝒀\[t,1\]\)−Gt→tb\(𝒀\[t,1\]\)⊤λtb,φ,l\(𝒀\[t,1\]\)=−∫t1Gt→sb\(𝒀\[t,1\]\)⊤∇ls\(Ys\)ds\.\\displaystyle G\_\{t\\to 1\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\lambda\_\{1\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\-G\_\{t\\to t\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\lambda\_\{t\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\int\_\{t\}^\{1\}G\_\{t\\to s\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla l\_\{s\}\(Y\_\{s\}\)\\mathrm\{d\}s\.Then, \([196b](https://arxiv.org/html/2609.05727#A4.E196.2)\) and \([193b](https://arxiv.org/html/2609.05727#A4.E193.2)\) prove \([197](https://arxiv.org/html/2609.05727#A4.E197)\)\. If the path is generated by \([163](https://arxiv.org/html/2609.05727#A4.E163)\), the chain rule together with \([194](https://arxiv.org/html/2609.05727#A4.E194)\) gives \([198](https://arxiv.org/html/2609.05727#A4.E198)\)\. ∎

We next characterize the conditional mean of a pathwise adjoint under a possibly different path\-generating SDE\.

###### Proposition 113\.

Consider another SDE

d​Ysb^,σ^​\(t,xt\)=b^s​\(Ysb^,σ^​\(t,xt\)\)​d​s\+σ^s​d​Ws,s∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{s\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(t,x\_\{t\}\)=\{\\hat\{b\}\}\_\{s\}\\left\(Y\_\{s\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(t,x\_\{t\}\)\\right\)\\mathrm\{d\}s\+\\hat\{\\sigma\}\_\{s\}\\mathrm\{d\}W\_\{s\},\\qquad s\\in\[t,1\],\(199a\)Ytb^,σ^​\(t,xt\)=xt\.\\displaystyle Y\_\{t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(t,x\_\{t\}\)=x\_\{t\}\.\(199b\)whose conditional path distribution and generator areℙ\[t,1\]\|tb^,σ^\(⋅\|xt\)\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(\\cdot\|x\_\{t\}\)and𝒜tb^,σ^\\mathcal\{A\}\_\{t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}, respectively\. Define the conditional\-mean adjoint by

λ¯t\(xt\):=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb^,σ^\(⋅\|xt\)\[λtb,φ,l\(𝒀\[t,1\]\)\],\\bar\{\\lambda\}\_\{t\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b,\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\],Under the standing regularity and uniqueness assumptions,λ¯t\\bar\{\\lambda\}\_\{t\}is the unique classical solution of

\(∂t\+𝒜tb^,σ^\)λ¯t\(x\)\+∇bt\(x\)⊤λ¯t\(x\)\+∇lt\(x\)=0,t∈\(0,1\),\\displaystyle\\left\(\\partial\_\{t\}\+\\mathcal\{A\}\_\{t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\\right\)\\bar\{\\lambda\}\_\{t\}\(x\)\+\\nabla b\_\{t\}\(x\)^\{\\top\}\\bar\{\\lambda\}\_\{t\}\(x\)\+\\nabla l\_\{t\}\(x\)=0,\\qquad t\\in\(0,1\),λ¯1​\(x\)=∇φ​\(x\)\.\\displaystyle\\bar\{\\lambda\}\_\{1\}\(x\)=\\nabla\\varphi\(x\)\.

###### Proof\.

For simplicity, denoteλt:=λtb,φ,l\\lambda\_\{t\}:=\\lambda\_\{t\}^\{b,\\varphi,l\}\. Then, we have the pathwise identity

λt\(𝒀\[t,1\]\)=λs\(𝒀\[t,1\]\)\+∫ts\(∇bu\(Yu\)⊤λu\(𝒀\[t,1\]\)\+∇lu\(Yu\)\)du\.\\lambda\_\{t\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\lambda\_\{s\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+\\int\_\{t\}^\{s\}\\left\(\\nabla b\_\{u\}\(Y\_\{u\}\)^\{\\top\}\\lambda\_\{u\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+\\nabla l\_\{u\}\(Y\_\{u\}\)\\right\)\\mathrm\{d\}u\.LetYb^,σ^Y^\{\\hat\{b\},\\hat\{\\sigma\}\}be the trajectory of the SDE \([199](https://arxiv.org/html/2609.05727#A4.E199)\)\. By the Markov property ofYb^,σ^Y^\{\\hat\{b\},\\hat\{\\sigma\}\}, taking the expectation gives

λ¯t​\(x\)=\\displaystyle\\bar\{\\lambda\}\_\{t\}\(x\)=𝔼Ys∼qs\|tb^,σ^\(⋅\|x\)\[λ¯s\(Ys\)\]\+𝔼𝒀\[t,s\]∼ℙ\[t,s\]\|tb^,σ^\(⋅\|x\)\[∫ts\(∇bu\(Yu\)⊤λ¯u\(Yu\)\+∇lu\(Yu\)\)du\]\.\\displaystyle\\mathbb\{E\}\_\{Y\_\{s\}\\sim q\_\{s\|t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(\\cdot\|x\)\}\\left\[\\bar\{\\lambda\}\_\{s\}\\left\(Y\_\{s\}\\right\)\\right\]\+\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,s\]\}\\sim\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(\\cdot\|x\)\}\\left\[\\int\_\{t\}^\{s\}\\left\(\\nabla b\_\{u\}\\left\(Y\_\{u\}\\right\)^\{\\top\}\\bar\{\\lambda\}\_\{u\}\\left\(Y\_\{u\}\\right\)\+\\nabla l\_\{u\}\\left\(Y\_\{u\}\\right\)\\right\)\\mathrm\{d\}u\\right\]\.Applying Dynkin’s formula in Lemma[94](https://arxiv.org/html/2609.05727#Thmtheorem94)componentwise to the vector\-valued functionλ¯s\\bar\{\\lambda\}\_\{s\}gives

𝔼Ys∼qs\|tb^,σ^\(⋅\|x\)\[λ¯s\(Ys\)\]=λ¯t\(x\)\+𝔼𝒀\[t,s\]∼ℙ\[t,s\]\|tb^,σ^\(⋅\|x\)\[∫ts\(∂u\+𝒜ub^,σ^\)λ¯u\(Yu\)du\]\.\\mathbb\{E\}\_\{Y\_\{s\}\\sim q\_\{s\|t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(\\cdot\|x\)\}\\left\[\\bar\{\\lambda\}\_\{s\}\\left\(Y\_\{s\}\\right\)\\right\]=\\bar\{\\lambda\}\_\{t\}\(x\)\+\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,s\]\}\\sim\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(\\cdot\|x\)\}\\left\[\\int\_\{t\}^\{s\}\\left\(\\partial\_\{u\}\+\\mathcal\{A\}\_\{u\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\\right\)\\bar\{\\lambda\}\_\{u\}\\left\(Y\_\{u\}\\right\)\\mathrm\{d\}u\\right\]\.Therefore, for every0<t<s<10<t<s<1andx∈ℝdx\\in\\mathbb\{R\}^\{d\}, we have

𝔼𝒀\[t,s\]∼ℙ\[t,s\]\|tb^,σ^\(⋅\|x\)\[∫ts\(\(∂u\+𝒜ub^,σ^\)λ¯u\(Yu\)\+∇bu\(Yu\)⊤λ¯u\(Yu\)\+∇lu\(Yu\)\)du\]=0\.\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,s\]\}\\sim\\mathbb\{P\}\_\{\[t,s\]\|t\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\(\\cdot\|x\)\}\\left\[\\int\_\{t\}^\{s\}\\left\(\\left\(\\partial\_\{u\}\+\\mathcal\{A\}\_\{u\}^\{\\hat\{b\},\\hat\{\\sigma\}\}\\right\)\\bar\{\\lambda\}\_\{u\}\\left\(Y\_\{u\}\\right\)\+\\nabla b\_\{u\}\\left\(Y\_\{u\}\\right\)^\{\\top\}\\bar\{\\lambda\}\_\{u\}\\left\(Y\_\{u\}\\right\)\+\\nabla l\_\{u\}\\left\(Y\_\{u\}\\right\)\\right\)\\mathrm\{d\}u\\right\]=0\.Takings↓ts\\downarrow tfinishes the proof\. ∎

#### D\.6\.2Endpoint\-Conditioned Sensitivity

The preceding sensitivity objects depend on the entire realized path\. For later endpoint\-based construction of sensitivity Stein kernels in Appendix[E\.4](https://arxiv.org/html/2609.05727#A5.SS4), we instead need a deterministic sensitivity operator that depends only on the endpoint pair\(xt,x1\)\(x\_\{t\},x\_\{1\}\)\. We therefore average the terminal state\-transition matrix over the conditional path distribution given the two endpoints\.

###### Definition 114\.

For the SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\), letℙ\[t,1\]\|t,1b,σ\(⋅\|xt,x1\)\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\},x\_\{1\}\)denote the conditional path distribution givenYt=xtY\_\{t\}=x\_\{t\}andY1=x1Y\_\{1\}=x\_\{1\}\. Define the endpoint\-conditioned state\-transition matrix by

K1\|tb,σ\(x1\|xt\):=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|t,1b,σ\(⋅\|xt,x1\)\[Gt→1b\(𝒀\[t,1\]\)\]\.K\_\{1\|t\}^\{b,\\sigma\}\(x\_\{1\}\|x\_\{t\}\):=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[G\_\{t\\to 1\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.

The following proposition converts pathwise sensitivity into an endpoint\-conditioned expectation\. Its terminal\-only specialization is used in Appendix[E\.4](https://arxiv.org/html/2609.05727#A5.SS4)to construct the posterior sensitivity kernel, whereas the generator\-form extension supports the Bolza realization with running objectives in Section[8\.4](https://arxiv.org/html/2609.05727#S8.SS4)\.

###### Proposition 115\.

We have

∇xt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb,σ\(⋅\|xt\)\[Jtφ,l\(𝒀\[t,1\]\)\]=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb,σ\(⋅\|xt\)\[λtb,φ,l\(𝒀\[t,1\]\)\]\.\\displaystyle\\nabla\_\{x\_\{t\}\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[J\_\{t\}^\{\\varphi,l\}\\left\(\\bm\{Y\}\_\{\[t,1\]\}\\right\)\\right\]=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b,\\varphi,l\}\\left\(\\bm\{Y\}\_\{\[t,1\]\}\\right\)\\right\]\.\(200\)Suppose additionally that the running objective takes the form

ls=\(∂s\+𝒜sb,σ\)hs,l\_\{s\}=\(\\partial\_\{s\}\+\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}\)h\_\{s\},where𝒜sb,σ\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}is the generator of \([163](https://arxiv.org/html/2609.05727#A4.E163)\), defined in \([165](https://arxiv.org/html/2609.05727#A4.E165)\), andh:\[t,1\]×ℝd→ℝh:\[t,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}satisfies the regularity assumptions in Appendix[D\.1\.2](https://arxiv.org/html/2609.05727#A4.SS1.SSS2)\. Then,

∇xt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb,σ\(⋅\|xt\)\[Jtφ,l\(𝒀\[t,1\]\)\]=𝔼Y1∼q1\|tb,σ\(⋅\|xt\)\[K1\|tb,σ\(Y1\|xt\)⊤∇\(φ\+h1\)\(Y1\)\]−∇ht\(xt\)\.\\displaystyle\\nabla\_\{x\_\{t\}\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[J\_\{t\}^\{\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]=\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[K\_\{1\|t\}^\{b,\\sigma\}\(Y\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla\(\\varphi\+h\_\{1\}\)\(Y\_\{1\}\)\\right\]\-\\nabla h\_\{t\}\(x\_\{t\}\)\.\(201\)In particular, ifls=0l\_\{s\}=0, then

∇xt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb,σ\(⋅\|xt\)\[φ\(Y1\)\]=𝔼Y1∼q1\|tb,σ\(⋅\|xt\)\[K1\|tb,σ\(Y1\|xt\)⊤∇φ\(Y1\)\]\.\\displaystyle\\nabla\_\{x\_\{t\}\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\varphi\\left\(Y\_\{1\}\\right\)\\right\]=\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[K\_\{1\|t\}^\{b,\\sigma\}\(Y\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla\\varphi\(Y\_\{1\}\)\\right\]\.\(202\)

###### Proof\.

Differentiate under the expectation and apply Lemma[112](https://arxiv.org/html/2609.05727#Thmtheorem112)to obtain \([200](https://arxiv.org/html/2609.05727#A4.E200)\)\. Ifls=\(∂s\+𝒜sb,σ\)hsl\_\{s\}=\(\\partial\_\{s\}\+\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}\)h\_\{s\}, Dynkin’s formula in Lemma[94](https://arxiv.org/html/2609.05727#Thmtheorem94)gives

𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb,σ\(⋅\|xt\)\[∫t1\(∂s\+𝒜sb,σ\)hs\(Ys\)ds\]=𝔼Y1∼q1\|tb,σ\(⋅\|xt\)\[h1\(Y1\)\]−ht\(xt\)\.\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\int\_\{t\}^\{1\}\(\\partial\_\{s\}\+\\mathcal\{A\}\_\{s\}^\{b,\\sigma\}\)h\_\{s\}\(Y\_\{s\}\)\\mathrm\{d\}s\\right\]=\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\[h\_\{1\}\(Y\_\{1\}\)\]\-h\_\{t\}\(x\_\{t\}\)\.Consequently, \([201](https://arxiv.org/html/2609.05727#A4.E201)\) is given by

∇xt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb,σ\(⋅\|xt\)\[Jtφ,l\(𝒀\[t,1\]\)\]\+∇ht\(xt\)\\displaystyle\\nabla\_\{x\_\{t\}\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[J\_\{t\}^\{\\varphi,l\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\+\\nabla h\_\{t\}\(x\_\{t\}\)=\\displaystyle=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tb,σ\(⋅\|xt\)\[Gt→1b\(𝒀\[t,1\]\)⊤∇\(φ\+h1\)\(Y1\)\]\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[G\_\{t\\to 1\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla\(\\varphi\+h\_\{1\}\)\(Y\_\{1\}\)\\right\]=\\displaystyle=𝔼Y1∼q1\|tb,σ\(⋅\|xt\)\[𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|t,1b,σ\(⋅\|xt,Y1\)\[Gt→1b\(𝒀\[t,1\]\)\]⊤∇\(φ\+h1\)\(Y1\)\]\\displaystyle\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\},Y\_\{1\}\)\}\\left\[G\_\{t\\to 1\}^\{b\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]^\{\\top\}\\nabla\(\\varphi\+h\_\{1\}\)\(Y\_\{1\}\)\\right\]=\\displaystyle=𝔼Y1∼q1\|tb,σ\(⋅\|xt\)\[K1\|tb,σ\(Y1\|xt\)⊤∇\(φ\+h1\)\(Y1\)\]\.\\displaystyle\\mathbb\{E\}\_\{Y\_\{1\}\\sim q\_\{1\|t\}^\{b,\\sigma\}\(\\cdot\|x\_\{t\}\)\}\\left\[K\_\{1\|t\}^\{b,\\sigma\}\(Y\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla\(\\varphi\+h\_\{1\}\)\(Y\_\{1\}\)\\right\]\.Takingls=0l\_\{s\}=0andhs=0h\_\{s\}=0gives \([202](https://arxiv.org/html/2609.05727#A4.E202)\)\. ∎

Therefore, we have obtained the endpoint\-conditioned sensitivity for SDE \([163](https://arxiv.org/html/2609.05727#A4.E163)\)\. Appendix[E\.4](https://arxiv.org/html/2609.05727#A5.SS4)specializes this operator to the posterior\-preserving SDE and verifies that a schedule\-dependent rescaling produces an exact posterior Stein kernel, which is called the posterior sensitivity kernel\.

## Appendix EPosterior Stein Kernels

This appendix develops the posterior Stein kernel, which provides the mechanism that converts the covariance\-form target \([45](https://arxiv.org/html/2609.05727#S8.E45)\) to the gradient\-form target \([48](https://arxiv.org/html/2609.05727#S8.E48)\)\. We prove that, for every sufficiently regular terminal observablef∈C1​\(ℝd,ℝ\)f\\in C^\{1\}\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}\), a posterior Stein kernelΛtρ\(⋅\|xt\):ℝd→ℝd×d\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\):\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}yields the covariance–gradient identity

CovX1∼pρ1\|t\(⋅\|xt\)\(vt\|1\(xt\|X1\),f\(X1\)\)=αt​κtβt2𝔼X1∼pρ1\|t\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇f\(X1\)\]\.\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p^\{\\rho\}\_\{1\|t\}\(\\cdot\|x\_\{t\}\)\}\\bigl\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),f\(X\_\{1\}\)\\bigr\)=\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p^\{\\rho\}\_\{1\|t\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]\.For Newton Matching, settingf=r~ρf=\\tilde\{r\}^\{\\rho\}in the covariance–gradient identity yields covariance\-form and gradient\-form sample\-wise targets with the same conditional mean and hence the same population minimizer under squared\-loss regression\. Their difference is a zero\-mean control variate, so any matrix\-weighted affine combination of the two remains population\-exact\. When the required conditional second moments exist, the matrix weight can be chosen to minimize the variance\.

Appendix[E\.1](https://arxiv.org/html/2609.05727#A5.SS1)first introduces the Stein identity for the Langevin Stein operator\. Appendix[E\.2](https://arxiv.org/html/2609.05727#A5.SS2)then uses this identity to construct a family of zero\-mean Stein control variates\. By specializing the construction to the posteriorp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\), Appendix[E\.3](https://arxiv.org/html/2609.05727#A5.SS3)defines posterior Stein kernels and proves the covariance–gradient identity displayed above\. Building on the adjoint calculus in Appendix[D\.6](https://arxiv.org/html/2609.05727#A4.SS6), Appendix[E\.4](https://arxiv.org/html/2609.05727#A5.SS4)constructs the posterior sensitivity kernel and its adjoint\-based evaluation used by exact Newton Matching in Section[8](https://arxiv.org/html/2609.05727#S8)\. Finally, Appendix[E\.5](https://arxiv.org/html/2609.05727#A5.SS5)records the Gaussian Stein kernels and posterior\-moment formulas used by the Gaussian\-kernel approximation in Section[9](https://arxiv.org/html/2609.05727#S9)\.

As in Appendix[D](https://arxiv.org/html/2609.05727#A4), all conclusions in this appendix apply to both the standard flow matching construction and the one\-sided interpolant setting\. The extension of Newton Matching to one\-sided interpolants is discussed in Appendix[H\.2](https://arxiv.org/html/2609.05727#A8.SS2)\.

### E\.1Langevin Stein Identities

In this subsection, we introduce the Langevin Stein operator\[[19](https://arxiv.org/html/2609.05727#bib.bib53),[32](https://arxiv.org/html/2609.05727#bib.bib35)\]and its columnwise extension to matrix\-valued test fields\. The Stein identity provides the zero\-mean property used to construct control variates, while the product rule, combined with the defining relation of a posterior Stein kernel, yields the covariance–gradient identity\.

###### Definition 116\.

Letppbe a positiveC1C^\{1\}density onℝd\\mathbb\{R\}^\{d\}, and letϕ:ℝd→ℝd\\phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}be a continuously differentiable vector field\. The Langevin Stein operator associated withppis

𝒮pϕ:ℝd→ℝ,\(𝒮pϕ\)\(x\):=∇⋅ϕ\(x\)\+ϕ\(x\)⊤∇logp\(x\)\.\\mathcal\{S\}\_\{p\}\\phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\},\\qquad\(\\mathcal\{S\}\_\{p\}\\phi\)\(x\):=\\nabla\\cdot\\phi\(x\)\+\\phi\(x\)^\{\\top\}\\nabla\\log p\(x\)\.
LetΦ:ℝd→ℝd×d\\Phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}beC1C^\{1\}, and writeΦ=\[ϕ1,…,ϕd\]\\Phi=\[\\phi\_\{1\},\\ldots,\\phi\_\{d\}\], where each columnϕj:ℝd→ℝd\\phi\_\{j\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}isC1C^\{1\}\. Define the Langevin Stein operator for the matrix\-valued test fieldΦ\\Phiby applying𝒮p\\mathcal\{S\}\_\{p\}columnwise:

𝒮pΦ:ℝd→ℝd,\(𝒮pΦ\)\(x\):=\[\(𝒮p​ϕ1\)​\(x\)\(𝒮p​ϕd\)​\(x\)\]=∇⋅Φ\(x\)\+Φ\(x\)⊤∇logp\(x\),\\mathcal\{S\}\_\{p\}\\Phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\},\\qquad\(\\mathcal\{S\}\_\{p\}\\Phi\)\(x\):=\\left\[\\begin\{matrix\}\(\\mathcal\{S\}\_\{p\}\\phi\_\{1\}\)\(x\)\\\\ \\vdots\\\\ \(\\mathcal\{S\}\_\{p\}\\phi\_\{d\}\)\(x\)\\end\{matrix\}\\right\]=\\nabla\\cdot\\Phi\(x\)\+\\Phi\(x\)^\{\\top\}\\nabla\\log p\(x\),where∇⋅Φ⁡\(x\)∈ℝd\\nabla\\cdot\\Phi\(x\)\\in\\mathbb\{R\}^\{d\}denotes the columnwise divergence:

\(∇⋅Φ\(x\)\)j=∑i=1d∂Φi​j​\(x\)∂xi,j=1,…,d\.\\bigl\(\\nabla\\cdot\\Phi\(x\)\\bigr\)\_\{j\}=\\sum\_\{i=1\}^\{d\}\\frac\{\\partial\\Phi\_\{ij\}\(x\)\}\{\\partial x\_\{i\}\},\\qquad j=1,\\ldots,d\.

Under the appropriate integrability and boundary conditions, the Langevin Stein operator yields the following zero\-mean property, which is called the Stein identity\.

###### Lemma 117\.

Use the notation in Definition[116](https://arxiv.org/html/2609.05727#Thmtheorem116)\. Assume that

𝔼X∼p​\[\|\(𝒮p​ϕ\)​\(X\)\|\]<∞,\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\left\|\(\\mathcal\{S\}\_\{p\}\\phi\)\(X\)\\right\|\\right\]<\\infty,\(203\)and that the boundary term vanishes:

limR→∞∫∂BRp⁡\(x\)​ϕ​\(x\)⊤​n​\(x\)​𝑑S​\(x\)=0,\\lim\_\{R\\to\\infty\}\\int\_\{\\partial B\_\{R\}\}p\(x\)\\phi\(x\)^\{\\top\}n\(x\)\\mathrm\{d\}S\(x\)=0,\(204\)whereBR:=\{x:‖x‖2≤R\}B\_\{R\}:=\\\{x:\\left\\\|x\\right\\\|\_\{2\}\\leq R\\\}andn⁡\(x\)n\(x\)is the outward unit normal vector on∂BR\\partial B\_\{R\}\. Then

𝔼X∼p​\[\(𝒮p​ϕ\)​\(X\)\]=0\.\\mathbb\{E\}\_\{X\\sim p\}\\left\[\(\\mathcal\{S\}\_\{p\}\\phi\)\(X\)\\right\]=0\.\(205\)Consequently, if every column ofΦ:ℝd→ℝd×d\\Phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}satisfies \([203](https://arxiv.org/html/2609.05727#A5.E203)\) and \([204](https://arxiv.org/html/2609.05727#A5.E204)\), then

𝔼X∼p​\[\(𝒮p​Φ\)​\(X\)\]=0∈ℝd\.\\mathbb\{E\}\_\{X\\sim p\}\\left\[\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\]=0\\in\\mathbb\{R\}^\{d\}\.\(206\)

###### Proof\.

Sinceppis positive and continuously differentiable,

p​\(x\)​\(𝒮p​ϕ\)​\(x\)\\displaystyle p\(x\)\(\\mathcal\{S\}\_\{p\}\\phi\)\(x\)=p\(x\)∇⋅ϕ\(x\)\+ϕ\(x\)⊤∇p\(x\)=∇⋅\(p\(x\)ϕ\(x\)\)\.\\displaystyle=p\(x\)\\nabla\\cdot\\phi\(x\)\+\\phi\(x\)^\{\\top\}\\nabla p\(x\)=\\nabla\\cdot\\left\(p\(x\)\\phi\(x\)\\right\)\.Integrating overBRB\_\{R\}and applying the divergence theorem gives

∫BR\(𝒮p​ϕ\)​\(x\)​p​\(x\)​𝑑x=∫∂BRp⁡\(x\)​ϕ​\(x\)⊤​n​\(x\)​𝑑S​\(x\)\.\\int\_\{B\_\{R\}\}\(\\mathcal\{S\}\_\{p\}\\phi\)\(x\)p\(x\)\\mathrm\{d\}x=\\int\_\{\\partial B\_\{R\}\}p\(x\)\\phi\(x\)^\{\\top\}n\(x\)\\mathrm\{d\}S\(x\)\.By \([203](https://arxiv.org/html/2609.05727#A5.E203)\) and \([204](https://arxiv.org/html/2609.05727#A5.E204)\), takingR→∞R\\to\\inftyproves \([205](https://arxiv.org/html/2609.05727#A5.E205)\)\. Applying \([205](https://arxiv.org/html/2609.05727#A5.E205)\) to every column ofΦ\\Phigives \([206](https://arxiv.org/html/2609.05727#A5.E206)\)\. ∎

In all subsequent applications, the densities and test fields are assumed to satisfy the integrability and boundary conditions in \([203](https://arxiv.org/html/2609.05727#A5.E203)\) and \([204](https://arxiv.org/html/2609.05727#A5.E204)\)\. Under these conditions, the Langevin Stein operator applied to an admissible test field has zero expectation and can therefore serve as a control variate\. We next record the product rule for the Langevin Stein operator\.

###### Lemma 118\.

Letf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}beC1C^\{1\}\. For a vector fieldϕ:ℝd→ℝd\\phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\},

𝒮p\(fϕ\)\(x\)=ϕ\(x\)⊤∇f\(x\)\+f\(x\)\(𝒮pϕ\)\(x\)\.\\mathcal\{S\}\_\{p\}\(f\\phi\)\(x\)=\\phi\(x\)^\{\\top\}\\nabla f\(x\)\+f\(x\)\(\\mathcal\{S\}\_\{p\}\\phi\)\(x\)\.\(207\)For a matrix\-valued test functionΦ:ℝd→ℝd×d\\Phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\},

𝒮p\(fΦ\)\(x\)=Φ\(x\)⊤∇f\(x\)\+f\(x\)\(𝒮pΦ\)\(x\)\.\\mathcal\{S\}\_\{p\}\(f\\Phi\)\(x\)=\\Phi\(x\)^\{\\top\}\\nabla f\(x\)\+f\(x\)\(\\mathcal\{S\}\_\{p\}\\Phi\)\(x\)\.

###### Proof\.

For the vector\-field identity,

𝒮p​\(f​ϕ\)​\(x\)\\displaystyle\\mathcal\{S\}\_\{p\}\(f\\phi\)\(x\)=∇⋅\(f\(x\)ϕ\(x\)\)\+f\(x\)ϕ\(x\)⊤∇logp\(x\)\\displaystyle=\\nabla\\cdot\(f\(x\)\\phi\(x\)\)\+f\(x\)\\phi\(x\)^\{\\top\}\\nabla\\log p\(x\)=ϕ\(x\)⊤∇f\(x\)\+f\(x\)∇⋅ϕ\(x\)\+f\(x\)ϕ\(x\)⊤∇logp\(x\)\\displaystyle=\\phi\(x\)^\{\\top\}\\nabla f\(x\)\+f\(x\)\\nabla\\cdot\\phi\(x\)\+f\(x\)\\phi\(x\)^\{\\top\}\\nabla\\log p\(x\)=ϕ\(x\)⊤∇f\(x\)\+f\(x\)\(𝒮pϕ\)\(x\)\.\\displaystyle=\\phi\(x\)^\{\\top\}\\nabla f\(x\)\+f\(x\)\(\\mathcal\{S\}\_\{p\}\\phi\)\(x\)\.The matrix\-valued identity follows by applying \([207](https://arxiv.org/html/2609.05727#A5.E207)\) to every column ofΦ\\Phi\. ∎

### E\.2Stein Control Variates

Appendix[E\.1](https://arxiv.org/html/2609.05727#A5.SS1)provides a class of zero\-mean control variates \([206](https://arxiv.org/html/2609.05727#A5.E206)\) through the Langevin Stein operator\. In this subsection, we provide a variance\-reduction result: the optimal linear coefficient in the Loewner order\.

###### Proposition 119\.

Consider a densityp∈𝒫p\\in\\mathscr\{P\}and an integrable functionψ:ℝd→ℝd\\psi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}which satisfies

𝔼X∼p​\[‖ψ⁡\(X\)‖2\]<∞\.\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\left\\\|\\psi\(X\)\\right\\\|\_\{2\}\\right\]<\\infty\.LetΦ:ℝd→ℝd×d\\Phi:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}be a regular matrix field whose columns satisfy \([203](https://arxiv.org/html/2609.05727#A5.E203)\) and \([204](https://arxiv.org/html/2609.05727#A5.E204)\)\. Then, for every coefficientΩ∈ℝd×d\\Omega\\in\\mathbb\{R\}^\{d\\times d\}, we have

𝔼X∼p​\[ψ⁡\(X\)\]=𝔼X∼p​\[ψ⁡\(X\)\+Ω⁡\(𝒮p​Φ\)​\(X\)\]\.\\mathbb\{E\}\_\{X\\sim p\}\[\\psi\(X\)\]=\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\psi\(X\)\+\\Omega\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\]\.\(208\)
Assume additionally that

𝔼X∼p​\[‖ψ⁡\(X\)‖22\]<∞,𝔼X∼p​\[‖\(𝒮p​Φ\)​\(X\)‖22\]<∞\.\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\left\\\|\\psi\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]<\\infty,\\qquad\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\left\\\|\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]<\\infty\.Then

Ω⋆=−CovX∼p⁡\(ψ⁡\(X\),\(𝒮p​Φ\)​\(X\)\)​VarX∼p​\(\(𝒮p​Φ\)​\(X\)\)†\\Omega^\{\\star\}=\-\\operatorname\{Cov\}\_\{X\\sim p\}\\left\(\\psi\(X\),\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)\\operatorname\{Var\}\_\{X\\sim p\}\\left\(\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)^\{\\dagger\}minimizes the variance in the Loewner order:

VarX∼p⁡\(ψ⁡\(X\)\+Ω⁡\(𝒮p​Φ\)​\(X\)\)−VarX∼p⁡\(ψ⁡\(X\)\+Ω⋆​\(𝒮p​Φ\)​\(X\)\)∈𝕊\+d\\displaystyle\\operatorname\{Var\}\_\{X\\sim p\}\\left\(\\psi\(X\)\+\\Omega\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)\-\\operatorname\{Var\}\_\{X\\sim p\}\\left\(\\psi\(X\)\+\\Omega^\{\\star\}\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)\\in\\mathbb\{S\}\_\{\+\}^\{d\}\(209\)for everyΩ∈ℝd×d\\Omega\\in\\mathbb\{R\}^\{d\\times d\}\. Here†denotes the Moore–Penrose pseudoinverse\.

###### Proof\.

Equation \([208](https://arxiv.org/html/2609.05727#A5.E208)\) follows directly from Lemma[117](https://arxiv.org/html/2609.05727#Thmtheorem117)\. Denote

Σ:=VarX∼p⁡\(ψ⁡\(X\)\),A:=VarX∼p⁡\(\(𝒮p​Φ\)​\(X\)\),B:=CovX∼p⁡\(ψ⁡\(X\),\(𝒮p​Φ\)​\(X\)\)\.\\Sigma:=\\operatorname\{Var\}\_\{X\\sim p\}\(\\psi\(X\)\),\\qquad A:=\\operatorname\{Var\}\_\{X\\sim p\}\(\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\),\\qquad B:=\\operatorname\{Cov\}\_\{X\\sim p\}\(\\psi\(X\),\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\)\.Then

VarX∼p⁡\(ψ⁡\(X\)\+Ω⁡\(𝒮p​Φ\)​\(X\)\)=Σ\+Ω​B⊤\+B​Ω⊤\+Ω​A​Ω⊤\.\\operatorname\{Var\}\_\{X\\sim p\}\\left\(\\psi\(X\)\+\\Omega\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)=\\Sigma\+\\Omega B^\{\\top\}\+B\\Omega^\{\\top\}\+\\Omega A\\Omega^\{\\top\}\.\(210\)
We next proveB=B​A†​AB=BA^\{\\dagger\}A\. It suffices to showker⁡\(A\)⊆ker⁡\(B\)\\ker\(A\)\\subseteq\\ker\(B\)\. For anyu∈ker⁡\(A\)u\\in\\ker\(A\),

VarX∼p⁡\(u⊤​\(𝒮p​Φ\)​\(X\)\)=u⊤​A​u=0\.\\operatorname\{Var\}\_\{X\\sim p\}\\left\(u^\{\\top\}\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)=u^\{\\top\}Au=0\.Since the Stein control variate has zero mean, we haveu⊤​\(𝒮p​Φ\)​\(X\)=0u^\{\\top\}\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)=0a\.s\. underX∼pX\\sim p\. Therefore,

B​u=CovX∼p⁡\(ψ⁡\(X\),u⊤​\(𝒮p​Φ\)​\(X\)\)=0,Bu=\\operatorname\{Cov\}\_\{X\\sim p\}\\left\(\\psi\(X\),u^\{\\top\}\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)=0,which provesker⁡\(A\)⊆ker⁡\(B\)\\ker\(A\)\\subseteq\\ker\(B\)and hence

B=B​A†​A\.B=BA^\{\\dagger\}A\.\(211\)Combining \([210](https://arxiv.org/html/2609.05727#A5.E210)\) and \([211](https://arxiv.org/html/2609.05727#A5.E211)\) gives

VarX∼p⁡\(ψ⁡\(X\)\+Ω⁡\(𝒮p​Φ\)​\(X\)\)=Σ−B​A†​B⊤\+\(Ω\+B​A†\)​A​\(Ω\+B​A†\)⊤,\\displaystyle\\operatorname\{Var\}\_\{X\\sim p\}\\left\(\\psi\(X\)\+\\Omega\(\\mathcal\{S\}\_\{p\}\\Phi\)\(X\)\\right\)=\\Sigma\-BA^\{\\dagger\}B^\{\\top\}\+\(\\Omega\+BA^\{\\dagger\}\)A\(\\Omega\+BA^\{\\dagger\}\)^\{\\top\},which proves \([209](https://arxiv.org/html/2609.05727#A5.E209)\)\. ∎

Using Proposition[119](https://arxiv.org/html/2609.05727#Thmtheorem119), Appendix[E\.3](https://arxiv.org/html/2609.05727#A5.SS3)applies this construction to the posterior density and combines it with posterior Stein kernels\.

### E\.3Covariance–Gradient Identity of Posterior Stein Kernels

Recall from Proposition[5](https://arxiv.org/html/2609.05727#Thmtheorem5)that, in Newton Matching, the posterior covariance

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[f\(X1\)\(X1−Mtρ\(xt\)\)\]\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(X\_\{1\}\)\\left\(X\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\\right\]\(212\)is used to represent canonical tangent directions, where

Mtρ\(xt\):=𝔼X1∼p1\|tρ\(⋅\|xt\)\[X1\]M\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[X\_\{1\}\]is the posterior mean\. Note thatvt\|1​\(xt\|X1\)v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)is affine inX1X\_\{1\}\. Using the results in Appendices[E\.1](https://arxiv.org/html/2609.05727#A5.SS1)and[E\.2](https://arxiv.org/html/2609.05727#A5.SS2), we equip the posteriorp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)with a Stein kernel\[[10](https://arxiv.org/html/2609.05727#bib.bib52)\]\. Then, the resulting posterior Stein kernel transforms \([212](https://arxiv.org/html/2609.05727#A5.E212)\) into the gradient form

𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇f\(X1\)\]\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]\.
###### Definition 120\.

Consider a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}\. For anyt∈\(0,1\)t\\in\(0,1\)andxt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}, a matrix\-valuedC1C^\{1\}functionΛtρ\(⋅\|xt\):ℝd→ℝd×d\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\):\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}is called a posterior Stein kernel forp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)if its columns satisfy \([203](https://arxiv.org/html/2609.05727#A5.E203)\)–\([204](https://arxiv.org/html/2609.05727#A5.E204)\) and for anyx1∈ℝdx\_\{1\}\\in\\mathbb\{R\}^\{d\},

\(𝒮p1\|tρ\(⋅\|xt\)Λtρ\(⋅\|xt\)\)\(x1\)=−\(x1−Mtρ\(xt\)\)\.\\left\(\\mathcal\{S\}\_\{p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\right\)\(x\_\{1\}\)=\-\\left\(x\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.\(213\)Equivalently,

∇x1⋅\(p1\|tρ​\(x1\|xt\)​Λtρ​\(x1\|xt\)\)=−p1\|tρ​\(x1\|xt\)​\(x1−Mtρ​\(xt\)\)\.\\nabla\_\{x\_\{1\}\}\\cdot\\left\(p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\)=\-p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\left\(x\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.\(214\)

Note that \([214](https://arxiv.org/html/2609.05727#A5.E214)\) need not determine a unique matrix field\. Nevertheless, every admissible solution yields the same covariance–gradient identity below\. Appendix[E\.4](https://arxiv.org/html/2609.05727#A5.SS4)later constructs a distinguished solution by specializing the endpoint\-conditioned state\-transition matrix developed in Appendix[D\.6](https://arxiv.org/html/2609.05727#A4.SS6)to the posterior\-preserving SDE\.

###### Theorem 121\.

LetΛtρ\(⋅\|xt\)\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)be a posterior Stein kernel forp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\), and letf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}beC1C^\{1\}\. Assume that every column offΛtρ\(⋅\|xt\)f\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)satisfies \([203](https://arxiv.org/html/2609.05727#A5.E203)\)–\([204](https://arxiv.org/html/2609.05727#A5.E204)\)\. Then, we have the covariance–gradient identity

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇f\(X1\)\]\.\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]\.\(215\)

###### Proof\.

The Stein identity in Lemma[117](https://arxiv.org/html/2609.05727#Thmtheorem117)gives

𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(𝒮p1\|tρ\(⋅\|xt\)\(fΛtρ\(⋅\|xt\)\)\)\(X1\)\]=0\.\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\left\(\\mathcal\{S\}\_\{p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(f\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\right\)\\right\)\(X\_\{1\}\)\\right\]=0\.The product rule in Lemma[118](https://arxiv.org/html/2609.05727#Thmtheorem118)implies

𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇f\(X1\)\]=−𝔼X1∼p1\|tρ\(⋅\|xt\)\[f\(X1\)\(𝒮p1\|tρ\(⋅\|xt\)Λtρ\(⋅\|xt\)\)\(X1\)\]\.\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]=\-\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(X\_\{1\}\)\\left\(\\mathcal\{S\}\_\{p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\right\)\(X\_\{1\}\)\\right\]\.SinceΛtρ\(⋅\|xt\)\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)is a posterior Stein kernel, \([213](https://arxiv.org/html/2609.05727#A5.E213)\) implies \([215](https://arxiv.org/html/2609.05727#A5.E215)\)\. ∎

For a posterior Stein kernel, the product rule gives an exact affine family interpolating between the covariance and gradient forms\. Specifically, for every matrix fieldΩt​\(xt\)∈ℝd×d\\Omega\_\{t\}\(x\_\{t\}\)\\in\\mathbb\{R\}^\{d\\times d\}, we have

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)\\displaystyle\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[f\(X1\)\(X1−Mtρ\(xt\)\)\+Ωt\(xt\)\(𝒮p1\|tρ\(⋅\|xt\)\(fΛtρ\(⋅\|xt\)\)\)\(X1\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(X\_\{1\}\)\\left\(X\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\+\\Omega\_\{t\}\(x\_\{t\}\)\\left\(\\mathcal\{S\}\_\{p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(f\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\right\)\\right\)\(X\_\{1\}\)\\right\]=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(I−Ωt\(xt\)\)f\(X1\)\(X1−Mtρ\(xt\)\)\+Ωt\(xt\)Λtρ\(X1\|xt\)⊤∇f\(X1\)\],\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\left\(I\-\\Omega\_\{t\}\(x\_\{t\}\)\\right\)f\(X\_\{1\}\)\\left\(X\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\+\\Omega\_\{t\}\(x\_\{t\}\)\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\],which is a specialization of \([208](https://arxiv.org/html/2609.05727#A5.E208)\)\. Thus, the posterior covariance can be estimated by any affine combination of the covariance and gradient representations\. Proposition[119](https://arxiv.org/html/2609.05727#Thmtheorem119)gives the corresponding optimal linear coefficient whenever the required conditional second moments exist\.

### E\.4Posterior Sensitivity Kernel

This subsection specializes the endpoint\-conditioned sensitivity construction in Appendix[D\.6\.2](https://arxiv.org/html/2609.05727#A4.SS6.SSS2)to the posterior\-preserving SDE\. Rescaling the endpoint\-conditioned state\-transition matrix in Definition[114](https://arxiv.org/html/2609.05727#Thmtheorem114)byβt2/αt\\beta\_\{t\}^\{2\}/\\alpha\_\{t\}yields the posterior sensitivity kernel, which satisfies the defining equation of a posterior Stein kernel\. Its action on an endpoint gradient can be evaluated via the backward adjoint ODE\.

For a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}, recall the posterior\-preserving SDE from Definition[97](https://arxiv.org/html/2609.05727#Thmtheorem97):

d​Ysρ​\(t,xt\)=bsρ​\(Ysρ​\(t,xt\)\)​d​s\+2​κs​d​Ws,s∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{s\}^\{\\rho\}\(t,x\_\{t\}\)=b\_\{s\}^\{\\rho\}\\left\(Y\_\{s\}^\{\\rho\}\(t,x\_\{t\}\)\\right\)\\mathrm\{d\}s\+\\sqrt\{2\\kappa\_\{s\}\}\\mathrm\{d\}W\_\{s\},\\qquad s\\in\[t,1\],\(216a\)Ytρ​\(t,xt\)=xt,\\displaystyle Y\_\{t\}^\{\\rho\}\(t,x\_\{t\}\)=x\_\{t\},\(216b\)where

bsρ​\(x\)=2​vsρ​\(x\)−α˙sαs​x,vρ=𝒞⁡\(ρ\)∈𝒱can\.b\_\{s\}^\{\\rho\}\(x\)=2v\_\{s\}^\{\\rho\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}x,\\qquad v^\{\\rho\}=\\mathcal\{C\}\(\\rho\)\\in\\mathscr\{V\}\_\{\\mathrm\{can\}\}\.For brevity, writeℙρ:=ℙbρ,2​κ\\mathbb\{P\}^\{\\rho\}:=\\mathbb\{P\}^\{b^\{\\rho\},\\sqrt\{2\\kappa\}\}andK1\|tρ:=K1\|tbρ,2​κK\_\{1\|t\}^\{\\rho\}:=K\_\{1\|t\}^\{b^\{\\rho\},\\sqrt\{2\\kappa\}\}for the corresponding path distribution and endpoint\-conditioned state\-transition matrix\.

###### Theorem 122\.

Define

Λtρ​\(x1\|xt\):=βt2αt​K1\|tρ​\(x1\|xt\)\.\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\):=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\.\(217\)ThenΛtρ\(⋅\|xt\)\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)is a posterior Stein kernel forp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\), which we call the posterior sensitivity kernel\. In particular, for every regular test functionf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}such that the columns offΛtρ\(⋅\|xt\)f\\Lambda\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)satisfy the integrability and boundary conditions in \([203](https://arxiv.org/html/2609.05727#A5.E203)\)–\([204](https://arxiv.org/html/2609.05727#A5.E204)\), we have

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇f\(X1\)\]\.\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]\.\(218\)

###### Proof\.

The required integrability and boundary conditions are included in the standing regularity assumptions\. We therefore focus on verifying the defining equation \([214](https://arxiv.org/html/2609.05727#A5.E214)\)\.

By Theorem[98](https://arxiv.org/html/2609.05727#Thmtheorem98), the terminal transition density of \([216](https://arxiv.org/html/2609.05727#A5.E216)\) is the posteriorp1\|tρ\(⋅\|xt\)p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\. Therefore, for every compactly supportedC1C^\{1\}terminal test functionff,

𝔼X1∼p1\|tρ\(⋅\|xt\)\[f\(X1\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[f\(X\_\{1\}\)\]=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[Jtf,0\(𝒀\[t,1\]\)\]\.\\displaystyle=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[J\_\{t\}^\{f,0\}\\left\(\\bm\{Y\}\_\{\[t,1\]\}\\right\)\\right\]\.Taking the gradient with respect toxtx\_\{t\}and applying Proposition[115](https://arxiv.org/html/2609.05727#Thmtheorem115)gives

∫ℝdf⁡\(x1\)​∇xtp1\|tρ​\(x1\|xt\)​d​x1=\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\_\{1\}\)\\nabla\_\{x\_\{t\}\}p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\mathrm\{d\}x\_\{1\}=𝔼X1∼p1\|tρ\(⋅\|xt\)\[K1\|tρ\(X1\|xt\)⊤∇f\(X1\)\]\\displaystyle\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[K\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]=\\displaystyle=∫ℝdK1\|tρ\(x1\|xt\)⊤∇f\(x1\)p1\|tρ\(x1\|xt\)dx1\.\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\mathrm\{d\}x\_\{1\}\.
Ifffis supported in a ballBRB\_\{R\}, then the relevant terms are integrable under the standing regularity assumptions\. In particular,

∫ℝd‖K1\|tρ\(x1\|xt\)⊤∇f\(x1\)p1\|tρ\(x1\|xt\)‖2dx1<∞,\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\\\|K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\\\|\_\{2\}\\mathrm\{d\}x\_\{1\}<\\infty,∫ℝd‖f⁡\(x1\)​∇x1⋅\(p1\|tρ​\(x1\|xt\)​K1\|tρ​\(x1\|xt\)\)‖2​d​x1<∞\.\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\\\|f\(x\_\{1\}\)\\nabla\_\{x\_\{1\}\}\\cdot\\left\(p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\)\\right\\\|\_\{2\}\\mathrm\{d\}x\_\{1\}<\\infty\.Integration by parts therefore gives

∫ℝdf⁡\(x1\)​∇xtp1\|tρ​\(x1\|xt\)​d​x1=\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\_\{1\}\)\\nabla\_\{x\_\{t\}\}p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\mathrm\{d\}x\_\{1\}=∫ℝdK1\|tρ\(x1\|xt\)⊤∇f\(x1\)p1\|tρ\(x1\|xt\)dx1\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\mathrm\{d\}x\_\{1\}=\\displaystyle=−∫ℝdf\(x1\)∇x1⋅\(p1\|tρ\(x1\|xt\)K1\|tρ\(x1\|xt\)\)dx1\.\\displaystyle\-\\int\_\{\\mathbb\{R\}^\{d\}\}f\(x\_\{1\}\)\\nabla\_\{x\_\{1\}\}\\cdot\\left\(p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\)\\mathrm\{d\}x\_\{1\}\.By the arbitrariness of compactly supportedff, we have

∇xtp1\|tρ\(x1\|xt\)=−∇x1⋅\(p1\|tρ\(x1\|xt\)K1\|tρ\(x1\|xt\)\)\.\\nabla\_\{x\_\{t\}\}p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)=\-\\nabla\_\{x\_\{1\}\}\\cdot\\left\(p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)K\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\)\.\(219\)
The posterior score satisfies

∇xt​log​p1\|tρ​\(x1\|xt\)=αtβt2​\(x1−Mtρ​\(xt\)\)\.\\nabla\_\{x\_\{t\}\}\\log p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)=\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\left\(x\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.Multiplying \([219](https://arxiv.org/html/2609.05727#A5.E219)\) byβt2αt\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}and using \([217](https://arxiv.org/html/2609.05727#A5.E217)\) gives

∇x1⋅\(p1\|tρ​\(x1\|xt\)​Λtρ​\(x1\|xt\)\)=−p1\|tρ​\(x1\|xt\)​\(x1−Mtρ​\(xt\)\)\.\\nabla\_\{x\_\{1\}\}\\cdot\\left\(p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\)=\-p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\left\(x\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.This is exactly \([214](https://arxiv.org/html/2609.05727#A5.E214)\)\. Hence,Λtρ\\Lambda\_\{t\}^\{\\rho\}is a posterior Stein kernel by Definition[120](https://arxiv.org/html/2609.05727#Thmtheorem120), and \([218](https://arxiv.org/html/2609.05727#A5.E218)\) follows from Theorem[121](https://arxiv.org/html/2609.05727#Thmtheorem121)\. ∎

In practice, we do not explicitly construct the posterior sensitivity kernelΛtρ\\Lambda\_\{t\}^\{\\rho\}\. Instead, Lemma[112](https://arxiv.org/html/2609.05727#Thmtheorem112)and Proposition[115](https://arxiv.org/html/2609.05727#Thmtheorem115)allow its action on an endpoint gradient to be evaluated through a pathwise adjoint\.

###### Proposition 123\.

For everyt∈\(0,1\)t\\in\(0,1\)andxt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}, we have

Λtρ\(x1\|xt\)⊤∇f\(x1\)=βt2αt𝔼𝒀\[t,1\]∼ℙ\[t,1\]uni\(⋅\|xt,x1\)\[λtbρ,f,0\(𝒀\[t,1\]\)\],\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},f,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\],where the pathwise adjointλtbρ,f,0​\(𝐘\[t,1\]\)\\lambda\_\{t\}^\{b^\{\\rho\},f,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)satisfies the ODE \([196](https://arxiv.org/html/2609.05727#A4.E196)\), and the bridge distributionℙ\[t,1\]uni\(⋅\|xt,x1\)\\mathbb\{P\}\_\{\[t,1\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{1\}\)is defined in Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\.

###### Proof\.

By \([217](https://arxiv.org/html/2609.05727#A5.E217)\) and Definition[114](https://arxiv.org/html/2609.05727#Thmtheorem114), we have

Λtρ\(x1\|xt\)⊤∇f\(x1\)=βt2αt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|t,1ρ\(⋅\|xt,x1\)\[Gt→1bρ\(𝒀\[t,1\]\)⊤∇f\(x1\)\]\.\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{\\rho\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[G\_\{t\\to 1\}^\{b^\{\\rho\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)\\right\]\.Then, \([197](https://arxiv.org/html/2609.05727#A4.E197)\) implies

Λtρ\(x1\|xt\)⊤∇f\(x1\)=βt2αt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|t,1ρ\(⋅\|xt,x1\)\[λtbρ,f,0\(𝒀\[t,1\]\)\]\.\\Lambda\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(x\_\{1\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{\\rho\}\(\\cdot\|x\_\{t\},x\_\{1\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},f,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.Finally, Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)finishes the proof\. ∎

Proposition[123](https://arxiv.org/html/2609.05727#Thmtheorem123)gives

αt​κtβt2𝔼X1∼p1\|tρ\(⋅\|xt\)\[Λtρ\(X1\|xt\)⊤∇f\(X1\)\]\\displaystyle\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\Lambda\_\{t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\nabla f\(X\_\{1\}\)\\right\]=\\displaystyle=κt𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\[λtbρ,f,0\(𝒀\[t,1\]\)\]\\displaystyle\\kappa\_\{t\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},f,0\}\\left\(\\bm\{Y\}\_\{\[t,1\]\}\\right\)\\right\]=\\displaystyle=κt𝔼X1∼p1\|tρ\(⋅\|xt\)\[𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|t,1ρ\(⋅\|xt,X1\)\[λtbρ,f,0\(𝒀\[t,1\]\)\]\]\.\\displaystyle\\kappa\_\{t\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t,1\}^\{\\rho\}\(\\cdot\|x\_\{t\},X\_\{1\}\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},f,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\\right\]\.\(220\)Equation \([220](https://arxiv.org/html/2609.05727#A5.E220)\) admits two exact path\-sampling realizations\. Under the reverse construction, the full path is sampled by simulating the posterior\-preserving SDE fromYt=xtY\_\{t\}=x\_\{t\}\. Under the forward construction, the endpoint pair\(Xt,X1\)\(X\_\{t\},X\_\{1\}\)is generated first and then completed by sampling the universal bridge in Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)\. In either case, the pathwise adjoint is obtained by solving the ODE \([196](https://arxiv.org/html/2609.05727#A4.E196)\) along the sampled path\.

### E\.5Gaussian Stein Kernels and Posterior Moments

The Gaussian\-kernel approximation in Section[9\.2\.2](https://arxiv.org/html/2609.05727#S9.SS2.SSS2)uses the covariance matrix of a Gaussian surrogate in place of the exact posterior sensitivity kernel constructed above\. This subsection first records the Gaussian Stein identity, which identifies this covariance matrix as an exact Stein kernel for the surrogate distribution, and then derives exact formulas for the mean and covariance of the interpolant posterior that can be used to specify the surrogate\.

LetX∼𝒩⁡\(m,Σ\)X\\sim\\mathcal\{N\}\(m,\\Sigma\), wherem∈ℝdm\\in\\mathbb\{R\}^\{d\}andΣ∈𝕊\+⁣\+d\\Sigma\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}\. For every sufficiently regular functionf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}, the Gaussian Stein identity states that

𝔼X∼𝒩⁡\(m,Σ\)​\[\(X−m\)​f​\(X\)\]=Σ​𝔼X∼𝒩⁡\(m,Σ\)​\[∇f​\(X\)\]\.\\mathbb\{E\}\_\{X\\sim\\mathcal\{N\}\(m,\\Sigma\)\}\\left\[\(X\-m\)f\(X\)\\right\]=\\Sigma\\,\\mathbb\{E\}\_\{X\\sim\\mathcal\{N\}\(m,\\Sigma\)\}\\left\[\\nabla f\(X\)\\right\]\.Equivalently, the constant matrix fieldx↦Σx\\mapsto\\Sigmais a Stein kernel for𝒩⁡\(m,Σ\)\\mathcal\{N\}\(m,\\Sigma\)\. This property is specific to Gaussian distributions: for a non\-Gaussian posterior, its covariance matrix is generally not a posterior Stein kernel\.

This observation motivates approximating the posterior by a Gaussian for computational convenience and using the corresponding Stein kernel\. Consider a Gaussian surrogate

p^1\|tρ\(⋅\|xt\):=𝒩\(M^tρ\(xt\),Σ^tρ\(xt\)\),\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\):=\\mathcal\{N\}\\left\(\\widehat\{M\}\_\{t\}^\{\\rho\}\(x\_\{t\}\),\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\),whereM^tρ​\(xt\)∈ℝd\\widehat\{M\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\in\\mathbb\{R\}^\{d\}andΣ^tρ​\(xt\)∈𝕊\+⁣\+d\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}may be supplied by a trained model, estimated via statistical tools, or specified analytically\. Its score is given by

∇x1​log​p^1\|tρ​\(x1\|xt\)=−Σ^tρ​\(xt\)−1​\(x1−M^tρ​\(xt\)\)\.\\nabla\_\{x\_\{1\}\}\\log\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)=\-\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)^\{\-1\}\\left\(x\_\{1\}\-\\widehat\{M\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.Hence, the matrix field

Λ^tρ​\(x1\|xt\):=Σ^tρ​\(xt\)\\widehat\{\\Lambda\}\_\{t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\):=\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)is a Stein kernel for the surrogate posteriorp^1\|tρ\(⋅\|xt\)\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\), since

\(𝒮p^1\|tρ\(⋅\|xt\)Λ^tρ\(⋅\|xt\)\)\(x1\)=∇x1⋅Σ^tρ\(xt\)\+Σ^tρ\(xt\)⊤∇x1logp^1\|tρ\(x1\|xt\)=−\(x1−M^tρ\(xt\)\)\.\\displaystyle\\left\(\\mathcal\{S\}\_\{\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\widehat\{\\Lambda\}\_\{t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\right\)\(x\_\{1\}\)=\\nabla\_\{x\_\{1\}\}\\cdot\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)^\{\\top\}\\nabla\_\{x\_\{1\}\}\\log\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)=\-\\left\(x\_\{1\}\-\\widehat\{M\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)\.Therefore, we have

CovX1∼p^1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)=Σ^tρ\(xt\)𝔼X1∼p^1\|tρ\(⋅\|xt\)\[∇f\(X1\)\]\.\\operatorname\{Cov\}\_\{X\_\{1\}\\sim\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)=\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla f\(X\_\{1\}\)\\right\]\.A full plug\-in approximation consequently takes the form

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)≈Σ^tρ\(xt\)𝔼X1∼p^1\|tρ\(⋅\|xt\)\[∇f\(X1\)\]\.\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)\\approx\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim\\hat\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla f\(X\_\{1\}\)\\right\]\.By contrast, the Gaussian\-kernel approximation in Section[9\.2\.2](https://arxiv.org/html/2609.05727#S9.SS2.SSS2)retains the exact posterior density for endpoint sampling and replaces only the exact posterior Stein kernel byΣ^tρ​\(xt\)\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\. It therefore uses

CovX1∼p1\|tρ\(⋅\|xt\)\(X1,f\(X1\)\)≈Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇f\(X1\)\]\.\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},f\(X\_\{1\}\)\\right\)\\approx\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla f\(X\_\{1\}\)\\right\]\.Retaining the exact posterior density is required for the critical\-point consistency\. A full plug\-in approximation introduces an additional source of inexactness and is generally not critical\-point consistent, although it may reduce computational cost and remain useful in practice\.

The exact posterior moments provide natural choices for the parameters of the Gaussian surrogate\. We first consider the mean\. The posterior mean admits the equivalent representations

Mtρ​\(xt\)\\displaystyle M\_\{t\}^\{\\rho\}\(x\_\{t\}\)=xtαt\+βt2αt∇logptρ\(xt\)\\displaystyle=\\frac\{x\_\{t\}\}\{\\alpha\_\{t\}\}\+\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}\\nabla\\log p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\(221a\)=βt2αt​κt​\(vtρ​\(xt\)−β˙tβt​xt\)\.\\displaystyle=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\\right\)\.\(221b\)Accordingly, the surrogate mean can be chosen to match the exact posterior mean:

M^tρ​\(xt\):=Mtρ​\(xt\)=βt2αt​κt​\(vtρ​\(xt\)−β˙tβt​xt\)\.\\widehat\{M\}\_\{t\}^\{\\rho\}\(x\_\{t\}\):=M\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\\kappa\_\{t\}\}\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}x\_\{t\}\\right\)\.For fixed\(t,xt\)\(t,x\_\{t\}\), this choice requires only a single evaluation of the canonical anchorvtρv\_\{t\}^\{\\rho\}\.

The covariance matrix of the exact posterior admits equivalent representations in terms of the Jacobian of the canonical velocity and the Hessian oflog⁡ptρ​\(xt\)\\log p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\. Define

Σtρ\(xt\):=CovX1∼p1\|tρ\(⋅\|xt\)\(X1,X1\)\.\\Sigma\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(X\_\{1\},X\_\{1\}\\right\)\.Then

Σtρ​\(xt\)\\displaystyle\\Sigma\_\{t\}^\{\\rho\}\(x\_\{t\}\)=βt4αt2​κt​\[∇vtρ​\(xt\)−β˙tβt​I\]\\displaystyle=\\frac\{\\beta\_\{t\}^\{4\}\}\{\\alpha\_\{t\}^\{2\}\\kappa\_\{t\}\}\\left\[\\nabla v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}I\\right\]\(222a\)=βt2αt2​I\+βt4αt2​∇2​log⁡ptρ​\(xt\)\.\\displaystyle=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\+\\frac\{\\beta\_\{t\}^\{4\}\}\{\\alpha\_\{t\}^\{2\}\}\\nabla^\{2\}\\log p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\.\(222b\)
To obtain these formulas, we start from the definition of the posterior mean:

Mtρ​\(xt\)=∫ℝdx1​p1\|tρ​\(x1\|xt\)​d​x1\.M\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\int\_\{\\mathbb\{R\}^\{d\}\}x\_\{1\}p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\,\\mathrm\{d\}x\_\{1\}\.Under the standing regularity assumptions, differentiation with respect toxtx\_\{t\}may be interchanged with integration\. Hence,

∇Mtρ​\(xt\)\\displaystyle\\nabla M\_\{t\}^\{\\rho\}\(x\_\{t\}\)=∫ℝdx1​\(∇xtp1\|tρ​\(x1\|xt\)\)⊤​d​x1\\displaystyle=\\int\_\{\\mathbb\{R\}^\{d\}\}x\_\{1\}\\left\(\\nabla\_\{x\_\{t\}\}p\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\)\\right\)^\{\\top\}\\mathrm\{d\}x\_\{1\}=𝔼X1∼p1\|tρ\(⋅\|xt\)\[X1∇xtlogp1\|tρ\(X1\|xt\)⊤\]\\displaystyle=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[X\_\{1\}\\nabla\_\{x\_\{t\}\}\\log p\_\{1\|t\}^\{\\rho\}\(X\_\{1\}\|x\_\{t\}\)^\{\\top\}\\right\]=αtβt2𝔼X1∼p1\|tρ\(⋅\|xt\)\[X1\(X1−Mtρ\(xt\)\)⊤\]\\displaystyle=\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[X\_\{1\}\\left\(X\_\{1\}\-M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\right\)^\{\\top\}\\right\]=αtβt2​Σtρ​\(xt\)\.\\displaystyle=\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\Sigma\_\{t\}^\{\\rho\}\(x\_\{t\}\)\.Therefore,

Σtρ\(xt\)=βt2αt∇Mtρ\(xt\)\.\\Sigma\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}\}\\nabla M\_\{t\}^\{\\rho\}\(x\_\{t\}\)\.Differentiating the two representations ofMtρM\_\{t\}^\{\\rho\}in \([221](https://arxiv.org/html/2609.05727#A5.E221)\) with respect toxtx\_\{t\}and substituting them into this identity gives \([222a](https://arxiv.org/html/2609.05727#A5.E222.1)\) and \([222b](https://arxiv.org/html/2609.05727#A5.E222.2)\)\.

A simple isotropic surrogate is obtained by omitting the Hessian correction in \([222b](https://arxiv.org/html/2609.05727#A5.E222.2)\):

Σ^tρ​\(xt\)=βt2αt2​I\.\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\.\(223\)More refined surrogates may incorporate estimates based on either representation in \([222](https://arxiv.org/html/2609.05727#A5.E222)\)\. For the Gaussian\-kernel approximation in Section[9\.2\.2](https://arxiv.org/html/2609.05727#S9.SS2.SSS2), however, any choice that is positive definite preserves critical\-point consistency\.

## Appendix FProofs for Approximate Newton Matching

This appendix provides the proofs of the results in Section[9](https://arxiv.org/html/2609.05727#S9)\. We use the notation, targets, and assumptions introduced there and follow the same organization\. Appendix[F\.1](https://arxiv.org/html/2609.05727#A6.SS1)characterizes the population\-stationary points of the two covariance approximations in Section[9\.1](https://arxiv.org/html/2609.05727#S9.SS1)and establishes their critical\-point consistency\. Appendix[F\.2](https://arxiv.org/html/2609.05727#A6.SS2)proves the critical\-point consistency for the reference\-adjoint and Gaussian\-kernel approximations in Section[9\.2](https://arxiv.org/html/2609.05727#S9.SS2)\. Finally, Appendix[F\.3](https://arxiv.org/html/2609.05727#A6.SS3)establishes the decomposition underlying approximate regularization and proves the reward\-ascent and limiting results for the unregularized iteration in Section[9\.3](https://arxiv.org/html/2609.05727#S9.SS3)\.

### F\.1Approximate Covariance Forms with Critical\-Point Consistency

See[37](https://arxiv.org/html/2609.05727#Thmtheorem37)

###### Proof\.

Letv⋆v^\{\\star\}be a population\-stationary point of the lossℒρ,η,Bdir​\-​lin\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}in \([112](https://arxiv.org/html/2609.05727#S9.E112)\)\. Proposition[47](https://arxiv.org/html/2609.05727#Thmtheorem47)implies that for a\.e\.ttandxtx\_\{t\},

vt⋆​\(xt\)=\\displaystyle v\_\{t\}^\{\\star\}\(x\_\{t\}\)=𝔼X1∼p1\|tbase\(⋅\|xt\)\[\(1−ητ\)vtρ\(xt\)\+ητvtbase\(xt\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\Big\[\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\+\(eτ⁡\(r⁡\(X1\)−Bt​\(xt\)\)−1\)\(\(1−ητ\)vtρ\(xt\)\+ητvt\|1\(xt\|X1\)−vt⋆\(xt\)\)\]\.\\displaystyle\\qquad\\qquad\+\\left\(e^\{\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\)\}\-1\\right\)\\left\(\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\star\}\(x\_\{t\}\)\\right\)\\Big\]\.Sincevtbase\(xt\)=𝔼X1∼p1\|tbase\(⋅\|xt\)\[vt\|1\(xt\|X1\)\]v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\}\[v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\], we have

0=\\displaystyle 0=𝔼X1∼p1\|tbase\(⋅\|xt\)\[vt⋆\(xt\)−\(\(1−ητ\)vtρ\(xt\)\+ητvt\|1\(xt\|X1\)\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\Big\[v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-\\left\(\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\)−\(eτ⁡\(r⁡\(X1\)−Bt​\(xt\)\)−1\)\(\(1−ητ\)vtρ\(xt\)\+ητvt\|1\(xt\|X1\)−vt⋆\(xt\)\)\]\\displaystyle\\qquad\\qquad\-\\left\(e^\{\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\)\}\-1\\right\)\\left\(\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\star\}\(x\_\{t\}\)\\right\)\\Big\]=\\displaystyle=e−τ​Bt​\(xt\)​𝔼X1∼p1\|tbase\(⋅\|xt\)​\[eτ​r​\(X1\)​\(vt⋆​\(xt\)−\(1−ητ\)​vtρ​\(xt\)−ητ​vt\|1​\(xt\|X1\)\)\]\.\\displaystyle e^\{\-\\tau B\_\{t\}\(x\_\{t\}\)\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\left\[e^\{\\tau r\(X\_\{1\}\)\}\\left\(v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\)\\right\]\.By \([113a](https://arxiv.org/html/2609.05727#S9.E113.1)\), we can rewrite the first\-order stationarity condition as

0=\\displaystyle 0=𝔼X1∼p1\|tπ\(⋅\|xt\)​\[vt⋆​\(xt\)−\(1−ητ\)​vtρ​\(xt\)−ητ​vt\|1​\(xt\|X1\)\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\pi\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\left\[v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]=\\displaystyle=vt⋆​\(xt\)−\(1−ητ\)​vtρ​\(xt\)−ητ​vtπ​\(xt\)\.\\displaystyle v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\pi\}\(x\_\{t\}\)\.Therefore, the unique population\-stationary point is

v⋆=\(1−ητ\)​vρ\+ητ​vπ\.v^\{\\star\}=\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v^\{\\rho\}\+\\frac\{\\eta\}\{\\tau\}v^\{\\pi\}\.For anyη∈\(0,τ\]\\eta\\in\(0,\\tau\], we have

v⋆=vρ⇔vπ=vρ⇔π=ρ,v^\{\\star\}=v^\{\\rho\}\\quad\\Leftrightarrow\\quad v^\{\\pi\}=v^\{\\rho\}\\quad\\Leftrightarrow\\quad\\pi=\\rho,which establishes critical\-point consistency\. ∎

See[38](https://arxiv.org/html/2609.05727#Thmtheorem38)

###### Proof\.

Letv⋆v^\{\\star\}be a population\-stationary point of the lossℒρ,η,Bspl​\-​lin\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{spl\}\\text\{\-\}\\mathrm\{lin\}\}in \([115](https://arxiv.org/html/2609.05727#S9.E115)\)\. Proposition[47](https://arxiv.org/html/2609.05727#Thmtheorem47)implies that for a\.e\.ttandxtx\_\{t\},

vt⋆​\(xt\)=\\displaystyle v\_\{t\}^\{\\star\}\(x\_\{t\}\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[\(1−ητ\)vtρ\(xt\)\+ητvtbase\(xt\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\Big\[\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)−\(e−τ⁡\(r⁡\(X1\)−Bt​\(xt\)\)−1\)\(vtρ\(xt\)−ητ\(vt\|1\(xt\|X1\)−vtbase\(xt\)\)−vt⋆\(xt\)\)\]\.\\displaystyle\-\\left\(e^\{\-\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\)\}\-1\\right\)\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)\-v\_\{t\}^\{\\star\}\(x\_\{t\}\)\\right\)\\Big\]\.Thus,

0=\\displaystyle 0=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt⋆\(xt\)−\(\(1−ητ\)vtρ\(xt\)\+ητvtbase\(xt\)\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\Big\[v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-\\left\(\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)−\(e−τ⁡\(r⁡\(X1\)−Bt​\(xt\)\)−1\)\(vtρ\(xt\)−ητ\(vt\|1\(xt\|X1\)−vtbase\(xt\)\)−vt⋆\(xt\)\)\]\\displaystyle\\qquad\-\\left\(e^\{\-\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\)\}\-1\\right\)\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)\-v\_\{t\}^\{\\star\}\(x\_\{t\}\)\\right\)\\Big\]=\\displaystyle=𝔼X1∼p1\|tρ\(⋅\|xt\)\[vt⋆\(xt\)−\(vtρ\(xt\)−ητvt\|1\(xt\|X1\)\+ητvtbase\(xt\)\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\Big\[v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)−\(e−τ⁡\(r⁡\(X1\)−Bt​\(xt\)\)−1\)\(vtρ\(xt\)−ητ\(vt\|1\(xt\|X1\)−vtbase\(xt\)\)−vt⋆\(xt\)\)\]\\displaystyle\\qquad\-\\left\(e^\{\-\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(x\_\{t\}\)\)\}\-1\\right\)\\left\(v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}\\left\(v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)\-v\_\{t\}^\{\\star\}\(x\_\{t\}\)\\right\)\\Big\]=\\displaystyle=eτ​Bt​\(xt\)​𝔼X1∼p1\|tρ\(⋅\|xt\)​\[e−τ​r​\(X1\)​\(vt⋆​\(xt\)−vtρ​\(xt\)\+ητ​vt\|1​\(xt\|X1\)−ητ​vtbase​\(xt\)\)\]\.\\displaystyle e^\{\\tau B\_\{t\}\(x\_\{t\}\)\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\Big\[e^\{\-\\tau r\(X\_\{1\}\)\}\\left\(v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\)\\Big\]\.By \([113b](https://arxiv.org/html/2609.05727#S9.E113.2)\), we can rewrite the first\-order stationarity condition as

0=\\displaystyle 0=𝔼X1∼p1\|tρ−\(⋅\|xt\)​\[vt⋆​\(xt\)−vtρ​\(xt\)\+ητ​vt\|1​\(xt\|X1\)−ητ​vtbase​\(xt\)\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho^\{\-\}\}\(\\cdot\|x\_\{t\}\)\\end\{subarray\}\}\\left\[v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\\right\]=\\displaystyle=vt⋆​\(xt\)−vtρ​\(xt\)\+ητ​vtρ−​\(xt\)−ητ​vtbase​\(xt\)\.\\displaystyle v\_\{t\}^\{\\star\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\rho^\{\-\}\}\(x\_\{t\}\)\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\.Therefore, the population\-stationary point is

v⋆=vρ\+ητ​\(vbase−vρ−\)\.v^\{\\star\}=v^\{\\rho\}\+\\frac\{\\eta\}\{\\tau\}\\left\(v^\{\\mathrm\{base\}\}\-v^\{\\rho^\{\-\}\}\\right\)\.For anyη∈\(0,τ\]\\eta\\in\(0,\\tau\], we have

v⋆=vρ⇔vbase=vρ−⇔ρbase=ρ−⇔π=ρ,v^\{\\star\}=v^\{\\rho\}\\quad\\Leftrightarrow\\quad v^\{\\mathrm\{base\}\}=v^\{\\rho^\{\-\}\}\\quad\\Leftrightarrow\\quad\\rho^\{\\mathrm\{base\}\}=\\rho^\{\-\}\\quad\\Leftrightarrow\\quad\\pi=\\rho,which establishes critical\-point consistency\. ∎

### F\.2Approximate Gradient Forms with Critical\-Point Consistency

See[39](https://arxiv.org/html/2609.05727#Thmtheorem39)

###### Proof\.

It is straightforward to verify that the unique population minimizer of the training loss \([118](https://arxiv.org/html/2609.05727#S9.E118)\) isv^\\hat\{v\}in \([117](https://arxiv.org/html/2609.05727#S9.E117)\)\. We next prove thatv^ref​\-​adj=vρ\\hat\{v\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}=v^\{\\rho\}if and only ifρ=π\\rho=\\pi\.

Define

λ¯tρ\(x\):=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|x\)\[λtbρ,r,lρ,μ\(𝒀\[t,1\]\)\],λ¯tμ\(x\):=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|x\)\[λtbμ,r,0\(𝒀\[t,1\]\)\]\.\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\(x\):=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\],\\quad\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\(x\):=\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\)\}\\left\[\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.Applying Proposition[113](https://arxiv.org/html/2609.05727#Thmtheorem113)toλ¯tρ\\bar\{\\lambda\}\_\{t\}^\{\\rho\}andλ¯tμ\\bar\{\\lambda\}\_\{t\}^\{\\mu\}gives

\(∂t\+ℒtρ\)λ¯tρ\(x\)\+∇btρ\(x\)⊤λ¯tρ\(x\)\+∇ltρ,μ\(x\)=0,t∈\(0,1\),\\displaystyle\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\}\)\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\(x\)\+\\nabla b\_\{t\}^\{\\rho\}\(x\)^\{\\top\}\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\(x\)\+\\nabla l\_\{t\}^\{\\rho,\\mu\}\(x\)=0,\\qquad t\\in\(0,1\),\(224a\)λ¯1ρ​\(x\)=∇r​\(x\),\\displaystyle\\bar\{\\lambda\}\_\{1\}^\{\\rho\}\(x\)=\\nabla r\(x\),\(224b\)and

\(∂t\+ℒtρ\)λ¯tμ\(x\)\+∇btμ\(x\)⊤λ¯tμ\(x\)=0,t∈\(0,1\),\\displaystyle\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\}\)\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\(x\)\+\\nabla b\_\{t\}^\{\\mu\}\(x\)^\{\\top\}\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\(x\)=0,\\qquad t\\in\(0,1\),\(225a\)λ¯1μ​\(x\)=∇r​\(x\),\\displaystyle\\bar\{\\lambda\}\_\{1\}^\{\\mu\}\(x\)=\\nabla r\(x\),\(225b\)respectively\.

According to Proposition[18](https://arxiv.org/html/2609.05727#Thmtheorem18), we have

ρ=π⇔λ¯tρ≡vtρ−vtμτ​κtfor allt∈\(0,1\)\.\\rho=\\pi\\quad\\Leftrightarrow\\quad\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}\\text\{ for all \}t\\in\(0,1\)\.By \([117](https://arxiv.org/html/2609.05727#S9.E117)\), we have

v^ref​\-​adj=vρ⇔λ¯tμ≡vtρ−vtμτ​κtfor allt∈\(0,1\)\.\\hat\{v\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\}\}=v^\{\\rho\}\\quad\\Leftrightarrow\\quad\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}\\text\{ for all \}t\\in\(0,1\)\.Therefore, we need to prove that

λ¯tρ≡vtρ−vtμτ​κtfor allt∈\(0,1\)⇔λ¯tμ≡vtρ−vtμτ​κtfor allt∈\(0,1\)\.\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}\\text\{ for all \}t\\in\(0,1\)\\quad\\Leftrightarrow\\quad\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}\\text\{ for all \}t\\in\(0,1\)\.
Assume thatλ¯tρ≡vtρ−vtμτ​κt\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}for allt∈\(0,1\)t\\in\(0,1\)\. Substituting this into \([224](https://arxiv.org/html/2609.05727#A6.E224)\) implies that

\(∂t\+ℒtρ\)λ¯tρ\(x\)\+∇btμ\(x\)⊤λ¯tρ\(x\)=0\.\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\}\)\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\(x\)\+\\nabla b\_\{t\}^\{\\mu\}\(x\)^\{\\top\}\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\(x\)=0\.In other words,λ¯tρ\\bar\{\\lambda\}\_\{t\}^\{\\rho\}satisfies the same equation \([225](https://arxiv.org/html/2609.05727#A6.E225)\) asλ¯tμ\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\. Therefore,λ¯tρ≡λ¯tμ\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\\equiv\\bar\{\\lambda\}\_\{t\}^\{\\mu\}for allt∈\(0,1\)t\\in\(0,1\); hence,λ¯tμ≡vtρ−vtμτ​κt\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}for allt∈\(0,1\)t\\in\(0,1\)\.

Assume thatλ¯tμ≡vtρ−vtμτ​κt\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}for allt∈\(0,1\)t\\in\(0,1\)\. Substituting this into \([225](https://arxiv.org/html/2609.05727#A6.E225)\) implies that

\(∂t\+ℒtρ\)λ¯tμ\(x\)\+∇btρ\(x\)⊤λ¯tμ\(x\)\+∇ltρ,μ\(x\)=0\.\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\}\)\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\(x\)\+\\nabla b\_\{t\}^\{\\rho\}\(x\)^\{\\top\}\\bar\{\\lambda\}\_\{t\}^\{\\mu\}\(x\)\+\\nabla l\_\{t\}^\{\\rho,\\mu\}\(x\)=0\.For the same reason,λ¯tρ≡vtρ−vtμτ​κt\\bar\{\\lambda\}\_\{t\}^\{\\rho\}\\equiv\\frac\{v\_\{t\}^\{\\rho\}\-v\_\{t\}^\{\\mu\}\}\{\\tau\\kappa\_\{t\}\}for allt∈\(0,1\)t\\in\(0,1\)in this case\. ∎

See[41](https://arxiv.org/html/2609.05727#Thmtheorem41)

###### Proof\.

It is straightforward to verify that the unique population minimizer of the training loss \([122](https://arxiv.org/html/2609.05727#S9.E122)\) isv^\\hat\{v\}in \([121b](https://arxiv.org/html/2609.05727#S9.E121.2)\)\. SinceΣ^tρ​\(xt\)∈𝕊\+⁣\+d\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\in\\mathbb\{S\}\_\{\+\+\}^\{d\}, we have

v^=vρ⇔𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r~ρ\(X1\)\]=0for allt∈\(0,1\),xt∈ℝd\.\\hat\{v\}=v^\{\\rho\}\\quad\\Leftrightarrow\\quad\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\]=0\\text\{ for all \}t\\in\(0,1\),\\,x\_\{t\}\\in\\mathbb\{R\}^\{d\}\.
Assume thatρ=π\\rho=\\pi\. Then,r~ρ≡const\\tilde\{r\}^\{\\rho\}\\equiv\\mathrm\{const\}, which gives𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r~ρ\(X1\)\]=0\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\]=0\.

Assume that𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r~ρ\(X1\)\]=0\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\[\\nabla\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\]=0\. Then, we have

∫ℝdpt\|1\(xt\|x1\)ρ\(x1\)∇r~ρ\(x1\)dx1=0\.\\int\_\{\\mathbb\{R\}^\{d\}\}p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\rho\(x\_\{1\}\)\\nabla\\tilde\{r\}^\{\\rho\}\(x\_\{1\}\)\\mathrm\{d\}x\_\{1\}=0\.For any interior timet∈\(0,1\)t\\in\(0,1\), the left\-hand side is a nondegenerate scaled Gaussian convolution\. Injectivity gives

ρ\(x\)∇r~ρ\(x\)≡0\.\\rho\(x\)\\nabla\\tilde\{r\}^\{\\rho\}\(x\)\\equiv 0\.Sinceρ\\rhois positive, we haver~ρ≡const\\tilde\{r\}^\{\\rho\}\\equiv\\mathrm\{const\}, which impliesρ=π\\rho=\\pi\. ∎

### F\.3Regularization Trade\-Offs

See[42](https://arxiv.org/html/2609.05727#Thmtheorem42)

###### Proof\.

By Bayes’ rule, for every interior\(t,xt\)\(t,x\_\{t\}\), we have

Vtρ​\[log⁡ρρbase\]​\(xt\)\\displaystyle V\_\{t\}^\{\\rho\}\\left\[\\log\\frac\{\\rho\}\{\\rho^\{\\mathrm\{base\}\}\}\\right\]\(x\_\{t\}\)=logptρ​\(xt\)ptbase​\(xt\)\+KL\(p1\|tρ\(⋅\|xt\)∥p1\|tbase\(⋅\|xt\)\)\.\\displaystyle=\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\}\+\\operatorname\{KL\}\\left\(p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\\middle\\\|p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|x\_\{t\}\)\\right\)\.Then,

Γρ,r~ρ=Γρ,r−1τ​Γρ,log⁡\(ρ/ρbase\),κt​∇xt​log⁡ptρ​\(xt\)ptbase​\(xt\)=vtρ​\(xt\)−vtbase​\(xt\),\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}=\\Gamma^\{\\rho,r\}\-\\frac\{1\}\{\\tau\}\\Gamma^\{\\rho,\\log\(\\rho/\\rho^\{\\mathrm\{base\}\}\)\},\\qquad\\kappa\_\{t\}\\nabla\_\{x\_\{t\}\}\\log\\frac\{p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\{p\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\)\}=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\_\{t\}\),which proves the conclusion\. ∎

See[43](https://arxiv.org/html/2609.05727#Thmtheorem43)

###### Proof\.

The value\-ascent certificate is a specialization of Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12)withf:=rf:=r\. Ifρ=π\\rho=\\pibutrris nonconstant, then the strict value ascent implies thatq≠ρq\\neq\\rho; hence,Γρ,r≠0\\Gamma^\{\\rho,r\}\\neq 0even atρ=π\\rho=\\pi\. Therefore, the approximation is not critical\-point consistent\. ∎

See[44](https://arxiv.org/html/2609.05727#Thmtheorem44)

###### Proof\.

We have

𝒟ρk,ηk,r​\(x\)=ηk2​∫01κt​‖∇Vtρk​\[r\]​\(Φ1→tρk,ηk,r​\(x\)\)‖22​𝑑t≥0\.\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},r\}\(x\)=\\eta\_\{k\}^\{2\}\\int\_\{0\}^\{1\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\_\{k\}\}\[r\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho\_\{k\},\\eta\_\{k\},r\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\geq 0\.DenoteAε:=\{x∈ℝd:r⁡\(x\)<rmax−ε\}A\_\{\\varepsilon\}:=\\\{x\\in\\mathbb\{R\}^\{d\}:r\(x\)<r\_\{\\max\}\-\\varepsilon\\\}\. DenoteSk:=∑i=0k−1𝒟ρi,ηi,rS\_\{k\}:=\\sum\_\{i=0\}^\{k\-1\}\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},r\}\. Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12)implies

ρk​\(x\)∝ρ0​\(x\)​exp⁡\(∑i=0k−1ηi​r​\(x\)−Sk​\(x\)\)\.\\rho\_\{k\}\(x\)\\propto\\rho\_\{0\}\(x\)\\exp\\left\(\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}r\(x\)\-S\_\{k\}\(x\)\\right\)\.For anyδ∈\(0,ε\)\\delta\\in\(0,\\varepsilon\), we have

ρk​\(Aε\)=\\displaystyle\\rho\_\{k\}\(A\_\{\\varepsilon\}\)=∫Aερ0​\(x\)​exp⁡\(∑i=0k−1ηi​r​\(x\)−Sk​\(x\)\)​𝑑x∫ℝdρ0​\(x\)​exp⁡\(∑i=0k−1ηi​r​\(x\)−Sk​\(x\)\)​𝑑x\\displaystyle\\frac\{\\int\_\{A\_\{\\varepsilon\}\}\\rho\_\{0\}\(x\)\\exp\\left\(\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}r\(x\)\-S\_\{k\}\(x\)\\right\)\\mathrm\{d\}x\}\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\_\{0\}\(x\)\\exp\\left\(\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}r\(x\)\-S\_\{k\}\(x\)\\right\)\\mathrm\{d\}x\}≤\\displaystyle\\leq∫Aερ0​\(x\)​exp⁡\(∑i=0k−1ηi​\(rmax−ε\)−Sk​\(x\)\)​𝑑x∫Aδcρ0​\(x\)​exp⁡\(∑i=0k−1ηi​\(rmax−δ\)−Sk​\(x\)\)​𝑑x\\displaystyle\\frac\{\\int\_\{A\_\{\\varepsilon\}\}\\rho\_\{0\}\(x\)\\exp\\left\(\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}\(r\_\{\\max\}\-\\varepsilon\)\-S\_\{k\}\(x\)\\right\)\\mathrm\{d\}x\}\{\\int\_\{A\_\{\\delta\}^\{\\mathrm\{c\}\}\}\\rho\_\{0\}\(x\)\\exp\\left\(\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}\(r\_\{\\max\}\-\\delta\)\-S\_\{k\}\(x\)\\right\)\\mathrm\{d\}x\}≤\\displaystyle\\leqexp⁡\(\(δ−ε\)​∑i=0k−1ηi\)​\(∫Aδcρ0​\(x\)​exp⁡\(−Sk​\(x\)\)​dx\)−1\.\\displaystyle\\exp\\left\(\(\\delta\-\\varepsilon\)\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}\\right\)\\left\(\\int\_\{A\_\{\\delta\}^\{\\mathrm\{c\}\}\}\\rho\_\{0\}\(x\)\\exp\\left\(\-S\_\{k\}\(x\)\\right\)\\mathrm\{d\}x\\right\)^\{\-1\}\.By the definition ofrmaxr\_\{\\max\}and the continuity ofrr, we haveρ0​\(Aδc\)\>0\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\>0\. Letρ0\|Aδc\\rho\_\{0\|A\_\{\\delta\}^\{\\mathrm\{c\}\}\}denote the normalized restriction ofρ0\\rho\_\{0\}toAδcA\_\{\\delta\}^\{\\mathrm\{c\}\}\. By Jensen’s inequality, we have

∫Aδcρ0​\(x\)​exp⁡\(−Sk​\(x\)\)​dx=\\displaystyle\\int\_\{A\_\{\\delta\}^\{\\mathrm\{c\}\}\}\\rho\_\{0\}\(x\)\\exp\\left\(\-S\_\{k\}\(x\)\\right\)\\mathrm\{d\}x=ρ0​\(Aδc\)​𝔼X∼ρ0\|Aδc​\[exp⁡\(−Sk​\(X\)\)\]\\displaystyle\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\|A\_\{\\delta\}^\{\\mathrm\{c\}\}\}\}\\left\[\\exp\\left\(\-S\_\{k\}\(X\)\\right\)\\right\]≥\\displaystyle\\geqρ0​\(Aδc\)​exp⁡\(−𝔼X∼ρ0\|Aδc​\[Sk​\(X\)\]\)\\displaystyle\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\\exp\\left\(\-\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\|A\_\{\\delta\}^\{\\mathrm\{c\}\}\}\}\\left\[S\_\{k\}\(X\)\\right\]\\right\)≥\\displaystyle\\geqρ0​\(Aδc\)​exp⁡\(−𝔼X∼ρ0​\[Sk​\(X\)\]ρ0​\(Aδc\)\)\.\\displaystyle\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\\exp\\left\(\-\\frac\{\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\}\}\\left\[S\_\{k\}\(X\)\\right\]\}\{\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\}\\right\)\.Therefore, we have

ρk​\(Aε\)≤\\displaystyle\\rho\_\{k\}\(A\_\{\\varepsilon\}\)\\leq1ρ0​\(Aδc\)​exp⁡\(\(δ−ε\)​∑i=0k−1ηi\+1ρ0​\(Aδc\)​∑i=0k−1𝔼X∼ρ0​\[𝒟ρi,ηi,r​\(X\)\]\)\\displaystyle\\frac\{1\}\{\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\}\\exp\\left\(\(\\delta\-\\varepsilon\)\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}\+\\frac\{1\}\{\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\}\\sum\_\{i=0\}^\{k\-1\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\}\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},r\}\(X\)\\right\]\\right\)≤\\displaystyle\\leq1ρ0​\(Aδc\)​exp⁡\(\(∑i=0k−1ηi\)​\(∑i=0k−1𝔼X∼ρ0​\[𝒟ρi,ηi,r​\(X\)\]ρ0​\(Aδc\)​\(∑i=0k−1ηi\)\+δ−ε\)\)\.\\displaystyle\\frac\{1\}\{\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\}\\exp\\left\(\\left\(\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}\\right\)\\left\(\\frac\{\\sum\_\{i=0\}^\{k\-1\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\}\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},r\}\(X\)\\right\]\}\{\\rho\_\{0\}\(A\_\{\\delta\}^\{\\mathrm\{c\}\}\)\\left\(\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}\\right\)\}\+\\delta\-\\varepsilon\\right\)\\right\)\.By condition \([130](https://arxiv.org/html/2609.05727#S9.E130)\), the Stolz–Cesàro theorem implies

limk→∞∑i=0k−1𝔼X∼ρ0​\[𝒟ρi,ηi,r​\(X\)\]∑i=0k−1ηi=limk→∞1ηk​𝔼X∼ρ0​\[𝒟ρk,ηk,r​\(X\)\]=0\.\\lim\_\{k\\to\\infty\}\\frac\{\\sum\_\{i=0\}^\{k\-1\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\}\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{i\},\\eta\_\{i\},r\}\(X\)\\right\]\}\{\\sum\_\{i=0\}^\{k\-1\}\\eta\_\{i\}\}=\\lim\_\{k\\to\\infty\}\\frac\{1\}\{\\eta\_\{k\}\}\\mathbb\{E\}\_\{X\\sim\\rho\_\{0\}\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},r\}\(X\)\\right\]=0\.Sinceδ−ε<0\\delta\-\\varepsilon<0is a fixed negative number, we havelimk→∞ρk​\(Aε\)=0\\lim\_\{k\\to\\infty\}\\rho\_\{k\}\(A\_\{\\varepsilon\}\)=0\. ∎

To prove Proposition[45](https://arxiv.org/html/2609.05727#Thmtheorem45), we separate the terminal\-density error into two components: a residual exponential tilt caused by the mismatchτ^−τ\\hat\{\\tau\}\-\\tau, and an accumulated path\-dissipation correction\. The former can be controlled uniformly for bounded rewards through Hoeffding’s lemma\[[26](https://arxiv.org/html/2609.05727#bib.bib59), Equation \(4\.16\)\]\.

###### Lemma 124\(Hoeffding’s lemma\)\.

LetX∼pX\\sim pbe a bounded real\-valued random variable satisfying

a≤X≤ba\.s\.a\\leq X\\leq b\\qquad\\text\{a\.s\.\}Denotem:=𝔼X∼p​\[X\]m:=\\mathbb\{E\}\_\{X\\sim p\}\[X\]\. Then, for anyτ∈ℝ\\tau\\in\\mathbb\{R\},

𝔼X∼p​\[eτ⁡\(X−m\)\]≤exp⁡\(τ2​\(b−a\)28\)\.\\mathbb\{E\}\_\{X\\sim p\}\\left\[e^\{\\tau\(X\-m\)\}\\right\]\\leq\\exp\\left\(\\frac\{\\tau^\{2\}\(b\-a\)^\{2\}\}\{8\}\\right\)\.

###### Proof\.

The case wherea=ba=bis immediate\. Assume thata<ba<b\. For allx∈\[a,b\]x\\in\[a,b\],

eτ​x≤b−xb−a​eτ​a\+x−ab−a​eτ​b\.e^\{\\tau x\}\\leq\\frac\{b\-x\}\{b\-a\}e^\{\\tau a\}\+\\frac\{x\-a\}\{b\-a\}e^\{\\tau b\}\.Taking the expectation overppgives

𝔼X∼p​\[eτ⁡\(X−m\)\]≤b−mb−a​eτ⁡\(a−m\)\+m−ab−a​eτ⁡\(b−m\)=eL⁡\(τ⁡\(b−a\)\),\\mathbb\{E\}\_\{X\\sim p\}\\left\[e^\{\\tau\(X\-m\)\}\\right\]\\leq\\frac\{b\-m\}\{b\-a\}e^\{\\tau\(a\-m\)\}\+\\frac\{m\-a\}\{b\-a\}e^\{\\tau\(b\-m\)\}=e^\{L\(\\tau\(b\-a\)\)\},\(226\)where

L⁡\(y\):=y⁡\(a−m\)b−a\+log⁡\(1−\(ey−1\)​\(a−m\)b−a\)\.L\(y\):=\\frac\{y\(a\-m\)\}\{b\-a\}\+\\log\\left\(1\-\\frac\{\(e^\{y\}\-1\)\(a\-m\)\}\{b\-a\}\\right\)\.Then,

L⁡\(0\)=0,L′​\(0\)=0,L′′​\(y\)=\(m−a\)​\(b−m\)​ey​\(\(m−a\)​ey\+b−m\)−2∈\[0,14\]\.L\(0\)=0,\\qquad L^\{\\prime\}\(0\)=0,\\qquad L^\{\\prime\\prime\}\(y\)=\(m\-a\)\(b\-m\)e^\{y\}\\left\(\(m\-a\)e^\{y\}\+b\-m\\right\)^\{\-2\}\\in\\left\[0,\\frac\{1\}\{4\}\\right\]\.By the Taylor expansion, for everyy∈ℝy\\in\\mathbb\{R\}, there existszzbetween00andyysuch that

L⁡\(y\)=12​L′′​\(z\)​y2≤y28\.L\(y\)=\\frac\{1\}\{2\}L^\{\\prime\\prime\}\(z\)y^\{2\}\\leq\\frac\{y^\{2\}\}\{8\}\.Substituting this back into \([226](https://arxiv.org/html/2609.05727#A6.E226)\) finishes the proof\. ∎

Based on Hoeffding’s lemma, the following lemma bounds the KL divergences in both directions when a probability density is exponentially tilted by a bounded reward\.

###### Lemma 125\.

Letr:ℝd→ℝr:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}be a bounded reward function satisfying

‖r‖∞:=supx∈ℝd\|r⁡\(x\)\|<∞\.\\left\\\|r\\right\\\|\_\{\\infty\}:=\\sup\_\{x\\in\\mathbb\{R\}^\{d\}\}\\left\|r\(x\)\\right\|<\\infty\.Letμ∈𝒫\\mu\\in\\mathscr\{P\}andπ∈𝒫\\pi\\in\\mathscr\{P\}be two probability densities satisfying

π⁡\(x\)∝μ⁡\(x\)​eτ​r​\(x\)\\pi\(x\)\\propto\\mu\(x\)e^\{\\tau r\(x\)\}for some constantτ∈ℝ\\tau\\in\\mathbb\{R\}\. Then,

max⁡\{KL⁡\(π∥μ\),KL⁡\(μ∥π\)\}≤12​τ2​‖r‖∞2\.\\max\\left\\\{\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\mu\\right\),\\operatorname\{KL\}\\left\(\\mu\\middle\\\|\\pi\\right\)\\right\\\}\\leq\\frac\{1\}\{2\}\\tau^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\.

###### Proof\.

Since

π⁡\(x\)=μ⁡\(x\)​eτ​r​\(x\)𝔼X∼μ​\[eτ​r​\(X\)\],−‖r‖∞≤r⁡\(x\)≤‖r‖∞,\\pi\(x\)=\\frac\{\\mu\(x\)e^\{\\tau r\(x\)\}\}\{\\mathbb\{E\}\_\{X\\sim\\mu\}\\left\[e^\{\\tau r\(X\)\}\\right\]\},\\qquad\-\\left\\\|r\\right\\\|\_\{\\infty\}\\leq r\(x\)\\leq\\left\\\|r\\right\\\|\_\{\\infty\},applying Hoeffding’s lemma in Lemma[124](https://arxiv.org/html/2609.05727#Thmtheorem124)gives

KL⁡\(μ∥π\)=\\displaystyle\\operatorname\{KL\}\\left\(\\mu\\middle\\\|\\pi\\right\)=𝔼X∼μ​\[log⁡μ⁡\(X\)π⁡\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\mu\}\\left\[\\log\\frac\{\\mu\(X\)\}\{\\pi\(X\)\}\\right\]=\\displaystyle=log⁡𝔼X∼μ​\[eτ​r​\(X\)\]−τ​𝔼X∼μ​\[r⁡\(X\)\]\\displaystyle\\log\\mathbb\{E\}\_\{X\\sim\\mu\}\\left\[e^\{\\tau r\(X\)\}\\right\]\-\\tau\\mathbb\{E\}\_\{X\\sim\\mu\}\\left\[r\(X\)\\right\]=\\displaystyle=log⁡𝔼X∼μ​\[eτ⁡\(r⁡\(X\)−𝔼Y∼μ​\[r⁡\(Y\)\]\)\]\\displaystyle\\log\\mathbb\{E\}\_\{X\\sim\\mu\}\\left\[e^\{\\tau\\left\(r\(X\)\-\\mathbb\{E\}\_\{Y\\sim\\mu\}\[r\(Y\)\]\\right\)\}\\right\]≤\\displaystyle\\leqlog⁡\(exp⁡\(τ2​\(2​‖r‖∞\)28\)\)\\displaystyle\\log\\left\(\\exp\\left\(\\frac\{\\tau^\{2\}\\left\(2\\left\\\|r\\right\\\|\_\{\\infty\}\\right\)^\{2\}\}\{8\}\\right\)\\right\)=\\displaystyle=12​τ2​‖r‖∞2\.\\displaystyle\\frac\{1\}\{2\}\\tau^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\.Then, the same argument gives

KL⁡\(π∥μ\)≤12​τ2​‖r‖∞2\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\mu\\right\)\\leq\\frac\{1\}\{2\}\\tau^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}sinceμ⁡\(x\)∝π⁡\(x\)​e−τ​r​\(x\)\\mu\(x\)\\propto\\pi\(x\)e^\{\-\\tau r\(x\)\}\. ∎

See[45](https://arxiv.org/html/2609.05727#Thmtheorem45)

###### Proof\.

DenoteS⁡\(x\):=∑k=0K−1𝒟ρk,ηk,r​\(x\)≥0S\(x\):=\\sum\_\{k=0\}^\{K\-1\}\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},r\}\(x\)\\geq 0\. Then,

𝔼X∼π​\[S⁡\(X\)\]=∑k=0K−1𝔼X∼π​\[𝒟ρk,ηk,r​\(X\)\]≤C​∑k=0K−1ηk2≤C​ηmax​τ^\.\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[S\(X\)\\right\]=\\sum\_\{k=0\}^\{K\-1\}\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{\\rho\_\{k\},\\eta\_\{k\},r\}\(X\)\\right\]\\leq C\\sum\_\{k=0\}^\{K\-1\}\\eta\_\{k\}^\{2\}\\leq C\\eta\_\{\\mathrm\{max\}\}\\hat\{\\tau\}\.Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12)implies

ρK​\(x\)∝π⁡\(x\)​exp⁡\(\(τ^−τ\)​r​\(x\)−S⁡\(x\)\)\.\\rho\_\{K\}\(x\)\\propto\\pi\(x\)\\exp\\left\(\\left\(\\hat\{\\tau\}\-\\tau\\right\)r\(x\)\-S\(x\)\\right\)\.
We first investigate the forward KL\. Let

π^​\(x\):=π⁡\(x\)​e\(τ^−τ\)​r​\(x\)𝔼X∼π​\[e\(τ^−τ\)​r​\(X\)\]\.\\hat\{\\pi\}\(x\):=\\frac\{\\pi\(x\)e^\{\\left\(\\hat\{\\tau\}\-\\tau\\right\)r\(x\)\}\}\{\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[e^\{\\left\(\\hat\{\\tau\}\-\\tau\\right\)r\(X\)\}\\right\]\}\.Then, Lemma[125](https://arxiv.org/html/2609.05727#Thmtheorem125)implies

KL⁡\(π∥π^\)≤12​\(τ^−τ\)2​‖r‖∞2\.\\displaystyle\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\hat\{\\pi\}\\right\)\\leq\\frac\{1\}\{2\}\\left\(\\hat\{\\tau\}\-\\tau\\right\)^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\.Since

ρK​\(x\)=π^​\(x\)​e−S⁡\(x\)𝔼X∼π^​\[e−S⁡\(X\)\],S⁡\(x\)≥0,\\rho\_\{K\}\(x\)=\\frac\{\\hat\{\\pi\}\(x\)e^\{\-S\(x\)\}\}\{\\mathbb\{E\}\_\{X\\sim\\hat\{\\pi\}\}\\left\[e^\{\-S\(X\)\}\\right\]\},\\qquad S\(x\)\\geq 0,the forward KL can be bounded by

KL⁡\(π∥ρK\)=\\displaystyle\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\_\{K\}\\right\)=𝔼X∼π​\[log⁡π⁡\(X\)ρK​\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\log\\frac\{\\pi\(X\)\}\{\\rho\_\{K\}\(X\)\}\\right\]=\\displaystyle=𝔼X∼π​\[log⁡π^​\(X\)ρK​\(X\)\]\+𝔼X∼π​\[log⁡π⁡\(X\)π^​\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\log\\frac\{\\hat\{\\pi\}\(X\)\}\{\\rho\_\{K\}\(X\)\}\\right\]\+\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\log\\frac\{\\pi\(X\)\}\{\\hat\{\\pi\}\(X\)\}\\right\]≤\\displaystyle\\leqlog⁡𝔼X∼π^​\[e−S⁡\(X\)\]\+𝔼X∼π​\[S⁡\(X\)\]\+12​\(τ^−τ\)2​‖r‖∞2\\displaystyle\\log\\mathbb\{E\}\_\{X\\sim\\hat\{\\pi\}\}\\left\[e^\{\-S\(X\)\}\\right\]\+\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[S\(X\)\\right\]\+\\frac\{1\}\{2\}\\left\(\\hat\{\\tau\}\-\\tau\\right\)^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}≤\\displaystyle\\leqC​ηmax​τ^\+12​\(τ^−τ\)2​‖r‖∞2\.\\displaystyle C\\eta\_\{\\mathrm\{max\}\}\\hat\{\\tau\}\+\\frac\{1\}\{2\}\\left\(\\hat\{\\tau\}\-\\tau\\right\)^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\.
We then bound the reverse KL\. Let

π~​\(x\):=π⁡\(x\)​e−S⁡\(x\)𝔼X∼π​\[e−S⁡\(X\)\]\.\\tilde\{\\pi\}\(x\):=\\frac\{\\pi\(x\)e^\{\-S\(x\)\}\}\{\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[e^\{\-S\(X\)\}\\right\]\}\.Since

ρK​\(x\)=π~​\(x\)​e\(τ^−τ\)​r​\(x\)𝔼X∼π~​\[e\(τ^−τ\)​r​\(X\)\],\\rho\_\{K\}\(x\)=\\frac\{\\tilde\{\\pi\}\(x\)e^\{\(\\hat\{\\tau\}\-\\tau\)r\(x\)\}\}\{\\mathbb\{E\}\_\{X\\sim\\tilde\{\\pi\}\}\\left\[e^\{\(\\hat\{\\tau\}\-\\tau\)r\(X\)\}\\right\]\},Lemma[125](https://arxiv.org/html/2609.05727#Thmtheorem125)implies

KL⁡\(ρK∥π~\)≤12​\(τ^−τ\)2​‖r‖∞2\.\\operatorname\{KL\}\\left\(\\rho\_\{K\}\\middle\\\|\\tilde\{\\pi\}\\right\)\\leq\\frac\{1\}\{2\}\\left\(\\hat\{\\tau\}\-\\tau\\right\)^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\.The reverse KL can be decomposed as

KL⁡\(ρK∥π\)=\\displaystyle\\operatorname\{KL\}\\left\(\\rho\_\{K\}\\middle\\\|\\pi\\right\)=𝔼X∼ρK​\[log⁡ρK​\(X\)π⁡\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\rho\_\{K\}\}\\left\[\\log\\frac\{\\rho\_\{K\}\(X\)\}\{\\pi\(X\)\}\\right\]=\\displaystyle=𝔼X∼ρK​\[log⁡ρK​\(X\)π~​\(X\)\]\+𝔼X∼ρK​\[log⁡π~​\(X\)π⁡\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim\\rho\_\{K\}\}\\left\[\\log\\frac\{\\rho\_\{K\}\(X\)\}\{\\tilde\{\\pi\}\(X\)\}\\right\]\+\\mathbb\{E\}\_\{X\\sim\\rho\_\{K\}\}\\left\[\\log\\frac\{\\tilde\{\\pi\}\(X\)\}\{\\pi\(X\)\}\\right\]≤\\displaystyle\\leq12​\(τ^−τ\)2​‖r‖∞2−𝔼X∼ρK​\[S⁡\(X\)\]−log⁡𝔼X∼π​\[e−S⁡\(X\)\]\\displaystyle\\frac\{1\}\{2\}\\left\(\\hat\{\\tau\}\-\\tau\\right\)^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\-\\mathbb\{E\}\_\{X\\sim\\rho\_\{K\}\}\\left\[S\(X\)\\right\]\-\\log\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[e^\{\-S\(X\)\}\\right\]≤\\displaystyle\\leq12​\(τ^−τ\)2​‖r‖∞2−log⁡𝔼X∼π​\[e−S⁡\(X\)\]\.\\displaystyle\\frac\{1\}\{2\}\\left\(\\hat\{\\tau\}\-\\tau\\right\)^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\-\\log\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[e^\{\-S\(X\)\}\\right\]\.By Jensen’s inequality, we have

log⁡𝔼X∼π​\[e−S⁡\(X\)\]≥𝔼X∼π​\[−S⁡\(X\)\]≥−C​ηmax​τ^\.\\log\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[e^\{\-S\(X\)\}\\right\]\\geq\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\-S\(X\)\\right\]\\geq\-C\\eta\_\{\\mathrm\{max\}\}\\hat\{\\tau\}\.Therefore, we have

KL⁡\(ρK∥π\)≤\\displaystyle\\operatorname\{KL\}\\left\(\\rho\_\{K\}\\middle\\\|\\pi\\right\)\\leq12​\(τ^−τ\)2​‖r‖∞2\+C​ηmax​τ^\.\\displaystyle\\frac\{1\}\{2\}\\left\(\\hat\{\\tau\}\-\\tau\\right\)^\{2\}\\left\\\|r\\right\\\|\_\{\\infty\}^\{2\}\+C\\eta\_\{\\mathrm\{max\}\}\\hat\{\\tau\}\.This finishes the proof\. ∎

## Appendix GCanonical Updates with Time\-Dependent Stepsizes

Section[5](https://arxiv.org/html/2609.05727#S5)studies the finite\-stepsize canonical retraction\. The corresponding tangential update is

v¯t=vtρ\+η​Γtρ,f,η∈\(0,∞\)​is constant\.\\bar\{v\}\_\{t\}=v\_\{t\}^\{\\rho\}\+\\eta\\Gamma\_\{t\}^\{\\rho,f\},\\qquad\\eta\\in\(0,\\infty\)\\text\{ is constant\}\.Sinceη​Γρ,f∈Tvρ​𝒱can\\eta\\Gamma^\{\\rho,f\}\\in T\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}, composing this velocity\-space addition with the canonical projection implements the canonical retraction introduced in Section[5\.1](https://arxiv.org/html/2609.05727#S5.SS1)\.

This appendix replaces the constant stepsizeη\\etaby a time\-dependent modulation:

v¯t=vtρ\+ηt​Γtρ,f,η∈C1​\(\[0,1\],\(0,∞\)\)​is time\-dependent\.\\bar\{v\}\_\{t\}=v\_\{t\}^\{\\rho\}\+\\eta\_\{t\}\\Gamma\_\{t\}^\{\\rho,f\},\\qquad\\eta\\in C^\{1\}\(\[0,1\];\(0,\\infty\)\)\\text\{ is time\-dependent\}\.AlthoughΓρ,f\\Gamma^\{\\rho,f\}is a tangent vector inTvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}, its time\-dependent modulation\(ηt​Γtρ,f\)t∈\[0,1\]\\left\(\\eta\_\{t\}\\Gamma\_\{t\}^\{\\rho,f\}\\right\)\_\{t\\in\[0,1\]\}need not belong toTvρ​𝒱canT\_\{v^\{\\rho\}\}\\mathscr\{V\}\_\{\\mathrm\{can\}\}\. The resulting velocity\-space addition followed by canonicalization is therefore generally not a retraction in the geometric sense\. Nevertheless, the resulting velocity field generates a well\-defined probability path and terminal density\. We derive an exact formula for this terminal density in Appendix[G\.1](https://arxiv.org/html/2609.05727#A7.SS1)\. Then, Appendix[G\.2](https://arxiv.org/html/2609.05727#A7.SS2)shows that the monotonicity condition

η˙t≥0,t∈\(0,1\),\\dot\{\\eta\}\_\{t\}\\geq 0,\\qquad t\\in\(0,1\),is sufficient to extend the finite\-stepsize value\-ascent certificate in Section[5\.2](https://arxiv.org/html/2609.05727#S5.SS2)to this new setting\. Furthermore, if we setf=r~ρf=\\tilde\{r\}^\{\\rho\}and use a nondecreasingη∈C1​\(\[0,1\],\(0,τ\]\)\\eta\\in C^\{1\}\(\[0,1\];\(0,\\tau\]\)for Newton Matching, the reverse\-KL descent guarantee remains valid\.

### G\.1Terminal\-Density Characterization

Since the canonical projection preserves terminal densities, the terminal density of the canonical updates with time\-dependent stepsizes is determined entirely by the ambient velocity field before canonicalization\. We therefore analyze the density path generated by

v¯tρ,η,f=vtρ\+ηt​Γtρ,f,\\bar\{v\}\_\{t\}^\{\\rho,\\eta,f\}=v\_\{t\}^\{\\rho\}\+\\eta\_\{t\}\\Gamma\_\{t\}^\{\\rho,f\},and recover the same terminal density as the field obtained after canonicalization\.

This subsection derives the resulting terminal\-density formula in two steps\. First, Appendix[G\.1\.1](https://arxiv.org/html/2609.05727#A7.SS1.SSS1)establishes a general terminal\-density formula for velocity\-space additive updates with generic gradient fields:

v¯tρ,Ψ=vtρ\+κt∇Ψt\.\\bar\{v\}\_\{t\}^\{\\rho,\\Psi\}=v\_\{t\}^\{\\rho\}\+\\kappa\_\{t\}\\nabla\\Psi\_\{t\}\.This calculation uses neither the posterior\-value structure nor tangency to the canonical manifold\. Appendix[G\.1\.2](https://arxiv.org/html/2609.05727#A7.SS1.SSS2)then choosesΨt=ηt​Vtρ​\[f\]\\Psi\_\{t\}=\\eta\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\. This specialization provides the terminal\-density formula for the updatev¯tρ,η,f=vtρ\+ηt​Γtρ,f\\bar\{v\}\_\{t\}^\{\\rho,\\eta,f\}=v\_\{t\}^\{\\rho\}\+\\eta\_\{t\}\\Gamma\_\{t\}^\{\\rho,f\}\.

#### G\.1\.1Additive Updates with Generic Gradients

We derive the general formula for the terminal density of a velocity\-space additive update with a generic gradient field\. LetΨt:ℝd→ℝ\\Psi\_\{t\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}be a time\-dependent potential\. Define

v¯tρ,Ψ\(x\):=vtρ\(x\)\+κt∇Ψt\(x\),\\bar\{v\}\_\{t\}^\{\\rho,\\Psi\}\(x\):=v\_\{t\}^\{\\rho\}\(x\)\+\\kappa\_\{t\}\\nabla\\Psi\_\{t\}\(x\),\(227\)and

ℜtρ,Ψ\(x\):=\(∂t\+ℒtρ\)Ψt\(x\)\+κt‖∇Ψt\(x\)‖22\.\\mathfrak\{R\}\_\{t\}^\{\\rho,\\Psi\}\(x\):=\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\}\)\\Psi\_\{t\}\(x\)\+\\kappa\_\{t\}\\left\\\|\\nabla\\Psi\_\{t\}\(x\)\\right\\\|\_\{2\}^\{2\}\.\(228\)Letνtρ,Ψ\\nu\_\{t\}^\{\\rho,\\Psi\}andΦρ,Ψ\\Phi^\{\\rho,\\Psi\}denote the density path and the ODE flow generated byv¯tρ,Ψ\\bar\{v\}\_\{t\}^\{\\rho,\\Psi\}, respectively\.

###### Proposition 126\.

Define

G~tρ,Ψ​\(x\):=ptρ​\(x\)​eΨt​\(x\)\.\\widetilde\{G\}\_\{t\}^\{\\rho,\\Psi\}\(x\):=p\_\{t\}^\{\\rho\}\(x\)e^\{\\Psi\_\{t\}\(x\)\}\.Then, we have the following transport identity:

∂tG~tρ,Ψ\+∇⋅\(G~tρ,Ψ​v¯tρ,Ψ\)=G~tρ,Ψ​ℜtρ,Ψ\.\\partial\_\{t\}\\widetilde\{G\}\_\{t\}^\{\\rho,\\Psi\}\+\\nabla\\cdot\\left\(\\widetilde\{G\}\_\{t\}^\{\\rho,\\Psi\}\\bar\{v\}\_\{t\}^\{\\rho,\\Psi\}\\right\)=\\widetilde\{G\}\_\{t\}^\{\\rho,\\Psi\}\\mathfrak\{R\}\_\{t\}^\{\\rho,\\Psi\}\.

###### Proof\.

Expand the left\-hand side usingG~tρ,Ψ=ptρ​eΨt\\widetilde\{G\}\_\{t\}^\{\\rho,\\Psi\}=p\_\{t\}^\{\\rho\}e^\{\\Psi\_\{t\}\}and \([227](https://arxiv.org/html/2609.05727#A7.E227)\)\. The continuity equation cancels the terms withoutΨt\\Psi\_\{t\}, and the remaining factor is \([228](https://arxiv.org/html/2609.05727#A7.E228)\)\. ∎

###### Proposition 127\.

LetYtY\_\{t\}be the characteristic

d​Ytd​t=v¯tρ,Ψ​\(Yt\)\.\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=\\bar\{v\}\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\)\.Then, we have

dd​t​\[log⁡ptρ​\(Yt\)νtρ,Ψ​\(Yt\)\+Ψt​\(Yt\)\]=ℜtρ,Ψ​\(Yt\)\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\[\\log\\frac\{p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\}\{\\nu\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\)\}\+\\Psi\_\{t\}\(Y\_\{t\}\)\\right\]=\\mathfrak\{R\}\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\)\.\(229\)Consequently, the updated terminal density is

ν1ρ,Ψ​\(x\)=\\displaystyle\\nu\_\{1\}^\{\\rho,\\Psi\}\(x\)=ρ⁡\(x\)​exp⁡\(Ψ1​\(x\)−Ψ0​\(Φ1→0ρ,Ψ​\(x\)\)−∫01ℜtρ,Ψ​\(Φ1→tρ,Ψ​\(x\)\)​𝑑t\)\.\\displaystyle\\rho\(x\)\\exp\\left\(\\Psi\_\{1\}\(x\)\-\\Psi\_\{0\}\\left\(\\Phi\_\{1\\to 0\}^\{\\rho,\\Psi\}\(x\)\\right\)\-\\int\_\{0\}^\{1\}\\mathfrak\{R\}\_\{t\}^\{\\rho,\\Psi\}\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\Psi\}\(x\)\\right\)\\mathrm\{d\}t\\right\)\.

###### Proof\.

By Proposition[126](https://arxiv.org/html/2609.05727#Thmtheorem126), we have

dd​tlogG~tρ,Ψ\(Yt\)=−∇⋅v¯tρ,Ψ\(Yt\)\+ℜtρ,Ψ\(Yt\)\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log\\widetilde\{G\}\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\)=\-\\nabla\\cdot\\bar\{v\}\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\)\+\\mathfrak\{R\}\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\)\.By the continuity equation, we have

dd​tlogνtρ,Ψ\(Yt\)=−∇⋅v¯tρ,Ψ\(Yt\),\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\log\\nu\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\)=\-\\nabla\\cdot\\bar\{v\}\_\{t\}^\{\\rho,\\Psi\}\(Y\_\{t\}\),which gives \([229](https://arxiv.org/html/2609.05727#A7.E229)\)\. Integrating fromt=0t=0tot=1t=1finishes the proof\. ∎

#### G\.1\.2Additive Updates with Posterior\-Value Gradients

Here, we specialize the generic gradient field to the posterior\-value gradient, i\.e\., choosingΨt=ηt​Vtρ​\[f\]\\Psi\_\{t\}=\\eta\_\{t\}V\_\{t\}^\{\\rho\}\[f\]in \([227](https://arxiv.org/html/2609.05727#A7.E227)\), where the time\-dependent stepsizeη∈C1​\(\[0,1\],ℝ\)\\eta\\in C^\{1\}\(\[0,1\];\\mathbb\{R\}\)\. Consider the following update:

v¯tρ,η,f​\(x\):=vtρ​\(x\)\+ηt​Γtρ,f​\(x\),\\displaystyle\\bar\{v\}\_\{t\}^\{\\rho,\\eta,f\}\(x\):=v\_\{t\}^\{\\rho\}\(x\)\+\\eta\_\{t\}\\Gamma\_\{t\}^\{\\rho,f\}\(x\),\(230a\)q:=𝒯⁡\(v¯ρ,η,f\)\.\\displaystyle q:=\\mathcal\{T\}\(\\bar\{v\}^\{\\rho,\\eta,f\}\)\.\(230b\)HereΓtρ,f\(x\)=κt∇Vtρ\[f\]\(x\)\\Gamma\_\{t\}^\{\\rho,f\}\(x\)=\\kappa\_\{t\}\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\)\. Letνtρ,η,f\\nu\_\{t\}^\{\\rho,\\eta,f\}andΦρ,η,f\\Phi^\{\\rho,\\eta,f\}denote the density path and the ODE flow generated byv¯tρ,η,f\\bar\{v\}\_\{t\}^\{\\rho,\\eta,f\}, respectively\.

###### Proposition 128\.

Along the characteristic

d​Ytd​t=v¯tρ,η,f​\(Yt\),\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=\\bar\{v\}\_\{t\}^\{\\rho,\\eta,f\}\(Y\_\{t\}\),we have

dd​t​\[log⁡ptρ​\(Yt\)νtρ,η,f​\(Yt\)\+ηt​\(Vtρ​\[f\]​\(Yt\)−𝔼ρ​\[f\]\)\]=η˙t​\(Vtρ​\[f\]​\(Yt\)−𝔼ρ​\[f\]\)\+ηt2​κt​‖∇Vtρ​\[f\]​\(Yt\)‖22\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\[\\log\\frac\{p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\}\{\\nu\_\{t\}^\{\\rho,\\eta,f\}\(Y\_\{t\}\)\}\+\\eta\_\{t\}\\left\(V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\)\\right\]=\\dot\{\\eta\}\_\{t\}\\left\(V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\-\\mathbb\{E\}\_\{\\rho\}\[f\]\\right\)\+\\eta\_\{t\}^\{2\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\.\(231\)Consequently,

q⁡\(x\)=ρ⁡\(x\)​exp⁡\(η1​\(f⁡\(x\)−𝔼X∼ρ​\[f⁡\(X\)\]\)−𝒟ρ,η,f​\(x\)\),q\(x\)=\\rho\(x\)\\exp\\left\(\\eta\_\{1\}\\left\(f\(x\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(x\)\\right\),where

𝒟ρ,η,f​\(x\):=\\displaystyle\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(x\):=∫01\[η˙t​\(Vtρ​\[f\]​\(Φ1→tρ,η,f​\(x\)\)−𝔼X∼ρ​\[f⁡\(X\)\]\)\+ηt2​κt​‖∇Vtρ​\[f\]​\(Φ1→tρ,η,f​\(x\)\)‖22\]​𝑑t\.\\displaystyle\\int\_\{0\}^\{1\}\\left\[\\dot\{\\eta\}\_\{t\}\\left\(V\_\{t\}^\{\\rho\}\[f\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,f\}\(x\)\\right\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\+\\eta\_\{t\}^\{2\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,f\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}t\.

###### Proof\.

According to Theorem[98](https://arxiv.org/html/2609.05727#Thmtheorem98), we have\(∂t\+ℒtρ\)Vtρ\[f\]=0\(\\partial\_\{t\}\+\\mathcal\{L\}\_\{t\}^\{\\rho\}\)V\_\{t\}^\{\\rho\}\[f\]=0\. Substituting this into \([228](https://arxiv.org/html/2609.05727#A7.E228)\) gives

ℜtρ,η​Vρ​\[f\]​\(x\)=η˙t​Vtρ​\[f\]​\(x\)\+κt​ηt2​‖∇Vtρ​\[f\]​\(x\)‖22\.\\mathfrak\{R\}\_\{t\}^\{\\rho,\\eta V^\{\\rho\}\[f\]\}\(x\)=\\dot\{\\eta\}\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(x\)\+\\kappa\_\{t\}\\eta\_\{t\}^\{2\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(x\)\\right\\\|\_\{2\}^\{2\}\.By Proposition[127](https://arxiv.org/html/2609.05727#Thmtheorem127),

dd​t​\[log⁡ptρ​\(Yt\)νtρ,η,f​\(Yt\)\+ηt​Vtρ​\[f\]​\(Yt\)\]=η˙t​Vtρ​\[f\]​\(Yt\)\+ηt2​κt​‖∇Vtρ​\[f\]​\(Yt\)‖22,\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\[\\log\\frac\{p\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\}\{\\nu\_\{t\}^\{\\rho,\\eta,f\}\(Y\_\{t\}\)\}\+\\eta\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\]=\\dot\{\\eta\}\_\{t\}V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\+\\eta\_\{t\}^\{2\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(Y\_\{t\}\)\\right\\\|\_\{2\}^\{2\},which gives \([231](https://arxiv.org/html/2609.05727#A7.E231)\)\. Integrating fromt=0t=0tot=1t=1finishes the proof\. ∎

Whenηt≡η\\eta\_\{t\}\\equiv\\etais constant, the term involvingη˙t\\dot\{\\eta\}\_\{t\}vanishes, and𝒟ρ,η,f\\mathcal\{D\}\_\{\\rho,\\eta,f\}reduces exactly to the nonnegative path dissipation introduced in Section[5\.2](https://arxiv.org/html/2609.05727#S5.SS2)\. However, whenη\\etais time\-dependent, the term

η˙t​\(Vtρ​\[f\]​\(Φ1→tρ,η,f​\(x\)\)−𝔼X∼ρ​\[f⁡\(X\)\]\)\\dot\{\\eta\}\_\{t\}\\left\(V\_\{t\}^\{\\rho\}\[f\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,f\}\(x\)\\right\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)need not be pointwise nonnegative\. The terminal\-density formula in Proposition[128](https://arxiv.org/html/2609.05727#Thmtheorem128)alone therefore does not yet imply value ascent\. The next subsection shows that a nondecreasing time\-dependent stepsize is sufficient for value ascent and, in the Newton Matching specialization, for reverse\-KL descent under the additional conditionη1≤τ\\eta\_\{1\}\\leq\\tau\.

### G\.2Monotonicity Guarantees

Appendix[G\.1](https://arxiv.org/html/2609.05727#A7.SS1)presents the terminal\-density characterization for general additive updates with time\-dependent stepsizes\. We now identify conditions under which the monotonicity guarantees for constant stepsizes extend to this setting\. A nondecreasing time\-dependent stepsize yields value ascent for generalff; forf=r~ρf=\\tilde\{r\}^\{\\rho\}, the additional conditionη1≤τ\\eta\_\{1\}\\leq\\tauyields reverse\-KL descent\.

Appendix[G\.2\.1](https://arxiv.org/html/2609.05727#A7.SS2.SSS1)first derives an identity for the expectation of the pathwise correction under the updated density and uses it to prove value ascent\. Appendix[G\.2\.2](https://arxiv.org/html/2609.05727#A7.SS2.SSS2)then specializes tof=r~ρf=\\tilde\{r\}^\{\\rho\}, yielding the time\-dependent\-stepsize counterparts of the Newton Matching terminal\-density formula and three\-point identity\. Under the additional boundη1≤τ\\eta\_\{1\}\\leq\\tau, the latter implies one\-stage reverse\-KL descent\.

#### G\.2\.1Value\-Ascent Certificate

The terminal\-density formula in Appendix[G\.1\.2](https://arxiv.org/html/2609.05727#A7.SS1.SSS2)contains a pathwise correction that need not be nonnegative pointwise\. To establish value ascent, it suffices to show that its expectation under the updated densityqqis nonnegative\. We first derive an exact identity for this expectation and then combine it with the terminal\-density formula to obtain the time\-dependent\-stepsize counterpart of the decomposition in Section[5\.2](https://arxiv.org/html/2609.05727#S5.SS2)\.

###### Proposition 129\.

Assume

∀t∈\(0,1\),ηt\>0\\forall t\\in\(0,1\),\\,\\eta\_\{t\}\>0and the boundary conditions required for the integration by parts below\. Then, we have

𝔼X∼q​\[𝒟ρ,η,f​\(X\)\]=η1​∫01\[η˙tηt2​KL⁡\(νtρ,η,f∥ptρ\)\+ηt​κt​𝔼X∼νtρ,η,f​\[‖∇Vtρ​\[f\]​\(X\)‖22\]\]​𝑑t\.\\displaystyle\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]=\\eta\_\{1\}\\int\_\{0\}^\{1\}\\left\[\\frac\{\\dot\{\\eta\}\_\{t\}\}\{\\eta\_\{t\}^\{2\}\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}^\{\\rho,\\eta,f\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)\+\\eta\_\{t\}\\kappa\_\{t\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}^\{\\rho,\\eta,f\}\}\\left\[\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]\\right\]\\mathrm\{d\}t\.In particular, if we additionally assume

∀t∈\(0,1\),η˙t≥0,\\forall t\\in\(0,1\),\\,\\dot\{\\eta\}\_\{t\}\\geq 0,then

𝔼X∼q​\[𝒟ρ,η,f​\(X\)\]≥0\.\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]\\geq 0\.

###### Proof\.

For simplicity, denoteνt=νtρ,η,f\\nu\_\{t\}=\\nu\_\{t\}^\{\\rho,\\eta,f\}and

𝒟t​\(x\):=∫0t\[η˙s​\(Vsρ​\[f\]​\(Φt→sρ,η,f​\(x\)\)−𝔼X∼ρ​\[f⁡\(X\)\]\)\+ηs2​κs​‖∇Vsρ​\[f\]​\(Φt→sρ,η,f​\(x\)\)‖22\]​𝑑s\.\\mathcal\{D\}\_\{t\}\(x\):=\\int\_\{0\}^\{t\}\\left\[\\dot\{\\eta\}\_\{s\}\\left\(V\_\{s\}^\{\\rho\}\[f\]\\left\(\\Phi\_\{t\\to s\}^\{\\rho,\\eta,f\}\(x\)\\right\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\+\\eta\_\{s\}^\{2\}\\kappa\_\{s\}\\left\\\|\\nabla V\_\{s\}^\{\\rho\}\[f\]\\left\(\\Phi\_\{t\\to s\}^\{\\rho,\\eta,f\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\\mathrm\{d\}s\.Integrating \([231](https://arxiv.org/html/2609.05727#A7.E231)\) froms=0s=0tos=ts=tgives

νt​\(x\)=ptρ​\(x\)​exp⁡\(ηt​\(Vtρ​\[f\]​\(x\)−𝔼X∼ρ​\[f⁡\(X\)\]\)−𝒟t​\(x\)\)\.\\nu\_\{t\}\(x\)=p\_\{t\}^\{\\rho\}\(x\)\\exp\\left\(\\eta\_\{t\}\\left\(V\_\{t\}^\{\\rho\}\[f\]\(x\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\-\\mathcal\{D\}\_\{t\}\(x\)\\right\)\.Therefore, we have

𝔼X∼νt​\[𝒟t​\(X\)\]=−KL⁡\(νt∥ptρ\)\+ηt​\(𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼X∼ρ​\[f⁡\(X\)\]\)\.\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\mathcal\{D\}\_\{t\}\(X\)\\right\]=\-\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)\+\\eta\_\{t\}\\left\(\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\.Then, taking the expectation of \([231](https://arxiv.org/html/2609.05727#A7.E231)\) gives

dd​t​𝔼X∼νt​\[𝒟t​\(X\)\]=η˙t​\(𝔼X∼νt​\[Vtρ​\[f\]​\(X\)\]−𝔼X∼ρ​\[f⁡\(X\)\]\)\+ηt2​κt​𝔼X∼νt​\[‖∇Vtρ​\[f\]​\(X\)‖22\]\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\mathcal\{D\}\_\{t\}\(X\)\\right\]=\\dot\{\\eta\}\_\{t\}\\left\(\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\+\\eta\_\{t\}^\{2\}\\kappa\_\{t\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Therefore,

dd​t​\(1ηt​𝔼X∼νt​\[𝒟t​\(X\)\]\)=\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(\\frac\{1\}\{\\eta\_\{t\}\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\mathcal\{D\}\_\{t\}\(X\)\\right\]\\right\)=1ηt​dd​t​𝔼X∼νt​\[𝒟t​\(X\)\]−η˙tηt2​𝔼X∼νt​\[𝒟t​\(X\)\]\\displaystyle\\frac\{1\}\{\\eta\_\{t\}\}\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\mathcal\{D\}\_\{t\}\(X\)\\right\]\-\\frac\{\\dot\{\\eta\}\_\{t\}\}\{\\eta\_\{t\}^\{2\}\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\mathcal\{D\}\_\{t\}\(X\)\\right\]=\\displaystyle=η˙tηt2​KL⁡\(νt∥ptρ\)\+ηt​κt​𝔼X∼νt​\[‖∇Vtρ​\[f\]​\(X\)‖22\]\.\\displaystyle\\frac\{\\dot\{\\eta\}\_\{t\}\}\{\\eta\_\{t\}^\{2\}\}\\operatorname\{KL\}\\left\(\\nu\_\{t\}\\middle\\\|p\_\{t\}^\{\\rho\}\\right\)\+\\eta\_\{t\}\\kappa\_\{t\}\\mathbb\{E\}\_\{X\\sim\\nu\_\{t\}\}\\left\[\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[f\]\(X\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Integrating fromt=0t=0tot=1t=1finishes the proof\. ∎

The following theorem provides the value\-ascent certificate\.

###### Theorem 130\.

Consider the ideal stage \([230](https://arxiv.org/html/2609.05727#A7.E230)\)\. For any test densityp∈𝒫p\\in\\mathscr\{P\}, we have

KL⁡\(p∥q\)=KL⁡\(p∥ρ\)−η1​\(𝔼X∼p​\[f⁡\(X\)\]−𝔼X∼ρ​\[f⁡\(X\)\]\)\+𝔼X∼p​\[𝒟ρ,η,f​\(X\)\]\.\\operatorname\{KL\}\\left\(p\\middle\\\|q\\right\)=\\operatorname\{KL\}\\left\(p\\middle\\\|\\rho\\right\)\-\\eta\_\{1\}\(\\mathbb\{E\}\_\{X\\sim p\}\[f\(X\)\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\)\+\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]\.\(232\)In particular, whenη1\>0\\eta\_\{1\}\>0, lettingp=qp=qgives

𝔼X∼q​\[f⁡\(X\)\]−𝔼X∼ρ​\[f⁡\(X\)\]=1η1​KL⁡\(q∥ρ\)\+1η1​𝔼X∼q​\[𝒟ρ,η,f​\(X\)\]\.\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]=\\frac\{1\}\{\\eta\_\{1\}\}\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)\+\\frac\{1\}\{\\eta\_\{1\}\}\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]\.\(233\)
Consequently, for any nondecreasing time\-dependent stepsizeη\\etawhich satisfies

∀t∈\(0,1\),ηt\>0,η˙t≥0,\\forall t\\in\(0,1\),\\,\\eta\_\{t\}\>0,\\,\\dot\{\\eta\}\_\{t\}\\geq 0,we have the value\-ascent certificate:

𝔼X∼q​\[f⁡\(X\)\]≥𝔼X∼ρ​\[f⁡\(X\)\]\.\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]\\geq\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\.Furthermore,

𝔼X∼q\[f\(X\)\]=𝔼X∼ρ\[f\(X\)\]⇔ρ=q⇔f≡const\.\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]=\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\quad\\Leftrightarrow\\quad\\rho=q\\quad\\Leftrightarrow\\quad f\\equiv\\mathrm\{const\}\.

###### Proof\.

According to Proposition[128](https://arxiv.org/html/2609.05727#Thmtheorem128), we have

KL⁡\(p∥q\)=\\displaystyle\\operatorname\{KL\}\\left\(p\\middle\\\|q\\right\)=𝔼X∼p​\[log⁡p⁡\(X\)\]−𝔼X∼p​\[log⁡q⁡\(X\)\]\\displaystyle\\mathbb\{E\}\_\{X\\sim p\}\[\\log p\(X\)\]\-\\mathbb\{E\}\_\{X\\sim p\}\[\\log q\(X\)\]=\\displaystyle=𝔼X∼p​\[log⁡p⁡\(X\)\]−𝔼X∼p​\[log⁡ρ⁡\(X\)\+η1​\(f⁡\(X\)−𝔼X∼ρ​\[f⁡\(X\)\]\)−𝒟ρ,η,f​\(X\)\],\\displaystyle\\mathbb\{E\}\_\{X\\sim p\}\[\\log p\(X\)\]\-\\mathbb\{E\}\_\{X\\sim p\}\[\\log\\rho\(X\)\+\\eta\_\{1\}\\left\(f\(X\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\],which gives \([232](https://arxiv.org/html/2609.05727#A7.E232)\)\. Settingp=qp=qgives \([233](https://arxiv.org/html/2609.05727#A7.E233)\)\.

If we further assume thatη\\etais nondecreasing, Proposition[129](https://arxiv.org/html/2609.05727#Thmtheorem129)implies

𝔼X∼q​\[𝒟ρ,η,f​\(X\)\]≥0\.\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]\\geq 0\.Therefore, we have𝔼X∼q​\[f⁡\(X\)\]≥𝔼X∼ρ​\[f⁡\(X\)\]\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]\\geq\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\.

The remaining task is to prove the sufficiency and necessity of the equality\. First, if𝔼X∼q​\[f⁡\(X\)\]=𝔼X∼ρ​\[f⁡\(X\)\]\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]=\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]holds, then \([233](https://arxiv.org/html/2609.05727#A7.E233)\) impliesKL⁡\(q∥ρ\)=0\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)=0; hence, we haveρ=q\\rho=q\. Second,ρ=q\\rho=qimplies𝔼X∼q​\[𝒟ρ,η,f​\(X\)\]=0\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,f\}\(X\)\\right\]=0\. Therefore, we have∇Vtρ​\[f\]a\.e\.0\\nabla V\_\{t\}^\{\\rho\}\[f\]\\mathrel\{\\mathop\{\\vbox\{\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\}\\kern 1\.50696pt\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\} \}\}\\limits^\{\\mathrm\{a\.e\.\}\}\}0\. Since𝔼ρ​\[\|f\|\]<∞\\mathbb\{E\}\_\{\\rho\}\[\\left\|f\\right\|\]<\\inftyand the convolution kernel is a nondegenerate Gaussian,Vtρ​\[f\]V\_\{t\}^\{\\rho\}\[f\]is smooth; hence,Vtρ​\[f\]a\.e\.ctV\_\{t\}^\{\\rho\}\[f\]\\mathrel\{\\mathop\{\\vbox\{\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\}\\kern 1\.50696pt\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\} \}\}\\limits^\{\\mathrm\{a\.e\.\}\}\}c\_\{t\}is constant\. In other words, we have

∫ℝd\(f⁡\(x1\)−ct\)​ρ​\(x1\)​pt\|1​\(xt\|x1\)​d​x1=0\\int\_\{\\mathbb\{R\}^\{d\}\}\(f\(x\_\{1\}\)\-c\_\{t\}\)\\rho\(x\_\{1\}\)p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\,\\mathrm\{d\}x\_\{1\}=0for a\.e\.xt∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}\. Since the Gaussian convolution is injective andff,ρ\\rhoare continuous, we have\(f⁡\(x1\)−ct\)​ρ​\(x1\)≡0\(f\(x\_\{1\}\)\-c\_\{t\}\)\\rho\(x\_\{1\}\)\\equiv 0\. Therefore,f≡constf\\equiv\\mathrm\{const\}\. Third, iff≡constf\\equiv\\mathrm\{const\}, then𝔼X∼q​\[f⁡\(X\)\]=f⁡\(0\)=𝔼X∼ρ​\[f⁡\(X\)\]\\mathbb\{E\}\_\{X\\sim q\}\[f\(X\)\]=f\(0\)=\\mathbb\{E\}\_\{X\\sim\\rho\}\[f\(X\)\]\. ∎

#### G\.2\.2Finite\-Stepsize KL Descent

Section[6](https://arxiv.org/html/2609.05727#S6)identifiesτ​Γρ,r~ρ\\tau\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}as the canonical Newton direction, while Section[7\.1\.1](https://arxiv.org/html/2609.05727#S7.SS1.SSS1)proves finite\-stepsize reverse\-KL descent for the associated canonical retraction with a constant stepsizeη∈\(0,τ\]\\eta\\in\(0,\\tau\]\. Here, we instead scale the tangentΓtρ,r~ρ\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}by a time\-dependent stepsizeηt\\eta\_\{t\}\. The results below characterize the resulting single\-stage canonical update and establish reverse\-KL descent for a nondecreasing time\-dependent stepsize satisfyingη1≤τ\\eta\_\{1\}\\leq\\tau\.

###### Proposition 131\.

For any terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}and time\-dependent stepsizeη∈C1​\(\[0,1\],ℝ\)\\eta\\in C^\{1\}\(\[0,1\];\\mathbb\{R\}\), we define the path dissipation𝒟ρ,η,r~ρ:ℝd→ℝ\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}by

𝒟ρ,η,r~ρ​\(x\):=∫01\(η˙t​\(Vtρ​\[r~ρ\]​\(Φ1→tρ,η,r~ρ​\(x\)\)−𝔼ρ​\[r~ρ\]\)\+ηt2​κt​‖∇Vtρ​\[r~ρ\]​\(Φ1→tρ,η,r~ρ​\(x\)\)‖22\)​𝑑t,\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\):=\\int\_\{0\}^\{1\}\\left\(\\dot\{\\eta\}\_\{t\}\\left\(V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)\-\\mathbb\{E\}\_\{\\rho\}\\left\[\\tilde\{r\}^\{\\rho\}\\right\]\\right\)\+\\eta\_\{t\}^\{2\}\\kappa\_\{t\}\\left\\\|\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\left\(\\Phi\_\{1\\to t\}^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\)\\mathrm\{d\}t,whereΦρ,η,r~ρ\\Phi^\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}is the ODE flow generated by the velocity fieldvρ\+η​Γρ,r~ρv^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\. Then, for an ideal Newton step \([33](https://arxiv.org/html/2609.05727#S6.E33)\) whereq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\\left\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\\right\), we have

q⁡\(x\)\\displaystyle q\(x\)=ρ​\(x\)1−η1/τ​πμ,τ,r​\(x\)η1/τ​exp⁡\(η1τ​KL⁡\(ρ∥πμ,τ,r\)−𝒟ρ,η,r~ρ​\(x\)\)\.\\displaystyle=\\rho\(x\)^\{1\-\\eta\_\{1\}/\\tau\}\\pi\_\{\\mu,\\tau,r\}\(x\)^\{\\eta\_\{1\}/\\tau\}\\exp\\left\(\\frac\{\\eta\_\{1\}\}\{\\tau\}\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\_\{\\mu,\\tau,r\}\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)\.

###### Proof\.

By Proposition[128](https://arxiv.org/html/2609.05727#Thmtheorem128), lettingf=r~ρf=\\tilde\{r\}^\{\\rho\}gives

q⁡\(x\)=\\displaystyle q\(x\)=ρ⁡\(x\)​exp⁡\(η1​\(r~ρ​\(x\)−𝔼X∼ρ​\[r~ρ​\(X\)\]\)−𝒟ρ,η,r~ρ​\(x\)\)\\displaystyle\\rho\(x\)\\exp\\left\(\\eta\_\{1\}\\left\(\\tilde\{r\}^\{\\rho\}\(x\)\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[\\tilde\{r\}^\{\\rho\}\(X\)\]\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\)=\\displaystyle=ρ⁡\(x\)​exp⁡\(η1τ​\(log⁡πμ,τ,r​\(x\)ρ⁡\(x\)−𝔼X∼ρ​\[log⁡πμ,τ,r​\(X\)ρ⁡\(X\)\]\)−𝒟ρ,η,r~ρ​\(x\)\),\\displaystyle\\rho\(x\)\\exp\\left\(\\frac\{\\eta\_\{1\}\}\{\\tau\}\\left\(\\log\\frac\{\\pi\_\{\\mu,\\tau,r\}\(x\)\}\{\\rho\(x\)\}\-\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\log\\frac\{\\pi\_\{\\mu,\\tau,r\}\(X\)\}\{\\rho\(X\)\}\\right\]\\right\)\-\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(x\)\\right\),which finishes the proof\. ∎

The following three\-point identity is obtained by specializing Theorem[130](https://arxiv.org/html/2609.05727#Thmtheorem130)tof=r~ρf=\\tilde\{r\}^\{\\rho\}\. Whenηt≡η\\eta\_\{t\}\\equiv\\eta, it reduces to the constant\-stepsize identity in Section[7\.1\.1](https://arxiv.org/html/2609.05727#S7.SS1.SSS1)\.

###### Theorem 132\.

Letq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\), whereη∈C1​\(\[0,1\],ℝ\)\\eta\\in C^\{1\}\(\[0,1\];\\mathbb\{R\}\)is a time\-dependent stepsize\. Then, for any densityp∈𝒫p\\in\\mathscr\{P\}, we have

KL⁡\(p∥q\)=\(1−η1τ\)​KL⁡\(p∥ρ\)\+η1τ​\(KL⁡\(p∥π\)−KL⁡\(ρ∥π\)\)\+𝔼X∼p​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(p\\middle\\\|q\\right\)=\\left\(1\-\\frac\{\\eta\_\{1\}\}\{\\tau\}\\right\)\\operatorname\{KL\}\\left\(p\\middle\\\|\\rho\\right\)\+\\frac\{\\eta\_\{1\}\}\{\\tau\}\\left\(\\operatorname\{KL\}\\left\(p\\middle\\\|\\pi\\right\)\-\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\right\)\+\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.Consequently, whenη1\>0\\eta\_\{1\}\>0, applyingp=qp=qgives

KL⁡\(ρ∥π\)−KL⁡\(q∥π\)=\(τη1−1\)​KL⁡\(q∥ρ\)\+τη1​𝔼X∼q​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\-\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\left\(\\frac\{\\tau\}\{\\eta\_\{1\}\}\-1\\right\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)\+\\frac\{\\tau\}\{\\eta\_\{1\}\}\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.\(234\)Applyingp=πp=\\pigives

KL⁡\(π∥ρ\)−KL⁡\(π∥q\)=η1τ​\(KL⁡\(π∥ρ\)\+KL⁡\(ρ∥π\)\)−𝔼X∼π​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\-\\operatorname\{KL\}\\left\(\\pi\\middle\\\|q\\right\)=\\frac\{\\eta\_\{1\}\}\{\\tau\}\(\\operatorname\{KL\}\\left\(\\pi\\middle\\\|\\rho\\right\)\+\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\)\-\\mathbb\{E\}\_\{X\\sim\\pi\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.Applyingp=ρp=\\rhogives

KL⁡\(ρ∥q\)=𝔼X∼ρ​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\operatorname\{KL\}\\left\(\\rho\\middle\\\|q\\right\)=\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.

###### Proof\.

By Theorem[130](https://arxiv.org/html/2609.05727#Thmtheorem130), lettingf=r~ρf=\\tilde\{r\}^\{\\rho\}gives

KL⁡\(p∥q\)=\\displaystyle\\operatorname\{KL\}\\left\(p\\middle\\\|q\\right\)=KL⁡\(p∥ρ\)−η1​\(𝔼X∼p​\[r~ρ​\(X\)\]−𝔼X∼ρ​\[r~ρ​\(X\)\]\)\+𝔼X∼p​\[𝒟ρ,η,r~ρ​\(X\)\]\\displaystyle\\operatorname\{KL\}\\left\(p\\middle\\\|\\rho\\right\)\-\\eta\_\{1\}\(\\mathbb\{E\}\_\{X\\sim p\}\[\\tilde\{r\}^\{\\rho\}\(X\)\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\[\\tilde\{r\}^\{\\rho\}\(X\)\]\)\+\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]=\\displaystyle=KL⁡\(p∥ρ\)−η1τ​\(𝔼X∼p​\[log⁡π⁡\(X\)ρ⁡\(X\)\]−𝔼X∼ρ​\[log⁡π⁡\(X\)ρ⁡\(X\)\]\)\+𝔼X∼p​\[𝒟ρ,η,r~ρ​\(X\)\]\\displaystyle\\operatorname\{KL\}\\left\(p\\middle\\\|\\rho\\right\)\-\\frac\{\\eta\_\{1\}\}\{\\tau\}\\left\(\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\log\\frac\{\\pi\(X\)\}\{\\rho\(X\)\}\\right\]\-\\mathbb\{E\}\_\{X\\sim\\rho\}\\left\[\\log\\frac\{\\pi\(X\)\}\{\\rho\(X\)\}\\right\]\\right\)\+\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]=\\displaystyle=\(1−η1τ\)​KL⁡\(p∥ρ\)\+η1τ​\(KL⁡\(p∥π\)−KL⁡\(ρ∥π\)\)\+𝔼X∼p​\[𝒟ρ,η,r~ρ​\(X\)\]\.\\displaystyle\\left\(1\-\\frac\{\\eta\_\{1\}\}\{\\tau\}\\right\)\\operatorname\{KL\}\\left\(p\\middle\\\|\\rho\\right\)\+\\frac\{\\eta\_\{1\}\}\{\\tau\}\\left\(\\operatorname\{KL\}\\left\(p\\middle\\\|\\pi\\right\)\-\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\right\)\+\\mathbb\{E\}\_\{X\\sim p\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\.Applyingp=ρ,π,qp=\\rho,\\pi,qfinishes the proof\. ∎

###### Theorem 133\.

Letq=𝒯⁡\(vρ\+η​Γρ,r~ρ\)q=\\mathcal\{T\}\(v^\{\\rho\}\+\\eta\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\)\. Then, for every nondecreasing time\-dependent stepsizeη\\etawhich satisfies

∀t∈\(0,1\),ηt∈\(0,τ\],η˙t≥0,\\forall t\\in\(0,1\),\\qquad\\eta\_\{t\}\\in\(0,\\tau\],\\qquad\\dot\{\\eta\}\_\{t\}\\geq 0,the reverse KL decreases:

KL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.Furthermore, sinceπ,ρ,q\\pi,\\rho,qare positive densities onℝd\\mathbb\{R\}^\{d\}, we have

KL\(q∥π\)=KL\(ρ∥π\)⇔ρ=q=π\.\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\\quad\\Leftrightarrow\\quad\\rho=q=\\pi\.

###### Proof\.

Sinceηt\>0\\eta\_\{t\}\>0andη˙t≥0\\dot\{\\eta\}\_\{t\}\\geq 0for allt∈\(0,1\)t\\in\(0,1\), Proposition[129](https://arxiv.org/html/2609.05727#Thmtheorem129)implies𝔼X∼q​\[𝒟ρ,η,r~ρ​\(X\)\]≥0\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]\\geq 0\. Sinceη1∈\(0,τ\]\\eta\_\{1\}\\in\(0,\\tau\], \([234](https://arxiv.org/html/2609.05727#A7.E234)\) impliesKL⁡\(q∥π\)≤KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)\\leq\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\.

The implicationρ=q=π⇒KL⁡\(q∥π\)=KL⁡\(ρ∥π\)\\rho=q=\\pi\\Rightarrow\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)is immediate\. For the opposite direction, assume thatKL⁡\(q∥π\)=KL⁡\(ρ∥π\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\pi\\right\)=\\operatorname\{KL\}\\left\(\\rho\\middle\\\|\\pi\\right\)\. Proposition[129](https://arxiv.org/html/2609.05727#Thmtheorem129)and \([234](https://arxiv.org/html/2609.05727#A7.E234)\) imply that\(τη1−1\)​KL⁡\(q∥ρ\)=0\\left\(\\frac\{\\tau\}\{\\eta\_\{1\}\}\-1\\right\)\\operatorname\{KL\}\\left\(q\\middle\\\|\\rho\\right\)=0and𝔼X∼q​\[𝒟ρ,η,r~ρ​\(X\)\]=0\\mathbb\{E\}\_\{X\\sim q\}\\left\[\\mathcal\{D\}\_\{\\rho,\\eta,\\tilde\{r\}^\{\\rho\}\}\(X\)\\right\]=0\. Therefore,∇Vtρ​\[r~ρ\]a\.e\.0\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\mathrel\{\\mathop\{\\vbox\{\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\}\\kern 1\.50696pt\\hbox to12\.0695pt\{\\leaders\{\\hrule height=0\.45pt\}\{\\hfill\}\} \}\}\\limits^\{\\mathrm\{a\.e\.\}\}\}0\. Since𝔼ρ​\[\|r~ρ\|\]<∞\\mathbb\{E\}\_\{\\rho\}\[\\left\|\\tilde\{r\}^\{\\rho\}\\right\|\]<\\inftyand the convolution kernel is a nondegenerate Gaussian,Vtρ​\[r~ρ\]V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]is smooth; hence,Vtρ​\[r~ρ\]≡ctV\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\\equiv c\_\{t\}is constant\. Then, Proposition[18](https://arxiv.org/html/2609.05727#Thmtheorem18)impliesρ=π\\rho=\\pi; hence,Γρ,r~ρ=0\\Gamma^\{\\rho,\\tilde\{r\}^\{\\rho\}\}=0, which givesq=ρq=\\rho\. ∎

Theorem[133](https://arxiv.org/html/2609.05727#Thmtheorem133)establishes one\-stage reverse\-KL descent for any nondecreasing time\-dependent stepsize satisfyingηt∈\(0,τ\]\\eta\_\{t\}\\in\(0,\\tau\]\.

## Appendix HExtensions to Alternative Coordinates and Interpolants

This appendix develops two complementary representation\-level extensions of Newton Matching\. Appendix[H\.1](https://arxiv.org/html/2609.05727#A8.SS1)changes the prediction coordinate: it rewrites regression objectives for exact and approximate tangential updates and for canonicalization in score and drift coordinates\. Appendix[H\.2](https://arxiv.org/html/2609.05727#A8.SS2)instead shows that Newton Matching extends beyond the standard CFM construction adopted in the main text, which uses a Gaussian source and a linear interpolant\. As a representative alternative, it develops the corresponding canonical representations and regression objectives for a one\-sided interpolant with deterministic initial stateX0=0X\_\{0\}=0\. In both extensions, the density\-space objective, Fisher–Rao geometry, and Newton direction remain unchanged; only the prediction coordinate or interpolant construction used to realize a Newton Matching stage is modified\.

### H\.1Newton Matching under Score and Drift Coordinates

Both exact and approximate Newton Matching are formulated through velocity\-field regression\. At one stage, the tangential\-update and canonicalization targets share the generic conditional\-expectation representation

v^t​\(xt\)=\\displaystyle\\hat\{v\}\_\{t\}\(x\_\{t\}\)=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ^\(⋅\|xt\)\[𝒴vel\(t,𝒀\[t,1\]\)\],\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}^\{\\mathrm\{vel\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\],\(235\)where𝒴vel\\mathcal\{Y\}^\{\\mathrm\{vel\}\}is a sample\-wise velocity target andρ^∈𝒫\\hat\{\\rho\}\\in\\mathscr\{P\}denotes the terminal density associated with the conditional path distribution\. For endpoint\-based sampling, \([235](https://arxiv.org/html/2609.05727#A8.E235)\) reduces to

v^t\(xt\)=𝔼X1∼p1\|tρ^\(⋅\|xt\)\[𝒴^vel\(t,xt,X1\)\]\.\\hat\{v\}\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\widehat\{\\mathcal\{Y\}\}^\{\\mathrm\{vel\}\}\(t,x\_\{t\},X\_\{1\}\)\\right\]\.The corresponding velocity\-coordinate regression loss is

ℒvel​\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ^​\[‖vtθ​\(Yt\)−𝒴vel​\(t,𝒀\[t,1\]\)‖22\]\.\\mathcal\{L\}^\{\\mathrm\{vel\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\mathcal\{Y\}^\{\\mathrm\{vel\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.\(236\)HereΞ\[t,1\]ρ^\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}follows the requirement in \([82](https://arxiv.org/html/2609.05727#S8.E82)\)\. By definition, the population minimizer of \([236](https://arxiv.org/html/2609.05727#A8.E236)\) is exactly the field in \([235](https://arxiv.org/html/2609.05727#A8.E235)\)\. When the target involves stop\-gradient operations, the corresponding regression loss can be defined analogously, following the construction in Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4)\.

We now express this regression template in score and drift coordinates\.

#### H\.1\.1Score Coordinate

##### Relation between score and velocity\.

Under Gaussian source and linear interpolant, the conditional score is

st\|1​\(xt\|x1\):=∇xt​log​pt\|1​\(xt\|x1\)=−xt−αt​x1βt2,t∈\[0,1\)\.s\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\):=\\nabla\_\{x\_\{t\}\}\\log p\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\-\\frac\{x\_\{t\}\-\\alpha\_\{t\}x\_\{1\}\}\{\\beta\_\{t\}^\{2\}\},\\qquad t\\in\[0,1\)\.It is related to the conditional velocity by

vt\|1​\(xt\|x1\)=α˙tαt​xt\+κt​st\|1​\(xt\|x1\),t∈\(0,1\)\.v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\}\+\\kappa\_\{t\}s\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\),\\qquad t\\in\(0,1\)\.Given a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}, define itscanonical scoresρs^\{\\rho\}by

stρ​\(xt\):=∇xt​log​ptρ​\(xt\)\.s\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}^\{\\rho\}\(x\_\{t\}\)\.It admits the conditional\-expectation representation

stρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[st\|1\(xt\|X1\)\],t∈\[0,1\)\.s\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[s\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\],\\qquad t\\in\[0,1\)\.We then have the following velocity–score relation

vtρ​\(xt\)=α˙tαt​xt\+κt​stρ​\(xt\),t∈\(0,1\)\.v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\}\+\\kappa\_\{t\}s\_\{t\}^\{\\rho\}\(x\_\{t\}\),\\qquad t\\in\(0,1\)\.\(237\)

##### Tangential updates in score coordinates\.

The relation above defines an affine change of prediction coordinates at every interior time\. For the velocity fieldv^\\hat\{v\}in \([235](https://arxiv.org/html/2609.05727#A8.E235)\), define its score\-coordinate representation by

s^t​\(xt\):=1κt​\(v^t​\(xt\)−α˙tαt​xt\),t∈\(0,1\)\.\\hat\{s\}\_\{t\}\(x\_\{t\}\):=\\frac\{1\}\{\\kappa\_\{t\}\}\\left\(\\hat\{v\}\_\{t\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\}\\right\),\\qquad t\\in\(0,1\)\.Whenv^\\hat\{v\}is canonical,s^\\hat\{s\}coincides with the marginal score of the associated canonical probability path\. For a general noncanonical fieldv^\\hat\{v\}, however,s^\\hat\{s\}is only its score\-coordinate representation\.

Applying the same coordinate transformation to the sample\-wise velocity target, define

𝒴score​\(t,𝒀\[t,1\]\):=1κt​\(𝒴vel​\(t,𝒀\[t,1\]\)−α˙tαt​Yt\)\.\\mathcal\{Y\}^\{\\mathrm\{score\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\):=\\frac\{1\}\{\\kappa\_\{t\}\}\\left\(\\mathcal\{Y\}^\{\\mathrm\{vel\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\)\.SinceYt=xtY\_\{t\}=x\_\{t\}under the conditional path distribution, \([235](https://arxiv.org/html/2609.05727#A8.E235)\) becomes

s^t​\(xt\)=\\displaystyle\\hat\{s\}\_\{t\}\(x\_\{t\}\)=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ^\(⋅\|xt\)\[𝒴score\(t,𝒀\[t,1\]\)\]\.\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}^\{\\mathrm\{score\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.
Suppose that the velocity model is parameterized through a score model as

vtθ​\(x\)=α˙tαt​x\+κt​stθ​\(x\)\.v\_\{t\}^\{\\theta\}\(x\)=\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\+\\kappa\_\{t\}s\_\{t\}^\{\\theta\}\(x\)\.Then, the velocity loss in \([236](https://arxiv.org/html/2609.05727#A8.E236)\) can be written as

ℒvel​\(θ\)=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{vel\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ^​\[‖κt​stθ​\(Yt\)\+α˙tαt​Yt−𝒴vel​\(t,𝒀\[t,1\]\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\left\\\|\\kappa\_\{t\}s\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\-\\mathcal\{Y\}^\{\\mathrm\{vel\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\]=\\displaystyle=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ^​\[κt2​‖stθ​\(Yt\)−𝒴score​\(t,𝒀\[t,1\]\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\kappa\_\{t\}^\{2\}\\left\\\|s\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\mathcal\{Y\}^\{\\mathrm\{score\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.More generally, consider the weighted score\-coordinate regression loss

ℒωscore​\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ^​\[ωt​\(Yt\)​‖stθ​\(Yt\)−𝒴score​\(t,𝒀\[t,1\]\)‖22\],\\mathcal\{L\}\_\{\\omega\}^\{\\mathrm\{score\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\omega\_\{t\}\(Y\_\{t\}\)\\left\\\|s\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\mathcal\{Y\}^\{\\mathrm\{score\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\],where the positive weightωt​\(xt\)\>0\\omega\_\{t\}\(x\_\{t\}\)\>0preserves the same population minimizer and may be chosen to mitigate endpoint singularities\. The choiceωt≡κt2\\omega\_\{t\}\\equiv\\kappa\_\{t\}^\{2\}makesℒωscore​\(θ\)\\mathcal\{L\}\_\{\\omega\}^\{\\mathrm\{score\}\}\(\\theta\)andℒvel​\(θ\)\\mathcal\{L\}^\{\\mathrm\{vel\}\}\(\\theta\)equal sample\-wise\.

##### Examples of tangential updates\.

The coordinate transformation above applies directly to the exact and approximate tangential\-update losses\. In exact Newton Matching, the tangential update in score coordinates is

s^t=stρ\+ηκtΓtρ,r~ρ=stρ\+η∇Vtρ\[r~ρ\]\.\\hat\{s\}\_\{t\}=s\_\{t\}^\{\\rho\}\+\\frac\{\\eta\}\{\\kappa\_\{t\}\}\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}=s\_\{t\}^\{\\rho\}\+\\eta\\nabla V\_\{t\}^\{\\rho\}\[\\tilde\{r\}^\{\\rho\}\]\.For example, the covariance\-form of exact Newton Matching in \([56](https://arxiv.org/html/2609.05727#S8.E56)\)–\([57](https://arxiv.org/html/2609.05727#S8.E57)\) becomes

ℒρ,η,B,ωcov,score​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B,\\omega\}^\{\\mathrm\{cov\},\\mathrm\{score\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[ωt\(Xt\)∥stθ\(Xt\)−st\|1\(Xt\|X1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\Big\[\\omega\_\{t\}\(X\_\{t\}\)\\big\\\|s\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-s\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−η\(r~ρ\(X1\)−Bt\(Xt\)\)\(st\|1\(Xt\|X1\)−stρ\(Xt\)\)∥22\]\.\\displaystyle\\qquad\\qquad\-\\eta\\left\(\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(s\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-s\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\)\\big\\\|\_\{2\}^\{2\}\\Big\]\.Herer~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)can be computed through the ODE\-based or SDE\-based methods introduced in Section[8\.3](https://arxiv.org/html/2609.05727#S8.SS3), with the canonical velocityvρv^\{\\rho\}recovered from the canonical score through \([237](https://arxiv.org/html/2609.05727#A8.E237)\)\.

The Bolza realization of exact Newton Matching in \([94](https://arxiv.org/html/2609.05727#S8.E94)\)–\([95](https://arxiv.org/html/2609.05727#S8.E95)\) becomes

ℒρ,η,ωgrad,Bolza,score\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ\[ωt\(Yt\)∥\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{score\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\rho\}\\end\{subarray\}\}\\bigg\[\\omega\_\{t\}\(Y\_\{t\}\)\\Big\\\|stθ​\(Yt\)−\(1−ητ\)​st\|1​\(Yt\|Y1\)\\displaystyle s\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)s\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητstμ\(Yt\)−ηλtbρ,r,lρ,μ\(𝒀\[t,1\]\)∥22\]\.\\displaystyle\-\\frac\{\\eta\}\{\\tau\}s\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-\\eta\\lambda\_\{t\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.Under the reverse construction, the conditional path𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)can be sampled from the posterior\-preserving SDE

d​Yuρ=\(2​κu​suρ​\(Yuρ\)\+α˙uαu​Yuρ\)​d​u\+2​κu​d​Wu,u∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{u\}^\{\\rho\}=\\left\(2\\kappa\_\{u\}s\_\{u\}^\{\\rho\}\(Y\_\{u\}^\{\\rho\}\)\+\\frac\{\\dot\{\\alpha\}\_\{u\}\}\{\\alpha\_\{u\}\}Y\_\{u\}^\{\\rho\}\\right\)\\mathrm\{d\}u\+\\sqrt\{2\\kappa\_\{u\}\}\\mathrm\{d\}W\_\{u\},\\qquad u\\in\[t,1\],Ytρ=xt\.\\displaystyle Y\_\{t\}^\{\\rho\}=x\_\{t\}\.The pathwise adjoint satisfies

dd​uλubρ,r,lρ,μ\(𝒀\[t,1\]\)=−∇\(2κusuρ\(Yuρ\)\+α˙uαuYuρ\)⊤λubρ,r,lρ,μ\(𝒀\[t,1\]\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}u\}\\lambda\_\{u\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla\\left\(2\\kappa\_\{u\}s\_\{u\}^\{\\rho\}\(Y\_\{u\}^\{\\rho\}\)\+\\frac\{\\dot\{\\alpha\}\_\{u\}\}\{\\alpha\_\{u\}\}Y\_\{u\}^\{\\rho\}\\right\)^\{\\top\}\\lambda\_\{u\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+κuτ∇\(‖suρ\(Yu\)−suμ\(Yu\)‖22\),u∈\[t,1\],\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\+\\frac\{\\kappa\_\{u\}\}\{\\tau\}\\nabla\\left\(\\left\\\|s\_\{u\}^\{\\rho\}\(Y\_\{u\}\)\-s\_\{u\}^\{\\mu\}\(Y\_\{u\}\)\\right\\\|\_\{2\}^\{2\}\\right\),\\qquad u\\in\[t,1\],λ1bρ,r,lρ,μ​\(𝒀\[t,1\]\)=∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\.In particular, whenμ=1\\mu=1, we use

st1​\(xt\)=0\.s\_\{t\}^\{1\}\(x\_\{t\}\)=0\.
As an example of applying the stop\-gradient trick in score coordinates, the Gaussian\-kernel approximation objective in \([142](https://arxiv.org/html/2609.05727#S9.E142)\) becomes

ℒη,Σ^,ωGau​\-​ker,score,sg\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1sg⁡\(θ\)\[ωt\(Xt\)∥stθ\(Xt\)−st\|1\(Xt\|X1\)\\displaystyle\\mathcal\{L\}\_\{\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{score\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\Bigg\[\\omega\_\{t\}\(X\_\{t\}\)\\bigg\\\|s\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-s\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−ηαtβt2Σ^t\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\-\\eta\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}\(X\_\{t\}\)\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.

##### Score canonicalization\.

We demonstrate how to perform the explicit canonicalization in \([106](https://arxiv.org/html/2609.05727#S8.E106)\) for score coordinates\. We provide two options\.

Given a noncanonical velocity fieldv^\\hat\{v\}with score\-coordinate representations^\\hat\{s\}, let

q=𝒯⁡\(v^\)q=\\mathcal\{T\}\(\\hat\{v\}\)denote its terminal density\. A direct score\-coordinate realization for recovering the canonical scoresqs^\{q\}is

ℒωcan,score​\(θ\):=𝔼t∼U⁡\(0,1\),X1∼q,X0∼p0,Xt=αt​X1\+βt​X0​\[ωt​\(Xt\)​‖stθ​\(Xt\)−st\|1​\(Xt\|X1\)‖22\]\.\\mathcal\{L\}^\{\\mathrm\{can\},\\mathrm\{score\}\}\_\{\\omega\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim q,\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\omega\_\{t\}\(X\_\{t\}\)\\left\\\|s\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-s\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.The endpointX1∼qX\_\{1\}\\sim qcan be generated from the ODE

d​Ytd​t=κt​s^t​\(Yt\)\+α˙tαt​Yt,Y0∼p0,\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=\\kappa\_\{t\}\\hat\{s\}\_\{t\}\(Y\_\{t\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\},\\qquad Y\_\{0\}\\sim p\_\{0\},withX1:=Y1X\_\{1\}:=Y\_\{1\}\.

Alternatively, we can generalize the notion of the terminal\-density map to score coordinates\. Given a score fields^\\hat\{s\}, consider the SDE

d​Yt=\(2​κt​s^t​\(Yt\)\+α˙tαt​Yt\)​d​t\+2​κt​d​Wt,t∈\(0,1\),\\displaystyle\\mathrm\{d\}Y\_\{t\}=\\left\(2\\kappa\_\{t\}\\hat\{s\}\_\{t\}\(Y\_\{t\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\)\\mathrm\{d\}t\+\\sqrt\{2\\kappa\_\{t\}\}\\mathrm\{d\}W\_\{t\},\\qquad t\\in\(0,1\),Y0∼p0,\\displaystyle Y\_\{0\}\\sim p\_\{0\},and define

q^:=𝒯score​\(s^\)\\hat\{q\}:=\\mathcal\{T\}^\{\\mathrm\{score\}\}\(\\hat\{s\}\)to be its terminal density\. We again setX1:=Y1X\_\{1\}:=Y\_\{1\}\. For a canonical scoresρs^\{\\rho\}, this score\-coordinate terminal\-density map agrees with the original terminal\-density map:

𝒯score​\(sρ\)=𝒯⁡\(vρ\)=ρ\.\\mathcal\{T\}^\{\\mathrm\{score\}\}\(s^\{\\rho\}\)=\\mathcal\{T\}\(v^\{\\rho\}\)=\\rho\.By contrast, for a general noncanonical velocity fieldv^\\hat\{v\}and its score\-coordinate representations^\\hat\{s\}, the identity

𝒯score​\(s^\)=𝒯​\(v^\)\\mathcal\{T\}^\{\\mathrm\{score\}\}\(\\hat\{s\}\)=\\mathcal\{T\}\(\\hat\{v\}\)need not hold\. Nevertheless,q^\\hat\{q\}is a well\-defined terminal density induced by the score fields^\\hat\{s\}, and endpoint samplesX1∼q^X\_\{1\}\\sim\\hat\{q\}can be used in the standard conditional score regression\. The corresponding SDE\-based canonicalization loss is

ℒ^ωcan,score​\(θ\):=𝔼t∼U⁡\(0,1\),X1∼q^,X0∼p0,Xt=αt​X1\+βt​X0​\[ωt​\(Xt\)​‖stθ​\(Xt\)−st\|1​\(Xt\|X1\)‖22\]\.\\widehat\{\\mathcal\{L\}\}^\{\\mathrm\{can\},\\mathrm\{score\}\}\_\{\\omega\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\hat\{q\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\omega\_\{t\}\(X\_\{t\}\)\\left\\\|s\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-s\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Its population minimizer is exactly the canonical scorestq^​\(xt\)s\_\{t\}^\{\\hat\{q\}\}\(x\_\{t\}\)of the SDE\-induced terminal densityq^\\hat\{q\}\.

More broadly, the canonical\-score assignmentρ↦sρ\\rho\\mapsto s^\{\\rho\}and the score\-coordinate terminal\-density map𝒯score\\mathcal\{T\}^\{\\mathrm\{score\}\}suggest a score\-space counterpart to the pair\(𝒞,𝒯\)\(\\mathcal\{C\},\\mathcal\{T\}\)\. Developing this viewpoint into a complete score\-space formulation, including an appropriate canonical manifold, transported density\-space geometry, canonical retraction, Newton direction, and convergence theory, would require further careful analysis\. We leave this extension, together with its drift\-coordinate analogue, to future work\.

#### H\.1\.2Drift Coordinate

Having rewritten the common regression structure in score coordinates, we now carry out the analogous construction in drift coordinates\. For simplicity, we consider the diffusion coefficient2​κt\\sqrt\{2\\kappa\_\{t\}\}from the posterior\-preserving SDE\.

##### Relation between drift and velocity\.

For a marginal probability path\(pt\)t∈\[0,1\]\(p\_\{t\}\)\_\{t\\in\[0,1\]\}associated with velocityvvand scoress, define the drift by

bt​\(xt\):=vt​\(xt\)\+κt​st​\(xt\)\.b\_\{t\}\(x\_\{t\}\):=v\_\{t\}\(x\_\{t\}\)\+\\kappa\_\{t\}s\_\{t\}\(x\_\{t\}\)\.The corresponding SDE

d​Yt=bt​\(Yt\)​d​t\+2​κt​d​Wt\\mathrm\{d\}Y\_\{t\}=b\_\{t\}\(Y\_\{t\}\)\\mathrm\{d\}t\+\\sqrt\{2\\kappa\_\{t\}\}\\mathrm\{d\}W\_\{t\}shares the same marginal probability path as the ODE ofvv\.

Under Gaussian source and linear interpolant, the conditional drift is

bt\|1​\(xt\|x1\):=vt\|1​\(xt\|x1\)\+κt​st\|1​\(xt\|x1\)=2​vt\|1​\(xt\|x1\)−α˙tαt​xt,t∈\(0,1\)\.b\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\):=v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\+\\kappa\_\{t\}s\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=2v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\},\\qquad t\\in\(0,1\)\.Given a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}, define itscanonical driftbρb^\{\\rho\}by

btρ​\(xt\):=vtρ​\(xt\)\+κt​stρ​\(xt\)=2​vtρ​\(xt\)−α˙tαt​xt,t∈\(0,1\)\.b\_\{t\}^\{\\rho\}\(x\_\{t\}\):=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\kappa\_\{t\}s\_\{t\}^\{\\rho\}\(x\_\{t\}\)=2v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\},\\qquad t\\in\(0,1\)\.It admits the conditional\-expectation representation

btρ\(xt\)=𝔼X1∼p1\|tρ\(⋅\|xt\)\[bt\|1\(xt\|X1\)\]\.b\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[b\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]\.Equivalently, the canonical velocity can be recovered from the canonical drift through

vtρ​\(xt\)=12​\(btρ​\(xt\)\+α˙tαt​xt\),t∈\(0,1\)\.v\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\frac\{1\}\{2\}\\left\(b\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\}\\right\),\\qquad t\\in\(0,1\)\.\(238\)

##### Tangential updates in drift coordinates\.

The relation above defines another affine change of prediction coordinates at every interior time\. For the velocity fieldv^\\hat\{v\}in \([235](https://arxiv.org/html/2609.05727#A8.E235)\), define its drift\-coordinate representation by

b^t​\(xt\):=2​v^t​\(xt\)−α˙tαt​xt,t∈\(0,1\)\.\\hat\{b\}\_\{t\}\(x\_\{t\}\):=2\\hat\{v\}\_\{t\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\},\\qquad t\\in\(0,1\)\.Whenv^\\hat\{v\}is canonical,b^\\hat\{b\}coincides with the marginal drift of the associated canonical probability path\. For a general noncanonical fieldv^\\hat\{v\}, however,b^\\hat\{b\}is only its drift\-coordinate representation\.

Applying the same coordinate transformation to the sample\-wise velocity target, define

𝒴drift​\(t,𝒀\[t,1\]\):=2​𝒴vel​\(t,𝒀\[t,1\]\)−α˙tαt​Yt\.\\mathcal\{Y\}^\{\\mathrm\{drift\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\):=2\\mathcal\{Y\}^\{\\mathrm\{vel\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\.SinceYt=xtY\_\{t\}=x\_\{t\}under the conditional path distribution, \([235](https://arxiv.org/html/2609.05727#A8.E235)\) becomes

b^t​\(xt\)=\\displaystyle\\hat\{b\}\_\{t\}\(x\_\{t\}\)=𝔼𝒀\[t,1\]∼ℙ\[t,1\]\|tρ^\(⋅\|xt\)\[𝒴drift\(t,𝒀\[t,1\]\)\]\.\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\mathcal\{Y\}^\{\\mathrm\{drift\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\]\.
Suppose that the velocity model is parameterized through a drift model as

vtθ​\(x\)=12​\(btθ​\(x\)\+α˙tαt​x\)\.v\_\{t\}^\{\\theta\}\(x\)=\\frac\{1\}\{2\}\\left\(b\_\{t\}^\{\\theta\}\(x\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\\right\)\.Then, the velocity loss in \([236](https://arxiv.org/html/2609.05727#A8.E236)\) can be written as

ℒvel​\(θ\)=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{vel\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ^​\[‖12​\(btθ​\(Yt\)\+α˙tαt​Yt\)−𝒴vel​\(t,𝒀\[t,1\]\)‖22\]\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\left\\\|\\frac\{1\}\{2\}\\left\(b\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\)\-\\mathcal\{Y\}^\{\\mathrm\{vel\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\]=\\displaystyle=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ^​\[14​‖btθ​\(Yt\)−𝒴drift​\(t,𝒀\[t,1\]\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\frac\{1\}\{4\}\\left\\\|b\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\mathcal\{Y\}^\{\\mathrm\{drift\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.More generally, consider the weighted drift\-coordinate regression loss

ℒωdrift​\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ^​\[ωt​\(Yt\)​‖btθ​\(Yt\)−𝒴drift​\(t,𝒀\[t,1\]\)‖22\],\\mathcal\{L\}\_\{\\omega\}^\{\\mathrm\{drift\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\end\{subarray\}\}\\left\[\\omega\_\{t\}\(Y\_\{t\}\)\\left\\\|b\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\mathcal\{Y\}^\{\\mathrm\{drift\}\}\(t,\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\],where the positive weightωt​\(xt\)\>0\\omega\_\{t\}\(x\_\{t\}\)\>0preserves the same population minimizer\. The choiceωt≡14\\omega\_\{t\}\\equiv\\frac\{1\}\{4\}makesℒωdrift​\(θ\)\\mathcal\{L\}\_\{\\omega\}^\{\\mathrm\{drift\}\}\(\\theta\)andℒvel​\(θ\)\\mathcal\{L\}^\{\\mathrm\{vel\}\}\(\\theta\)equal sample\-wise\.

##### Examples of tangential updates\.

As in score coordinates, the transformation above applies directly to the exact and approximate tangential\-update losses\. In exact Newton Matching, the tangential update under the drift coordinate is

b^t=btρ\+2​η​Γtρ,r~ρ\.\\hat\{b\}\_\{t\}=b\_\{t\}^\{\\rho\}\+2\\eta\\Gamma\_\{t\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.For example, the covariance\-form of exact Newton Matching in \([56](https://arxiv.org/html/2609.05727#S8.E56)\)–\([57](https://arxiv.org/html/2609.05727#S8.E57)\) becomes

ℒρ,η,B,ωcov,drift​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B,\\omega\}^\{\\mathrm\{cov\},\\mathrm\{drift\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[ωt\(Xt\)∥btθ\(Xt\)−bt\|1\(Xt\|X1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\Big\[\\omega\_\{t\}\(X\_\{t\}\)\\big\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−η\(r~ρ\(X1\)−Bt\(Xt\)\)\(bt\|1\(Xt\|X1\)−btρ\(Xt\)\)∥22\]\.\\displaystyle\\qquad\\qquad\-\\eta\\left\(\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-b\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\)\\big\\\|\_\{2\}^\{2\}\\Big\]\.Herer~ρ​\(X1\)\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)can be computed through the ODE\-based or SDE\-based methods introduced in Section[8\.3](https://arxiv.org/html/2609.05727#S8.SS3), with the canonical velocityvρv^\{\\rho\}recovered from the drift through \([238](https://arxiv.org/html/2609.05727#A8.E238)\)\.

The Bolza realization of exact Newton Matching in \([94](https://arxiv.org/html/2609.05727#S8.E94)\)–\([95](https://arxiv.org/html/2609.05727#S8.E95)\) becomes

ℒρ,η,ωgrad,Bolza,drift\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]ρ\[ωt\(Yt\)∥\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{drift\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\rho\}\\end\{subarray\}\}\\bigg\[\\omega\_\{t\}\(Y\_\{t\}\)\\Big\\\|btθ​\(Yt\)−\(1−ητ\)​bt\|1​\(Yt\|Y1\)\\displaystyle b\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)b\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητbtμ\(Yt\)−2ηκtλtbρ,r,lρ,μ\(𝒀\[t,1\]\)∥22\]\.\\displaystyle\-\\frac\{\\eta\}\{\\tau\}b\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-2\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.Under the reverse construction, the conditional path𝒀\[t,1\]∼ℙ\[t,1\]\|tρ\(⋅\|Xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)can be sampled directly from the posterior\-preserving SDE

d​Yuρ=buρ​\(Yuρ\)​d​u\+2​κu​d​Wu,u∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{u\}^\{\\rho\}=b\_\{u\}^\{\\rho\}\(Y\_\{u\}^\{\\rho\}\)\\mathrm\{d\}u\+\\sqrt\{2\\kappa\_\{u\}\}\\mathrm\{d\}W\_\{u\},\\qquad u\\in\[t,1\],Ytρ=xt\.\\displaystyle Y\_\{t\}^\{\\rho\}=x\_\{t\}\.The pathwise adjoint satisfies

dd​uλubρ,r,lρ,μ\(𝒀\[t,1\]\)=−∇buρ\(Yuρ\)⊤λubρ,r,lρ,μ\(𝒀\[t,1\]\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}u\}\\lambda\_\{u\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla b\_\{u\}^\{\\rho\}\(Y\_\{u\}^\{\\rho\}\)^\{\\top\}\\lambda\_\{u\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+14​τ​κu∇\(‖buρ\(Yu\)−buμ\(Yu\)‖22\),u∈\[t,1\],\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\+\\frac\{1\}\{4\\tau\\kappa\_\{u\}\}\\nabla\\left\(\\left\\\|b\_\{u\}^\{\\rho\}\(Y\_\{u\}\)\-b\_\{u\}^\{\\mu\}\(Y\_\{u\}\)\\right\\\|\_\{2\}^\{2\}\\right\),\\qquad u\\in\[t,1\],λ1bρ,r,lρ,μ​\(𝒀\[t,1\]\)=∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{b^\{\\rho\},r,l^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\.In particular, whenμ=1\\mu=1, we use

bt1​\(xt\)=α˙tαt​xt\.b\_\{t\}^\{1\}\(x\_\{t\}\)=\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\}\.
As an example of applying the stop\-gradient trick in drift coordinates, the Gaussian\-kernel approximation objective in \([142](https://arxiv.org/html/2609.05727#S9.E142)\) becomes

ℒη,Σ^Gau​\-​ker,drift,sg\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1sg⁡\(θ\)\[∥btθ\(Xt\)−bt\|1\(Xt\|X1\)\\displaystyle\\mathcal\{L\}\_\{\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−2ηαt​κtβt2Σ^t\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\-2\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}\(X\_\{t\}\)\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.

##### Drift canonicalization\.

We demonstrate how to perform the explicit canonicalization in \([106](https://arxiv.org/html/2609.05727#S8.E106)\) for drift coordinates\. We provide two options\.

Given a noncanonical velocity fieldv^\\hat\{v\}with drift\-coordinate representationb^\\hat\{b\}, let

q=𝒯⁡\(v^\)q=\\mathcal\{T\}\(\\hat\{v\}\)denote its terminal density\. A direct drift\-coordinate realization for recovering the canonical driftbqb^\{q\}is

ℒωcan,drift​\(θ\):=𝔼t∼U⁡\(0,1\),X1∼q,X0∼p0,Xt=αt​X1\+βt​X0​\[ωt​\(Xt\)​‖btθ​\(Xt\)−bt\|1​\(Xt\|X1\)‖22\]\.\\mathcal\{L\}^\{\\mathrm\{can\},\\mathrm\{drift\}\}\_\{\\omega\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim q,\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\omega\_\{t\}\(X\_\{t\}\)\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.The endpointX1∼qX\_\{1\}\\sim qcan be generated from the ODE

d​Ytd​t=12​\(b^t​\(Yt\)\+α˙tαt​Yt\),Y0∼p0,\\frac\{\\mathrm\{d\}Y\_\{t\}\}\{\\mathrm\{d\}t\}=\\frac\{1\}\{2\}\\left\(\\hat\{b\}\_\{t\}\(Y\_\{t\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\),\\qquad Y\_\{0\}\\sim p\_\{0\},withX1:=Y1X\_\{1\}:=Y\_\{1\}\.

Alternatively, we can generalize the notion of the terminal\-density map to drift coordinates\. Given a drift fieldb^\\hat\{b\}, consider the SDE

d​Yt=b^t​\(Yt\)​d​t\+2​κt​d​Wt,t∈\(0,1\),\\displaystyle\\mathrm\{d\}Y\_\{t\}=\\hat\{b\}\_\{t\}\(Y\_\{t\}\)\\mathrm\{d\}t\+\\sqrt\{2\\kappa\_\{t\}\}\\mathrm\{d\}W\_\{t\},\\qquad t\\in\(0,1\),Y0∼p0,\\displaystyle Y\_\{0\}\\sim p\_\{0\},and define

q^:=𝒯drift​\(b^\)\\hat\{q\}:=\\mathcal\{T\}^\{\\mathrm\{drift\}\}\(\\hat\{b\}\)to be its terminal density\. We again setX1:=Y1X\_\{1\}:=Y\_\{1\}\. For a canonical driftbρb^\{\\rho\}, this drift\-coordinate terminal\-density map agrees with the original terminal\-density map:

𝒯drift​\(bρ\)=𝒯⁡\(vρ\)=ρ\.\\mathcal\{T\}^\{\\mathrm\{drift\}\}\(b^\{\\rho\}\)=\\mathcal\{T\}\(v^\{\\rho\}\)=\\rho\.By contrast, for a general noncanonical velocity fieldv^\\hat\{v\}and its drift\-coordinate representationb^\\hat\{b\}, the identity

𝒯drift​\(b^\)=𝒯​\(v^\)\\mathcal\{T\}^\{\\mathrm\{drift\}\}\(\\hat\{b\}\)=\\mathcal\{T\}\(\\hat\{v\}\)need not hold\. Nevertheless,q^\\hat\{q\}is a well\-defined terminal density induced by the drift fieldb^\\hat\{b\}, and endpoint samplesX1∼q^X\_\{1\}\\sim\\hat\{q\}can be used in the standard conditional drift regression\. The corresponding SDE\-based canonicalization loss is

ℒ^ωcan,drift​\(θ\):=𝔼t∼U⁡\(0,1\),X1∼q^,X0∼p0,Xt=αt​X1\+βt​X0​\[ωt​\(Xt\)​‖btθ​\(Xt\)−bt\|1​\(Xt\|X1\)‖22\]\.\\widehat\{\\mathcal\{L\}\}^\{\\mathrm\{can\},\\mathrm\{drift\}\}\_\{\\omega\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\hat\{q\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\omega\_\{t\}\(X\_\{t\}\)\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Its population minimizer is exactly the canonical driftbtq^​\(xt\)b\_\{t\}^\{\\hat\{q\}\}\(x\_\{t\}\)of the SDE\-induced terminal densityq^\\hat\{q\}\.

### H\.2Newton Matching under One\-Sided Interpolants

In this subsection, we extend Newton Matching beyond the standard CFM construction adopted in the main text, which uses a Gaussian source and a linear interpolant\. Specifically, we consider a one\-sided interpolant\[[2](https://arxiv.org/html/2609.05727#bib.bib50)\]with deterministic initial stateX0=0X\_\{0\}=0\. For suitable interpolant schedules, this one\-sided construction also admits a Schrödinger half\-bridge interpretation with a linear\-Gaussian reference\[[38](https://arxiv.org/html/2609.05727#bib.bib49)\]\. The constructions in Appendices[D](https://arxiv.org/html/2609.05727#A4)and[E](https://arxiv.org/html/2609.05727#A5)remain applicable in this setting\.

#### H\.2\.1Canonical Velocity, Score, and Drift

The linear interpolantXt=αt​X1\+βt​X0X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}in this paper is associated with a standard Gaussian source and the interpolant schedule satisfies the boundary conditionsα0=0\\alpha\_\{0\}=0,β0=1\\beta\_\{0\}=1,α1=1\\alpha\_\{1\}=1, andβ1=0\\beta\_\{1\}=0\. For the one\-sided interpolant considered here, the source is instead the deterministic endpointX0=0X\_\{0\}=0\. Thus, for a terminal densityρ∈𝒫\\rho\\in\\mathscr\{P\}, we consider

Xt=α~t​X1\+β~t​Z,X1∼ρ,Z∼𝒩⁡\(0,I\),X\_\{t\}=\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\+\\widetilde\{\\beta\}\_\{t\}Z,\\qquad X\_\{1\}\\sim\\rho,\\qquad Z\\sim\\mathcal\{N\}\(0,I\),\(239\)with the boundary conditions

α~0=0,β~0=0,α~1=1,β~1=0\.\\widetilde\{\\alpha\}\_\{0\}=0,\\qquad\\widetilde\{\\beta\}\_\{0\}=0,\\qquad\\widetilde\{\\alpha\}\_\{1\}=1,\\qquad\\widetilde\{\\beta\}\_\{1\}=0\.We assume thatα~\\widetilde\{\\alpha\}andβ~\\widetilde\{\\beta\}are continuous on\[0,1\]\[0,1\], differentiable on\(0,1\)\(0,1\), and strictly positive at every interior time\. Define

κ~t:=β~tα~t​\(α~˙t​β~t−α~t​β~˙t\),\\widetilde\{\\kappa\}\_\{t\}:=\\frac\{\\widetilde\{\\beta\}\_\{t\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}\\left\(\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}\\widetilde\{\\beta\}\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}\\dot\{\\widetilde\{\\beta\}\}\_\{t\}\\right\),and assumeκ~t\>0\\widetilde\{\\kappa\}\_\{t\}\>0for everyt∈\(0,1\)t\\in\(0,1\)\.

For every interior timet∈\(0,1\)t\\in\(0,1\), define

p~t\|1​\(xt\|x1\):=𝒩⁡\(xt,α~t​x1,β~t2​I\),p~tρ​\(xt\):=∫ℝdρ⁡\(x1\)​p~t\|1​\(xt\|x1\)​d​x1\.\\widetilde\{p\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\):=\\mathcal\{N\}\(x\_\{t\};\\widetilde\{\\alpha\}\_\{t\}x\_\{1\},\\widetilde\{\\beta\}\_\{t\}^\{2\}I\),\\qquad\\widetilde\{p\}\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\(x\_\{1\}\)\\widetilde\{p\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\\mathrm\{d\}x\_\{1\}\.The corresponding endpoint posterior is

p~1\|tρ​\(x1\|xt\):=ρ⁡\(x1\)​p~t\|1​\(xt\|x1\)p~tρ​\(xt\)\.\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(x\_\{1\}\|x\_\{t\}\):=\\frac\{\\rho\(x\_\{1\}\)\\widetilde\{p\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\}\{\\widetilde\{p\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\}\.
The conditional velocity associated with \([239](https://arxiv.org/html/2609.05727#A8.E239)\) is

v~t\|1​\(xt\|x1\)=β~˙tβ~t​xt\+α~t​κ~tβ~t2​x1\.\\widetilde\{v\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\frac\{\\dot\{\\widetilde\{\\beta\}\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}\}x\_\{t\}\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}x\_\{1\}\.Under \([239](https://arxiv.org/html/2609.05727#A8.E239)\), we have

v~t\|1​\(Xt\|X1\)=α~˙t​X1\+β~˙t​Z\.\\widetilde\{v\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)=\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}X\_\{1\}\+\\dot\{\\widetilde\{\\beta\}\}\_\{t\}Z\.Under this one\-sided setting, we define thecanonical velocity fieldofρ\\rhoby

v~tρ\(xt\):=𝔼X1∼p~1\|tρ\(⋅\|xt\)\[v~t\|1\(xt\|X1\)\]=α~˙tα~txt\+κ~t∇logp~tρ\(xt\)\.\\widetilde\{v\}\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\widetilde\{v\}\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]=\\frac\{\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}x\_\{t\}\+\\widetilde\{\\kappa\}\_\{t\}\\nabla\\log\\widetilde\{p\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\.
The conditional score is

s~t\|1​\(xt\|x1\):=∇xt​log​p~t\|1​\(xt\|x1\)=−xt−α~t​x1β~t2\.\\widetilde\{s\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\):=\\nabla\_\{x\_\{t\}\}\\log\\widetilde\{p\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\-\\frac\{x\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}x\_\{1\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\.Thecanonical scoreis

s~tρ\(xt\):=𝔼X1∼p~1\|tρ\(⋅\|xt\)\[s~t\|1\(xt\|X1\)\]=∇xtlogp~tρ\(xt\)\.\\widetilde\{s\}\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\widetilde\{s\}\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]=\\nabla\_\{x\_\{t\}\}\\log\\widetilde\{p\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\.
Define the conditional drift as

b~t\|1​\(xt\|x1\):=v~t\|1​\(xt\|x1\)\+κ~t​∇xt​log⁡p~t\|1​\(xt\|x1\)=v~t\|1​\(xt\|x1\)−κ~tβ~t2​\(xt−α~t​x1\),\\widetilde\{b\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\):=\\widetilde\{v\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\+\\widetilde\{\\kappa\}\_\{t\}\\nabla\_\{x\_\{t\}\}\\log\\widetilde\{p\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\widetilde\{v\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\-\\frac\{\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\(x\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}x\_\{1\}\),and its marginalization gives thecanonical drift

b~tρ\(xt\):=𝔼X1∼p~1\|tρ\(⋅\|xt\)\[b~t\|1\(xt\|X1\)\]=v~tρ\(xt\)\+κ~t∇logp~tρ\(xt\)=2v~tρ\(xt\)−α~˙tα~txt\.\\widetilde\{b\}\_\{t\}^\{\\rho\}\(x\_\{t\}\):=\\mathbb\{E\}\_\{X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\widetilde\{b\}\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\)\\right\]=\\widetilde\{v\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+\\widetilde\{\\kappa\}\_\{t\}\\nabla\\log\\widetilde\{p\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)=2\\widetilde\{v\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-\\frac\{\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}x\_\{t\}\.
In particular, forμ=1\\mu=1, we use

s~t1​\(x\)=0,v~t1​\(x\)=b~t1​\(x\)=α~˙tα~t​x\.\\widetilde\{s\}\_\{t\}^\{1\}\(x\)=0,\\qquad\\widetilde\{v\}\_\{t\}^\{1\}\(x\)=\\widetilde\{b\}\_\{t\}^\{1\}\(x\)=\\frac\{\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}x\.
Consider the posterior\-preserving SDE

d​Yu=b~uρ​\(Yu\)​d​u\+2​κ~u​d​Wu,u∈\[t,1\],\\displaystyle\\mathrm\{d\}Y\_\{u\}=\\widetilde\{b\}\_\{u\}^\{\\rho\}\(Y\_\{u\}\)\\mathrm\{d\}u\+\\sqrt\{2\\widetilde\{\\kappa\}\_\{u\}\}\\mathrm\{d\}W\_\{u\},\\qquad u\\in\[t,1\],\(240a\)Yt=xt\.\\displaystyle Y\_\{t\}=x\_\{t\}\.\(240b\)Then, we have

Y1∼p~1\|tρ\(⋅\|xt\)\.Y\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\.The path distribution is accordingly denoted by

𝒀\[t,1\]:=\(Yu\)u∈\[t,1\]∼ℙ~\[t,1\]\|tρ\(⋅\|xt\)\.\\bm\{Y\}\_\{\[t,1\]\}:=\(Y\_\{u\}\)\_\{u\\in\[t,1\]\}\\sim\\widetilde\{\\mathbb\{P\}\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\.Conditioning further on an endpointYu=xuY\_\{u\}=x\_\{u\}, wheret<u≤1t<u\\leq 1, gives the universal bridge in Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102):

𝒀\[t,u\]\|\(Yt=xt,Yu=xu\)∼ℙ~\[t,u\]\|t,uρ\(⋅\|xt,xu\)=ℙ~\[t,u\]uni\(⋅\|xt,xu\)\.\\bm\{Y\}\_\{\[t,u\]\}\|\(Y\_\{t\}=x\_\{t\},Y\_\{u\}=x\_\{u\}\)\\sim\\widetilde\{\\mathbb\{P\}\}\_\{\[t,u\]\|t,u\}^\{\\rho\}\(\\cdot\|x\_\{t\},x\_\{u\}\)=\\widetilde\{\\mathbb\{P\}\}\_\{\[t,u\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{u\}\)\.Here, we use the notationℙ~\[t,u\]uni\(⋅\|xt,xu\)\\widetilde\{\\mathbb\{P\}\}\_\{\[t,u\]\}^\{\\mathrm\{uni\}\}\(\\cdot\|x\_\{t\},x\_\{u\}\)to emphasize that the Gaussian coefficients in Theorem[102](https://arxiv.org/html/2609.05727#Thmtheorem102)are evaluated withα~\\widetilde\{\\alpha\}andβ~\\widetilde\{\\beta\}\.

Whenever the specific construction is immaterial, we use the generic notation

\(Xt,X1\)∼Ξ~t,1ρ^⇒X1\|Xt∼p~1\|tρ^\(⋅\|Xt\)for endpoint sampling,\(X\_\{t\},X\_\{1\}\)\\sim\\widetilde\{\\Xi\}\_\{t,1\}^\{\\hat\{\\rho\}\}\\quad\\Rightarrow\\quad X\_\{1\}\|X\_\{t\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|X\_\{t\}\)\\qquad\\text\{for endpoint sampling\},and

𝒀\[t,1\]∼Ξ~\[t,1\]ρ^⇒Y\[t,1\]\|Yt∼ℙ~\[t,1\]\|tρ^\(⋅\|Yt\)for path sampling\.\\bm\{Y\}\_\{\[t,1\]\}\\sim\\widetilde\{\\Xi\}\_\{\[t,1\]\}^\{\\hat\{\\rho\}\}\\quad\\Rightarrow\\quad Y\_\{\[t,1\]\}\|Y\_\{t\}\\sim\\widetilde\{\\mathbb\{P\}\}\_\{\[t,1\]\|t\}^\{\\hat\{\\rho\}\}\(\\cdot\|Y\_\{t\}\)\\qquad\\text\{for path sampling\}\.Here, the terminal densityρ^\\hat\{\\rho\}is typicallyρ\\rhoor, in certain cases,ρbase\\rho^\{\\mathrm\{base\}\}\. The marginal distribution ofXtX\_\{t\}orYtY\_\{t\}is left implicit\.

#### H\.2\.2Training Losses

The regression constructions in Sections[8](https://arxiv.org/html/2609.05727#S8)and[9](https://arxiv.org/html/2609.05727#S9), together with their score\-coordinate and drift\-coordinate reformulations in Appendix[H\.1](https://arxiv.org/html/2609.05727#A8.SS1), carry over to the one\-sided setting by replacing\(αt,βt,κt,ptρ,p1\|tρ,ℙ\[t,1\]\|tρ\)\(\\alpha\_\{t\},\\beta\_\{t\},\\kappa\_\{t\},p\_\{t\}^\{\\rho\},p\_\{1\|t\}^\{\\rho\},\\mathbb\{P\}\_\{\[t,1\]\|t\}^\{\\rho\}\)with\(α~t,β~t,κ~t,p~tρ,p~1\|tρ,ℙ~\[t,1\]\|tρ\)\(\\widetilde\{\\alpha\}\_\{t\},\\widetilde\{\\beta\}\_\{t\},\\widetilde\{\\kappa\}\_\{t\},\\widetilde\{p\}\_\{t\}^\{\\rho\},\\widetilde\{p\}\_\{1\|t\}^\{\\rho\},\\widetilde\{\\mathbb\{P\}\}\_\{\[t,1\]\|t\}^\{\\rho\}\)\.

The forward construction in the one\-sided setting is given by

t∼U⁡\(0,1\),X1∼ρ,Z∼𝒩⁡\(0,I\),Xt=α~t​X1\+β~t​Z\.t\\sim\\operatorname\{U\}\(0,1\),\\qquad X\_\{1\}\\sim\\rho,\\qquad Z\\sim\\mathcal\{N\}\(0,I\),\\qquad X\_\{t\}=\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\+\\widetilde\{\\beta\}\_\{t\}Z\.In particular,X1∼ρX\_\{1\}\\sim\\rhocan be sampled from the SDE

d​Yt=b~tρ​\(Yt\)​d​t\+2​κ~t​d​Wt,t∈\[0,1\],\\displaystyle\\mathrm\{d\}Y\_\{t\}=\\widetilde\{b\}\_\{t\}^\{\\rho\}\(Y\_\{t\}\)\\mathrm\{d\}t\+\\sqrt\{2\\widetilde\{\\kappa\}\_\{t\}\}\\mathrm\{d\}W\_\{t\},\\qquad t\\in\[0,1\],Y0=0,\\displaystyle Y\_\{0\}=0,withX1:=Y1X\_\{1\}:=Y\_\{1\}\.

The reverse construction is given by

t∼U\(0,1\),Xt∼p^t,X1∼p~1\|tρ\(⋅\|Xt\)or𝒀\[t,1\]∼ℙ~\[t,1\]\|tρ\(⋅\|Xt\)\.t\\sim\\operatorname\{U\}\(0,1\),\\qquad X\_\{t\}\\sim\\hat\{p\}\_\{t\},\\qquad X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\\text\{ or \}\\bm\{Y\}\_\{\[t,1\]\}\\sim\\widetilde\{\\mathbb\{P\}\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)\.In particular,X1∼p~1\|tρ\(⋅\|Xt\)X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)or𝒀\[t,1\]∼ℙ~\[t,1\]\|tρ\(⋅\|Xt\)\\bm\{Y\}\_\{\[t,1\]\}\\sim\\widetilde\{\\mathbb\{P\}\}\_\{\[t,1\]\|t\}^\{\\rho\}\(\\cdot\|X\_\{t\}\)can both be sampled from SDE \([240](https://arxiv.org/html/2609.05727#A8.E240)\)\.

Below we provide some examples of performing exact or approximate tangential updates in the one\-sided setting\.

The covariance form of exact Newton Matching in \([56](https://arxiv.org/html/2609.05727#S8.E56)\)–\([57](https://arxiv.org/html/2609.05727#S8.E57)\) can be generalized to the one\-sided setting as:

ℒ~ρ,η,B,ωcov,vel​\(θ\):=\\displaystyle\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,B,\\omega\}^\{\\mathrm\{cov\},\\mathrm\{vel\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξ~t,1ρ\[ωt\(Xt\)∥vtθ\(Xt\)−v~t\|1\(Xt\|X1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\widetilde\{\\Xi\}\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\Big\[\\omega\_\{t\}\(X\_\{t\}\)\\big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\widetilde\{v\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−η\(r~ρ\(X1\)−Bt\(Xt\)\)\(v~t\|1\(Xt\|X1\)−v~tρ\(Xt\)\)∥22\],\\displaystyle\\qquad\-\\eta\\left\(\\tilde\{r\}^\{\\rho\}\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(\\widetilde\{v\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\widetilde\{v\}\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\right\)\\big\\\|\_\{2\}^\{2\}\\Big\],whose unique population minimizer is

v~ρ\+η​Γ~ρ,r~ρ\.\\widetilde\{v\}^\{\\rho\}\+\\eta\\widetilde\{\\Gamma\}^\{\\rho,\\tilde\{r\}^\{\\rho\}\}\.Here we define

Γ~tρ,f\(xt\):=CovX1∼p~1\|tρ\(⋅\|xt\)\(v~t\|1\(xt\|X1\),f\(X1\)\)=κ~t∇xt𝔼X1∼p~1\|tρ\(⋅\|xt\)\[f\(X1\)\]\.\\widetilde\{\\Gamma\}^\{\\rho,f\}\_\{t\}\(x\_\{t\}\):=\\operatorname\{Cov\}\_\{X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\(\\tilde\{v\}\_\{t\|1\}\(x\_\{t\}\|X\_\{1\}\),f\(X\_\{1\}\)\\right\)=\\widetilde\{\\kappa\}\_\{t\}\\nabla\_\{x\_\{t\}\}\\mathbb\{E\}\_\{X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[f\(X\_\{1\}\)\\right\]\.
The Bolza realization of exact Newton Matching in \([94](https://arxiv.org/html/2609.05727#S8.E94)\)–\([95](https://arxiv.org/html/2609.05727#S8.E95)\) can be generalized to the one\-sided setting with drift coordinates as:

ℒ~ρ,η,ωgrad,Bolza,drift\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ~\[t,1\]ρ\[ωt\(Yt\)∥\\displaystyle\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{drift\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\widetilde\{\\Xi\}\_\{\[t,1\]\}^\{\\rho\}\\end\{subarray\}\}\\bigg\[\\omega\_\{t\}\(Y\_\{t\}\)\\Big\\\|btθ​\(Yt\)−\(1−ητ\)​b~t\|1​\(Yt\|Y1\)\\displaystyle b\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)\\widetilde\{b\}\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητb~tμ\(Yt\)−2ηκ~tλtb~ρ,r,l~ρ,μ\(𝒀\[t,1\]\)∥22\]\.\\displaystyle\-\\frac\{\\eta\}\{\\tau\}\\widetilde\{b\}\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-2\\eta\\widetilde\{\\kappa\}\_\{t\}\\lambda\_\{t\}^\{\\widetilde\{b\}^\{\\rho\},r,\\widetilde\{l\}^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.The pathwise adjoint satisfies

dd​uλub~ρ,r,l~ρ,μ\(𝒀\[t,1\]\)=−∇b~uρ\(Yuρ\)⊤λub~ρ,r,l~ρ,μ\(𝒀\[t,1\]\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}u\}\\lambda\_\{u\}^\{\\widetilde\{b\}^\{\\rho\},r,\\widetilde\{l\}^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla\\widetilde\{b\}\_\{u\}^\{\\rho\}\(Y\_\{u\}^\{\\rho\}\)^\{\\top\}\\lambda\_\{u\}^\{\\widetilde\{b\}^\{\\rho\},r,\\widetilde\{l\}^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\+14​τ​κ~u∇\(‖b~uρ\(Yu\)−b~uμ\(Yu\)‖22\),u∈\[t,1\],\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\+\\frac\{1\}\{4\\tau\\widetilde\{\\kappa\}\_\{u\}\}\\nabla\\left\(\\left\\\|\\widetilde\{b\}\_\{u\}^\{\\rho\}\(Y\_\{u\}\)\-\\widetilde\{b\}\_\{u\}^\{\\mu\}\(Y\_\{u\}\)\\right\\\|\_\{2\}^\{2\}\\right\),\\qquad u\\in\[t,1\],λ1b~ρ,r,l~ρ,μ​\(𝒀\[t,1\]\)=∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{\\widetilde\{b\}^\{\\rho\},r,\\widetilde\{l\}^\{\\rho,\\mu\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\\nabla r\(Y\_\{1\}\)\.
The Gaussian\-kernel approximation with the stop\-gradient trick in \([142](https://arxiv.org/html/2609.05727#S9.E142)\) can be generalized to the one\-sided setting with score coordinates as:

ℒ~η,Σ^,ωGau​\-​ker,score,sg\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξ~t,1sg⁡\(θ\)\[ωt\(Xt\)∥stθ\(Xt\)−s~t\|1\(Xt\|X1\)\\displaystyle\\widetilde\{\\mathcal\{L\}\}\_\{\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{score\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\widetilde\{\\Xi\}\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\Bigg\[\\omega\_\{t\}\(X\_\{t\}\)\\bigg\\\|s\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\widetilde\{s\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−ηα~tβ~t2Σ^t\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+α~tτ​β~t2\(Xt−α~tX1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\-\\eta\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}\(X\_\{t\}\)\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\tau\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.
The canonicalization losses can be generalized accordingly\.

## Appendix IDetailed Derivations for Existing Works

This appendix provides the detailed derivations underlying the correspondences stated in Section[10](https://arxiv.org/html/2609.05727#S10)\.

### I\.1Tilt Matching

Explicit Tilt Matching \(ETM\) and Implicit Tilt Matching \(ITM\) are organized along the same exponential\-tilt continuation path, but differ in the population\-level update realized at each stage\. Following the notation of Section[10\.1](https://arxiv.org/html/2609.05727#S10.SS1), we relabel the current continuation density, its canonical velocity, and the inverse\-temperature increment byρbase\\rho^\{\\mathrm\{base\}\},vbase=𝒞⁡\(ρbase\)v^\{\\mathrm\{base\}\}=\\mathcal\{C\}\(\\rho^\{\\mathrm\{base\}\}\), andτ\\tau, respectively\. We analyze the two methods separately\.

#### I\.1\.1Explicit Tilt Matching

Using our notation, the ETM loss is

ℒETM​\(θ\):=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{ETM\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),X1∼ρbase,X0∼p0,Xt=αt​X1\+βt​X0\[∥vtθ\(Xt\)−vtbase\(Xt\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)−τ\(vt\|1\(Xt\|X1\)r\(X1\)−vtbase\(Xt\)r\(X1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\-\\tau\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)r\(X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)r\(X\_\{1\}\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.Rearranging the ETM target gives

vtθ​\(Xt\)−vtbase​\(Xt\)−τ⁡\(vt\|1​\(Xt\|X1\)​r​\(X1\)−vtbase​\(Xt\)​r​\(X1\)\)\\displaystyle v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\-\\tau\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)r\(X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)r\(X\_\{1\}\)\\right\)=\\displaystyle=vtθ​\(Xt\)−vt\|1​\(Xt\|X1\)\+vt\|1​\(Xt\|X1\)−vtbase​\(Xt\)−τ​r​\(X1\)​\(vt\|1​\(Xt\|X1\)−vtbase​\(Xt\)\)\\displaystyle v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\-\\tau r\(X\_\{1\}\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\)=\\displaystyle=vtθ​\(Xt\)−vt\|1​\(Xt\|X1\)−τ⁡\(r⁡\(X1\)−1τ\)​\(vt\|1​\(Xt\|X1\)−vtbase​\(Xt\)\)\.\\displaystyle v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\tau\\left\(r\(X\_\{1\}\)\-\\frac\{1\}\{\\tau\}\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\)\.Comparing the ETM loss with our unregularized covariance\-form loss \([128](https://arxiv.org/html/2609.05727#S9.E128)\) yields the exact sample\-wise correspondence

ℒETM​\(θ\)=ℒρ,η,Bno​\-​reg​\-​cov​\(θ\)\|ρ=ρbase,η=τ,Bt≡1τunder the forward construction\.\\mathcal\{L\}^\{\\mathrm\{ETM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{no\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\}\}\(\\theta\)\\right\|\_\{\\rho=\\rho^\{\\mathrm\{base\}\},\\,\\eta=\\tau,\\,B\_\{t\}\\equiv\\frac\{1\}\{\\tau\}\}\\qquad\\text\{under the forward construction\.\}Therefore, ETM is the covariance form of the full\-step unregularized update at the base anchor with the forward construction and a constant baseline\. It is not critical\-point consistent wheneverrris nonconstant\. Its unique population minimizer is

v¯ETM=vbase\+τ​Γρbase,r\.\\bar\{v\}^\{\\mathrm\{ETM\}\}=v^\{\\mathrm\{base\}\}\+\\tau\\Gamma^\{\\rho^\{\\mathrm\{base\}\},r\}\.Theorem[12](https://arxiv.org/html/2609.05727#Thmtheorem12)gives the terminal densityqETM:=𝒯⁡\(v¯ETM\)q^\{\\mathrm\{ETM\}\}:=\\mathcal\{T\}\(\\bar\{v\}^\{\\mathrm\{ETM\}\}\)as

qETM​\(x\)=πρbase,τ,r​\(x\)​exp⁡\(KL⁡\(ρbase∥πρbase,τ,r\)−𝒟ρbase,τ,r​\(x\)\)\.q^\{\\mathrm\{ETM\}\}\(x\)=\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\(x\)\\exp\\left\(\\operatorname\{KL\}\\left\(\\rho^\{\\mathrm\{base\}\}\\middle\\\|\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\\right\)\-\\mathcal\{D\}\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\(x\)\\right\)\.
The original analysis in Tilt Matching attributes the mismatch betweenv¯ETM\\bar\{v\}^\{\\mathrm\{ETM\}\}andvπρbase,τ,rv^\{\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\}to the discretization error defined in\[[39](https://arxiv.org/html/2609.05727#bib.bib32), Section 3\.1\]\. Newton Matching further separates this mismatch into two mechanisms through the decomposition

v¯ETM−vπρbase,τ,r=\(v¯ETM−vqETM\)\+\(vqETM−vπρbase,τ,r\)\.\\bar\{v\}^\{\\mathrm\{ETM\}\}\-v^\{\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\}=\\left\(\\bar\{v\}^\{\\mathrm\{ETM\}\}\-v^\{q^\{\\mathrm\{ETM\}\}\}\\right\)\+\\left\(v^\{q^\{\\mathrm\{ETM\}\}\}\-v^\{\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\}\\right\)\.The first term is the velocity\-levelcanonicality errorsincev¯ETM∉𝒱can\\bar\{v\}^\{\\mathrm\{ETM\}\}\\notin\\mathscr\{V\}\_\{\\mathrm\{can\}\}in general\. The second term arises from the terminal\-density mismatch induced by thepath dissipation𝒟ρbase,τ,r\\mathcal\{D\}\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\.

#### I\.1\.2Implicit Tilt Matching

Using our notation, the ITM loss is

ℒcITM​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{c\}^\{\\mathrm\{ITM\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),X1∼ρbase,X0∼p0,Xt=αt​X1\+βt​X0\[∥c\(Xt\)\(vtθ\(Xt\)−vtbase\(Xt\)\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\mathrm\{base\}\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\bigg\[\\Big\\\|c\(X\_\{t\}\)\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\)\+\(eτ​r​\(X1\)−c\(Xt\)\)\(sg\(vtθ\(Xt\)\)−vt\|1\(Xt\|X1\)\)∥22\],\\displaystyle\\qquad\\quad\+\\left\(e^\{\\tau r\(X\_\{1\}\)\}\-c\(X\_\{t\}\)\\right\)\\left\(\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\],wherec⁡\(xt\)\>0c\(x\_\{t\}\)\>0\. Define the corresponding baseline and weight by

Btc​\(xt\):=1τ​log⁡c⁡\(xt\),ωtc​\(xt\):=c​\(xt\)2\.B\_\{t\}^\{c\}\(x\_\{t\}\):=\\frac\{1\}\{\\tau\}\\log c\(x\_\{t\}\),\\qquad\\omega\_\{t\}^\{c\}\(x\_\{t\}\):=c\(x\_\{t\}\)^\{2\}\.The ITM loss can be rewritten as

ℒcITM​\(θ\)=\\displaystyle\\mathcal\{L\}\_\{c\}^\{\\mathrm\{ITM\}\}\(\\theta\)=𝔼t∼U\(0,1\),Xt∼ptbase,X1∼p1\|tbase\(⋅\|Xt\)\[ωtc\(Xt\)∥vtθ\(Xt\)−vtbase\(Xt\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{t\}\\sim p\_\{t\}^\{\\mathrm\{base\}\},\\,X\_\{1\}\\sim p\_\{1\|t\}^\{\\mathrm\{base\}\}\(\\cdot\|X\_\{t\}\)\\end\{subarray\}\}\\bigg\[\\omega\_\{t\}^\{c\}\(X\_\{t\}\)\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\+\(eτ⁡\(r⁡\(X1\)−Btc​\(Xt\)\)−1\)\(sg\(vtθ\(Xt\)\)−vt\|1\(Xt\|X1\)\)∥22\]\.\\displaystyle\\qquad\\quad\+\\left\(e^\{\\tau\(r\(X\_\{1\}\)\-B\_\{t\}^\{c\}\(X\_\{t\}\)\)\}\-1\\right\)\\left\(\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.
To match the ITM weighting exactly, we generalize the direct\-linearization loss \([112](https://arxiv.org/html/2609.05727#S9.E112)\) by introducing a positive weightωt​\(xt\)\>0\\omega\_\{t\}\(x\_\{t\}\)\>0:

ℒρ,η,B,ωdir​\-​lin\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1base\[ωt\(Xt\)∥vtθ\(Xt\)−\[\(1−ητ\)vtρ\(Xt\)\+ητvtbase\(Xt\)\]\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B,\\omega\}^\{\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\mathrm\{base\}\}\\end\{subarray\}\}\\bigg\[\\omega\_\{t\}\(X\_\{t\}\)\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\[\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\\right\]−\(eτ⁡\(r⁡\(X1\)−Bt​\(Xt\)\)−1\)\[\(1−ητ\)vtρ\(Xt\)\+ητvt\|1\(Xt\|X1\)−sg\(vtθ\(Xt\)\)\]∥22\]\.\\displaystyle\\quad\-\\left\(e^\{\\tau\(r\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\)\}\-1\\right\)\\left\[\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\frac\{\\eta\}\{\\tau\}v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\\right\]\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(241\)Here,Ξt,1base:=Ξt,1ρbase\\Xi\_\{t,1\}^\{\\mathrm\{base\}\}:=\\Xi\_\{t,1\}^\{\\rho^\{\\mathrm\{base\}\}\}satisfies the posterior requirement in \([49](https://arxiv.org/html/2609.05727#S8.E49)\)\. At the function\-space level, the positive weightωt​\(xt\)\\omega\_\{t\}\(x\_\{t\}\)does not change the pointwise population\-stationarity condition\. Comparing the ITM lossℒcITM\\mathcal\{L\}\_\{c\}^\{\\mathrm\{ITM\}\}with \([241](https://arxiv.org/html/2609.05727#A9.E241)\) yields the exact sample\-wise correspondence

ℒcITM​\(θ\)=ℒρ,η,B,ωdir​\-​lin​\(θ\)\|ρ=ρbase,η=τ,Bt=Btc,ωt=ωtcunder the forward construction\.\\mathcal\{L\}\_\{c\}^\{\\mathrm\{ITM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B,\\omega\}^\{\\mathrm\{dir\}\\text\{\-\}\\mathrm\{lin\}\}\(\\theta\)\\right\|\_\{\\rho=\\rho^\{\\mathrm\{base\}\},\\,\\eta=\\tau,\\,B\_\{t\}=B\_\{t\}^\{c\},\\,\\omega\_\{t\}=\\omega\_\{t\}^\{c\}\}\\qquad\\text\{under the forward construction\.\}Therefore, ITM is equivalent to the full\-step update of the direct\-linearization approximation under the forward construction, with specific choices of the baseline and weighting\. Its unique population\-stationary point is

vITM=vπρbase,τ,r\.v^\{\\mathrm\{ITM\}\}=v^\{\\pi\_\{\\rho^\{\\mathrm\{base\}\},\\tau,r\}\}\.Consequently, ITM is critical\-point consistent\.

### I\.2DiffusionNFT

Using our notation, the DiffusionNFT loss is

ℒρ,cNFT​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}\(\\theta\):=𝔼t∼U\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0\[r\(X1\)‖\(1−c\)vtρ\(Xt\)\+cvtθ\(Xt\)−vt\|1\(Xt\|X1\)‖22\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\\\ X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\bigg\[r\(X\_\{1\}\)\\left\\\|\(1\-c\)v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+cv\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\+\(1−r\(X1\)\)‖\(1\+c\)vtρ\(Xt\)−cvtθ\(Xt\)−vt\|1\(Xt\|X1\)‖22\],\\displaystyle\\qquad\\qquad\\quad\+\(1\-r\(X\_\{1\}\)\)\\left\\\|\(1\+c\)v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-cv\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\bigg\],wherec\>0c\>0is a user\-specified scalar value\. To expose the regression target without changing the sample\-wise gradient with respect toθ\\theta, define

utθ​\(xt\):=vtθ​\(xt\)−vtρ​\(xt\),utρ​\(xt,x1\):=vtρ​\(xt\)−vt\|1​\(xt\|x1\)\.u\_\{t\}^\{\\theta\}\(x\_\{t\}\):=v\_\{t\}^\{\\theta\}\(x\_\{t\}\)\-v\_\{t\}^\{\\rho\}\(x\_\{t\}\),\\qquad u\_\{t\}^\{\\rho\}\(x\_\{t\},x\_\{1\}\):=v\_\{t\}^\{\\rho\}\(x\_\{t\}\)\-v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\.Then, the DiffusionNFT lossℒρ,cNFT\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}can be rewritten as

ℒρ,cNFT​\(θ\)=\\displaystyle\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0\[r\(X1\)‖cutθ\(Xt\)\+utρ\(Xt,X1\)‖22\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\bigg\[r\(X\_\{1\}\)\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\+\(1−r\(X1\)\)‖cutθ\(Xt\)−utρ\(Xt,X1\)‖22\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\+\(1\-r\(X\_\{1\}\)\)\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\bigg\]\.
Since

‖c​utθ​\(Xt\)\+utρ​\(Xt,X1\)‖22−‖c​utθ​\(Xt\)−utρ​\(Xt,X1\)‖22=4​c​utθ​\(Xt\)⊤​utρ​\(Xt,X1\),\\displaystyle\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\-\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}=4cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)^\{\\top\}u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\),and

‖c​utθ​\(Xt\)−utρ​\(Xt,X1\)‖22=c2​‖utθ​\(Xt\)‖22\+‖utρ​\(Xt,X1\)‖22−2​c​utθ​\(Xt\)⊤​utρ​\(Xt,X1\),\\displaystyle\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}=c^\{2\}\\left\\\|u\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\+\\left\\\|u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\-2cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)^\{\\top\}u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\),we have

r⁡\(X1\)​‖c​utθ​\(Xt\)\+utρ​\(Xt,X1\)‖22\+\(1−r⁡\(X1\)\)​‖c​utθ​\(Xt\)−utρ​\(Xt,X1\)‖22\\displaystyle r\(X\_\{1\}\)\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\+\(1\-r\(X\_\{1\}\)\)\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=c2​‖utθ​\(Xt\)‖22\+‖utρ​\(Xt,X1\)‖22\+2​c​\(2​r​\(X1\)−1\)​utθ​\(Xt\)⊤​utρ​\(Xt,X1\)\\displaystyle c^\{2\}\\left\\\|u\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\+\\left\\\|u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\+2c\(2r\(X\_\{1\}\)\-1\)u\_\{t\}^\{\\theta\}\(X\_\{t\}\)^\{\\top\}u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)=\\displaystyle=‖c​utθ​\(Xt\)\+\(2​r​\(X1\)−1\)​utρ​\(Xt,X1\)‖22\+\(1−\(2​r​\(X1\)−1\)2\)​‖utρ​\(Xt,X1\)‖22\.\\displaystyle\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+\(2r\(X\_\{1\}\)\-1\)u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\+\\left\(1\-\(2r\(X\_\{1\}\)\-1\)^\{2\}\\right\)\\left\\\|u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\.Here, the second term\(1−\(2​r​\(X1\)−1\)2\)​‖utρ​\(Xt,X1\)‖22\\left\(1\-\(2r\(X\_\{1\}\)\-1\)^\{2\}\\right\)\\left\\\|u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}is independent ofθ\\thetaand therefore does not affect the sample\-wise gradient\. The first term satisfies

‖c​utθ​\(Xt\)\+\(2​r​\(X1\)−1\)​utρ​\(Xt,X1\)‖22\\displaystyle\\left\\\|cu\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+\(2r\(X\_\{1\}\)\-1\)u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=c2​‖utθ​\(Xt\)\+2c​\(r⁡\(X1\)−12\)​utρ​\(Xt,X1\)‖22\\displaystyle c^\{2\}\\left\\\|u\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+\\frac\{2\}\{c\}\\left\(r\(X\_\{1\}\)\-\\frac\{1\}\{2\}\\right\)u\_\{t\}^\{\\rho\}\(X\_\{t\},X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=c2​‖vtθ​\(Xt\)−vtρ​\(Xt\)\+2c​\(r⁡\(X1\)−12\)​\(vtρ​\(Xt\)−vt\|1​\(Xt\|X1\)\)‖22\\displaystyle c^\{2\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\+\\frac\{2\}\{c\}\\left\(r\(X\_\{1\}\)\-\\frac\{1\}\{2\}\\right\)\\left\(v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=c2​‖vtθ​\(Xt\)−vt\|1​\(Xt\|X1\)\+2c​\(r⁡\(X1\)−c\+12\)​\(vtρ​\(Xt\)−vt\|1​\(Xt\|X1\)\)‖22\.\\displaystyle c^\{2\}\\left\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\frac\{2\}\{c\}\\left\(r\(X\_\{1\}\)\-\\frac\{c\+1\}\{2\}\\right\)\\left\(v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\.
Consequently, the DiffusionNFT lossℒρ,cNFT\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}can be written as

ℒρ,cNFT​\(θ\)=\\displaystyle\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}\(\\theta\)=c2𝔼t∼U⁡\(0,1\),X1∼ρ,X0∼p0,Xt=αt​X1\+βt​X0\[∥vtθ\(Xt\)−vt\|1\(Xt\|X1\)\\displaystyle c^\{2\}\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho,\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+2c\(r\(X1\)−c\+12\)\(vtρ\(Xt\)−vt\|1\(Xt\|X1\)\)∥22\]\+const,\\displaystyle\\qquad\+\\frac\{2\}\{c\}\\left\(r\(X\_\{1\}\)\-\\frac\{c\+1\}\{2\}\\right\)\\left\(v\_\{t\}^\{\\rho\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\+\\mathrm\{const\},where the additive constant is independent ofθ\\theta\. Comparing the DiffusionNFT lossℒρ,cNFT\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}with \([128](https://arxiv.org/html/2609.05727#S9.E128)\) yields the exact sample\-wise correspondence up to thisθ\\theta\-independent additive term:

ℒρ,cNFT​\(θ\)=c2⋅ℒρ,η,Bno​\-​reg​\-​cov​\(θ\)\|η=2c,Bt≡c\+12\+constunder the forward construction\.\\mathcal\{L\}\_\{\\rho,c\}^\{\\mathrm\{NFT\}\}\(\\theta\)=c^\{2\}\\cdot\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{no\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\}\}\(\\theta\)\\right\|\_\{\\eta=\\frac\{2\}\{c\},\\,B\_\{t\}\\equiv\\frac\{c\+1\}\{2\}\}\+\\mathrm\{const\}\\qquad\\text\{under the forward construction\.\}Therefore, DiffusionNFT is the covariance form of the unregularized update under the forward construction, stepsizeη=2c\\eta=\\frac\{2\}\{c\}, and constant baselineBt≡c\+12B\_\{t\}\\equiv\\frac\{c\+1\}\{2\}\. Its unique population minimizer is

v¯NFT=vρ\+2c​Γρ,r\.\\bar\{v\}^\{\\mathrm\{NFT\}\}=v^\{\\rho\}\+\\frac\{2\}\{c\}\\Gamma^\{\\rho,r\}\.It is not critical\-point consistent wheneverrris nonconstant\.

### I\.3Reinforce Adjoint Matching

Using our notation, the Reinforce Adjoint Matching \(RAM\) loss is

ℒRAM​\(θ\)=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{RAM\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),X1∼ρsg⁡\(θ\),X0∼p0,Xt=αt​X1\+βt​X0\[∥vtθ\(Xt\)−vtbase\(Xt\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)−r\(X1\)\(vt\|1\(Xt\|X1\)−sg\(vtθ\(Xt\)\)\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\-r\(X\_\{1\}\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.Here,X1∼ρsg⁡\(θ\)X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\}denotes an endpoint generated by the ODE of the current modelvθv^\{\\theta\}, with the dependence of the sampling procedure onθ\\thetastopped during differentiation\.

From the fixed\-point perspective of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), the approximate\-regularization update \([123](https://arxiv.org/html/2609.05727#S9.E123)\) induces the fixed\-point condition

vρ=vρ\+η​Γ^ρ,appr​\-​reg\.v^\{\\rho\}=v^\{\\rho\}\+\\eta\\widehat\{\\Gamma\}^\{\\rho,\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\.Accordingly, the covariance\-form approximate\-regularization loss \([124](https://arxiv.org/html/2609.05727#S9.E124)\) admits the following stop\-gradient form:

ℒρ,η,Bappr​\-​reg​\-​cov,sg\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1sg⁡\(θ\)\[∥vtθ\(Xt\)−\(1−ητ\)vt\|1\(Xt\|X1\)\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−ητvtbase\(Xt\)−η\(r\(X1\)−Bt\(Xt\)\)\(vt\|1\(Xt\|X1\)−sg\(vtθ\)\(Xt\)\)\+ητκtK^t\(Xt\)∥22\]\.\\displaystyle\\quad\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\-\\eta\\left\(r\(X\_\{1\}\)\-B\_\{t\}\(X\_\{t\}\)\\right\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\operatorname\{sg\}\(v\_\{t\}^\{\\theta\}\)\(X\_\{t\}\)\\right\)\+\\frac\{\\eta\}\{\\tau\}\\kappa\_\{t\}\\widehat\{K\}\_\{t\}\(X\_\{t\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(242\)Here,Ξt,1sg⁡\(θ\)\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}denotes the joint distribution constructed from the current modelvθv^\{\\theta\}, with its dependence onθ\\thetaplaced under stop\-gradient\. Under the specialization

ℒρ,η,Bappr​\-​reg​\-​cov,sg​\(θ\)\|η=τ=1,K^t≡0,Bt≡0\\displaystyle\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\eta=\\tau=1,\\,\\widehat\{K\}\_\{t\}\\equiv 0,\\,B\_\{t\}\\equiv 0\}=\\displaystyle=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1sg⁡\(θ\)​\[‖vtθ​\(Xt\)−vtbase​\(Xt\)−r⁡\(X1\)​\(vt\|1​\(Xt\|X1\)−sg⁡\(vtθ​\(Xt\)\)\)‖22\],\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\bigg\[\\Big\\\|v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(X\_\{t\}\)\-r\(X\_\{1\}\)\\left\(v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\operatorname\{sg\}\\left\(v\_\{t\}^\{\\theta\}\(X\_\{t\}\)\\right\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\],the fixed\-point condition reduces to

vρ=v^appr​\-​reg\|η=τ=1,K^t≡0=vbase\+Γρ,r\.v^\{\\rho\}=\\left\.\\hat\{v\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\}\\right\|\_\{\\eta=\\tau=1,\\,\\widehat\{K\}\_\{t\}\\equiv 0\}=v^\{\\mathrm\{base\}\}\+\\Gamma^\{\\rho,r\}\.
ComparingℒRAM\\mathcal\{L\}^\{\\mathrm\{RAM\}\}with the loss above yields the exact sample\-wise correspondence

ℒRAM​\(θ\)=ℒρ,η,Bappr​\-​reg​\-​cov,sg​\(θ\)\|η=τ=1,K^t≡0,Bt≡0under the forward construction\.\\mathcal\{L\}^\{\\mathrm\{RAM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\rho,\\eta,B\}^\{\\mathrm\{appr\}\\text\{\-\}\\mathrm\{reg\}\\text\{\-\}\\mathrm\{cov\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\eta=\\tau=1,\\,\\widehat\{K\}\_\{t\}\\equiv 0,\\,B\_\{t\}\\equiv 0\}\\qquad\\text\{under the forward construction\.\}Therefore, RAM is the stop\-gradient covariance\-form realization of the fixed\-point condition \([146](https://arxiv.org/html/2609.05727#S10.E146)\) associated with the full\-step approximate\-regularization update, using the forward construction and a zero baseline\. Equivalently, RAM uses the stop\-gradient trick to enforce the fixed\-point condition

vρ=vbase\+Γρ,r\.v^\{\\rho\}=v^\{\\mathrm\{base\}\}\+\\Gamma^\{\\rho,r\}\.RAM is not critical\-point consistent in general\.

### I\.4Adjoint Matching

We first translate the SOC formulation used by Basic Adjoint Matching \(BAM\) and Lean Adjoint Matching \(LAM\) into our notation\. In Adjoint Matching\[[14](https://arxiv.org/html/2609.05727#bib.bib30)\], the SOC problem for reward\-based fine\-tuning is

minu\\displaystyle\\min\_\{u\}\\,\\,𝔼⁡\[∫0112​‖ut​\(Yt\)‖22​𝑑t−r⁡\(Y1\)\],\\displaystyle\\mathbb\{E\}\\left\[\\int\_\{0\}^\{1\}\\frac\{1\}\{2\}\\left\\\|u\_\{t\}\\left\(Y\_\{t\}\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\-r\\left\(Y\_\{1\}\\right\)\\right\],s\.t\.\\displaystyle\\mathrm\{s\.t\.\}\\,\\,d​Yt=\(btbase​\(Yt\)\+2​κt​ut​\(Yt\)\)​d​t\+2​κt​d​Wt,\\displaystyle\\mathrm\{d\}Y\_\{t\}=\\left\(b\_\{t\}^\{\\mathrm\{base\}\}\\left\(Y\_\{t\}\\right\)\+\\sqrt\{2\\kappa\_\{t\}\}u\_\{t\}\\left\(Y\_\{t\}\\right\)\\right\)\\mathrm\{d\}t\+\\sqrt\{2\\kappa\_\{t\}\}\\mathrm\{d\}W\_\{t\},Y0∼p0=𝒩⁡\(0,I\),\\displaystyle Y\_\{0\}\\sim p\_\{0\}=\\mathcal\{N\}\(0,I\),where the inverse temperature is set toτ=1\\tau=1\. Here, the base drift is

btbase​\(x\)=2​vtbase​\(x\)−α˙tαt​x\.b\_\{t\}^\{\\mathrm\{base\}\}\(x\)=2v\_\{t\}^\{\\mathrm\{base\}\}\(x\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\.
Under a parameterized velocity modelvθv^\{\\theta\}, Adjoint Matching rewrites the control as

utθ​\(x\)=2κt​\(vtθ​\(x\)−vtbase​\(x\)\)\.u\_\{t\}^\{\\theta\}\(x\)=\\sqrt\{\\frac\{2\}\{\\kappa\_\{t\}\}\}\\left\(v\_\{t\}^\{\\theta\}\(x\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(x\)\\right\)\.The SOC problem then becomes

minθ\\displaystyle\\min\_\{\\theta\}\\,\\,𝔼⁡\[∫011κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)‖22​𝑑t−r⁡\(Y1\)\],\\displaystyle\\mathbb\{E\}\\left\[\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\-r\\left\(Y\_\{1\}\\right\)\\right\],\(243a\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}\\,\\,d​Yt=\(2​vtθ​\(Yt\)−α˙tαt​Yt\)​d​t\+2​κt​d​Wt,t∈\(0,1\),\\displaystyle\\mathrm\{d\}Y\_\{t\}=\\left\(2v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}Y\_\{t\}\\right\)\\mathrm\{d\}t\+\\sqrt\{2\\kappa\_\{t\}\}\\mathrm\{d\}W\_\{t\},\\qquad t\\in\(0,1\),\(243b\)Y0∼p0=𝒩⁡\(0,I\)\.\\displaystyle Y\_\{0\}\\sim p\_\{0\}=\\mathcal\{N\}\(0,I\)\.\(243c\)
Whenvθv^\{\\theta\}is canonical, the SDE \([243b](https://arxiv.org/html/2609.05727#A9.E243.2)\) under the “memoryless noise schedule”σt=2​κt\\sigma\_\{t\}=\\sqrt\{2\\kappa\_\{t\}\}\[[14](https://arxiv.org/html/2609.05727#bib.bib30)\]coincides exactly with the posterior\-preserving SDE \([176](https://arxiv.org/html/2609.05727#A4.E176)\)\. In the stop\-gradient implementations of BAM and LAM, the current modelvθv^\{\\theta\}is used as a self\-anchor and treated as canonical, as discussed in Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4)\. The corresponding sampling scheme is

X0∼p0,𝒀\[0,1\]∼ℙ\[0,1\]\|0θ\(⋅\|X0\)⇒𝒀\[0,1\]∼ℙ\[0,1\]θ\.X\_\{0\}\\sim p\_\{0\},\\,\\bm\{Y\}\_\{\[0,1\]\}\\sim\\mathbb\{P\}^\{\\theta\}\_\{\[0,1\]\|0\}\(\\cdot\|X\_\{0\}\)\\quad\\Rightarrow\\quad\\bm\{Y\}\_\{\[0,1\]\}\\sim\\mathbb\{P\}^\{\\theta\}\_\{\[0,1\]\}\.\(244\)For everyt∈\[0,1\)t\\in\[0,1\), we denoteℙ\[t,1\]θ\\mathbb\{P\}^\{\\theta\}\_\{\[t,1\]\}by the path distribution of𝒀\[t,1\]\\bm\{Y\}\_\{\[t,1\]\}under the current modelvθv^\{\\theta\}\. Using this notation, the losses of both BAM and LAM take the form

ℒ⁡\(θ\)=12​𝔼𝒀\[0,1\]∼ℙ\[0,1\]sg⁡\(θ\)​\[∫01‖2κt​\(vtθ​\(Yt\)−vtbase​\(Yt\)\)\+2​κt​sg⁡\(λtθ​\(𝒀\[0,1\]\)\)‖22​𝑑t\],\\mathcal\{L\}\(\\theta\)=\\frac\{1\}\{2\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[0,1\]\}\\sim\\mathbb\{P\}^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[0,1\]\}\}\\left\[\\int\_\{0\}^\{1\}\\left\\\|\\sqrt\{\\frac\{2\}\{\\kappa\_\{t\}\}\}\\left\(v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\\right\)\+\\sqrt\{2\\kappa\_\{t\}\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\theta\}\(\\bm\{Y\}\_\{\[0,1\]\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\right\],where the pathwise adjointλtθ​\(𝒀\[0,1\]\)\\lambda\_\{t\}^\{\\theta\}\(\\bm\{Y\}\_\{\[0,1\]\}\)satisfies the basic or lean adjoint ODE in BAM and LAM, respectively\. The full\-trajectory implementation corresponds to the multi\-time supervision scheme for the gradient\-form \(under the reverse construction\) introduced in Section[8\.6\.2](https://arxiv.org/html/2609.05727#S8.SS6.SSS2)\. For ease of presentation, we rewrite the full\-trajectory loss in the equivalent single\-time supervision form:

ℒ⁡\(θ\)=\\displaystyle\\mathcal\{L\}\(\\theta\)=12​𝔼𝒀\[0,1\]∼ℙ\[0,1\]sg⁡\(θ\)​\[∫01‖2κt​\(vtθ​\(Yt\)−vtbase​\(Yt\)\)\+2​κt​sg⁡\(λtθ​\(𝒀\[0,1\]\)\)‖22​𝑑t\]\\displaystyle\\frac\{1\}\{2\}\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[0,1\]\}\\sim\\mathbb\{P\}^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[0,1\]\}\}\\left\[\\int\_\{0\}^\{1\}\\left\\\|\\sqrt\{\\frac\{2\}\{\\kappa\_\{t\}\}\}\\left\(v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\\right\)\+\\sqrt\{2\\kappa\_\{t\}\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\theta\}\(\\bm\{Y\}\_\{\[0,1\]\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\right\]=\\displaystyle=𝔼𝒀\[0,1\]∼ℙ\[0,1\]sg⁡\(θ\)​\[∫011κt​‖\(vtθ​\(Yt\)−vtbase​\(Yt\)\)\+κt​sg⁡\(λtθ​\(𝒀\[t,1\]\)\)‖22​𝑑t\]\\displaystyle\\mathbb\{E\}\_\{\\bm\{Y\}\_\{\[0,1\]\}\\sim\\mathbb\{P\}^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[0,1\]\}\}\\left\[\\int\_\{0\}^\{1\}\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|\\left\(v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\\right\)\+\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\mathrm\{d\}t\\right\]=\\displaystyle=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼ℙ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)\+κt​sg⁡\(λtθ​\(𝒀\[t,1\]\)\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\mathbb\{P\}^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\+\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Here,ℙ\[t,1\]sg⁡\(θ\)\\mathbb\{P\}^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}is a particular choice ofΞ\[t,1\]sg⁡\(θ\)\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}for which theYtY\_\{t\}\-marginal is given by the marginal probability path of the current modelvθv^\{\\theta\}\. The notationΞ\[t,1\]sg⁡\(θ\)\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}imposes the same posterior\-compatibility requirement asΞ\[t,1\]ρ\\Xi\_\{\[t,1\]\}^\{\\rho\}in \([82](https://arxiv.org/html/2609.05727#S8.E82)\), withvθv^\{\\theta\}treated as canonical in place ofvρv^\{\\rho\}and all resulting dependence onθ\\thetaheld fixed during differentiation\. For simplicity, we use the following more general form of the loss throughout the remainder of this subsection:

ℒ⁡\(θ\)=\\displaystyle\\mathcal\{L\}\(\\theta\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)\+κt​sg⁡\(λtθ​\(𝒀\[t,1\]\)\)‖22\]\.\\displaystyle\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\+\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\.
#### I\.4\.1Basic Adjoint Matching

The BAM loss is

ℒBAM​\(θ\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)\+κt​sg⁡\(λtBAM,θ​\(𝒀\[t,1\]\)\)‖22\],\\mathcal\{L\}^\{\\mathrm\{BAM\}\}\(\\theta\)=\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\+\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\mathrm\{BAM\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\],where the pathwise basic adjointλsBAM,θ​\(𝒀\[t,1\]\)\\lambda\_\{s\}^\{\\mathrm\{BAM\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)satisfies the ODE

dd​s​λsBAM,θ​\(𝒀\[t,1\]\)=−∇Ys\(2​vsθ​\(Ys\)−α˙sαs​Ys\)⊤​λsBAM,θ​\(𝒀\[t,1\]\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{\\mathrm\{BAM\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla\_\{Y\_\{s\}\}\\left\(2v\_\{s\}^\{\\theta\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}\\right\)^\{\\top\}\\lambda\_\{s\}^\{\\mathrm\{BAM\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)−∇Ys\(12​‖2κs​\(vsθ​\(Ys\)−vsbase​\(Ys\)\)‖22\),s∈\(t,1\),\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\-\\nabla\_\{Y\_\{s\}\}\\left\(\\frac\{1\}\{2\}\\left\\\|\\sqrt\{\\frac\{2\}\{\\kappa\_\{s\}\}\}\\left\(v\_\{s\}^\{\\theta\}\(Y\_\{s\}\)\-v\_\{s\}^\{\\mathrm\{base\}\}\(Y\_\{s\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\),\\qquad s\\in\(t,1\),λ1BAM,θ​\(𝒀\[t,1\]\)=−∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{\\mathrm\{BAM\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla r\(Y\_\{1\}\)\.Since\[[14](https://arxiv.org/html/2609.05727#bib.bib30)\]setsτ=1\\tau=1, comparing this ODE with \([140](https://arxiv.org/html/2609.05727#S9.E140)\) gives

λtBAM,θ​\(𝒀\[t,1\]\)=−λtBolza,θ​\(𝒀\[t,1\]\)\|τ=1,μ=ρbase\.\\lambda\_\{t\}^\{\\mathrm\{BAM\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\left\.\\lambda\_\{t\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\|\_\{\\tau=1,\\,\\mu=\\rho^\{\\mathrm\{base\}\}\}\.Therefore, the BAM loss can be rewritten as

ℒBAM​\(θ\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)−κt​sg⁡\(λtBolza,θ​\(𝒀\[t,1\]\)\|τ=1,μ=ρbase\)‖22\]\.\\mathcal\{L\}^\{\\mathrm\{BAM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\\\ \\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\\end\{subarray\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\-\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\left\.\\lambda\_\{t\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\|\_\{\\tau=1,\\,\\mu=\\rho^\{\\mathrm\{base\}\}\}\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\.
To make the correspondence exact, we generalize the stop\-gradient loss \([139](https://arxiv.org/html/2609.05727#S9.E139)\) of the Bolza realization by introducing a positive weightωt​\(xt\)\>0\\omega\_\{t\}\(x\_\{t\}\)\>0:

ℒη,ωgrad,Bolza,sg​\(θ\):=\\displaystyle\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{sg\}\}\(\\theta\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)\[ωt\(Yt\)∥vtθ\(Yt\)−\(1−ητ\)vt\|1\(Yt\|Y1\)\\displaystyle\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\bigg\[\\omega\_\{t\}\(Y\_\{t\}\)\\Big\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητvtμ\(Yt\)−ηκtsg\(λtBolza,θ\(𝒀\[t,1\]\)\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-\\eta\\kappa\_\{t\}\\operatorname\{sg\}\\left\(\\lambda\_\{t\}^\{\\mathrm\{Bolza\},\\theta\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(245\)Comparing the BAM lossℒBAM\\mathcal\{L\}^\{\\mathrm\{BAM\}\}withℒη,ωgrad,Bolza,sg\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{sg\}\}yields the exact sample\-wise correspondence

ℒBAM​\(θ\)=ℒη,ωgrad,Bolza,sg​\(θ\)\|τ=η=1,ωt≡1κt,μ=ρbaseunder the reverse construction\.\\mathcal\{L\}^\{\\mathrm\{BAM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{grad,Bolza\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\tau=\\eta=1,\\,\\omega\_\{t\}\\equiv\\frac\{1\}\{\\kappa\_\{t\}\},\\,\\mu=\\rho^\{\\mathrm\{base\}\}\}\\qquad\\text\{under the reverse construction\}\.Therefore, BAM is the stop\-gradient Bolza realization of the fixed\-point condition associated with the full\-step exact tangential update, using the reverse construction and a specific weighting\. According to Propositions[18](https://arxiv.org/html/2609.05727#Thmtheorem18)and[48](https://arxiv.org/html/2609.05727#Thmtheorem48), the stop\-gradient loss has𝒞⁡\(πρbase,1,r\)\\mathcal\{C\}\(\\pi\_\{\\rho^\{\\mathrm\{base\}\},1,r\}\)as its unique canonical population\-stationary point\.

#### I\.4\.2Lean Adjoint Matching

The LAM loss is

ℒLAM​\(θ\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)\+κt​λtLAM​\(𝒀\[t,1\]\)‖22\],\\mathcal\{L\}^\{\\mathrm\{LAM\}\}\(\\theta\)=\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\+\\kappa\_\{t\}\\lambda\_\{t\}^\{\\mathrm\{LAM\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\\\|\_\{2\}^\{2\}\\right\],where the pathwise lean adjoint satisfies the ODE

dd​s​λsLAM​\(𝒀\[t,1\]\)=−∇Ys\(2​vsbase​\(Ys\)−α˙sαs​Ys\)⊤​λsLAM​\(𝒀\[t,1\]\),s∈\(t,1\),\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}s\}\\lambda\_\{s\}^\{\\mathrm\{LAM\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla\_\{Y\_\{s\}\}\\left\(2v\_\{s\}^\{\\mathrm\{base\}\}\(Y\_\{s\}\)\-\\frac\{\\dot\{\\alpha\}\_\{s\}\}\{\\alpha\_\{s\}\}Y\_\{s\}\\right\)^\{\\top\}\\lambda\_\{s\}^\{\\mathrm\{LAM\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\),\\qquad s\\in\(t,1\),λ1LAM​\(𝒀\[t,1\]\)=−∇r​\(Y1\)\.\\displaystyle\\lambda\_\{1\}^\{\\mathrm\{LAM\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\nabla r\(Y\_\{1\}\)\.Comparing this ODE with \([116](https://arxiv.org/html/2609.05727#S9.E116)\) yields the pathwise identity

λtLAM​\(𝒀\[t,1\]\)=−λtbμ,r,0​\(𝒀\[t,1\]\)\|μ=ρbase\.\\lambda\_\{t\}^\{\\mathrm\{LAM\}\}\(\\bm\{Y\}\_\{\[t,1\]\}\)=\-\\left\.\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\|\_\{\\mu=\\rho^\{\\mathrm\{base\}\}\}\.Therefore, the LAM loss can be rewritten as

ℒLAM​\(θ\)=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)​\[1κt​‖vtθ​\(Yt\)−vtbase​\(Yt\)−κt​λtbμ,r,0​\(𝒀\[t,1\]\)\|μ=ρbase‖22\]\.\\mathcal\{L\}^\{\\mathrm\{LAM\}\}\(\\theta\)=\\mathbb\{E\}\_\{t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi^\{\\operatorname\{sg\}\(\\theta\)\}\_\{\[t,1\]\}\}\\left\[\\frac\{1\}\{\\kappa\_\{t\}\}\\left\\\|v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-v\_\{t\}^\{\\mathrm\{base\}\}\(Y\_\{t\}\)\-\\left\.\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\right\|\_\{\\mu=\\rho^\{\\mathrm\{base\}\}\}\\right\\\|\_\{2\}^\{2\}\\right\]\.
To match the LAM weighting exactly, we generalize the stop\-gradient loss \([141](https://arxiv.org/html/2609.05727#S9.E141)\) of the reference\-adjoint approximation by introducing a positive weightωt​\(xt\)\>0\\omega\_\{t\}\(x\_\{t\}\)\>0:

ℒη,ωref​\-​adj,sg\(θ\):=𝔼t∼U⁡\(0,1\),𝒀\[t,1\]∼Ξ\[t,1\]sg⁡\(θ\)\[ωt\(Yt\)∥\\displaystyle\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\\bm\{Y\}\_\{\[t,1\]\}\\sim\\Xi\_\{\[t,1\]\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\bigg\[\\omega\_\{t\}\(Y\_\{t\}\)\\Big\\\|vtθ​\(Yt\)−\(1−ητ\)​vt\|1​\(Yt\|Y1\)\\displaystyle v\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\-\\left\(1\-\\frac\{\\eta\}\{\\tau\}\\right\)v\_\{t\|1\}\(Y\_\{t\}\|Y\_\{1\}\)−ητvtμ\(Yt\)−ηκtλtbμ,r,0\(𝒀\[t,1\]\)∥22\]\.\\displaystyle\-\\frac\{\\eta\}\{\\tau\}v\_\{t\}^\{\\mu\}\(Y\_\{t\}\)\-\\eta\\kappa\_\{t\}\\lambda\_\{t\}^\{b^\{\\mu\},r,0\}\(\\bm\{Y\}\_\{\[t,1\]\}\)\\Big\\\|\_\{2\}^\{2\}\\bigg\]\.\(246\)Comparing the LAM lossℒLAM\\mathcal\{L\}^\{\\mathrm\{LAM\}\}withℒη,ωref​\-​adj,sg\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\},\\mathrm\{sg\}\}yields the exact sample\-wise correspondence

ℒLAM​\(θ\)=ℒη,ωref​\-​adj,sg​\(θ\)\|τ=η=1,ωt≡1κt,μ=ρbaseunder the reverse construction\.\\mathcal\{L\}^\{\\mathrm\{LAM\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\eta,\\omega\}^\{\\mathrm\{ref\}\\text\{\-\}\\mathrm\{adj\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\tau=\\eta=1,\\,\\omega\_\{t\}\\equiv\\frac\{1\}\{\\kappa\_\{t\}\},\\,\\mu=\\rho^\{\\mathrm\{base\}\}\}\\qquad\\text\{under the reverse construction\}\.Therefore, LAM is the stop\-gradient realization of the full\-step reference\-adjoint approximation update, using the reverse construction and a specific weighting\. According to Propositions[39](https://arxiv.org/html/2609.05727#Thmtheorem39)and[48](https://arxiv.org/html/2609.05727#Thmtheorem48), the stop\-gradient loss has𝒞⁡\(πρbase,1,r\)\\mathcal\{C\}\(\\pi\_\{\\rho^\{\\mathrm\{base\}\},1,r\}\)as its unique canonical population\-stationary point\.

### I\.5Flow Sampling

Using our notation, the Flow Sampling \(FS\) loss can be written as

ℒFS\(θ\):=𝔼t∼U⁡\(0,1\),X1∼ρsg⁡\(θ\),X0∼p0,Xt=αt​X1\+βt​X0\[‖btθ\(Xt\)−vt\|1\(Xt\|X1\)−κtαt∇r\(X1\)‖22\]\.\\mathcal\{L\}^\{\\mathrm\{FS\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\\left\[\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-\\frac\{\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\nabla r\(X\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Here,X1∼ρsg⁡\(θ\)X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\}denotes the endpointX1:=Y1X\_\{1\}:=Y\_\{1\}generated by

d​Yt=btθ​\(Yt\)​d​t\+σt​d​Wt,Y0∼p0,\\mathrm\{d\}Y\_\{t\}=b\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\\,\\mathrm\{d\}t\+\\sigma\_\{t\}\\,\\mathrm\{d\}W\_\{t\},\\qquad Y\_\{0\}\\sim p\_\{0\},\(247\)with the dependence of the sampling construction onθ\\thetaplaced under stop\-gradient\.

Since

bt\|1​\(xt\|x1\)=2​vt\|1​\(xt\|x1\)−α˙tαt​xt,b\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=2v\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}x\_\{t\},we have

vt\|1\(Xt\|X1\)\+κtαt∇r\(X1\)=\\displaystyle v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\frac\{\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\nabla r\(X\_\{1\}\)=bt\|1\(Xt\|X1\)−vt\|1\(Xt\|X1\)\+α˙tαtXt\+κtαt∇r\(X1\)\\displaystyle b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\-v\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}X\_\{t\}\+\\frac\{\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\nabla r\(X\_\{1\}\)=\\displaystyle=bt\|1\(Xt\|X1\)\+κtαt∇r\(X1\)\+α˙tαtXt−\(α˙tX1\+β˙tβt\(Xt−αtX1\)\)\\displaystyle b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\frac\{\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\nabla r\(X\_\{1\}\)\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}X\_\{t\}\-\\left\(\\dot\{\\alpha\}\_\{t\}X\_\{1\}\+\\frac\{\\dot\{\\beta\}\_\{t\}\}\{\\beta\_\{t\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)=\\displaystyle=bt\|1​\(Xt\|X1\)\+κtαt​\(∇r​\(X1\)\+αtβt2​\(Xt−αt​X1\)\)\.\\displaystyle b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\frac\{\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\.Therefore, the FS loss can be equivalently rewritten as

ℒFS​\(θ\):=𝔼t∼U⁡\(0,1\),X1∼ρsg⁡\(θ\),X0∼p0,Xt=αt​X1\+βt​X0\\displaystyle\\mathcal\{L\}^\{\\mathrm\{FS\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,X\_\{0\}\\sim p\_\{0\},\\,X\_\{t\}=\\alpha\_\{t\}X\_\{1\}\+\\beta\_\{t\}X\_\{0\}\\end\{subarray\}\}\[∥btθ\(Xt\)−bt\|1\(Xt\|X1\)\\displaystyle\\Bigg\[\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−κtαt\(∇r\(X1\)\+αtβt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\-\\frac\{\\kappa\_\{t\}\}\{\\alpha\_\{t\}\}\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.
By the drift\-coordinate extension in Appendix[H\.1\.2](https://arxiv.org/html/2609.05727#A8.SS1.SSS2), the Gaussian\-kernel approximation \([122](https://arxiv.org/html/2609.05727#S9.E122)\) takes the form

ℒρ,η,Σ^Gau​\-​ker,drift\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1ρ\[∥btθ\(Xt\)−bt\|1\(Xt\|X1\)\\displaystyle\\mathcal\{L\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−2ηαt​κtβt2Σ^tρ\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\\qquad\-2\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.\(248\)At a canonical anchor, the associated fixed\-point condition is

btρ\(xt\)=btρ\(xt\)\+2ηαt​κtβt2Σ^tρ\(xt\)𝔼X1∼p1\|tρ\(⋅\|xt\)\[∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(xt−αtX1\)\]\.b\_\{t\}^\{\\rho\}\(x\_\{t\}\)=b\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+2\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim p\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\]\.By the stop\-gradient construction of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), this fixed\-point condition admits the following stop\-gradient realization:

ℒη,Σ^Gau​\-​ker,drift,sg\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξt,1sg⁡\(θ\)\[∥btθ\(Xt\)−bt\|1\(Xt\|X1\)\\displaystyle\\mathcal\{L\}\_\{\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\Xi\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\Bigg\[\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-b\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−2ηαt​κtβt2Σ^t\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+αtτ​βt2\(Xt−αtX1\)\)∥22\]\.\\displaystyle\\qquad\-2\\eta\\frac\{\\alpha\_\{t\}\\kappa\_\{t\}\}\{\\beta\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}\(X\_\{t\}\)\\left\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\alpha\_\{t\}\}\{\\tau\\beta\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\alpha\_\{t\}X\_\{1\}\\right\)\\right\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.\(249\)
With the isotropic choiceΣ^t​\(Xt\)=βt2αt2​I\\widehat\{\\Sigma\}\_\{t\}\(X\_\{t\}\)=\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I, comparing the two losses yields the exact sample\-wise correspondence

ℒFS​\(θ\)=ℒη,Σ^Gau​\-​ker,drift,sg​\(θ\)\|μ=1,τ=1,η=12,Σ^t≡βt2αt2​Iunder the forward construction\.\\mathcal\{L\}^\{\\mathrm\{FS\}\}\(\\theta\)=\\left\.\\mathcal\{L\}\_\{\\eta,\\widehat\{\\Sigma\}\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\mu=1,\\,\\tau=1,\\,\\eta=\\frac\{1\}\{2\},\\,\\widehat\{\\Sigma\}\_\{t\}\\equiv\\frac\{\\beta\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}I\}\\qquad\\text\{under the forward construction\}\.Therefore, FS is the stop\-gradient realization of the damped Gaussian\-kernel approximation update with stepsizeη=12\\eta=\\frac\{1\}\{2\}, using the drift coordinate and the forward construction with SDE\-based sampling\. According to Propositions[41](https://arxiv.org/html/2609.05727#Thmtheorem41)and[48](https://arxiv.org/html/2609.05727#Thmtheorem48), the stop\-gradient loss has𝒞⁡\(π1,1,r\)\\mathcal\{C\}\(\\pi\_\{1,1,r\}\)as its unique canonical population\-stationary point\.

### I\.6Adjoint Sampling

Using our notation, Adjoint Sampling \(AS\) considers the SDE

d​Yt=btθ​\(Yt\)​d​t\+σt​d​Wt,t∈\(0,1\),\\displaystyle\\mathrm\{d\}Y\_\{t\}=b\_\{t\}^\{\\theta\}\(Y\_\{t\}\)\\mathrm\{d\}t\+\\sigma\_\{t\}\\mathrm\{d\}W\_\{t\},\\qquad t\\in\(0,1\),\(250a\)Y0=0\.\\displaystyle Y\_\{0\}=0\.\(250b\)Letρθ\\rho^\{\\theta\}denote the terminal density induced by this SDE\. AS then uses the following stop\-gradient loss:

ℒAS\(θ\):=𝔼t∼U\(0,1\),X1∼ρsg⁡\(θ\),Xt∼𝒩⁡\(γ~tγ~1​X1,γ~t​\(γ~1−γ~t\)γ~1​I\)\[12​σt2‖1σtbtθ\(Xt\)\+σt∇\(log𝒩\(X1;0,γ~1I\)−τr\(X1\)\)‖22\],\\displaystyle\\mathcal\{L\}^\{\\mathrm\{AS\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,\\\\ X\_\{t\}\\sim\\mathcal\{N\}\\left\(\\frac\{\\widetilde\{\\gamma\}\_\{t\}\}\{\\widetilde\{\\gamma\}\_\{1\}\}X\_\{1\},\\frac\{\\widetilde\{\\gamma\}\_\{t\}\(\\widetilde\{\\gamma\}\_\{1\}\-\\widetilde\{\\gamma\}\_\{t\}\)\}\{\\widetilde\{\\gamma\}\_\{1\}\}I\\right\)\\end\{subarray\}\}\\left\[\\frac\{1\}\{2\\sigma\_\{t\}^\{2\}\}\\left\\\|\\frac\{1\}\{\\sigma\_\{t\}\}b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+\\sigma\_\{t\}\\nabla\\left\(\\log\\mathcal\{N\}\\left\(X\_\{1\};0,\\widetilde\{\\gamma\}\_\{1\}I\\right\)\-\\tau r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\],where

γ~t:=∫0tσu2​𝑑u\.\\widetilde\{\\gamma\}\_\{t\}:=\\int\_\{0\}^\{t\}\\sigma\_\{u\}^\{2\}\\,\\mathrm\{d\}u\.The reciprocal sampling kernel used by AS,

Xt\|X1∼𝒩⁡\(γ~tγ~1​X1,γ~t​\(γ~1−γ~t\)γ~1​I\),X\_\{t\}\|X\_\{1\}\\sim\\mathcal\{N\}\\left\(\\frac\{\\widetilde\{\\gamma\}\_\{t\}\}\{\\widetilde\{\\gamma\}\_\{1\}\}X\_\{1\},\\frac\{\\widetilde\{\\gamma\}\_\{t\}\(\\widetilde\{\\gamma\}\_\{1\}\-\\widetilde\{\\gamma\}\_\{t\}\)\}\{\\widetilde\{\\gamma\}\_\{1\}\}I\\right\),can be recovered from the one\-sided interpolant setting\. Using the notation of Appendix[H\.2](https://arxiv.org/html/2609.05727#A8.SS2), we define the AS\-specific schedule

α~t:=γ~tγ~1,β~t:=γ~t​\(γ~1−γ~t\)γ~1,κ~t:=β~tα~t​\(α~˙t​β~t−α~t​β~˙t\)\.\\widetilde\{\\alpha\}\_\{t\}:=\\frac\{\\widetilde\{\\gamma\}\_\{t\}\}\{\\widetilde\{\\gamma\}\_\{1\}\},\\qquad\\widetilde\{\\beta\}\_\{t\}:=\\sqrt\{\\frac\{\\widetilde\{\\gamma\}\_\{t\}\(\\widetilde\{\\gamma\}\_\{1\}\-\\widetilde\{\\gamma\}\_\{t\}\)\}\{\\widetilde\{\\gamma\}\_\{1\}\}\},\\qquad\\widetilde\{\\kappa\}\_\{t\}:=\\frac\{\\widetilde\{\\beta\}\_\{t\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}\\left\(\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}\\widetilde\{\\beta\}\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}\\dot\{\\widetilde\{\\beta\}\}\_\{t\}\\right\)\.\(251\)This schedule satisfies

σt=2​κ~t\.\\sigma\_\{t\}=\\sqrt\{2\\widetilde\{\\kappa\}\_\{t\}\}\.The sampling rule used by AS is equivalent to the forward construction under the one\-sided interpolant setting:

X1∼ρsg⁡\(θ\),Xt∼𝒩\(α~tX1,β~t2I\)⇔X1∼ρsg⁡\(θ\),Z∼𝒩\(0,I\),Xt=α~tX1\+β~tZ\.X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,X\_\{t\}\\sim\\mathcal\{N\}\\left\(\\widetilde\{\\alpha\}\_\{t\}X\_\{1\},\\widetilde\{\\beta\}\_\{t\}^\{2\}I\\right\)\\quad\\Leftrightarrow\\quad X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,Z\\sim\\mathcal\{N\}\(0,I\),\\,X\_\{t\}=\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\+\\widetilde\{\\beta\}\_\{t\}Z\.Accordingly, the AS loss can be rewritten as

ℒAS\(θ\):=𝔼t∼U\(0,1\),X1∼ρsg⁡\(θ\),Z∼𝒩\(0,I\),Xt=α~t​X1\+β~t​Z\[12​σt2‖1σtbtθ\(Xt\)\+σt∇\(log𝒩\(X1;0,γ~1I\)−τr\(X1\)\)‖22\]\.\\displaystyle\\mathcal\{L\}^\{\\mathrm\{AS\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,Z\\sim\\mathcal\{N\}\(0,I\),\\\\ X\_\{t\}=\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\+\\widetilde\{\\beta\}\_\{t\}Z\\end\{subarray\}\}\\left\[\\frac\{1\}\{2\\sigma\_\{t\}^\{2\}\}\\left\\\|\\frac\{1\}\{\\sigma\_\{t\}\}b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+\\sigma\_\{t\}\\nabla\\left\(\\log\\mathcal\{N\}\\left\(X\_\{1\};0,\\widetilde\{\\gamma\}\_\{1\}I\\right\)\-\\tau r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\\right\]\.We have

12​σt2‖1σtbtθ\(Xt\)\+σt∇\(log𝒩\(X1;0,γ~1I\)−τr\(X1\)\)‖22\\displaystyle\\frac\{1\}\{2\\sigma\_\{t\}^\{2\}\}\\left\\\|\\frac\{1\}\{\\sigma\_\{t\}\}b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+\\sigma\_\{t\}\\nabla\\left\(\\log\\mathcal\{N\}\\left\(X\_\{1\};0,\\widetilde\{\\gamma\}\_\{1\}I\\right\)\-\\tau r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=12​σt4‖btθ\(Xt\)\+σt2\(−1γ~1X1−τ∇r\(X1\)\)‖22\\displaystyle\\frac\{1\}\{2\\sigma\_\{t\}^\{4\}\}\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\+\\sigma\_\{t\}^\{2\}\\left\(\-\\frac\{1\}\{\\widetilde\{\\gamma\}\_\{1\}\}X\_\{1\}\-\\tau\\nabla r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=18​κ~t2‖btθ\(Xt\)−2κ~t\(1γ~1X1\+τ∇r\(X1\)\)‖22\.\\displaystyle\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-2\\widetilde\{\\kappa\}\_\{t\}\\left\(\\frac\{1\}\{\\widetilde\{\\gamma\}\_\{1\}\}X\_\{1\}\+\\tau\\nabla r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\.
Moreover, the schedule \([251](https://arxiv.org/html/2609.05727#A9.E251)\) satisfies

α~˙t=2​κ~tγ~1,β~˙t=γ~1−2​γ~t2​β~t​γ~1⋅2​κ~t=κ~tβ~t​\(1−2​α~t\)\.\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}=\\frac\{2\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\gamma\}\_\{1\}\},\\qquad\\dot\{\\widetilde\{\\beta\}\}\_\{t\}=\\frac\{\\widetilde\{\\gamma\}\_\{1\}\-2\\widetilde\{\\gamma\}\_\{t\}\}\{2\\widetilde\{\\beta\}\_\{t\}\\widetilde\{\\gamma\}\_\{1\}\}\\cdot 2\\widetilde\{\\kappa\}\_\{t\}=\\frac\{\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}\}\\left\(1\-2\\widetilde\{\\alpha\}\_\{t\}\\right\)\.It follows that

b~t\|1​\(xt\|x1\)=α~˙t​x1\+\(β~˙t−κ~tβ~t\)​xt−α~t​x1β~t=α~˙t​x1−2​α~t​κ~tβ~t2​\(xt−α~t​x1\)\.\\displaystyle\\widetilde\{b\}\_\{t\|1\}\(x\_\{t\}\|x\_\{1\}\)=\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}x\_\{1\}\+\\left\(\\dot\{\\widetilde\{\\beta\}\}\_\{t\}\-\\frac\{\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}\}\\right\)\\frac\{x\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}x\_\{1\}\}\{\\widetilde\{\\beta\}\_\{t\}\}=\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}x\_\{1\}\-\\frac\{2\\widetilde\{\\alpha\}\_\{t\}\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}x\_\{1\}\\right\)\.Thus,

18​κ~t2‖btθ\(Xt\)−2κ~t\(1γ~1X1\+τ∇r\(X1\)\)‖22\\displaystyle\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-2\\widetilde\{\\kappa\}\_\{t\}\\left\(\\frac\{1\}\{\\widetilde\{\\gamma\}\_\{1\}\}X\_\{1\}\+\\tau\\nabla r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=18​κ~t2‖btθ\(Xt\)−\(α~˙tX1\+2κ~tτ∇r\(X1\)\)‖22\\displaystyle\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\(\\dot\{\\widetilde\{\\alpha\}\}\_\{t\}X\_\{1\}\+2\\widetilde\{\\kappa\}\_\{t\}\\tau\\nabla r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}=\\displaystyle=18​κ~t2‖btθ\(Xt\)−\(b~t\|1\(Xt\|X1\)\+2​α~t​κ~tβ~t2\(Xt−α~tX1\)\+2κ~tτ∇r\(X1\)\)‖22\.\\displaystyle\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\\left\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\left\(\\widetilde\{b\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)\+\\frac\{2\\widetilde\{\\alpha\}\_\{t\}\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\+2\\widetilde\{\\kappa\}\_\{t\}\\tau\\nabla r\(X\_\{1\}\)\\right\)\\right\\\|\_\{2\}^\{2\}\.Therefore, the AS loss can be rewritten as

ℒAS​\(θ\)\\displaystyle\\mathcal\{L\}^\{\\mathrm\{AS\}\}\(\\theta\)=𝔼t∼U⁡\(0,1\),X1∼ρsg⁡\(θ\),Z∼𝒩⁡\(0,I\),Xt=α~t​X1\+β~t​Z\[18​κ~t2∥btθ\(Xt\)−b~t\|1\(Xt\|X1\)\\displaystyle=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,X\_\{1\}\\sim\\rho^\{\\operatorname\{sg\}\(\\theta\)\},\\,Z\\sim\\mathcal\{N\}\(0,I\),\\,X\_\{t\}=\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\+\\widetilde\{\\beta\}\_\{t\}Z\\end\{subarray\}\}\\Bigg\[\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\widetilde\{b\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−2κ~tτ\(∇r\(X1\)\+α~tτ​β~t2\(Xt−α~tX1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\-2\\widetilde\{\\kappa\}\_\{t\}\\tau\\bigg\(\\nabla r\(X\_\{1\}\)\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\tau\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\\bigg\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.\(252\)
Using the one\-sided\-interpolant construction in Appendix[H\.2](https://arxiv.org/html/2609.05727#A8.SS2), we extend the drift\-coordinate Gaussian\-kernel approximation of Newton Matching in \([248](https://arxiv.org/html/2609.05727#A9.E248)\) to the one\-sided setting:

ℒ~ρ,η,Σ^,ωGau​\-​ker,drift\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξ~t,1ρ\[ωt\(Xt\)∥btθ\(Xt\)−b~t\|1\(Xt\|X1\)\\displaystyle\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\widetilde\{\\Xi\}\_\{t,1\}^\{\\rho\}\\end\{subarray\}\}\\Bigg\[\\omega\_\{t\}\(X\_\{t\}\)\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\widetilde\{b\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−2ηα~t​κ~tβ~t2Σ^tρ\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+α~tτ​β~t2\(Xt−α~tX1\)\)∥22\]\.\\displaystyle\\qquad\-2\\eta\\frac\{\\widetilde\{\\alpha\}\_\{t\}\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(X\_\{t\}\)\\bigg\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\tau\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\\bigg\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.\(253\)Here, the positive weightωt​\(xt\)\>0\\omega\_\{t\}\(x\_\{t\}\)\>0does not affect the population\-stationary point\. At a canonical driftb~ρ\\widetilde\{b\}^\{\\rho\}, the associated fixed\-point condition is

b~tρ\(xt\)=b~tρ\(xt\)\+2ηα~t​κ~tβ~t2Σ^tρ\(xt\)𝔼X1∼p~1\|tρ\(⋅\|xt\)\[∇r\(X1\)\+1τ∇logμ\(X1\)\+α~tτ​β~t2\(xt−α~tX1\)\]\.\\displaystyle\\widetilde\{b\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)=\\widetilde\{b\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\+2\\eta\\frac\{\\widetilde\{\\alpha\}\_\{t\}\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}^\{\\rho\}\(x\_\{t\}\)\\mathbb\{E\}\_\{X\_\{1\}\\sim\\widetilde\{p\}\_\{1\|t\}^\{\\rho\}\(\\cdot\|x\_\{t\}\)\}\\left\[\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\tau\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(x\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\\right\]\.By the stop\-gradient construction of Section[9\.4](https://arxiv.org/html/2609.05727#S9.SS4), this fixed\-point condition admits the following stop\-gradient realization:

ℒ~ρ,η,Σ^,ωGau​\-​ker,drift,sg\(θ\):=𝔼t∼U⁡\(0,1\),\(Xt,X1\)∼Ξ~t,1sg⁡\(θ\)\[ωt\(Xt\)∥btθ\(Xt\)−b~t\|1\(Xt\|X1\)\\displaystyle\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}\(\\theta\):=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}t\\sim\\operatorname\{U\}\(0,1\),\\,\(X\_\{t\},X\_\{1\}\)\\sim\\widetilde\{\\Xi\}\_\{t,1\}^\{\\operatorname\{sg\}\(\\theta\)\}\\end\{subarray\}\}\\Bigg\[\\omega\_\{t\}\(X\_\{t\}\)\\bigg\\\|b\_\{t\}^\{\\theta\}\(X\_\{t\}\)\-\\widetilde\{b\}\_\{t\|1\}\(X\_\{t\}\|X\_\{1\}\)−2ηα~t​κ~tβ~t2Σ^t\(Xt\)\(∇r\(X1\)\+1τ∇logμ\(X1\)\+α~tτ​β~t2\(Xt−α~tX1\)\)∥22\]\.\\displaystyle\\qquad\\qquad\-2\\eta\\frac\{\\widetilde\{\\alpha\}\_\{t\}\\widetilde\{\\kappa\}\_\{t\}\}\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\widehat\{\\Sigma\}\_\{t\}\(X\_\{t\}\)\\bigg\(\\nabla r\(X\_\{1\}\)\+\\frac\{1\}\{\\tau\}\\nabla\\log\\mu\(X\_\{1\}\)\+\\frac\{\\widetilde\{\\alpha\}\_\{t\}\}\{\\tau\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\\left\(X\_\{t\}\-\\widetilde\{\\alpha\}\_\{t\}X\_\{1\}\\right\)\\bigg\)\\bigg\\\|\_\{2\}^\{2\}\\Bigg\]\.\(254\)
Comparing this loss with \([252](https://arxiv.org/html/2609.05727#A9.E252)\) yields the exact sample\-wise correspondence

ℒAS​\(θ\)=ℒ~ρ,η,Σ^,ωGau​\-​ker,drift,sg​\(θ\)\|μ=1,α~t=γ~tγ~1,β~t=γ~t​\(γ~1−γ~t\)γ~1,η=τ,Σ^t≡β~t2α~t​I,ωt≡18​κ~t2\\mathcal\{L\}^\{\\mathrm\{AS\}\}\(\\theta\)=\\left\.\\widetilde\{\\mathcal\{L\}\}\_\{\\rho,\\eta,\\widehat\{\\Sigma\},\\omega\}^\{\\mathrm\{Gau\}\\text\{\-\}\\mathrm\{ker\},\\mathrm\{drift\},\\mathrm\{sg\}\}\(\\theta\)\\right\|\_\{\\mu=1,\\,\\widetilde\{\\alpha\}\_\{t\}=\\frac\{\\widetilde\{\\gamma\}\_\{t\}\}\{\\widetilde\{\\gamma\}\_\{1\}\},\\,\\widetilde\{\\beta\}\_\{t\}=\\sqrt\{\\frac\{\\widetilde\{\\gamma\}\_\{t\}\(\\widetilde\{\\gamma\}\_\{1\}\-\\widetilde\{\\gamma\}\_\{t\}\)\}\{\\widetilde\{\\gamma\}\_\{1\}\}\},\\,\\eta=\\tau,\\,\\widehat\{\\Sigma\}\_\{t\}\\equiv\\frac\{\\widetilde\{\\beta\}\_\{t\}^\{2\}\}\{\\widetilde\{\\alpha\}\_\{t\}\}I,\\,\\omega\_\{t\}\\equiv\\frac\{1\}\{8\\widetilde\{\\kappa\}\_\{t\}^\{2\}\}\}under the forward construction\.

Therefore, AS is the stop\-gradient realization of the full\-step Gaussian\-kernel approximation update under the forward construction with SDE\-based sampling, using the drift coordinate and the one\-sided interpolant\. The stop\-gradient loss has𝒞⁡\(π1,τ,r\)\\mathcal\{C\}\(\\pi\_\{1,\\tau,r\}\)as its unique canonical population\-stationary point\.

## References

- \[1\]T\. Akhound\-Sadegh, J\. Rector\-Brooks, J\. Bose, S\. Mittal, P\. Lemos, C\. Liu, M\. Sendera, S\. Ravanbakhsh, G\. Gidel, Y\. Bengio, N\. Malkin, and A\. Tong\(2024\)Iterated denoising energy matching for sampling from Boltzmann densities\.InInternational Conference on Machine Learning,Vol\.235,pp\. 760–786\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p5.1)\.
- \[2\]\(2025\)Stochastic interpolants: a unifying framework for flows and diffusions\.Journal of Machine Learning Research26\(209\),pp\. 1–80\.Cited by:[§H\.2](https://arxiv.org/html/2609.05727#A8.SS2.p1.1)\.
- \[3\]L\. Ambrosio, N\. Gigli, and G\. Savaré\(2005\)Gradient flows: in metric spaces and in the space of probability measures\.Springer\.Cited by:[§A\.1\.2](https://arxiv.org/html/2609.05727#A1.SS1.SSS2.p2.3.1),[§A\.1\.3](https://arxiv.org/html/2609.05727#A1.SS1.SSS3.p4.2.1)\.
- \[4\]N\. Ay, J\. Jost, H\. V\. Lê, and L\. Schwachhöfer\(2017\)Information geometry\.Vol\.64,Springer\.Cited by:[§A\.3\.1](https://arxiv.org/html/2609.05727#A1.SS3.SSS1.p1.1),[§4\.2\.3](https://arxiv.org/html/2609.05727#S4.SS2.SSS3.p2.1)\.
- \[5\]A\. Bergmeister, S\. Jegelka, N\. Nüsken, C\. Domingo\-Enrich, and J\. Pidstrigach\(2026\)Reinforce Adjoint Matching: scaling RL post\-training of diffusion and flow\-matching models\.arXiv preprint arXiv:2605\.10759\.Cited by:[item 8](https://arxiv.org/html/2609.05727#S1.I1.i8.p1.1),[§1](https://arxiv.org/html/2609.05727#S1.p5.1),[§10\.3](https://arxiv.org/html/2609.05727#S10.SS3.p1.1),[1st item](https://arxiv.org/html/2609.05727#S9.I5.i1.I1.i1.p1.1),[Remark 49](https://arxiv.org/html/2609.05727#Thmtheorem49.p1.1.1)\.
- \[6\]J\. Berner, L\. Richter, and K\. Ullrich\(2024\)An optimal control perspective on diffusion\-based generative modeling\.Transactions on Machine Learning Research\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[7\]K\. Black, M\. Janner, Y\. Du, I\. Kostrikov, and S\. Levine\(2024\)Training diffusion models with reinforcement learning\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p2.2),[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[8\]D\. Blessing, L\. Richter, J\. Berner, E\. Malitskiy, and G\. Neumann\(2026\)Bridge matching sampler: scalable sampling via generalized fixed\-point diffusion matching\.InInternational Conference on Machine Learning,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p5.1)\.
- \[9\]C\. Chi, Z\. Xu, S\. Feng, E\. Cousineau, Y\. Du, B\. Burchfiel, R\. Tedrake, and S\. Song\(2025\)Diffusion policy: visuomotor policy learning via action diffusion\.The International Journal of Robotics Research44\(10–11\),pp\. 1684–1704\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[10\]T\. A\. Courtade, M\. Fathi, and A\. Pananjady\(2019\)Existence of Stein kernels under a spectral gap, and discrepancy bounds\.Annales de l’Institut Henri Poincaré, Probabilités et Statistiques55\(2\),pp\. 777–790\.Cited by:[§E\.3](https://arxiv.org/html/2609.05727#A5.SS3.p1.3)\.
- \[11\]I\. Csiszár and G\. Tusnády\(1984\)Information geometry and alternating minimization procedures\.Statistics and Decisions,pp\. 205–237\.Note:Supplement no\. 1Cited by:[Remark 27](https://arxiv.org/html/2609.05727#Thmtheorem27.p2.1.1)\.
- \[12\]I\. Csiszár\(1975\)I\-divergence geometry of probability distributions and minimization problems\.The Annals of Probability3\(1\),pp\. 146–158\.Cited by:[Remark 27](https://arxiv.org/html/2609.05727#Thmtheorem27.p2.1.1)\.
- \[13\]V\. De Bortoli, J\. Thornton, J\. Heng, and A\. Doucet\(2021\)Diffusion Schrödinger bridge with applications to score\-based generative modeling\.InAdvances in Neural Information Processing Systems,Vol\.34,pp\. 17695–17709\.Cited by:[Remark 27](https://arxiv.org/html/2609.05727#Thmtheorem27.p2.1.1)\.
- \[14\]C\. Domingo\-Enrich, M\. Drozdzal, B\. Karrer, and R\. T\. Q\. Chen\(2025\)Adjoint matching: fine\-tuning flow and diffusion generative models with memoryless stochastic optimal control\.InInternational Conference on Learning Representations,Cited by:[§I\.4\.1](https://arxiv.org/html/2609.05727#A9.SS4.SSS1.p1.3),[§I\.4](https://arxiv.org/html/2609.05727#A9.SS4.p1.1),[§I\.4](https://arxiv.org/html/2609.05727#A9.SS4.p3.1),[item 8](https://arxiv.org/html/2609.05727#S1.I1.i8.p1.1),[§1](https://arxiv.org/html/2609.05727#S1.p5.1),[§10\.4](https://arxiv.org/html/2609.05727#S10.SS4.p1.1),[1st item](https://arxiv.org/html/2609.05727#S9.I2.i2.I1.i1.p1.1),[1st item](https://arxiv.org/html/2609.05727#S9.I4.i1.I1.i1.p1.1),[Remark 35](https://arxiv.org/html/2609.05727#Thmtheorem35.p2.1.1),[Remark 49](https://arxiv.org/html/2609.05727#Thmtheorem49.p1.1.1)\.
- \[15\]J\. C\. Duchi, S\. Shalev\-Shwartz, Y\. Singer, and A\. Tewari\(2010\)Composite objective mirror descent\.InConference on Learning Theory,pp\. 14–26\.Cited by:[Remark 20](https://arxiv.org/html/2609.05727#Thmtheorem20.p1.3.1)\.
- \[16\]P\. Esser, S\. Kulal, A\. Blattmann, R\. Entezari, J\. Müller, H\. Saini, Y\. Levi, D\. Lorenz, A\. Sauer, F\. Boesel, D\. Podell, T\. Dockhorn, Z\. English, and R\. Rombach\(2024\)Scaling rectified flow transformers for high\-resolution image synthesis\.InInternational Conference on Machine Learning,Vol\.235,pp\. 12606–12633\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[17\]J\. Fan, S\. Shen, C\. Cheng, Y\. Chen, C\. Liang, and G\. Liu\(2025\)Online reward\-weighted fine\-tuning of flow matching with Wasserstein regularization\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p5.1)\.
- \[18\]Y\. Fan, O\. Watkins, Y\. Du, H\. Liu, M\. Ryu, C\. Boutilier, P\. Abbeel, M\. Ghavamzadeh, K\. Lee, and K\. Lee\(2023\)DPOK: reinforcement learning for fine\-tuning text\-to\-image diffusion models\.InAdvances in Neural Information Processing Systems,Vol\.36,pp\. 79858–79885\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p2.2),[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[19\]J\. Gorham and L\. Mackey\(2015\)Measuring sample quality with Stein’s method\.InAdvances in Neural Information Processing Systems,Vol\.28,pp\. 226–234\.Cited by:[§E\.1](https://arxiv.org/html/2609.05727#A5.SS1.p1.1)\.
- \[20\]T\. Haarnoja, A\. Zhou, P\. Abbeel, and S\. Levine\(2018\)Soft actor\-critic: off\-policy maximum entropy deep reinforcement learning with a stochastic actor\.InInternational Conference on Machine Learning,Vol\.80,pp\. 1861–1870\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p2.2)\.
- \[21\]Y\. Han, M\. Razaviyayn, and R\. Xu\(2025\)Stochastic control for fine\-tuning diffusion models: optimality, regularity, and convergence\.InInternational Conference on Machine Learning,Vol\.267,pp\. 21844–21870\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[22\]A\. Havens, B\. Karrer, and N\. Shaul\(2026\)Flow sampling: learning to sample from unnormalized densities via denoising conditional processes\.InInternational Conference on Machine Learning,Cited by:[item 8](https://arxiv.org/html/2609.05727#S1.I1.i8.p1.1),[§1](https://arxiv.org/html/2609.05727#S1.p5.1),[§10\.5](https://arxiv.org/html/2609.05727#S10.SS5.p1.1),[1st item](https://arxiv.org/html/2609.05727#S9.I10.i1.I1.i1.p1.1),[Remark 49](https://arxiv.org/html/2609.05727#Thmtheorem49.p1.1.1)\.
- \[23\]A\. Havens, B\. K\. Miller, B\. Yan, C\. Domingo\-Enrich, A\. Sriram, D\. Levine, B\. Wood, B\. Hu, B\. Amos, B\. Karrer, X\. Fu, G\. Liu, and R\. T\. Q\. Chen\(2025\)Adjoint sampling: highly scalable diffusion samplers via adjoint matching\.InInternational Conference on Machine Learning,Vol\.267,pp\. 22204–22237\.Cited by:[item 8](https://arxiv.org/html/2609.05727#S1.I1.i8.p1.1),[§1](https://arxiv.org/html/2609.05727#S1.p5.1),[§10\.6](https://arxiv.org/html/2609.05727#S10.SS6.p1.1),[2nd item](https://arxiv.org/html/2609.05727#S9.I10.i1.I1.i2.p1.1),[Remark 49](https://arxiv.org/html/2609.05727#Thmtheorem49.p1.1.1)\.
- \[24\]J\. Ho, A\. Jain, and P\. Abbeel\(2020\)Denoising diffusion probabilistic models\.InAdvances in Neural Information Processing Systems,Vol\.33,pp\. 6840–6851\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[25\]J\. Ho, T\. Salimans, A\. Gritsenko, W\. Chan, M\. Norouzi, and D\. J\. Fleet\(2022\)Video diffusion models\.InAdvances in Neural Information Processing Systems,Vol\.35,pp\. 8633–8646\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[26\]W\. Hoeffding\(1963\)Probability inequalities for sums of bounded random variables\.Journal of the American Statistical Association58\(301\),pp\. 13–30\.Cited by:[§F\.3](https://arxiv.org/html/2609.05727#A6.SS3.p7.1)\.
- \[27\]M\. F\. Hutchinson\(1990\)A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines\.Communications in Statistics\-Simulation and Computation19\(2\),pp\. 433–450\.Cited by:[§8\.3\.1](https://arxiv.org/html/2609.05727#S8.SS3.SSS1.Px2.p3.1)\.
- \[28\]H\. Hwang, H\. Jeong, D\. K\. Shin, C\. Park, S\. Kweon, S\. Yoon, and F\. C\. Park\(2026\)Value gradient sampler: learning invariant value functions for equivariant diffusion sampling\.InInternational Conference on Artificial Intelligence and Statistics,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[29\]S\. Jafarpour and A\. D\. Lewis\(2014\)Time\-varying vector fields and their flows\.Springer\.Cited by:[§A\.1\.1](https://arxiv.org/html/2609.05727#A1.SS1.SSS1.p3.1.1),[§A\.1\.2](https://arxiv.org/html/2609.05727#A1.SS1.SSS2.p2.1.1)\.
- \[30\]S\. Kullback\(1967\)A lower bound for discrimination information in terms of variation\.IEEE Transactions on Information Theory13\(1\),pp\. 126–127\.Cited by:[§7\.1\.3](https://arxiv.org/html/2609.05727#S7.SS1.SSS3.p2.2)\.
- \[31\]S\. Lang\(1999\)Fundamentals of differential geometry\.Graduate Texts in Mathematics, Vol\.191,Springer,New York\.Cited by:[§2\.2](https://arxiv.org/html/2609.05727#S2.SS2.p1.1)\.
- \[32\]Z\. Li, S\. Tang, and N\. Azizan\(2026\)Reverse flow matching: a unified framework for online reinforcement learning with diffusion and flow policies\.InInternational Conference on Machine Learning,Cited by:[§E\.1](https://arxiv.org/html/2609.05727#A5.SS1.p1.1),[§1](https://arxiv.org/html/2609.05727#S1.p2.2),[§1](https://arxiv.org/html/2609.05727#S1.p5.1)\.
- \[33\]Y\. Lipman, R\. T\. Q\. Chen, H\. Ben\-Hamu, M\. Nickel, and M\. Le\(2023\)Flow matching for generative modeling\.InInternational Conference on Learning Representations,Cited by:[§A\.1\.3](https://arxiv.org/html/2609.05727#A1.SS1.SSS3.p3.1.1),[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[34\]G\. Liu, J\. Choi, Y\. Chen, B\. K\. Miller, and R\. T\. Q\. Chen\(2025\)Adjoint Schrödinger bridge sampler\.InAdvances in Neural Information Processing Systems,Vol\.38,pp\. 15673–15708\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p5.1)\.
- \[35\]J\. Liu, G\. Liu, J\. Liang, Y\. Li, J\. Liu, X\. Wang, P\. Wan, D\. Zhang, and W\. Ouyang\(2025\)Flow\-GRPO: training flow matching models via online RL\.InAdvances in Neural Information Processing Systems,Vol\.38,pp\. 40783–40818\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[36\]R\. Meyer and C\. Zhu\(2015\)Groupoids in categories with pretopology\.Theory and Applications of Categories30\(55\),pp\. 1906–1998\.Cited by:[§A\.3\.1](https://arxiv.org/html/2609.05727#A1.SS3.SSS1.p5.1.1)\.
- \[37\]F\. Noé, S\. Olsson, J\. Köhler, and H\. Wu\(2019\)Boltzmann generators: sampling equilibrium states of many\-body systems with deep learning\.Science365\(6457\),pp\. eaaw1147\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p2.2)\.
- \[38\]S\. Peluchetti\(2023\)Diffusion bridge mixture transports, Schrödinger bridge problems and generative modeling\.Journal of Machine Learning Research24\(374\),pp\. 1–51\.Cited by:[§H\.2](https://arxiv.org/html/2609.05727#A8.SS2.p1.1)\.
- \[39\]P\. Potaptchik, C\. Lee, and M\. S\. Albergo\(2026\)Tilt matching for scalable sampling and fine\-tuning\.InInternational Conference on Machine Learning,Cited by:[§I\.1\.1](https://arxiv.org/html/2609.05727#A9.SS1.SSS1.p2.1),[item 8](https://arxiv.org/html/2609.05727#S1.I1.i8.p1.1),[§1](https://arxiv.org/html/2609.05727#S1.p5.1),[§10\.1](https://arxiv.org/html/2609.05727#S10.SS1.p1.1),[§10\.1](https://arxiv.org/html/2609.05727#S10.SS1.p1.3),[1st item](https://arxiv.org/html/2609.05727#S9.I3.i1.I1.i1.p1.1),[1st item](https://arxiv.org/html/2609.05727#S9.I5.i2.I1.i1.p1.1),[1st item](https://arxiv.org/html/2609.05727#S9.I9.i1.I1.i1.p1.1)\.
- \[40\]L\. Richter and J\. Berner\(2024\)Improved sampling via learned diffusions\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[41\]R\. Rombach, A\. Blattmann, D\. Lorenz, P\. Esser, and B\. Ommer\(2022\)High\-resolution image synthesis with latent diffusion models\.InIEEE/CVF Conference on Computer Vision and Pattern Recognition,pp\. 10684–10695\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[42\]J\. Sohl\-Dickstein, E\. Weiss, N\. Maheswaranathan, and S\. Ganguli\(2015\)Deep unsupervised learning using nonequilibrium thermodynamics\.InInternational Conference on Machine Learning,Vol\.37,pp\. 2256–2265\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[43\]O\. Toeplitz\(1911\)Über allgemeine lineare Mittelbildungen\.Prace Matematyczno\-Fizyczne22\(1\),pp\. 113–119\.Cited by:[§B\.1](https://arxiv.org/html/2609.05727#A2.SS1.p1.1)\.
- \[44\]B\. Tzen and M\. Raginsky\(2019\)Theoretical guarantees for sampling and inference in generative models with latent diffusions\.InConference on Learning Theory,Vol\.99,pp\. 3084–3114\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[45\]M\. Uehara, Y\. Zhao, K\. Black, E\. Hajiramezanali, G\. Scalia, N\. L\. Diamant, A\. M\. Tseng, T\. Biancalani, and S\. Levine\(2024\)Fine\-tuning of continuous\-time diffusion models as entropy\-regularized control\.arXiv preprint arXiv:2402\.15194\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[46\]F\. Vargas, W\. S\. Grathwohl, and A\. Doucet\(2023\)Denoising diffusion samplers\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[47\]Z\. Wang, J\. J\. Hunt, and M\. Zhou\(2023\)Diffusion policies as an expressive policy class for offline reinforcement learning\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p2.2)\.
- \[48\]Z\. Wang, A\. Novikov, K\. Zolna, J\. T\. Springenberg, S\. Reed, B\. Shahriari, N\. Siegel, J\. Merel, C\. Gulcehre, N\. Heess, and N\. de Freitas\(2020\)Critic regularized regression\.InAdvances in Neural Information Processing Systems,Vol\.33,pp\. 7768–7778\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p2.2)\.
- \[49\]J\. L\. Watson, D\. Juergens, N\. R\. Bennett, B\. L\. Trippe, J\. Yim, H\. E\. Eisenach, W\. Ahern, A\. J\. Borst, R\. J\. Ragotte, L\. F\. Milles, B\. I\. M\. Wicky, N\. Hanikel, S\. J\. Pellock, A\. Courbet, W\. Sheffler, J\. Wang, P\. Venkatesh, I\. Sappington, S\. V\. Torres, A\. Lauko, V\. De Bortoli, E\. Mathieu, S\. Ovchinnikov, R\. Barzilay, T\. S\. Jaakkola, F\. DiMaio, M\. Baek, and D\. Baker\(2023\)De novo design of protein structure and function with RFdiffusion\.Nature620,pp\. 1089–1100\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[50\]S\. Xue, C\. Ge, S\. Zhang, Y\. Li, and Z\. Ma\(2026\)Advantage weighted matching: aligning RL with pretraining in diffusion models\.InInternational Conference on Machine Learning,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p5.1)\.
- \[51\]Z\. Xue, J\. Wu, Y\. Gao, F\. Kong, L\. Zhu, M\. Chen, Z\. Liu, W\. Liu, Q\. Guo, W\. Huang, and P\. Luo\(2025\)DanceGRPO: unleashing GRPO on visual generation\.arXiv preprint arXiv:2505\.07818\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[52\]J\. Yong and X\. Y\. Zhou\(1999\)Stochastic controls: hamiltonian systems and HJB equations\.Stochastic Modelling and Applied Probability, Vol\.43,Springer Science & Business Media\.Cited by:[§D\.6\.1](https://arxiv.org/html/2609.05727#A4.SS6.SSS1.p1.1)\.
- \[53\]W\. I\. Zangwill\(1969\)Nonlinear programming: a unified approach\.Prentice\-Hall,Englewood Cliffs, NJ\.Cited by:[Remark 27](https://arxiv.org/html/2609.05727#Thmtheorem27.p2.1.1)\.
- \[54\]C\. Zeni, R\. Pinsler, D\. Zügner, A\. Fowler, M\. Horton, X\. Fu, Z\. Wang, A\. Shysheya, J\. Crabbé, S\. Ueda, R\. Sordillo, L\. Sun, J\. Smith, B\. Nguyen, H\. Schulz, S\. Lewis, C\. Huang, Z\. Lu, Y\. Zhou, H\. Yang, H\. Hao, J\. Li, C\. Yang, W\. Li, R\. Tomioka, and T\. Xie\(2025\)A generative model for inorganic materials design\.Nature639,pp\. 624–632\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p1.1)\.
- \[55\]Q\. Zhang and Y\. Chen\(2022\)Path integral sampler: a stochastic control approach for sampling\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[56\]S\. Zhang, W\. Zhang, and Q\. Gu\(2025\)Energy\-weighted flow matching for offline reinforcement learning\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p5.1)\.
- \[57\]H\. Zhao, H\. Chen, J\. Zhang, D\. Yao, and W\. Tang\(2025\)Score as action: fine\-tuning diffusion generative models by continuous\-time reinforcement learning\.InInternational Conference on Machine Learning,Vol\.267,pp\. 77371–77389\.Cited by:[§1](https://arxiv.org/html/2609.05727#S1.p4.1)\.
- \[58\]K\. Zheng, H\. Chen, H\. Ye, H\. Wang, Q\. Zhang, K\. Jiang, H\. Su, S\. Ermon, J\. Zhu, and M\. Liu\(2026\)DiffusionNFT: online diffusion reinforcement with forward process\.InInternational Conference on Learning Representations,Cited by:[item 8](https://arxiv.org/html/2609.05727#S1.I1.i8.p1.1),[§1](https://arxiv.org/html/2609.05727#S1.p5.1),[§10\.2](https://arxiv.org/html/2609.05727#S10.SS2.p1.1),[2nd item](https://arxiv.org/html/2609.05727#S9.I5.i2.I1.i2.p1.1)\.
- \[59\]B\. Øksendal\(2003\)Stochastic differential equations: an introduction with applications\.6 edition,Universitext,Springer\.Cited by:[§D\.1\.3](https://arxiv.org/html/2609.05727#A4.SS1.SSS3.p1.1)\.