基于方差缩减序贯蒙特卡洛的特异性感知扩散引导方法

# Specificity-Aware Diffusion Steering via Variance-Reduced Sequential Monte Carlo
Source: [https://arxiv.org/html/2610.00395](https://arxiv.org/html/2610.00395)
Linrui MaHannes StärkRegina BarzilayAffiliation:CSAIL, Massachusetts Institute of Technology

###### Abstract

Inference\-time steering enables pretrained diffusion models to satisfy new constraints without full retraining\. However, specificity\-aware generation is difficult: repelling samples from a negative reference distribution can also erode the positive distribution where the two overlap\. The key challenge is to suppress negative mass while minimally distorting the positive distribution\. We address this problem by formulating specificity\-aware steering as a target\-design problem and deriving a target distribution from an overlap\-based objective\. The resulting target keeps the desired reference distribution only in regions where it is sufficiently preferred over the undesired reference distribution, giving a likelihood\-ratio interpretation of specificity\. To sample from the corresponding time\-dependent target path, we develop a Sequential Monte Carlo sampler with a variance\-minimized local proposal\. We further introduce a practical fixed\-noise optimization procedure with the Jacobian–vector products with the desired and undesired score fields\. Experiments on synthetic task, class\-contrastive generation, text\-to\-image tasks and peptide\-MHC \(p\-MHC\) binder show that the proposed method suppresses undesired regions more effectively, reduces mode shift, and improves sampling stability by decreasing the SMC weight collapse compared with negative\-guidance baselines\. Code is available at:[https://github\.com/WangLuran/Specificity\-Aware\-Diffusion\-Steering](https://github.com/WangLuran/Specificity-Aware-Diffusion-Steering)

## 1Introduction

Diffusion and score\-based models have become standard tools for high\-dimensional generation\[[1](https://arxiv.org/html/2610.00395#bib.bib19),[2](https://arxiv.org/html/2610.00395#bib.bib20)\]\. One important advantage of these models is that their sampling process can be steered at inference time, without retraining the underlying models\[[3](https://arxiv.org/html/2610.00395#bib.bib36)\]\[[4](https://arxiv.org/html/2610.00395#bib.bib37)\]\. This makes them suitable for settings where generation must satisfy both positive and negative constraints: samples should remain close to a desired distribution while avoiding regions associated with an undesired reference distribution\.\[[5](https://arxiv.org/html/2610.00395#bib.bib38)\]

Existing steering methods, including classifier guidance, classifier\-free guidance, negative prompting, and safe\-generation guidance, modify the denoising direction or score estimate to promote or suppress selected attributes\[[6](https://arxiv.org/html/2610.00395#bib.bib21)\]\[[7](https://arxiv.org/html/2610.00395#bib.bib22),[8](https://arxiv.org/html/2610.00395#bib.bib15)\]\[[9](https://arxiv.org/html/2610.00395#bib.bib25)\]\[[10](https://arxiv.org/html/2610.00395#bib.bib13)\]\. However, these methods usually define a steering rule rather than an explicit target distribution\. As a result, when the desired and undesired densities overlap, repulsive guidance can suppress valid modes and induce mode shift\.\[[11](https://arxiv.org/html/2610.00395#bib.bib39)\]\[[12](https://arxiv.org/html/2610.00395#bib.bib41)\]\[[13](https://arxiv.org/html/2610.00395#bib.bib40)\]\[[14](https://arxiv.org/html/2610.00395#bib.bib42)\]

To address this issue, we formulate specificity\-aware diffusion steering\. We introduce a specificity\-aware target distribution that preserves the desired density only in regions where it is sufficiently preferred over the undesired density\. To sample from the resulting time\-dependent target path, we use Sequential Monte Carlo \(SMC\)\. SMC provides a distributionally principled mechanism for diffusion steering through propagation and reweighting\[[15](https://arxiv.org/html/2610.00395#bib.bib26)\]\[[16](https://arxiv.org/html/2610.00395#bib.bib16)\]\[[17](https://arxiv.org/html/2610.00395#bib.bib17)\]\. However, its stability depends on the variance of incremental weights\[[18](https://arxiv.org/html/2610.00395#bib.bib35)\]\. We therefore derive a local proposal rule using forward–backward Gaussian kernel ratios, following the density\-ratio view of guided diffusion\[[19](https://arxiv.org/html/2610.00395#bib.bib12)\]\. The backward drift is chosen to cancel the leading fluctuation in the log\-weight, and the remaining leading\-order terms define a tractable variance objective for adapting the proposal\.

Our contributions are threefold\. First, we formulate negative guidance as sampling from a specificity\-aware target distribution derived from an overlap\-based objective, yielding a principled likelihood\-ratio gate that preserves positive mass while suppressing negative overlap\. Second, we develop a proposal\-aware SMC sampler with a gate\-augmented forward kernel and locally optimized backward kernel to reduce diffusion\-path weight degeneracy\. Third, we introduce a fixed\-noise optimization procedure using retained\-order weight surrogates and Jacobian–vector products to optimize ESS without retraining the base model\. We validate the method on synthetic dataset, class\-contrastive generation, and Stable Diffusion text\-to\-image benchmarks with both related and unrelated negative prompts as well as the peptide\-MHC \(p\-MHC\) binder design\. Across these settings, our method reduces mode shift, improves sampling stability, and better removes negative concepts than the baselines while preserving positive\-prompt quality\.

## 2Related Work

##### Diffusion guidance and negative guidance\.

Diffusion and score\-based generative models can be steered at inference time by modifying their reverse\-time score, denoising direction, or velocity field without retraining the base model\[[20](https://arxiv.org/html/2610.00395#bib.bib18),[1](https://arxiv.org/html/2610.00395#bib.bib19),[2](https://arxiv.org/html/2610.00395#bib.bib20)\]\. Such guidance can use classifiers, rewards, or compositions of learned score and flow fields\[[6](https://arxiv.org/html/2610.00395#bib.bib21),[7](https://arxiv.org/html/2610.00395#bib.bib22),[21](https://arxiv.org/html/2610.00395#bib.bib43),[22](https://arxiv.org/html/2610.00395#bib.bib44),[23](https://arxiv.org/html/2610.00395#bib.bib23),[24](https://arxiv.org/html/2610.00395#bib.bib24)\]\. Negative guidance further suppresses undesired concepts during sampling\[[8](https://arxiv.org/html/2610.00395#bib.bib15),[9](https://arxiv.org/html/2610.00395#bib.bib25),[25](https://arxiv.org/html/2610.00395#bib.bib14),[10](https://arxiv.org/html/2610.00395#bib.bib13)\]\. However, these methods typically define the resulting distribution only implicitly, and repulsive updates can remove valid modes when desired and undesired concepts overlap\. In contrast, we define an explicit specificity\-aware target that preserves probability mass favored by the desired reference while penalizing regions more characteristic of the undesired reference, and derive a sampler for its diffusion path\.

##### Feynman–Kac correction and density\-ratio steering\.

Feynman–Kac and Sequential Monte Carlo \(SMC\) methods approximate target distributions through propagation, reweighting, and resampling\[[15](https://arxiv.org/html/2610.00395#bib.bib26),[16](https://arxiv.org/html/2610.00395#bib.bib16)\]\. Recent work applies these ideas to diffusion guidance and model composition\[[17](https://arxiv.org/html/2610.00395#bib.bib17)\], as well as test\-time sampling from reward\-modified targets\[[26](https://arxiv.org/html/2610.00395#bib.bib27)\]\. Density\-ratio approaches instead characterize diffusion control through local transition\-kernel ratios, avoiding explicit estimation of higher\-order PDE quantities\[[19](https://arxiv.org/html/2610.00395#bib.bib12)\]\. We adopt this forward–backward kernel\-ratio view for proposal design and use local kernel ratios to reduce the variance of incremental SMC weights\.

##### SMC stability and proposal optimization\.

A central difficulty in SMC is weight degeneracy, which reduces effective sample size and can destabilize guided sampling\[[15](https://arxiv.org/html/2610.00395#bib.bib26),[16](https://arxiv.org/html/2610.00395#bib.bib16)\]\. Prior work has improved SMC proposals through proposal tuning, importance weight optimization, path\-space objectives, optimal\-control formulations, annealing schedules, prior shifts, and timestep adaptation\[[27](https://arxiv.org/html/2610.00395#bib.bib28),[28](https://arxiv.org/html/2610.00395#bib.bib29),[29](https://arxiv.org/html/2610.00395#bib.bib10),[30](https://arxiv.org/html/2610.00395#bib.bib30),[31](https://arxiv.org/html/2610.00395#bib.bib31),[32](https://arxiv.org/html/2610.00395#bib.bib11),[33](https://arxiv.org/html/2610.00395#bib.bib8),[34](https://arxiv.org/html/2610.00395#bib.bib9)\]\. These methods often fix part of the underlying dynamics, optimize within a restricted proposal family, or focus primarily on schedule adaptation\. Our approach instead jointly analyzes the local forward and backward Gaussian kernels to construct low\-variance proposals\.

##### pMHC\-specific binder design\.

Peptide–MHC \(pMHC\) targeting is a challenging specificity problem because different targets largely share the same MHC scaffold, while specificity is often determined by only a few exposed peptide residues\. Earlier TCR\-mimic antibody design preserved MHC interactions while redesigning peptide\-contacting residues\[[35](https://arxiv.org/html/2610.00395#bib.bib6)\]\. More recent generative approaches have designed peptide\-focused pMHC binders using explicit off\-target filtering, in silico cross\-panning, and large\-scale off\-target screening\[[36](https://arxiv.org/html/2610.00395#bib.bib2),[37](https://arxiv.org/html/2610.00395#bib.bib7),[38](https://arxiv.org/html/2610.00395#bib.bib3)\]\. These approaches are closely related to classical negative design, which favors the desired interaction while penalizing competing states\[[39](https://arxiv.org/html/2610.00395#bib.bib5)\]\. Our work instead studies pMHC specificity through inference\-time guidance of a pretrained binder generative model\.

![Refer to caption](https://arxiv.org/html/2610.00395v1/toy_1d_task1.png)
![Refer to caption](https://arxiv.org/html/2610.00395v1/toy_1d_task2.png)

Figure 1:Comparison of target distributions in two one\-dimensional examples\. Blue and orange denote the desired densitypAp^\{A\}and undesired densitypBp^\{B\}, respectively; green denotes the induced target\. The proposed target preserves regions wherepAp^\{A\}dominatespBp^\{B\}and removes regions with high undesired density\. By contrast, DNG can shift mass away from desired modes, while the negative\-prompt \(NP\) construction does not consistently suppress the undesired region\.

## 3Target Distribution Design

LetpAp^\{A\}andpBp^\{B\}denote the desired and undesired reference densities, respectively\. We seek a target that retains regions likely underpAp^\{A\}while suppressing regions supported bypBp^\{B\}\. For two densitiesqqandpp, define

OVL⁡\(q,p\):=∫min⁡\{q⁡\(x\),p⁡\(x\)\}​𝑑x\.\\mathrm\{OVL\}\(q,p\):=\\int\\min\\\{q\(x\),p\(x\)\\\}\\,dx\.\(1\)We consider the specificity objective

J⁡\(q\):=OVL⁡\(q,pA\)−γ​OVL​\(q,pB\),γ\>1\.J\(q\):=\\mathrm\{OVL\}\(q,p^\{A\}\)\-\\gamma\\,\\mathrm\{OVL\}\(q,p^\{B\}\),\\qquad\\gamma\>1\.\(2\)WhenJ⁡\(q\)J\(q\)has multiple maximizers, we select the one closest topAp^\{A\}in KL divergence\.

###### Proposition 1\(Optimal target and its soft relaxation\)\.

Let

𝒜γ:=\{x:pA​\(x\)\>γ​pB​\(x\)\}\.\\mathcal\{A\}\_\{\\gamma\}:=\\left\\\{x:p^\{A\}\(x\)\>\\gamma p^\{B\}\(x\)\\right\\\}\.Suppose thatγ\>1\\gamma\>1,pA​\(𝒜γ\)\>0p^\{A\}\(\\mathcal\{A\}\_\{\\gamma\}\)\>0, and the boundary\{x:pA​\(x\)=γ​pB​\(x\)\}\\\{x:p^\{A\}\(x\)=\\gamma p^\{B\}\(x\)\\\}has zeropAp^\{A\}\-mass\. Among all normalized densities maximizingJ⁡\(q\)J\(q\), the unique minimizer ofDKL\(q∥pA\)D\_\{\\mathrm\{KL\}\}\(q\\\|p^\{A\}\)is

q∗​\(x\)=pA\(x\)𝟏\{x∈𝒜γ\}∫𝒜γpA​\(u\)​𝑑u\.q^\{\*\}\(x\)=\\frac\{p^\{A\}\(x\)\\mathbf\{1\}\\\{x\\in\\mathcal\{A\}\_\{\\gamma\}\\\}\}\{\\displaystyle\\int\_\{\\mathcal\{A\}\_\{\\gamma\}\}p^\{A\}\(u\)\\,du\}\.\(3\)
For the time\-dependent marginalsptAp\_\{t\}^\{A\}andptBp\_\{t\}^\{B\}, define

st​\(x\):=log⁡ptA​\(x\)ptB​\(x\)−log⁡γ\.s\_\{t\}\(x\):=\\log\\frac\{p\_\{t\}^\{A\}\(x\)\}\{p\_\{t\}^\{B\}\(x\)\}\-\\log\\gamma\.\(4\)Under pointwise Bernoulli\-entropy regularization with inverse temperatureηt\>0\\eta\_\{t\}\>0, the hard gate𝟏\{st\(x\)\>0\}\\mathbf\{1\}\\\{s\_\{t\}\(x\)\>0\\\}is relaxed to

1−θt​\(x\)\\displaystyle 1\-\\theta\_\{t\}\(x\):=σ⁡\(ηt​st​\(x\)\)\\displaystyle:=\\sigma\\\!\\left\(\\eta\_\{t\}s\_\{t\}\(x\)\\right\)\(5\)=\(ptA​\(x\)\)ηtγηt​\(ptB​\(x\)\)ηt\+\(ptA​\(x\)\)ηt\.\\displaystyle=\\frac\{\\bigl\(p\_\{t\}^\{A\}\(x\)\\bigr\)^\{\\eta\_\{t\}\}\}\{\\gamma^\{\\eta\_\{t\}\}\\bigl\(p\_\{t\}^\{B\}\(x\)\\bigr\)^\{\\eta\_\{t\}\}\+\\bigl\(p\_\{t\}^\{A\}\(x\)\\bigr\)^\{\\eta\_\{t\}\}\}\.For anymt\>0m\_\{t\}\>0, the corresponding powered soft target is

qtη,m​\(x\)=ptA​\(x\)​\(1−θt​\(x\)\)mt∫ptA​\(u\)​\(1−θt​\(u\)\)mt​𝑑u\.q\_\{t\}^\{\\eta,m\}\(x\)=\\frac\{p\_\{t\}^\{A\}\(x\)\\bigl\(1\-\\theta\_\{t\}\(x\)\\bigr\)^\{m\_\{t\}\}\}\{\\displaystyle\\int p\_\{t\}^\{A\}\(u\)\\bigl\(1\-\\theta\_\{t\}\(u\)\\bigr\)^\{m\_\{t\}\}\\,du\}\.\(6\)For every fixedm\>0m\>0,qtη,mq\_\{t\}^\{\\eta,m\}converges to the corresponding hard target asηt→∞\\eta\_\{t\}\\to\\infty\.

The parameterηt\\eta\_\{t\}controls the sharpness of the likelihood\-ratio boundary, whilemtm\_\{t\}controls the strength of the soft reweighting\. Thus, the powered target preserves regions specific toptAp\_\{t\}^\{A\}and smoothly downweights regions supported byptBp\_\{t\}^\{B\}, without changing the limiting hard target\. Proofs and further discussion are provided in Appendices[A](https://arxiv.org/html/2610.00395#A1)and[A\.1](https://arxiv.org/html/2610.00395#A1.SS1)\.

## 4Local SMC Proposal Design

### 4\.1Local Forward–Backward Proposal Design

Letτ=t\+h\\tau=t\+h, withh\>0h\>0\. Suppose the current weighted particles approximateqτq\_\{\\tau\}\. To propagate them backward toqtq\_\{t\}, we introduce an auxiliary forward kernelKt​\(y∣x\)K\_\{t\}\(y\\mid x\)and sample from a backward kernelLτ​\(x∣y\)L\_\{\\tau\}\(x\\mid y\)\. The ideal incremental Radon–Nikodym weight is

ω⁡\(x,y\)=qt​\(x\)​Kt​\(y∣x\)qτ​\(y\)​Lτ​\(x∣y\)∝γt​\(x\)​Kt​\(y∣x\)γτ​\(y\)​Lτ​\(x∣y\)\.\\omega\(x,y\)=\\frac\{q\_\{t\}\(x\)K\_\{t\}\(y\\mid x\)\}\{q\_\{\\tau\}\(y\)L\_\{\\tau\}\(x\\mid y\)\}\\propto\\frac\{\\gamma\_\{t\}\(x\)K\_\{t\}\(y\\mid x\)\}\{\\gamma\_\{\\tau\}\(y\)L\_\{\\tau\}\(x\\mid y\)\}\.\(7\)Thus, the proposal kernels can be adapted while the target remains fixed, provided that the actual transition ratio is included in the weight\[[19](https://arxiv.org/html/2610.00395#bib.bib12),[40](https://arxiv.org/html/2610.00395#bib.bib1)\]\. We use this construction to motivate an auxiliary forward proposal and a matched backward drift that removes the leading stochastic variation of the Gaussian surrogate below\.

##### Two\-field forward proposal\.

Recall the powered soft target

qt​\(z\)∝γt​\(z\)\\displaystyle q\_\{t\}\(z\)\\propto\\gamma\_\{t\}\(z\):=ptA​\(z\)​\(1−θt​\(z\)\)m=ptA​\(z\)​eℛt​\(z\),\\displaystyle:=p\_\{t\}^\{A\}\(z\)\(1\-\\theta\_\{t\}\(z\)\)^\{m\}=p\_\{t\}^\{A\}\(z\)e^\{\\mathcal\{R\}\_\{t\}\(z\)\},\(8\)ℛt​\(z\)\\displaystyle\\mathcal\{R\}\_\{t\}\(z\):=m​log⁡\(1−θt​\(z\)\),\\displaystyle:=m\\log\(1\-\\theta\_\{t\}\(z\)\),The gate schedules are independent of the spatial variable\. Define

dt​\(z\)\\displaystyle d\_\{t\}\(z\):=∇z​log​ptA​\(z\)−∇z​log​ptB​\(z\),\\displaystyle:=\\nabla\_\{z\}\\log p\_\{t\}^\{A\}\(z\)\-\\nabla\_\{z\}\\log p\_\{t\}^\{B\}\(z\),\(9\)gt​\(z\)\\displaystyle g\_\{t\}\(z\):=∇zℛt\(z\)=ζt\(z\)dt\(z\),ζt\(z\):=mηtθt\(z\)\.\\displaystyle:=\\nabla\_\{z\}\\mathcal\{R\}\_\{t\}\(z\)=\\zeta\_\{t\}\(z\)d\_\{t\}\(z\),\\qquad\\zeta\_\{t\}\(z\):=m\\eta\_\{t\}\\theta\_\{t\}\(z\)\.Heredtd\_\{t\}is the desired–undesired score contrast, whilegtg\_\{t\}is the corresponding correction induced by the soft target\. Although these fields are pointwise collinear, their relative multiplierζt​\(z\)\\zeta\_\{t\}\(z\)varies across particles\.

The two references share the forward driftμrA=μrB\\mu\_\{r\}^\{A\}=\\mu\_\{r\}^\{B\}and the space\-independent scalar diffusion variance ratesr\>0s\_\{r\}\>0\. We adopt the decreasing\-time reverse\-drift convention

νrj​\(z\):=μrj​\(z\)−sr​∇z​log⁡prj​\(z\),j∈\{A,B\}\.\\nu\_\{r\}^\{j\}\(z\):=\\mu\_\{r\}^\{j\}\(z\)\-s\_\{r\}\\nabla\_\{z\}\\log p\_\{r\}^\{j\}\(z\),\\qquad j\\in\\\{A,B\\\}\.For each candidate\(ρ1,ρ2\)\(\\rho\_\{1\},\\rho\_\{2\}\), the coefficients are held fixed over the current interval\[t,τ\]\[t,\\tau\]\. We define

ur\(ρ1,ρ2\)​\(z\):=ρ1​dr​\(z\)\+ρ2​gr​\(z\),r∈\[t,τ\],u\_\{r\}^\{\(\\rho\_\{1\},\\rho\_\{2\}\)\}\(z\):=\\rho\_\{1\}d\_\{r\}\(z\)\+\\rho\_\{2\}g\_\{r\}\(z\),\\qquad r\\in\[t,\\tau\],whereρ1\\rho\_\{1\}controls a direct correction along the desired–undesired score contrast, whileρ2\\rho\_\{2\}modulates the reward\-gradient component induced by the soft target\. The resulting auxiliary forward drift is

bt\(ρ1,ρ2\)​\(z\)=μtA​\(z\)\+st​ut\(ρ1,ρ2\)​\(z\)\.b\_\{t\}^\{\(\\rho\_\{1\},\\rho\_\{2\}\)\}\(z\)=\\mu\_\{t\}^\{A\}\(z\)\+s\_\{t\}u\_\{t\}^\{\(\\rho\_\{1\},\\rho\_\{2\}\)\}\(z\)\.\(10\)
For arbitrary forward and backward driftsbbandaa, define

Ktb\(y∣x\):=𝒩\(y;x\+hbt\(x\),hstI\),Lτa\(x∣y\):=𝒩\(x;y−haτ\(y\),hsτI\)\.\\displaystyle K\_\{t\}^\{b\}\(y\\mid x\):=\\mathcal\{N\}\(y;x\+hb\_\{t\}\(x\),hs\_\{t\}I\),\\qquad L\_\{\\tau\}^\{a\}\(x\\mid y\):=\\mathcal\{N\}\(x;y\-ha\_\{\\tau\}\(y\),hs\_\{\\tau\}I\)\.\(11\)Conditioning on the current gate value, we define the operational Gaussian log\-weight surrogate

log⁡ωρ1,ρ2G​\(x,y\):=Δ​ℛG​\(x,y\)\+log⁡LτνA​\(x∣y\)KtμA​\(y∣x\)−log⁡Lτaρ1,ρ2​\(x∣y\)Ktb\(ρ1,ρ2\)​\(y∣x\)\.\\log\\omega\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\mathrm\{G\}\}\(x,y\):=\\Delta\\mathcal\{R\}^\{\\mathrm\{G\}\}\(x,y\)\+\\log\\frac\{L\_\{\\tau\}^\{\\nu^\{A\}\}\(x\\mid y\)\}\{K\_\{t\}^\{\\mu^\{A\}\}\(y\\mid x\)\}\-\\log\\frac\{L\_\{\\tau\}^\{a\_\{\\rho\_\{1\},\\rho\_\{2\}\}\}\(x\\mid y\)\}\{K\_\{t\}^\{b^\{\(\\rho\_\{1\},\\rho\_\{2\}\)\}\}\(y\\mid x\)\}\.\(12\)HereΔ​ℛG​\(x,y\)\\Delta\\mathcal\{R\}^\{\\mathrm\{G\}\}\(x,y\)is the one\-step gate contribution obtained from the same A/B reference Gaussian kernel\-ratio approximation\. Its construction and score\-based local spatial derivatives are specified in Appendix[B\.1](https://arxiv.org/html/2610.00395#A2.SS1)\. In numerical evaluation, we use the same notation for estimated gate values; their computation and any associated estimator\-state update follow the experimental protocol\. This defines a Gaussian weight surrogate, rather than an exact marginal RN weight\.

###### Proposition 2\(Two\-Field Local Reweighting Expansion\)\.

Assume the Gaussian kernels in \([11](https://arxiv.org/html/2610.00395#S4.E11)\), the Gaussian gate construction used in \([12](https://arxiv.org/html/2610.00395#S4.E12)\), and the regularity conditions in Appendix[B\.1](https://arxiv.org/html/2610.00395#A2.SS1)\. For the forward proposal \([10](https://arxiv.org/html/2610.00395#S4.E10)\), choose the matched backward drift

aρ1,ρ2⋆​\(y\)=ντA​\(y\)\+sτ​\(uτ\(ρ1,ρ2\)​\(y\)−gτ​\(y\)\)\.a\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\star\}\(y\)=\\nu\_\{\\tau\}^\{A\}\(y\)\+s\_\{\\tau\}\\left\(u\_\{\\tau\}^\{\(\\rho\_\{1\},\\rho\_\{2\}\)\}\(y\)\-g\_\{\\tau\}\(y\)\\right\)\.\(13\)This cancels theOp​\(h1/2\)O\_\{p\}\(h^\{1/2\}\)term in the Gaussian log\-weight surrogate\. Let

Δ​y:=x−y=−h​aρ1,ρ2⋆​\(y\)\+h​sτ​ξ,ξ∼𝒩⁡\(0,I\)\.\\Delta y:=x\-y=\-ha\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\star\}\(y\)\+\\sqrt\{hs\_\{\\tau\}\}\\,\\xi,\\qquad\\xi\\sim\\mathcal\{N\}\(0,I\)\.Then

log⁡ωρ1,ρ2G=h​αρ1,ρ2​\(y\)\+Δ​y⊤​Γρ1,ρ2​\(y\)​Δ​y\+Op​\(h3/2\),\\log\\omega\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\mathrm\{G\}\}=h\\alpha\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)\+\\Delta y^\{\\top\}\\Gamma\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)\\Delta y\+O\_\{p\}\(h^\{3/2\}\),\(14\)where

Γρ1,ρ2​\(y\)=−\(ρ1\+ρ2​ζτ​\(y\)\)​J​dτ​\(y\)\+χτ​\(y\)​\(ρ2−12\)​dτ​\(y\)​dτ​\(y\)⊤,\\displaystyle\\Gamma\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)=\-\\left\(\\rho\_\{1\}\+\\rho\_\{2\}\\zeta\_\{\\tau\}\(y\)\\right\)Jd\_\{\\tau\}\(y\)\+\\chi\_\{\\tau\}\(y\)\\left\(\\rho\_\{2\}\-\\frac\{1\}\{2\}\\right\)d\_\{\\tau\}\(y\)d\_\{\\tau\}\(y\)^\{\\top\},\(15\)with

J​dτ​\(y\):=∇ydτ​\(y\),χτ​\(y\):=m​ητ2​θτ​\(y\)​\(1−θτ​\(y\)\)\.Jd\_\{\\tau\}\(y\):=\\nabla\_\{y\}d\_\{\\tau\}\(y\),\\qquad\\chi\_\{\\tau\}\(y\):=m\\eta\_\{\\tau\}^\{2\}\\theta\_\{\\tau\}\(y\)\(1\-\\theta\_\{\\tau\}\(y\)\)\.The scalar coefficientαρ1,ρ2​\(y\)\\alpha\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)collects the gate and kernel contributions\. Its explicit expression and the complete derivation are given in Appendix[B\.1](https://arxiv.org/html/2610.00395#A2.SS1)\.The remainder is uniform over the fixed finite candidate grid used below\.

We present the fixed\-mmcase; the time\-varying\-mtm\_\{t\}extension is given in Appendix[B\.1](https://arxiv.org/html/2610.00395#A2.SS1)\. Conditioned on the current state, the scalar coefficient is independent of the mutation noise, while the quadratic term contributes both a conditional mean and transition\-level fluctuations\. TheJ​dJdterm captures curvature of the score contrast, and thed​d⊤dd^\{\\top\}term arises from the nonlinear gate\.

#### 4\.1\.1Fixed\-Displacement JVP Approximation

A naive candidate\-wise evaluation of \([14](https://arxiv.org/html/2610.00395#S4.E14)\) would require candidate\-dependent JVPs\. We instead fixρ1=ρ1,t\\rho\_\{1\}=\\rho\_\{1,t\}and reuse the displacement corresponding toρ2=0\\rho\_\{2\}=0\. Draw oneξi∼𝒩⁡\(0,I\)\\xi\_\{i\}\\sim\\mathcal\{N\}\(0,I\)per particle, independently of the current particle system, and define

a0,i⋆\\displaystyle a\_\{0,i\}^\{\\star\}:=ντA​\(yi\)\+sτ​\(ρ1,t​dτ​\(yi\)−gτ​\(yi\)\),\\displaystyle:=\\nu\_\{\\tau\}^\{A\}\(y\_\{i\}\)\+s\_\{\\tau\}\\left\(\\rho\_\{1,t\}d\_\{\\tau\}\(y\_\{i\}\)\-g\_\{\\tau\}\(y\_\{i\}\)\\right\),\(16\)Δ​y0,i\\displaystyle\\Delta y\_\{0,i\}:=−h​a0,i⋆\+h​sτ​ξi\.\\displaystyle:=\-ha\_\{0,i\}^\{\\star\}\+\\sqrt\{hs\_\{\\tau\}\}\\,\\xi\_\{i\}\.For any candidateρ2\\rho\_\{2\},

Δ​yi​\(ρ2\)=Δ​y0,i−h​sτ​ρ2​gτ​\(yi\)\.\\Delta y\_\{i\}\(\\rho\_\{2\}\)=\\Delta y\_\{0,i\}\-hs\_\{\\tau\}\\rho\_\{2\}g\_\{\\tau\}\(y\_\{i\}\)\.The difference isO⁡\(h\)O\(h\), whileΔ​y0,i=Op​\(h1/2\)\\Delta y\_\{0,i\}=O\_\{p\}\(h^\{1/2\}\)\. Replacing the candidate\-dependent displacement inside the quadratic term therefore changes the expansion only byOp​\(h3/2\)O\_\{p\}\(h^\{3/2\}\)\.

Define the fixed\-displacement log\-weight approximation by

log⁡ω^i​\(ρ2\)\\displaystyle\\log\\widehat\{\\omega\}\_\{i\}\(\\rho\_\{2\}\):=h​αρ1,t,ρ2​\(yi\)−\(ρ1,t\+ρ2​ζτ​\(yi\)\)​Δ​y0,i⊤​J​dτ​\(yi\)​Δ​y0,i\\displaystyle:=h\\alpha\_\{\\rho\_\{1,t\},\\rho\_\{2\}\}\(y\_\{i\}\)\-\\left\(\\rho\_\{1,t\}\+\\rho\_\{2\}\\zeta\_\{\\tau\}\(y\_\{i\}\)\\right\)\\Delta y\_\{0,i\}^\{\\top\}Jd\_\{\\tau\}\(y\_\{i\}\)\\Delta y\_\{0,i\}\+χτ​\(yi\)​\(ρ2−12\)​\(dτ​\(yi\)⊤​Δ​y0,i\)2\.\\displaystyle\\quad\+\\chi\_\{\\tau\}\(y\_\{i\}\)\\left\(\\rho\_\{2\}\-\\frac\{1\}\{2\}\\right\)\\left\(d\_\{\\tau\}\(y\_\{i\}\)^\{\\top\}\\Delta y\_\{0,i\}\\right\)^\{2\}\.\(17\)Forxi​\(ρ2\)=yi\+Δ​yi​\(ρ2\)x\_\{i\}\(\\rho\_\{2\}\)=y\_\{i\}\+\\Delta y\_\{i\}\(\\rho\_\{2\}\), this gives

log⁡ωρ1,t,ρ2G​\(xi​\(ρ2\),yi\)=log⁡ω^i​\(ρ2\)\+Op​\(h3/2\)\.\\log\\omega\_\{\\rho\_\{1,t\},\\rho\_\{2\}\}^\{\\mathrm\{G\}\}\\bigl\(x\_\{i\}\(\\rho\_\{2\}\),y\_\{i\}\\bigr\)=\\log\\widehat\{\\omega\}\_\{i\}\(\\rho\_\{2\}\)\+O\_\{p\}\(h^\{3/2\}\)\.All candidates reuseJ​dτ​\(yi\)​\[Δ​y0,i\]Jd\_\{\\tau\}\(y\_\{i\}\)\[\\Delta y\_\{0,i\}\], which requires one score JVP per reference model\. The remaining terms require only scalar coefficients and inner products\.

##### Relative weight\-variance minimization\.

For incoming weightsWiW\_\{i\}, letVi​\(ρ2\):=Wi​ω^iuse​\(ρ2\)V\_\{i\}\(\\rho\_\{2\}\):=W\_\{i\}\\widehat\{\\omega\}\_\{i\}^\{\\mathrm\{use\}\}\(\\rho\_\{2\}\)andV¯​\(ρ2\):=N−1​∑iVi​\(ρ2\)\\bar\{V\}\(\\rho\_\{2\}\):=N^\{\-1\}\\sum\_\{i\}V\_\{i\}\(\\rho\_\{2\}\)\. Hereω^iuse\\widehat\{\\omega\}\_\{i\}^\{\\mathrm\{use\}\}includes any prescribed stabilization, applied consistently in selection and reweighting\. We minimize the empirical relative variance which is the same as normalized cumulative ESS maximization:

𝒱^trel​\(ρ2\)\\displaystyle\\widehat\{\\mathcal\{V\}\}\_\{t\}^\{\\mathrm\{rel\}\}\(\\rho\_\{2\}\):=1N​∑i\(Vi​\(ρ2\)V¯​\(ρ2\)−1\)2,\\displaystyle:=\\frac\{1\}\{N\}\\sum\_\{i\}\\left\(\\frac\{V\_\{i\}\(\\rho\_\{2\}\)\}\{\\bar\{V\}\(\\rho\_\{2\}\)\}\-1\\right\)^\{2\},\(18\)ρ2,t⋆\\displaystyle\\rho\_\{2,t\}^\{\\star\}∈arg​minρ2∈ℛ2⁡𝒱^trel​\(ρ2\)=arg​maxρ2∈ℛ2⁡nESS^t​\(ρ2\)\.\\displaystyle\\in\\operatorname\*\{arg\\,min\}\_\{\\rho\_\{2\}\\in\\mathcal\{R\}\_\{2\}\}\\widehat\{\\mathcal\{V\}\}\_\{t\}^\{\\mathrm\{rel\}\}\(\\rho\_\{2\}\)=\\operatorname\*\{arg\\,max\}\_\{\\rho\_\{2\}\\in\\mathcal\{R\}\_\{2\}\}\\widehat\{\\mathrm\{nESS\}\}\_\{t\}\(\\rho\_\{2\}\)\.\(19\)This equivalence holds for arbitrary incoming weights\. For equal weights and small increments, incremental log\-weight variance is its leading\-order approximation; see Appendix[B\.2](https://arxiv.org/html/2610.00395#A2.SS2)\.

### 4\.2Continuous\-Time Interpretation

The local Gaussian surrogate in Proposition[2](https://arxiv.org/html/2610.00395#Thmproposition2)also admits a natural continuous\-time interpretation\. Consider the ideal reference setting with exact scores and gate values, and fix the proposal coefficients over the current step\. As shown in Appendix[B\.3](https://arxiv.org/html/2610.00395#A2.SS3), the finite\-step surrogate satisfies

log⁡ωρ1,ρ2G\\displaystyle\\log\\omega\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\mathrm\{G\}\}=log⁡ωρ1,ρ2cont\+h​sτ​\[ξ⊤​Γρ1,ρ2​\(y\)​ξ−tr⁡Γρ1,ρ2​\(y\)\]\+Op​\(h3/2\),\\displaystyle=\\log\\omega\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\mathrm\{cont\}\}\+hs\_\{\\tau\}\\left\[\\xi^\{\\top\}\\Gamma\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)\\xi\-\\operatorname\{tr\}\\Gamma\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)\\right\]\+O\_\{p\}\(h^\{3/2\}\),\(20\)whereωρ1,ρ2cont\\omega\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\mathrm\{cont\}\}is the corresponding continuous Feynman–Kac incremental weight along the matched backward process\.

For a fixed candidate andξ∼𝒩⁡\(0,I\)\\xi\\sim\\mathcal\{N\}\(0,I\),

𝔼ξ​\[ξ⊤​Γρ1,ρ2​\(y\)​ξ∣y\]=tr⁡Γρ1,ρ2​\(y\)\.\\mathbb\{E\}\_\{\\xi\}\\left\[\\xi^\{\\top\}\\Gamma\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)\\xi\\mid y\\right\]=\\operatorname\{tr\}\\Gamma\_\{\\rho\_\{1\},\\rho\_\{2\}\}\(y\)\.Thus, the finite Gaussian surrogate and the continuous formulation share the same conditionalO⁡\(h\)O\(h\)contribution\. The difference is that the continuous formulation contains the trace correction, whereas the finite\-step surrogate retains its realized Gaussian quadratic formξ⊤​Γ​ξ\\xi^\{\\top\}\\Gamma\\xi\. In this sense, the same mutation noise provides a one\-sample estimate of the continuous trace term while preserving transition\-level information from the finite step\. The full derivation and regularity conditions are given in Appendix[B\.3](https://arxiv.org/html/2610.00395#A2.SS3)\.

### 4\.3Relation to Variance\-Minimizing Proposals

For a fixed candidate𝝆=\(ρ1,ρ2\)\\boldsymbol\{\\rho\}=\(\\rho\_\{1\},\\rho\_\{2\}\), the variance of the finite Gaussian log\-weight surrogate decomposes as

VarY,ξ⁡\(log⁡ω^𝝆\)\\displaystyle\\operatorname\{Var\}\_\{Y,\\xi\}\\bigl\(\\log\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}\\bigr\)=𝔼Y​\[Varξ⁡\(log⁡ω^𝝆∣Y\)\]\+VarY⁡\(𝔼ξ​\[log⁡ω^𝝆∣Y\]\)\.\\displaystyle=\\mathbb\{E\}\_\{Y\}\\\!\\left\[\\operatorname\{Var\}\_\{\\xi\}\\\!\\left\(\\log\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}\\mid Y\\right\)\\right\]\+\\operatorname\{Var\}\_\{Y\}\\\!\\left\(\\mathbb\{E\}\_\{\\xi\}\[\\log\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}\\mid Y\]\\right\)\.\(21\)The first term captures transition\-level fluctuations, while the second captures variation of the conditional mean across particles\.

At the continuous level, after matching the target path and control family, our matched drift induces the same Feynman–Kac potential as the corresponding DriftLite\-VCG control\[[34](https://arxiv.org/html/2610.00395#bib.bib9)\]\. Its potential\-variance objective is the leading\-order coefficient of the second term above\. Our finite Gaussian construction instead evaluates the realized one\-step weight, retaining the quadratic trace estimator and the additional finite\-displacement corrections\. The benefit is therefore in the finite\-step weight construction and optimization objective, rather than in separate continuous control principles\.

This also connects to DAS\[[26](https://arxiv.org/html/2610.00395#bib.bib27)\]: for a fixed forward kernel, its ideal backward proposal minimizes the conditional transition variance\. Our coefficients additionally modify the forward kernel together with its matched backward proposal\. In practice, we selectρ2\\rho\_\{2\}by minimizing the empirical relative variance of the updated cumulative surrogate weights, equivalently maximizing cumulative ESS in \([19](https://arxiv.org/html/2610.00395#S4.E19)\)\. Detailed comparisons are given in Appendices[B\.4\.1](https://arxiv.org/html/2610.00395#A2.SS4.SSS1)and[B\.4\.2](https://arxiv.org/html/2610.00395#A2.SS4.SSS2)\.

Algorithm 1Common\-Noise Two\-Field SMC: One Stochastic Step1:Weighted particles

\{\(yi,Wi\)\}i=1N\\\{\(y\_\{i\},W\_\{i\}\)\\\}\_\{i=1\}^\{N\}and gate state; step

h=τ−th=\\tau\-t; fixed

ρ1,t\\rho\_\{1,t\}; grid

ℛ2\\mathcal\{R\}\_\{2\}; previous coefficient

ρ2prev\\rho\_\{2\}^\{\\mathrm\{prev\}\}
2:Evaluate the reference fields and local gate coefficients\.

3:Draw

ξi∼𝒩⁡\(0,I\)\\xi\_\{i\}\\sim\\mathcal\{N\}\(0,I\)independently and form

Δ​y0,i\\Delta y\_\{0,i\}by \([16](https://arxiv.org/html/2610.00395#S4.E16)\)\.

4:Compute the shared JVPs

J​dτ​\(yi\)​\[Δ​y0,i\]Jd\_\{\\tau\}\(y\_\{i\}\)\[\\Delta y\_\{0,i\}\]\.

5:Set

𝒞t=ℛ2\\mathcal\{C\}\_\{t\}=\\mathcal\{R\}\_\{2\}on optimization steps; otherwise

𝒞t=\{ρ2prev\}\\mathcal\{C\}\_\{t\}=\\\{\\rho\_\{2\}^\{\\mathrm\{prev\}\}\\\}\.

6:Form

ω^iuse​\(ρ2\)\\widehat\{\\omega\}\_\{i\}^\{\\mathrm\{use\}\}\(\\rho\_\{2\}\)for

ρ2∈𝒞t\\rho\_\{2\}\\in\\mathcal\{C\}\_\{t\}using \([17](https://arxiv.org/html/2610.00395#S4.E17)\) and the prescribed stabilization\.

7:Select

ρ2,t⋆∈arg​minρ2∈𝒞t⁡𝒱^trel​\(ρ2\)\\rho\_\{2,t\}^\{\\star\}\\in\\operatorname\*\{arg\\,min\}\_\{\\rho\_\{2\}\\in\\mathcal\{C\}\_\{t\}\}\\widehat\{\\mathcal\{V\}\}\_\{t\}^\{\\mathrm\{rel\}\}\(\\rho\_\{2\}\)by \([19](https://arxiv.org/html/2610.00395#S4.E19)\)\.

8:

xi←yi\+Δ​y0,i−h​sτ​ρ2,t⋆​gτ​\(yi\)x\_\{i\}\\leftarrow y\_\{i\}\+\\Delta y\_\{0,i\}\-hs\_\{\\tau\}\\rho\_\{2,t\}^\{\\star\}g\_\{\\tau\}\(y\_\{i\}\)\.⊳\\trianglerightReuse the same noise

9:Update the gate state using the selected transition\.

10:

log⁡Wi←log⁡Wi\+log⁡ω^iuse​\(ρ2,t⋆\)\\log W\_\{i\}\\leftarrow\\log W\_\{i\}\+\\log\\widehat\{\\omega\}\_\{i\}^\{\\mathrm\{use\}\}\(\\rho\_\{2,t\}^\{\\star\}\); normalize the weights\.

11:Resample particles and gate state jointly if triggered, retaining residual importance weights\.

12:return

\{\(xi,Wi\)\}i=1N\\\{\(x\_\{i\},W\_\{i\}\)\\\}\_\{i=1\}^\{N\}, gate state, and

ρ2,t⋆\\rho\_\{2,t\}^\{\\star\}\.

## 5Experiment

### 5\.1Synthetic Specificity on a 40\-Component Gaussian Mixture

![Refer to caption](https://arxiv.org/html/2610.00395v1/40_mog.png)Figure 2:Visualization on the 40\-MoG synthetic task\. The task setting containsAA\-only target modes,BB\-only negative modes, andA∩BA\\cap Bsuppressed modes\. The target distribution retains only theAA\-only modes\.In the first experiment, the 40\-MoG synthetic dataset\[[17](https://arxiv.org/html/2610.00395#bib.bib17)\]\[[19](https://arxiv.org/html/2610.00395#bib.bib12)\]is used to evaluate class\-contrastive sampling\. As shown in Fig\.[2](https://arxiv.org/html/2610.00395#S5.F2), the 40 Gaussian modes are randomly divided into three groups:AA\-only modes,BB\-only modes, and overlappingA∩BA\\cap Bmodes\. The desired target distribution retains theAA\-only modes while suppressing both theBB\-only andA∩BA\\cap Bmodes\.

The proposed sampler produces samples that are visually closest to the target distribution\. Dynamic Negative Guidance \(DNG\)\[[10](https://arxiv.org/html/2610.00395#bib.bib13)\]suppresses several negative regions, but it also induces mode shifts, especially for the modes in the lower\-right region\. Negative Guidance fails to represent the target distribution accurately and retains substantial mass near negative regions\. SSafe Denoiser \(SD\)\[[25](https://arxiv.org/html/2610.00395#bib.bib14)\]removes the negative modes more effectively than Negative Guidance, but its repulsion is driven by the absoluteBB\-density\. Consequently, it can over\-repel samples from regions whereBBis dense, causing missing or weakened positive modes that are surrounded by or close to negative modes, such as those in the lower\-left region\. This supports the use of a ratio\-based gate, which activates repulsion according to the relative evidencepB/pAp\_\{B\}/p\_\{A\}rather than the absolute negative density alone\.

### 5\.2CIFAR\-10 Specificity

![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_kl_0.png)
![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_kl_1.png)
![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_recall_012.png)
![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_recall_567.png)
![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_fid_0.png)\(a\)Removing Class 0
![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_fid_1.png)\(b\)Removing Class 1
![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_fid_012.png)\(c\)Keeping Classes 0,1,2
![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_fid_567.png)\(d\)Keeping Classes 5,6,7†

Figure 3:CIFAR\-10 specificity Pareto frontiers withN=10,240N=10\{,\}240samples\. Cols\. 1–2 show class removal \(§[5\.2\.1](https://arxiv.org/html/2610.00395#S5.SS2.SSS1)\); Cols\. 3–4 show multiclass selection \(§[5\.2\.2](https://arxiv.org/html/2610.00395#S5.SS2.SSS2)\)\. Bottom row: FID versus Wrong%\. Top row: KL versus Wrong% for class removal and Recall versus Wrong% for multiclass selection\.†Safe Denoiser is omitted because its Wrong% does not reach the plotted region across multiple kernel bandwidths; see App\.[D](https://arxiv.org/html/2610.00395#A4)\.We evaluate two CIFAR\-10 specificity tasks\. In class removal \(§[5\.2\.1](https://arxiv.org/html/2610.00395#S5.SS2.SSS1)\),pAp^\{A\}is an unconditional generator andpBp^\{B\}is a class\-only generator trained on one forget class\. In multiclass selection \(§[5\.2\.2](https://arxiv.org/html/2610.00395#S5.SS2.SSS2)\),pAp^\{A\}andpBp^\{B\}are overlapping multiclass generators, and the target is thepAp^\{A\}\-only modes\.

#### 5\.2\.1Class Removal

We remove planes \(c0\) and automobiles \(c1\)\. For each method, we sweep the guidance scale usingN=10,240N=10\{,\}240samples and report forget\-class leak \(Wrong%, ViT classifier\), keep\-class histogram KL to uniform, and FID to the keep\-class real subset\. Baselines include Composition, Safe Latent Diffusion \(SLD\)\[[8](https://arxiv.org/html/2610.00395#bib.bib15),[10](https://arxiv.org/html/2610.00395#bib.bib13)\], DNG, and SD\. Full protocols and per\-setting results are provided in App\.[C](https://arxiv.org/html/2610.00395#A3)\.

Our sampler gives the best Pareto trade\-off for both removed classes \(Fig\.[3](https://arxiv.org/html/2610.00395#S5.F3)\)\. The gain is consistent with the target design in §[A](https://arxiv.org/html/2610.00395#A1): DNG penalizes directly throughr⁡\(x\)=pB/pAr\(x\)=p^\{B\}/p^\{A\}, while our target usesr​\(x\)λr\(x\)^\{\\lambda\}, reducing over\-suppression near the positive–negative boundary\. In contrast, SD uses an effectively always\-active repulsive correction, which leads to a Wrong% floor and larger keep\-class distortion under stronger guidance\.

#### 5\.2\.2Multiclass Selection

We next consider overlapping five\-class generators\. ModelpAp^\{A\}covers\{0,1,2,3,4\}\\\{0,1,2,3,4\\\}, modelpBp^\{B\}covers\{3,4,5,6,7\}\\\{3,4,5,6,7\\\}, and the goal is to sample fromA∖BA\\setminus B\. The forward task keeps\{0,1,2\}\\\{0,1,2\\\}, while the reverse task keeps\{5,6,7\}\\\{5,6,7\\\}\. Because only three classes are retained, we report Recall\[[41](https://arxiv.org/html/2610.00395#bib.bib32)\]together with FID to measure both sample quality and mode coverage\. Further details are given in App\.[D](https://arxiv.org/html/2610.00395#A4)\.

Our sampler achieves the strongest FID–Recall Pareto frontier in both directions, with the largest gain on the harder forward task \(Fig\.[3](https://arxiv.org/html/2610.00395#S5.F3)\)\. The overlap between the two generators widens the transition region where the class densities are comparable, making excessive negative suppression more damaging\. The powered likelihood\-ratio gate better preserves valid modes near this boundary\.

### 5\.3Text\-to\-Image

We evaluate text\-to\-image specificity with Stable Diffusion v1\.4\[[42](https://arxiv.org/html/2610.00395#bib.bib34)\]on the five\-prompt DNG suite\[[10](https://arxiv.org/html/2610.00395#bib.bib13)\]\. Each positive prompt is paired with a*related*negative, testing suppression of semantically overlapping undesired content, and an*unrelated*negative, testing unnecessary interference with valid generation\. For each prompt and method, the same hyperparameter setting is used for both negative types\. Full prompts and settings are given in Appendix[E](https://arxiv.org/html/2610.00395#A5)\.

All methods use100100reverse steps\. We compare against CFG with negative prompting, DNG, Feynman–Kac Corrector \(FKC\), Feynman–Kac Steering \(FKS\), DriftLite, and DAS\. FKC, FKS, DriftLite, and DAS use our target distribution to isolate the effect of proposal design\. Each method is tuned for the overall margin\.

Table 1:Text\-to\-image specificity with SD v1\.4 and100100reverse steps, averaged over the 10 DNG prompt pairs\. For each prompt and method, a single hyperparameter setting is selected and used for both related and unrelated negatives\. CLIP\+is similarity to the positive prompt \(↑\\uparrow\), CLIP\-is similarity to the active negative prompt \(↓\\downarrow\), and margin is CLIP\+−\{\}\_\{\+\}\-CLIP\-\(↑\\uparrow\)\.Table[1](https://arxiv.org/html/2610.00395#S5.T1)shows that our method achieves the largest overall margin while maintaining competitive positive\-prompt alignment\. For related negatives, it obtains both the lowest CLIP\-and the largest margin, improving over DriftLite, the second\-best method, by6\.6%6\.6\\%\. For unrelated negatives, it again gives the lowest CLIP\-and the second\-highest margin, close to DAS\. These results indicate stronger selective suppression without uniformly weakening the positive prompt\.

Figure[4](https://arxiv.org/html/2610.00395#S5.F4)further shows that proposal optimization prevents the severe visual collapse observed with direct SMC while preserving image quality\. Additional results are provided in Appendix[E](https://arxiv.org/html/2610.00395#A5)\.

![Refer to caption](https://arxiv.org/html/2610.00395v1/particles_grid_rhoopt_25_28_vs_noopt_first4.png)Figure 4:Samples for “medieval feast\.” The first row uses proposal optimization and the second uses direct SMC\. Proposal optimization substantially reduces visual collapse\.
### 5\.4Specificity\-Aware pMHC Binder Design

We evaluate our method on a protein\-design task that requires specificity between two closely related targets\. The desired target is an HLA\-A\*02:01–β2\\beta\_\{2\}m complex presentingNLVPMVATV\. The undesired target has the same HLA\-A\*02:01–β2\\beta\_\{2\}m scaffold but presentsNLVFMVATV\. The peptides differ only at position 4, where Pro in the desired target is replaced by Phe in the undesired target\. Thus, successful discrimination must depend on the presented peptide rather than on a change in the MHC scaffold\.

We design an8080\-residue binder using the desired pMHC as the positive condition and the undesired pMHC as the negative condition\. The candidate and propagated weights use the same retained Proposition 2 expansion\. The complete target construction, weight terms, clipping rules, and hyperparameters are given in Appendix[F](https://arxiv.org/html/2610.00395#A6)\.

We use eight matched to generate 64 candidate binders\. We then retain one inverse\-folded sequence per backbone\. Generation uses200200reverse\-diffusion steps\. Each sequence is modeled with both the desired and undesired pMHC using Boltz\-2\.

##### Evaluation metric\.

For each Boltz\-2 prediction, we symmetrize the two entries of thepair\_chains\_iptmmatrix that correspond to the binder and peptide\. Letai,c,ka\_\{i,c,k\}be this binder–peptide iPTM for binderii, conditioncc, and predictionkk\. We average over theK=3K=3predictions for each condition and define the specificity margin

Ai,c=1K​∑k=1Kai,c,k,Δi=Ai,desired−Ai,undesired\.A\_\{i,c\}=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}a\_\{i,c,k\},\\qquad\\Delta\_\{i\}=A\_\{i,\\mathrm\{desired\}\}\-A\_\{i,\\mathrm\{undesired\}\}\.\(22\)It is a structural confidence measure and should not be interpreted as an experimental binding\-affinity measurement\.

##### Specificity results\.

We compare our method with target\-A\-only BoltzGen, fixed contrastive guidance, and Dynamic Negative Guidance \(DNG\)\. Every method is evaluated onN=64N=64binders generated from the same seed blocks and with the same desired\-minus\-undesired specificity metric\. All methods use three Boltz\-2 predictions per binder and condition\.

Table 2:Specificity on the pMHC binder\-design task\. Here,NNis the number of evaluated binders, Q75 is the7575th percentile, andΔ\>0\\Delta\>0is the fraction of binders with a positive specificity margin\. Higher values are better\.Our method achieves the largest mean, median, and Q75 specificity margins\. Relative to the strongest baseline on these statistics, target\-A\-only BoltzGen, it improves the mean by3\.9%3\.9\\%, the median by14\.8%14\.8\\%, and Q75 by16\.6%16\.6\\%\. Fixed guidance has the largest fraction of positive margins\. Thus, our method improves the central tendency and upper quartile, although it does not maximize the fraction of positive designs\.

##### Structural example\.

Figure[5](https://arxiv.org/html/2610.00395#S5.F5)shows predicted structures for a representative binder with peptide\-dependent recognition\. Its mean binder–peptide iPTM is

Adesired=0\.8673,Aundesired=0\.5174,Δ=0\.3499\.A\_\{\\mathrm\{desired\}\}=0\.8673,\\qquad A\_\{\\mathrm\{undesired\}\}=0\.5174,\\qquad\\Delta=0\.3499\.\(23\)The displayed desired\-target prediction has placed the binder directly against the peptide, including the mutated position\. The matched undesired\-target prediction places the binder away from the peptide\. Across the three predictions, the binder contacts the desired peptide in two cases and the undesired peptide in none\. The binder sequence and additional details are provided in Appendix[F](https://arxiv.org/html/2610.00395#A6)\.

Figure 5:Representative predicted complexes for a specificity\-selective binder\. The binder is shown in green, and the peptide is shown in red\. The remaining chains form the shared HLA\-A\*02:01–β2\\beta\_\{2\}m scaffold\. The desired target supports a peptide\-adjacent binder pose, whereas the displayed prediction for the undesired P4F target places the binder away from the peptide\.![Refer to caption](https://arxiv.org/html/2610.00395v1/new_wanted_visual.png)\(a\)Desired pMHC presentingNLVPMVATV\.
![Refer to caption](https://arxiv.org/html/2610.00395v1/new_unwanted_visual.png)\(b\)Undesired pMHC presentingNLVFMVATV\.

## 6Conclusion

We presented a specificity\-aware diffusion steering framework that preserves a desired distribution while suppressing an undesired one\. We first derive an explicit target from an overlap\-based objective\. Its softened likelihood\-ratio gate retains regions specific to the desired distribution without inducing the global mode shift of standard negative guidance\. We then construct an SMC sampler that locally minimizes the empirical relative variance of cumulative surrogate weights using the full displacement and one score Jacobian–vector product per reference model\.

Experiments on synthetic 40\-MoG, CIFAR\-10, text\-to\-image generation, and pMHC binder design show improved target fidelity, specificity, and particle diversity\. The framework is distributionally grounded, training\-free, and applicable when desired and undesired reference models are available\.

## 7Limitations

Our method assumes stochastic SDE inference and a local one\-step approximation; extending it to deterministic solvers, EDM parameterizations, and finite\-time flow maps is left for future work\. It also introduces application\-specific hyperparameters and additional compute from particle sampling with two reference models\. Evaluation is limited by the lack of a large\-scale text\-to\-image specificity benchmark\. For binder design, we study only one peptide pair with a modest number of candidates, and assess specificity using predicted iPTM rather than experimental affinity\. Larger\-scale studies with experimental on\- and off\-target validation are needed\.

## Acknowledgments

We sincerely thank Prof\. Tommi Jaakkola for valuable discussions and feedback on the formulation of the target distribution, the choice and interpretation of baselines, and the design about the sampling stability\. His insights on variance reduction and target distribution design were particularly helpful in shaping the theoretical motivation and positioning of this work\.

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## Appendix Contents

## Appendix ATarget Distribution Design

### A\.1Proof and Interpretation of the Target Distribution

See[1](https://arxiv.org/html/2610.00395#Thmproposition1)

###### Proof\.

We first prove the hard\-target result\. For fixedxx, write

a=pA​\(x\),b=pB​\(x\),t=q⁡\(x\),a=p^\{A\}\(x\),\\qquad b=p^\{B\}\(x\),\\qquad t=q\(x\),and define

Fx​\(t\)=min⁡\{t,a\}−γ​min⁡\{t,b\}\.F\_\{x\}\(t\)=\\min\\\{t,a\\\}\-\\gamma\\min\\\{t,b\\\}\.Then

J⁡\(q\)=∫Fx​\(q⁡\(x\)\)​𝑑x\.J\(q\)=\\int F\_\{x\}\(q\(x\)\)\\,dx\.
Forγ\>1\\gamma\>1, direct inspection gives

supt≥0Fx​\(t\)=\(a−γ​b\)\+\.\\sup\_\{t\\geq 0\}F\_\{x\}\(t\)=\(a\-\\gamma b\)\_\{\+\}\.Whena\>γ​ba\>\\gamma b, the supremum is attained fort≥at\\geq a; whena<γ​ba<\\gamma b, it is attained only att=0t=0\. Hence

J⁡\(q\)≤∫𝒜γ\(pA​\(x\)−γ​pB​\(x\)\)​𝑑x,J\(q\)\\leq\\int\_\{\\mathcal\{A\}\_\{\\gamma\}\}\\bigl\(p^\{A\}\(x\)\-\\gamma p^\{B\}\(x\)\\bigr\)\\,dx,where

𝒜γ=\{x:pA​\(x\)\>γ​pB​\(x\)\}\.\\mathcal\{A\}\_\{\\gamma\}=\\\{x:p^\{A\}\(x\)\>\\gamma p^\{B\}\(x\)\\\}\.
Let

ZA=∫𝒜γpA​\(x\)​𝑑x\.Z\_\{A\}=\\int\_\{\\mathcal\{A\}\_\{\\gamma\}\}p^\{A\}\(x\)\\,dx\.SinceZA\>0Z\_\{A\}\>0,

q∗​\(x\)=pA​\(x\)​𝟏𝒜γ​\(x\)ZAq^\{\*\}\(x\)=\\frac\{p^\{A\}\(x\)\\mathbf\{1\}\_\{\\mathcal\{A\}\_\{\\gamma\}\}\(x\)\}\{Z\_\{A\}\}satisfiesq∗​\(x\)≥pA​\(x\)q^\{\*\}\(x\)\\geq p^\{A\}\(x\)on𝒜γ\\mathcal\{A\}\_\{\\gamma\}and vanishes elsewhere, so it attains the upper bound and maximizesJJ\.

Now consider any maximizerqqwithDKL\(q∥pA\)<∞D\_\{\\mathrm\{KL\}\}\(q\\\|p^\{A\}\)<\\infty\. Pointwise optimality and the zero\-pAp^\{A\}\-mass boundary assumption imply that

q⁡\(x\)=pA​\(x\)​r​\(x\)​𝟏𝒜γ​\(x\),q\(x\)=p^\{A\}\(x\)r\(x\)\\mathbf\{1\}\_\{\\mathcal\{A\}\_\{\\gamma\}\}\(x\),where

r⁡\(x\)≥1,∫𝒜γpA​\(x\)​r​\(x\)​𝑑x=1\.r\(x\)\\geq 1,\\qquad\\int\_\{\\mathcal\{A\}\_\{\\gamma\}\}p^\{A\}\(x\)r\(x\)\\,dx=1\.Therefore,

DKL\(q∥pA\)=∫𝒜γpA\(x\)r\(x\)logr\(x\)dx\.D\_\{\\mathrm\{KL\}\}\(q\\\|p^\{A\}\)=\\int\_\{\\mathcal\{A\}\_\{\\gamma\}\}p^\{A\}\(x\)r\(x\)\\log r\(x\)\\,dx\.
Define

d​ν​\(x\)=pA​\(x\)​𝟏𝒜γ​\(x\)ZA​d​x\.d\\nu\(x\)=\\frac\{p^\{A\}\(x\)\\mathbf\{1\}\_\{\\mathcal\{A\}\_\{\\gamma\}\}\(x\)\}\{Z\_\{A\}\}\\,dx\.Then

𝔼ν​\[r\]=1ZA\.\\mathbb\{E\}\_\{\\nu\}\[r\]=\\frac\{1\}\{Z\_\{A\}\}\.Sinceu​log⁡uu\\log uis strictly convex, Jensen’s inequality yields

DKL\(q∥pA\)=ZA𝔼ν\[rlogr\]≥log1ZA,D\_\{\\mathrm\{KL\}\}\(q\\\|p^\{A\}\)=Z\_\{A\}\\mathbb\{E\}\_\{\\nu\}\[r\\log r\]\\geq\\log\\frac\{1\}\{Z\_\{A\}\},with equality if and only if

r⁡\(x\)=1ZAr\(x\)=\\frac\{1\}\{Z\_\{A\}\}almost everywhere\. Thusq∗q^\{\*\}is the unique KL\-minimizing maximizer\.

We next derive the soft relaxation\. Define

s⁡\(x\)=log⁡pA​\(x\)pB​\(x\)−log⁡γ\.s\(x\)=\\log\\frac\{p^\{A\}\(x\)\}\{p^\{B\}\(x\)\}\-\\log\\gamma\.The hard retention decision is

𝟏\{s\(x\)\>0\}\.\\mathbf\{1\}\\\{s\(x\)\>0\\\}\.Letπ⁡\(x\)∈\[0,1\]\\pi\(x\)\\in\[0,1\]denote a soft retention probability and consider

maxπ∈\[0,1\]⁡\{π​s​\(x\)\+1η​HB​\(π\)\},η\>0,\\max\_\{\\pi\\in\[0,1\]\}\\left\\\{\\pi s\(x\)\+\\frac\{1\}\{\\eta\}H\_\{\\mathrm\{B\}\}\(\\pi\)\\right\\\},\\qquad\\eta\>0,\(24\)where

HB​\(π\)=−π​log⁡π−\(1−π\)​log⁡\(1−π\)\.H\_\{\\mathrm\{B\}\}\(\\pi\)=\-\\pi\\log\\pi\-\(1\-\\pi\)\\log\(1\-\\pi\)\.The objective is strictly concave, and its first\-order condition gives

log⁡π1−π=η​s​\(x\)\.\\log\\frac\{\\pi\}\{1\-\\pi\}=\\eta s\(x\)\.Hence the unique optimizer is

rη​\(x\)=σ⁡\(η​s​\(x\)\)=pA​\(x\)ηpA​\(x\)η\+γη​pB​\(x\)η\.r\_\{\\eta\}\(x\)=\\sigma\(\\eta s\(x\)\)=\\frac\{p^\{A\}\(x\)^\{\\eta\}\}\{p^\{A\}\(x\)^\{\\eta\}\+\\gamma^\{\\eta\}p^\{B\}\(x\)^\{\\eta\}\}\.\(25\)Thusη\\etacontrols the sharpness of the likelihood\-ratio boundary\[[43](https://arxiv.org/html/2610.00395#bib.bib4)\]\.

Our powered target is

qη,m​\(x\)=pA​\(x\)​rη​\(x\)mZη,m,Zη,m=∫pA​\(u\)​rη​\(u\)m​𝑑u\.q\_\{\\eta,m\}\(x\)=\\frac\{p^\{A\}\(x\)r\_\{\\eta\}\(x\)^\{m\}\}\{Z\_\{\\eta,m\}\},\\qquad Z\_\{\\eta,m\}=\\int p^\{A\}\(u\)r\_\{\\eta\}\(u\)^\{m\}\\,du\.\(26\)The exponentmmcontrols the strength of the reweighting without changing the likelihood\-ratio boundary\.

To characterize this target variationally, observe that

log⁡qη,m​\(x\)pA​\(x\)=m​log⁡rη​\(x\)−log⁡Zη,m\.\\log\\frac\{q\_\{\\eta,m\}\(x\)\}\{p^\{A\}\(x\)\}=m\\log r\_\{\\eta\}\(x\)\-\\log Z\_\{\\eta,m\}\.Therefore, for anyq≪pAq\\ll p^\{A\},

DKL\(q∥qη,m\)\\displaystyle D\_\{\\mathrm\{KL\}\}\(q\\\|q\_\{\\eta,m\}\)=DKL\(q∥pA\)−m𝔼q\[logrη\(X\)\]\+logZη,m\.\\displaystyle=D\_\{\\mathrm\{KL\}\}\(q\\\|p^\{A\}\)\-m\\mathbb\{E\}\_\{q\}\[\\log r\_\{\\eta\}\(X\)\]\+\\log Z\_\{\\eta,m\}\.Equivalently,

𝔼q\[logrη\(X\)\]−1mDKL\(q∥pA\)=1mlogZη,m−1mDKL\(q∥qη,m\)\.\\mathbb\{E\}\_\{q\}\[\\log r\_\{\\eta\}\(X\)\]\-\\frac\{1\}\{m\}D\_\{\\mathrm\{KL\}\}\(q\\\|p^\{A\}\)=\\frac\{1\}\{m\}\\log Z\_\{\\eta,m\}\-\\frac\{1\}\{m\}D\_\{\\mathrm\{KL\}\}\(q\\\|q\_\{\\eta,m\}\)\.Since KL divergence is nonnegative,qη,mq\_\{\\eta,m\}is the unique solution of

maxq≪pA\{𝔼q\[logrη\(X\)\]−1mDKL\(q∥pA\)\}\.\\max\_\{q\\ll p^\{A\}\}\\left\\\{\\mathbb\{E\}\_\{q\}\[\\log r\_\{\\eta\}\(X\)\]\-\\frac\{1\}\{m\}D\_\{\\mathrm\{KL\}\}\(q\\\|p^\{A\}\)\\right\\\}\.\(27\)Thusmmcontrols the trade\-off between retention confidence and proximity topAp^\{A\}\.

Finally, for every fixedm\>0m\>0ands⁡\(x\)≠0s\(x\)\\neq 0,

rη\(x\)m=σ\(ηs\(x\)\)m⟶𝟏\{s\(x\)\>0\}r\_\{\\eta\}\(x\)^\{m\}=\\sigma\(\\eta s\(x\)\)^\{m\}\\longrightarrow\\mathbf\{1\}\\\{s\(x\)\>0\\\}asη→∞\\eta\\to\\infty\. Since0≤rη​\(x\)m≤10\\leq r\_\{\\eta\}\(x\)^\{m\}\\leq 1, dominated convergence gives

Zη,m⟶∫pA\(x\)𝟏\{s\(x\)\>0\}dx\.Z\_\{\\eta,m\}\\longrightarrow\\int p^\{A\}\(x\)\\mathbf\{1\}\\\{s\(x\)\>0\\\}\\,dx\.Under the assumptions of Proposition[1](https://arxiv.org/html/2610.00395#Thmproposition1),

qη,m⟶q∗q\_\{\\eta,m\}\\longrightarrow q^\{\*\}in total variation\. Hence every fixedm\>0m\>0preserves the same hard target\. ∎

### A\.2Relation to Existing Negative Guidance

##### Dynamic Negative Guidance\.

Dynamic Negative Guidance \(DNG\)\[[10](https://arxiv.org/html/2610.00395#bib.bib13)\]uses the negative\-class posterior\. With

R⁡\(x\)=pB​\(x\)pA​\(x\),c=p⁡\(B\)p⁡\(A\),R\(x\)=\\frac\{p^\{B\}\(x\)\}\{p^\{A\}\(x\)\},\\qquad c=\\frac\{p\(B\)\}\{p\(A\)\},and DNG exponentρ\>0\\rho\>0, its reweighting is

gDNG​\(x\)=\(1\+c​R​\(x\)\)−ρ\.g\_\{\\mathrm\{DNG\}\}\(x\)=\\bigl\(1\+cR\(x\)\\bigr\)^\{\-\\rho\}\.Our powered gate is

gours​\(x\)=rη​\(x\)m=\[1\+\(γ​R​\(x\)\)η\]−m\.g\_\{\\mathrm\{ours\}\}\(x\)=r\_\{\\eta\}\(x\)^\{m\}=\\left\[1\+\\bigl\(\\gamma R\(x\)\\bigr\)^\{\\eta\}\\right\]^\{\-m\}\.\(28\)
After matching the threshold and power,

c=γ,ρ=m,c=\\gamma,\\qquad\\rho=m,DNG is exactly the special caseη=1\\eta=1:

gDNG​\(x\)=gours​\(x\)when​η=1\.g\_\{\\mathrm\{DNG\}\}\(x\)=g\_\{\\mathrm\{ours\}\}\(x\)\\qquad\\text\{when \}\\eta=1\.Our formulation therefore separates two effects that are coupled in DNG:η\\etacontrols the sharpness of the likelihood\-ratio transition, whereasmmcontrols the strength of the reweighting\.

Forη\>1\\eta\>1, writingu=γ​R​\(x\)u=\\gamma R\(x\)gives

u<1⟹gours\>gDNG,u\>1⟹gours<gDNG\.u<1\\;\\Longrightarrow\\;g\_\{\\mathrm\{ours\}\}\>g\_\{\\mathrm\{DNG\}\},\\qquad u\>1\\;\\Longrightarrow\\;g\_\{\\mathrm\{ours\}\}<g\_\{\\mathrm\{DNG\}\}\.Thus the proposed gate preserves the same boundaryu=1u=1while producing a sharper distinction between positive\- and negative\-dominant regions\.

##### Safe Denoiser\.

Safe Denoiser\[[25](https://arxiv.org/html/2610.00395#bib.bib14)\]also modifies the base score along the contrastive direction

stA​\(x\)−stB​\(x\),s\_\{t\}^\{A\}\(x\)\-s\_\{t\}^\{B\}\(x\),but uses an unsafe\-density\-driven coefficient\.

For our target,

qtη,m​\(x\)∝ptA​\(x\)​\(1−θt​\(x\)\)m,q\_\{t\}^\{\\eta,m\}\(x\)\\propto p\_\{t\}^\{A\}\(x\)\\bigl\(1\-\\theta\_\{t\}\(x\)\\bigr\)^\{m\},where

θt​\(x\)=γηt​ptB​\(x\)ηtptA​\(x\)ηt\+γηt​ptB​\(x\)ηt\.\\theta\_\{t\}\(x\)=\\frac\{\\gamma^\{\\eta\_\{t\}\}p\_\{t\}^\{B\}\(x\)^\{\\eta\_\{t\}\}\}\{p\_\{t\}^\{A\}\(x\)^\{\\eta\_\{t\}\}\+\\gamma^\{\\eta\_\{t\}\}p\_\{t\}^\{B\}\(x\)^\{\\eta\_\{t\}\}\}\.Differentiating gives

∇x​log​qtη,m​\(x\)=stA​\(x\)\+m​ηt​θt​\(x\)​\(stA​\(x\)−stB​\(x\)\)\.\\nabla\_\{x\}\\log q\_\{t\}^\{\\eta,m\}\(x\)=s\_\{t\}^\{A\}\(x\)\+m\\eta\_\{t\}\\theta\_\{t\}\(x\)\\bigl\(s\_\{t\}^\{A\}\(x\)\-s\_\{t\}^\{B\}\(x\)\\bigr\)\.\(29\)
Hence both methods use the same contrastive score direction, but weight it differently\. Safe Denoiser uses a weight based primarily on the absolute unsafe density, whereas our coefficient

m​ηt​θt​\(x\)m\\eta\_\{t\}\\theta\_\{t\}\(x\)is determined by the relative likelihood of the positive and negative distributions\. The parameterηt\\eta\_\{t\}controls where the correction becomes active, whilemmcontrols its overall strength\.

## Appendix BMain Algorithm Derivation

We use the notation of Section[4](https://arxiv.org/html/2610.00395#S4)and write𝝆=\(ρ1,ρ2\)\\boldsymbol\{\\rho\}=\(\\rho\_\{1\},\\rho\_\{2\}\)\. The operational Gaussian weight isω𝝆G\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{G\}\}, its fixed\-displacement approximation isω^𝝆\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}, and the ideal normalized RN weight is denoted byω𝝆RN\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{RN\}\}when needed\. These objects are kept separate throughout\.

### B\.1Local Gaussian Weight and Proposal Expansion

##### Setup and regularity\.

Letτ=t\+h\\tau=t\+h,Δ​y=x−y\\Delta y=x\-y, and𝝆\\boldsymbol\{\\rho\}range over a fixed finite candidate set\. Each candidate is held fixed on\[t,τ\]\[t,\\tau\]\. The reference and proposal fields are smooth in time and space, with polynomially bounded derivatives through the orders used below \(two in time and three in space suffice\)\. The scheduless,c,η,ms,c,\\eta,mand their reciprocals are bounded on this interval, and the gate logits have the required moments\. We assume sufficiently high moments of the current state and of the local fields so that Taylor remainders areO⁡\(h3/2\)O\(h^\{3/2\}\)inL2L^\{2\}, uniformly over candidates\. For fixed states, the corresponding statements hold conditionally on those states\. These assumptions concern stochastic steps withs\>0s\>0\. HereOL2​\(ha\)O\_\{L^\{2\}\}\(h^\{a\}\)denotes a remainder withL2L^\{2\}normO⁡\(ha\)O\(h^\{a\}\)\.

Unless shown otherwise, quantities are evaluated at\(τ,y\)\(\\tau,y\)\. Write

d=dτ​\(y\),g=ζ​d,ζ=mτ​ητ​θ,χ=mτ​ητ2​θ​\(1−θ\),u=ρ1​d\+ρ2​g,v=u−g\.d=d\_\{\\tau\}\(y\),\\quad g=\\zeta d,\\quad\\zeta=m\_\{\\tau\}\\eta\_\{\\tau\}\\theta,\\quad\\chi=m\_\{\\tau\}\\eta\_\{\\tau\}^\{2\}\\theta\(1\-\\theta\),\\quad u=\\rho\_\{1\}d\+\\rho\_\{2\}g,\\quad v=u\-g\.The local score\-based derivatives are

∇θ=−ητ​θ​\(1−θ\)​d,J​g=ζ​J​d−χ​d​d⊤,J​u=\(ρ1\+ρ2​ζ\)​J​d−ρ2​χ​d​d⊤\.\\nabla\\theta=\-\\eta\_\{\\tau\}\\theta\(1\-\\theta\)d,\\qquad Jg=\\zeta Jd\-\\chi dd^\{\\top\},\\qquad Ju=\(\\rho\_\{1\}\+\\rho\_\{2\}\\zeta\)Jd\-\\rho\_\{2\}\\chi dd^\{\\top\}\.\(30\)For the exact gate, these follow by differentiation\. With an estimated current gate, they specify the smooth local extension used in the proposal expansion; they do not differentiate an estimator through its sampling history\. A different derivative convention requires a corresponding change to the local expansion\.

##### Gaussian gate contribution\.

The current gate valueθ∈\(0,1\)\\theta\\in\(0,1\)is supplied by the gate evaluation\. Because the reference forward kernels coincide, their contributions cancel in the A/B contrast\. For the Gaussian kernels in \([11](https://arxiv.org/html/2610.00395#S4.E11)\), define

DG​\(x,y\)\\displaystyle D^\{\\mathrm\{G\}\}\(x,y\):=log⁡LτνB​\(x∣y\)LτνA​\(x∣y\)=h​αA​B−d⊤​Δ​y,\\displaystyle:=\\log\\frac\{L\_\{\\tau\}^\{\\nu^\{B\}\}\(x\\mid y\)\}\{L\_\{\\tau\}^\{\\nu^\{A\}\}\(x\\mid y\)\}=h\\alpha\_\{AB\}\-d^\{\\top\}\\Delta y,\(31\)αA​B\\displaystyle\\alpha\_\{AB\}:=‖ντA​\(y\)‖2−‖ντB​\(y\)‖22​sτ\.\\displaystyle:=\\frac\{\\\|\\nu\_\{\\tau\}^\{A\}\(y\)\\\|^\{2\}\-\\\|\\nu\_\{\\tau\}^\{B\}\(y\)\\\|^\{2\}\}\{2s\_\{\\tau\}\}\.The equality is exact for these equal\-covariance Gaussian kernels: it follows by expanding the difference of their squared residuals\.

The resulting change in the gate log odds is

SG:=log⁡ctcτ\+\(ηtητ−1\)​\(log⁡θ1−θ−log⁡cτ\)\+ηt​DG\.S^\{\\mathrm\{G\}\}:=\\log\\frac\{c\_\{t\}\}\{c\_\{\\tau\}\}\+\\left\(\\frac\{\\eta\_\{t\}\}\{\\eta\_\{\\tau\}\}\-1\\right\)\\left\(\\log\\frac\{\\theta\}\{1\-\\theta\}\-\\log c\_\{\\tau\}\\right\)\+\\eta\_\{t\}D^\{\\mathrm\{G\}\}\.\(32\)Multiplying the current gate odds byeSGe^\{S^\{\\mathrm\{G\}\}\}gives the one\-step gate contribution used in the main text:

Δ​ℛG​\(x,y\):=−mt​log⁡\[1\+θ⁡\(eSG−1\)\]\+\(mt−mτ\)​log⁡\(1−θ\)\.\\Delta\\mathcal\{R\}^\{\\mathrm\{G\}\}\(x,y\):=\-m\_\{t\}\\log\\left\[1\+\\theta\(e^\{S^\{\\mathrm\{G\}\}\}\-1\)\\right\]\+\(m\_\{t\}\-m\_\{\\tau\}\)\\log\(1\-\\theta\)\.\(33\)For fixedmm, the last term vanishes\. This defines a local weight contribution conditioned on the current gate; it does not prescribe a recursive estimator for later gate values\.

Expanding the schedules atτ\\taugives

SG\\displaystyle S^\{\\mathrm\{G\}\}=h​αS−ητ​d⊤​Δ​y\+OL2​\(h3/2\),\\displaystyle=h\\alpha\_\{S\}\-\\eta\_\{\\tau\}d^\{\\top\}\\Delta y\+O\_\{L^\{2\}\}\(h^\{3/2\}\),\(34\)αS\\displaystyle\\alpha\_\{S\}:=ητ​αA​B−∂τlog⁡cτ−η˙τητ​\(log⁡θ1−θ−log⁡cτ\)\.\\displaystyle:=\\eta\_\{\\tau\}\\alpha\_\{AB\}\-\\partial\_\{\\tau\}\\log c\_\{\\tau\}\-\\frac\{\\dot\{\\eta\}\_\{\\tau\}\}\{\\eta\_\{\\tau\}\}\\left\(\\log\\frac\{\\theta\}\{1\-\\theta\}\-\\log c\_\{\\tau\}\\right\)\.Usinglog⁡\[1\+θ⁡\(eS−1\)\]=θ​S\+12​θ​\(1−θ\)​S2\+O⁡\(S3\)\\log\[1\+\\theta\(e^\{S\}\-1\)\]=\\theta S\+\\tfrac\{1\}\{2\}\\theta\(1\-\\theta\)S^\{2\}\+O\(S^\{3\}\), we obtain

Δ​ℛG\\displaystyle\\Delta\\mathcal\{R\}^\{\\mathrm\{G\}\}=h​αℛ\+g⊤​Δ​y−χ2​\(d⊤​Δ​y\)2\+OL2​\(h3/2\),\\displaystyle=h\\alpha\_\{\\mathcal\{R\}\}\+g^\{\\top\}\\Delta y\-\\frac\{\\chi\}\{2\}\(d^\{\\top\}\\Delta y\)^\{2\}\+O\_\{L^\{2\}\}\(h^\{3/2\}\),\(35\)αℛ\\displaystyle\\alpha\_\{\\mathcal\{R\}\}:=−mτ​θ​αS−m˙τ​log⁡\(1−θ\)\.\\displaystyle:=\-m\_\{\\tau\}\\theta\\alpha\_\{S\}\-\\dot\{m\}\_\{\\tau\}\\log\(1\-\\theta\)\.Settingm˙τ=0\\dot\{m\}\_\{\\tau\}=0recovers the fixed\-mmcoefficient\.

###### Proof of Proposition[2](https://arxiv.org/html/2610.00395#Thmproposition2)\.

Leta=a𝝆​\(y\)a=a\_\{\\boldsymbol\{\\rho\}\}\(y\)and define the kernel contribution

𝒦𝝆:=log⁡LτνAKtμA−log⁡LτaKtb𝝆,\\mathcal\{K\}\_\{\\boldsymbol\{\\rho\}\}:=\\log\\frac\{L\_\{\\tau\}^\{\\nu^\{A\}\}\}\{K\_\{t\}^\{\\mu^\{A\}\}\}\-\\log\\frac\{L\_\{\\tau\}^\{a\}\}\{K\_\{t\}^\{b^\{\\boldsymbol\{\\rho\}\}\}\},with all kernels evaluated at\(x,y\)\(x,y\)\. Their common covariance terms cancel\. Sincebt𝝆​\(x\)=μtA​\(x\)\+st​ut𝝆​\(x\)b\_\{t\}^\{\\boldsymbol\{\\rho\}\}\(x\)=\\mu\_\{t\}^\{A\}\(x\)\+s\_\{t\}u\_\{t\}^\{\\boldsymbol\{\\rho\}\}\(x\), exact Gaussian algebra gives

𝒦𝝆\\displaystyle\\mathcal\{K\}\_\{\\boldsymbol\{\\rho\}\}=\[a−ντA​\(y\)sτ−ut𝝆​\(x\)\]⊤​Δ​y\\displaystyle=\\left\[\\frac\{a\-\\nu\_\{\\tau\}^\{A\}\(y\)\}\{s\_\{\\tau\}\}\-u\_\{t\}^\{\\boldsymbol\{\\rho\}\}\(x\)\\right\]^\{\\top\}\\Delta y\+h⁡\[−μtA​\(x\)⊤​ut𝝆​\(x\)−st2​‖ut𝝆​\(x\)‖2\+‖a‖2−‖ντA​\(y\)‖22​sτ\]\.\\displaystyle\\quad\+h\\left\[\-\\mu\_\{t\}^\{A\}\(x\)^\{\\top\}u\_\{t\}^\{\\boldsymbol\{\\rho\}\}\(x\)\-\\frac\{s\_\{t\}\}\{2\}\\\|u\_\{t\}^\{\\boldsymbol\{\\rho\}\}\(x\)\\\|^\{2\}\+\\frac\{\\\|a\\\|^\{2\}\-\\\|\\nu\_\{\\tau\}^\{A\}\(y\)\\\|^\{2\}\}\{2s\_\{\\tau\}\}\\right\]\.\(36\)Under the backward Gaussian proposal,Δ​y=Op​\(h1/2\)\\Delta y=O\_\{p\}\(h^\{1/2\}\)\. Expandingut𝝆​\(y\+Δ​y\)=u\+J​u​Δ​y\+O⁡\(h\+‖Δ​y‖2\)u\_\{t\}^\{\\boldsymbol\{\\rho\}\}\(y\+\\Delta y\)=u\+Ju\\,\\Delta y\+O\(h\+\\\|\\Delta y\\\|^\{2\}\)therefore yields

𝒦𝝆=h​αker\+\[−u\+a−ντAsτ\]⊤​Δ​y−Δ​y⊤​J​u​Δ​y\+OL2​\(h3/2\),\\mathcal\{K\}\_\{\\boldsymbol\{\\rho\}\}=h\\alpha\_\{\\mathrm\{ker\}\}\+\\left\[\-u\+\\frac\{a\-\\nu\_\{\\tau\}^\{A\}\}\{s\_\{\\tau\}\}\\right\]^\{\\top\}\\Delta y\-\\Delta y^\{\\top\}Ju\\,\\Delta y\+O\_\{L^\{2\}\}\(h^\{3/2\}\),\(37\)where

αker=−\(μtA\)⊤​u−st2​‖u‖2\+‖a‖2−‖ντA‖22​sτ\.\\alpha\_\{\\mathrm\{ker\}\}=\-\(\\mu\_\{t\}^\{A\}\)^\{\\top\}u\-\\frac\{s\_\{t\}\}\{2\}\\\|u\\\|^\{2\}\+\\frac\{\\\|a\\\|^\{2\}\-\\\|\\nu\_\{\\tau\}^\{A\}\\\|^\{2\}\}\{2s\_\{\\tau\}\}\.Adding \([35](https://arxiv.org/html/2610.00395#A2.E35)\), the linear coefficient becomes

g−u\+a−ντAsτ\.g\-u\+\\frac\{a\-\\nu\_\{\\tau\}^\{A\}\}\{s\_\{\\tau\}\}\.It vanishes fora=a𝝆⋆=ντA\+sτ​va=a\_\{\\boldsymbol\{\\rho\}\}^\{\\star\}=\\nu\_\{\\tau\}^\{A\}\+s\_\{\\tau\}v\. Moreover,

‖a𝝆⋆‖2−‖ντA‖22​sτ=\(ντA\)⊤​v\+sτ2​‖v‖2\.\\frac\{\\\|a\_\{\\boldsymbol\{\\rho\}\}^\{\\star\}\\\|^\{2\}\-\\\|\\nu\_\{\\tau\}^\{A\}\\\|^\{2\}\}\{2s\_\{\\tau\}\}=\(\\nu\_\{\\tau\}^\{A\}\)^\{\\top\}v\+\\frac\{s\_\{\\tau\}\}\{2\}\\\|v\\\|^\{2\}\.Thus the remaining coefficients are

α𝝆\\displaystyle\\alpha\_\{\\boldsymbol\{\\rho\}\}=αℛ−\(μtA\)⊤​u−st2​‖u‖2\+\(ντA\)⊤​v\+sτ2​‖v‖2,\\displaystyle=\\alpha\_\{\\mathcal\{R\}\}\-\(\\mu\_\{t\}^\{A\}\)^\{\\top\}u\-\\frac\{s\_\{t\}\}\{2\}\\\|u\\\|^\{2\}\+\(\\nu\_\{\\tau\}^\{A\}\)^\{\\top\}v\+\\frac\{s\_\{\\tau\}\}\{2\}\\\|v\\\|^\{2\},\(38\)Γ𝝆\\displaystyle\\Gamma\_\{\\boldsymbol\{\\rho\}\}=−J​u−χ2​d​d⊤=−\(ρ1\+ρ2​ζ\)​J​d\+χ⁡\(ρ2−12\)​d​d⊤\.\\displaystyle=\-Ju\-\\frac\{\\chi\}\{2\}dd^\{\\top\}=\-\(\\rho\_\{1\}\+\\rho\_\{2\}\\zeta\)Jd\+\\chi\\left\(\\rho\_\{2\}\-\\frac\{1\}\{2\}\\right\)dd^\{\\top\}\.These are \([38](https://arxiv.org/html/2610.00395#A2.E38)\) and \([15](https://arxiv.org/html/2610.00395#S4.E15)\)\. Consequently,

log⁡ω𝝆G=h​α𝝆\+Δ​y⊤​Γ𝝆​Δ​y\+OL2​\(h3/2\)\.\\log\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{G\}\}=h\\alpha\_\{\\boldsymbol\{\\rho\}\}\+\\Delta y^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\Delta y\+O\_\{L^\{2\}\}\(h^\{3/2\}\)\.The retained terms areOp​\(h\)O\_\{p\}\(h\), and the remainder is uniform over the fixed finite candidate set\. ∎

##### Fixed\-displacement evaluation and same\-noise selection\.

For fixedρ1\\rho\_\{1\}, letΔ​y0\\Delta y\_\{0\}be defined by \([16](https://arxiv.org/html/2610.00395#S4.E16)\)\. ThenΔ​yρ2=Δ​y0\+δρ2\\Delta y\_\{\\rho\_\{2\}\}=\\Delta y\_\{0\}\+\\delta\_\{\\rho\_\{2\}\}, whereδρ2=−h​sτ​ρ2​g=O⁡\(h\)\\delta\_\{\\rho\_\{2\}\}=\-hs\_\{\\tau\}\\rho\_\{2\}g=O\(h\)\. Hence

Δ​yρ2⊤​Γ𝝆​Δ​yρ2−Δ​y0⊤​Γ𝝆​Δ​y0\\displaystyle\\Delta y\_\{\\rho\_\{2\}\}^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\Delta y\_\{\\rho\_\{2\}\}\-\\Delta y\_\{0\}^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\Delta y\_\{0\}=Δ​y0⊤​Γ𝝆​δρ2\+δρ2⊤​Γ𝝆​Δ​y0\+δρ2⊤​Γ𝝆​δρ2=OL2​\(h3/2\)\.\\displaystyle\\quad=\\Delta y\_\{0\}^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\delta\_\{\\rho\_\{2\}\}\+\\delta\_\{\\rho\_\{2\}\}^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\Delta y\_\{0\}\+\\delta\_\{\\rho\_\{2\}\}^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\delta\_\{\\rho\_\{2\}\}=O\_\{L^\{2\}\}\(h^\{3/2\}\)\.\(39\)All candidates therefore reuseJ​d​\[Δ​y0\]Jd\[\\Delta y\_\{0\}\], computed by subtracting one score JVP per reference model\. In particular,

‖maxρ2∈ℛ2⁡\|log⁡ωρ1,ρ2G​\(x⁡\(ρ2\),y\)−log⁡ω^​\(ρ2\)\|‖L2=O⁡\(h3/2\)\.\\left\\\|\\max\_\{\\rho\_\{2\}\\in\\mathcal\{R\}\_\{2\}\}\\left\|\\log\\omega\_\{\\rho\_\{1\},\\rho\_\{2\}\}^\{\\mathrm\{G\}\}\(x\(\\rho\_\{2\}\),y\)\-\\log\\widehat\{\\omega\}\(\\rho\_\{2\}\)\\right\|\\right\\\|\_\{L^\{2\}\}=O\(h^\{3/2\}\)\.\(40\)The selected error is bounded by this maximum, so the same expansion holds after choosingρ2\\rho\_\{2\}using the current noises\.

This samplewise statement does not imply conditional trace unbiasedness\. If𝝆⋆\\boldsymbol\{\\rho\}^\{\\star\}depends onξ\\xi, then generally

𝔼⁡\[ξ⊤​Γ𝝆⋆​ξ−tr⁡Γ𝝆⋆∣current particle system\]≠0\.\\mathbb\{E\}\\\!\\left\[\\xi^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}^\{\\star\}\}\\xi\-\\operatorname\{tr\}\\Gamma\_\{\\boldsymbol\{\\rho\}^\{\\star\}\}\\mid\\text\{current particle system\}\\right\]\\neq 0\.Likewise, conditioning on the selected coefficient changes the noise law, so the actual selected mutation need not have the nominal Gaussian candidate density\. The algorithm uses adaptive surrogate weights, not an exact RN correction for that selected law\.

##### Distinction from an oracle endpoint gate\.

Under the exact\-reference assumptions of Section[B\.3](https://arxiv.org/html/2610.00395#A2.SS3), setH=∇2​log​\(pτB/pτA\)​\(y\)=−J​dH=\\nabla^\{2\}\\log\(p\_\{\\tau\}^\{B\}/p\_\{\\tau\}^\{A\}\)\(y\)=\-Jd\. The log\-density Fokker–Planck equations and a Taylor expansion give

log⁡ptB​\(x\)ptA​\(x\)−log⁡pτB​\(y\)pτA​\(y\)=DG​\(x,y\)\+12​\[Δ​y⊤​H​Δ​y−h​sτ​tr⁡H\]\+OL2​\(h3/2\)\.\\log\\frac\{p\_\{t\}^\{B\}\(x\)\}\{p\_\{t\}^\{A\}\(x\)\}\-\\log\\frac\{p\_\{\\tau\}^\{B\}\(y\)\}\{p\_\{\\tau\}^\{A\}\(y\)\}=D^\{\\mathrm\{G\}\}\(x,y\)\+\\frac\{1\}\{2\}\\left\[\\Delta y^\{\\top\}H\\Delta y\-hs\_\{\\tau\}\\operatorname\{tr\}H\\right\]\+O\_\{L^\{2\}\}\(h^\{3/2\}\)\.\(41\)For an exact current gate, the corresponding reward difference is

ℛt​\(x\)−ℛτ​\(y\)−Δ​ℛG​\(x,y\)=ζ2​\[Δ​y⊤​J​d​Δ​y−h​sτ​tr⁡J​d\]\+OL2​\(h3/2\)\.\\mathcal\{R\}\_\{t\}\(x\)\-\\mathcal\{R\}\_\{\\tau\}\(y\)\-\\Delta\\mathcal\{R\}^\{\\mathrm\{G\}\}\(x,y\)=\\frac\{\\zeta\}\{2\}\\left\[\\Delta y^\{\\top\}Jd\\,\\Delta y\-hs\_\{\\tau\}\\operatorname\{tr\}Jd\\right\]\+O\_\{L^\{2\}\}\(h^\{3/2\}\)\.\(42\)This centered curvature term distinguishes the oracle endpoint weight from the defined Gaussian surrogate; it is not an omitted term in Proposition[2](https://arxiv.org/html/2610.00395#Thmproposition2)\.

### B\.2Variance Objective and Forward Kernel Matching

#### B\.2\.1Variance Objective

Here we show that normalized cumulative ESS maximization is our variance minimization used in section[4\.1\.1](https://arxiv.org/html/2610.00395#S4.SS1.SSS1)

nESS^t​\(ρ2\):=\(∑iVi​\(ρ2\)\)2N​∑iVi​\(ρ2\)2=\[1\+𝒱^trel​\(ρ2\)\]−1\.\\widehat\{\\mathrm\{nESS\}\}\_\{t\}\(\\rho\_\{2\}\):=\\frac\{\\bigl\(\\sum\_\{i\}V\_\{i\}\(\\rho\_\{2\}\)\\bigr\)^\{2\}\}\{N\\sum\_\{i\}V\_\{i\}\(\\rho\_\{2\}\)^\{2\}\}=\\left\[1\+\\widehat\{\\mathcal\{V\}\}\_\{t\}^\{\\mathrm\{rel\}\}\(\\rho\_\{2\}\)\\right\]^\{\-1\}\.

#### B\.2\.2Forward Kernel Matching

This subsection concerns fixed candidates, exact reference scores and gates, andY∼qτY\\sim q\_\{\\tau\}independent of the current Gaussian noise\. Writeqr=γr/Zrq\_\{r\}=\\gamma\_\{r\}/Z\_\{r\}, whereZr=∫γr​\(z\)​𝑑zZ\_\{r\}=\\int\\gamma\_\{r\}\(z\)\\,dz\. The forward\-propagated marginal is

q¯τ𝝆​\(y\):=∫qt​\(x\)​Ktb𝝆​\(y∣x\)​𝑑x\.\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\(y\):=\\int q\_\{t\}\(x\)K\_\{t\}^\{b^\{\\boldsymbol\{\\rho\}\}\}\(y\\mid x\)\\,dx\.\(43\)Define the adjoint generator and normalized residual by

ℒr,𝝆∗​f\\displaystyle\\mathcal\{L\}\_\{r,\\boldsymbol\{\\rho\}\}^\{\*\}f:=−∇⋅\(br𝝆f\)\+sr2∇⋅∇f,\\displaystyle:=\-\\nabla\\cdot\(b\_\{r\}^\{\\boldsymbol\{\\rho\}\}f\)\+\\frac\{s\_\{r\}\}\{2\}\\nabla\\cdot\\nabla f,\(44\)Rr,𝝆\\displaystyle R\_\{r,\\boldsymbol\{\\rho\}\}:=∂rqr−ℒr,𝝆∗​qrqr\.\\displaystyle:=\\frac\{\\partial\_\{r\}q\_\{r\}\-\\mathcal\{L\}\_\{r,\\boldsymbol\{\\rho\}\}^\{\*\}q\_\{r\}\}\{q\_\{r\}\}\.The corresponding unnormalized log\-weight potential is

A𝝆​\(r,⋅\):=−∂rγr−ℒr,𝝆∗​γrγr=−Rr,𝝆−∂rlog⁡Zr\.A\_\{\\boldsymbol\{\\rho\}\}\(r,\\cdot\):=\-\\frac\{\\partial\_\{r\}\\gamma\_\{r\}\-\\mathcal\{L\}\_\{r,\\boldsymbol\{\\rho\}\}^\{\*\}\\gamma\_\{r\}\}\{\\gamma\_\{r\}\}=\-R\_\{r,\\boldsymbol\{\\rho\}\}\-\\partial\_\{r\}\\log Z\_\{r\}\.\(45\)Under decay or no\-flux boundary conditions,𝔼qr​\[Rr,𝝆\]=0\\mathbb\{E\}\_\{q\_\{r\}\}\[R\_\{r,\\boldsymbol\{\\rho\}\}\]=0\. ThusA𝝆A\_\{\\boldsymbol\{\\rho\}\}andRr,𝝆R\_\{r,\\boldsymbol\{\\rho\}\}have the same variance across states\.

##### Conditional log\-weight moments\.

The continuous connection proved in Section[B\.3](https://arxiv.org/html/2610.00395#A2.SS3)gives

log⁡ω^𝝆​\(Y,ξ\)=h​A𝝆​\(τ,Y\)\+h​sτ​\[ξ⊤​Γ𝝆​\(Y\)​ξ−tr⁡Γ𝝆​\(Y\)\]\+OL2​\(h3/2\)\.\\log\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}\(Y,\\xi\)=hA\_\{\\boldsymbol\{\\rho\}\}\(\\tau,Y\)\+hs\_\{\\tau\}\\left\[\\xi^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\(Y\)\\xi\-\\operatorname\{tr\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\(Y\)\\right\]\+O\_\{L^\{2\}\}\(h^\{3/2\}\)\.For a fixed matrixMMand standard Gaussianξ\\xi,𝔼⁡\[ξ⊤​M​ξ\]=tr⁡M\\mathbb\{E\}\[\\xi^\{\\top\}M\\xi\]=\\operatorname\{tr\}MandVar⁡\(ξ⊤​M​ξ\)=2​‖sym⁡M‖F2\\operatorname\{Var\}\(\\xi^\{\\top\}M\\xi\)=2\\\|\\operatorname\{sym\}M\\\|\_\{F\}^\{2\}, wheresym⁡M=\(M\+M⊤\)/2\\operatorname\{sym\}M=\(M\+M^\{\\top\}\)/2\. Therefore

VarY⁡\(𝔼ξ​\[log⁡ω^𝝆∣Y\]\)\\displaystyle\\operatorname\{Var\}\_\{Y\}\\\!\\left\(\\mathbb\{E\}\_\{\\xi\}\[\\log\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}\\mid Y\]\\right\)=h2​Varqτ⁡\(A𝝆​\(τ,Y\)\)\+o⁡\(h2\),\\displaystyle=h^\{2\}\\operatorname\{Var\}\_\{q\_\{\\tau\}\}\(A\_\{\\boldsymbol\{\\rho\}\}\(\\tau,Y\)\)\+o\(h^\{2\}\),\(46\)𝔼Y​\[Varξ⁡\(log⁡ω^𝝆∣Y\)\]\\displaystyle\\mathbb\{E\}\_\{Y\}\\\!\\left\[\\operatorname\{Var\}\_\{\\xi\}\(\\log\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}\\mid Y\)\\right\]=2​h2​sτ2​𝔼qτ​‖sym⁡Γ𝝆​\(Y\)‖F2\+o⁡\(h2\)\.\\displaystyle=2h^\{2\}s\_\{\\tau\}^\{2\}\\mathbb\{E\}\_\{q\_\{\\tau\}\}\\\|\\operatorname\{sym\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\(Y\)\\\|\_\{F\}^\{2\}\+o\(h^\{2\}\)\.These are the two terms in \([21](https://arxiv.org/html/2610.00395#S4.E21)\)\. For exact scores,J​dJdandΓ𝝆\\Gamma\_\{\\boldsymbol\{\\rho\}\}are symmetric\.

##### Local forward\-marginal interpretation\.

A first\-order density expansion gives

q¯τ𝝆qτ=1−h​Rτ,𝝆\+oL2​\(qτ\)​\(h\)\.\\frac\{\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\}\{q\_\{\\tau\}\}=1\-hR\_\{\\tau,\\boldsymbol\{\\rho\}\}\+o\_\{L^\{2\}\(q\_\{\\tau\}\)\}\(h\)\.\(47\)For the KL expansion, assume positive density ratios and an integrable logarithmic Taylor remainder\. Precisely, forδh=q¯τ𝝆/qτ−1\\delta\_\{h\}=\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}/q\_\{\\tau\}\-1, assume

𝔼qτ​\|−log⁡\(1\+δh\)\+δh−12​δh2\|=o⁡\(h2\)\.\\mathbb\{E\}\_\{q\_\{\\tau\}\}\\left\|\-\\log\(1\+\\delta\_\{h\}\)\+\\delta\_\{h\}\-\\tfrac\{1\}\{2\}\\delta\_\{h\}^\{2\}\\right\|=o\(h^\{2\}\)\.Together with \([47](https://arxiv.org/html/2610.00395#A2.E47)\), this states the required density and tail control, rather than inferring it from pointwise smoothness alone\.

###### Proposition 3\(Conditional\-mean log\-weight variance and forward KL\)\.

Under the assumptions above,

VarY⁡\(𝔼ξ​\[log⁡ω^𝝆∣Y\]\)\\displaystyle\\operatorname\{Var\}\_\{Y\}\\\!\\left\(\\mathbb\{E\}\_\{\\xi\}\[\\log\\widehat\{\\omega\}\_\{\\boldsymbol\{\\rho\}\}\\mid Y\]\\right\)=h2​Varqτ⁡\(Rτ,𝝆\)\+o⁡\(h2\)\\displaystyle=h^\{2\}\\operatorname\{Var\}\_\{q\_\{\\tau\}\}\(R\_\{\\tau,\\boldsymbol\{\\rho\}\}\)\+o\(h^\{2\}\)\(48\)=2KL\(qτ∥q¯τ𝝆\)\+o\(h2\)\.\\displaystyle=2\\,\\mathrm\{KL\}\(q\_\{\\tau\}\\\|\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\)\+o\(h^\{2\}\)\.

###### Proof\.

The first equality follows from \([46](https://arxiv.org/html/2610.00395#A2.E46)\) and \([45](https://arxiv.org/html/2610.00395#A2.E45)\)\. Both marginals are normalized, so𝔼qτ​\[δh\]=0\\mathbb\{E\}\_\{q\_\{\\tau\}\}\[\\delta\_\{h\}\]=0\. The assumed Taylor control then yields

KL\(qτ∥q¯τ𝝆\)=12𝔼qτ\[δh2\]\+o\(h2\)=h22Varqτ\(Rτ,𝝆\)\+o\(h2\)\.\\mathrm\{KL\}\(q\_\{\\tau\}\\\|\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\)=\\tfrac\{1\}\{2\}\\mathbb\{E\}\_\{q\_\{\\tau\}\}\[\\delta\_\{h\}^\{2\}\]\+o\(h^\{2\}\)=\\frac\{h^\{2\}\}\{2\}\\operatorname\{Var\}\_\{q\_\{\\tau\}\}\(R\_\{\\tau,\\boldsymbol\{\\rho\}\}\)\+o\(h^\{2\}\)\.∎

Thus the conditional\-mean component admits a local forward\-matching interpretation\. The full log\-weight variance also contains the transition component in \([46](https://arxiv.org/html/2610.00395#A2.E46)\)\.

### B\.3Connection to the Continuous Path\-Space Formulation

Assume positive, smooth reference densities satisfying the common forward Fokker–Planck equation, exact gate values, and the moment conditions in Section[B\.1](https://arxiv.org/html/2610.00395#A2.SS1)\. For each fixed candidate,

∂rprj=−∇⋅\(μrjprj\)\+sr2∇⋅∇prj,j∈\{A,B\},\\partial\_\{r\}p\_\{r\}^\{j\}=\-\\nabla\\cdot\(\\mu\_\{r\}^\{j\}p\_\{r\}^\{j\}\)\+\\frac\{s\_\{r\}\}\{2\}\\nabla\\cdot\\nabla p\_\{r\}^\{j\},\\qquad j\\in\\\{A,B\\\},withμA=μB\\mu^\{A\}=\\mu^\{B\}\. Throughout this subsection the target and weights use the unnormalized conventionγr=prA​eℛr\\gamma\_\{r\}=p\_\{r\}^\{A\}e^\{\\mathcal\{R\}\_\{r\}\}\.

##### Matched continuous process\.

The potentialA𝝆A\_\{\\boldsymbol\{\\rho\}\}defined in \([45](https://arxiv.org/html/2610.00395#A2.E45)\) follows from the product rule:

A𝝆\\displaystyle A\_\{\\boldsymbol\{\\rho\}\}=−∂rℛ−\(νA\)⊤g\+s2\(∇⋅g\+∥g∥2\)\\displaystyle=\-\\partial\_\{r\}\\mathcal\{R\}\-\(\\nu^\{A\}\)^\{\\top\}g\+\\frac\{s\}\{2\}\(\\nabla\\cdot g\+\\\|g\\\|^\{2\}\)−s\[∇⋅u𝝆\+\(u𝝆\)⊤∇logγ\]\.\\displaystyle\\quad\-s\\left\[\\nabla\\cdot u^\{\\boldsymbol\{\\rho\}\}\+\(u^\{\\boldsymbol\{\\rho\}\}\)^\{\\top\}\\nabla\\log\\gamma\\right\]\.\(49\)The time and state arguments on the right are\(r,z\)\(r,z\)\. The matched drift satisfies

a𝝆⋆=νA\+s\(u𝝆−g\)=b𝝆−s∇logγ\.a\_\{\\boldsymbol\{\\rho\}\}^\{\\star\}=\\nu^\{A\}\+s\(u^\{\\boldsymbol\{\\rho\}\}\-g\)=b^\{\\boldsymbol\{\\rho\}\}\-s\\nabla\\log\\gamma\.Consequently,

∇⋅\(a𝝆⋆γ\)\+s2∇⋅∇γ=−ℒr,𝝆∗γ,\\nabla\\cdot\(a\_\{\\boldsymbol\{\\rho\}\}^\{\\star\}\\gamma\)\+\\frac\{s\}\{2\}\\nabla\\cdot\\nabla\\gamma=\-\\mathcal\{L\}\_\{r,\\boldsymbol\{\\rho\}\}^\{\*\}\\gamma,so−∂rγ\-\\partial\_\{r\}\\gammaequals this reverse\-time generator applied toγ\\gamma, plusA𝝆​γA\_\{\\boldsymbol\{\\rho\}\}\\gamma\.

In increasing reverse timev∈\[0,h\]v\\in\[0,h\], the corresponding process and incremental FK weight are

d​Zv\\displaystyle dZ\_\{v\}=−a𝝆⋆\(τ−v,Zv\)dv\+sτ−vdBv,Z0=y,\\displaystyle=\-a\_\{\\boldsymbol\{\\rho\}\}^\{\\star\}\(\\tau\-v,Z\_\{v\}\)\\,dv\+\\sqrt\{s\_\{\\tau\-v\}\}\\,dB\_\{v\},\\qquad Z\_\{0\}=y,\(50\)log⁡ω𝝆cont\\displaystyle\\log\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{cont\}\}:=∫0hA𝝆​\(τ−v,Zv\)​dv=h​A𝝆​\(τ,y\)\+OL2​\(h3/2\)\.\\displaystyle:=\\int\_\{0\}^\{h\}A\_\{\\boldsymbol\{\\rho\}\}\(\\tau\-v,Z\_\{v\}\)\\,dv=hA\_\{\\boldsymbol\{\\rho\}\}\(\\tau,y\)\+O\_\{L^\{2\}\}\(h^\{3/2\}\)\.The last estimate uses the spatial and temporal regularity ofA𝝆A\_\{\\boldsymbol\{\\rho\}\}, with polynomial growth and sufficient moments, and‖Zv−y‖Lp=O⁡\(v1/2\)\\\|Z\_\{v\}\-y\\\|\_\{L^\{p\}\}=O\(v^\{1/2\}\)for the required moments\. Normalizing the marginal path adds only the spatially constant potential∂rlog⁡Zr\\partial\_\{r\}\\log Z\_\{r\}\.

##### Identification of the local coefficient\.

At\(τ,y\)\(\\tau,y\), the two reference log\-density equations imply

αA​B=−∂τlogpτBpτA\+sτ2tr∇2logpτBpτA\.\\alpha\_\{AB\}=\-\\partial\_\{\\tau\}\\log\\frac\{p\_\{\\tau\}^\{B\}\}\{p\_\{\\tau\}^\{A\}\}\+\\frac\{s\_\{\\tau\}\}\{2\}\\operatorname\{tr\}\\nabla^\{2\}\\log\\frac\{p\_\{\\tau\}^\{B\}\}\{p\_\{\\tau\}^\{A\}\}\.\(51\)Substituting this into \([34](https://arxiv.org/html/2610.00395#A2.E34)\) and \([35](https://arxiv.org/html/2610.00395#A2.E35)\) gives

αℛ−sτ​χ2∥d∥2=−∂τℛτ\+sτ2∇⋅gτ\.\\alpha\_\{\\mathcal\{R\}\}\-\\frac\{s\_\{\\tau\}\\chi\}\{2\}\\\|d\\\|^\{2\}=\-\\partial\_\{\\tau\}\\mathcal\{R\}\_\{\\tau\}\+\\frac\{s\_\{\\tau\}\}\{2\}\\nabla\\cdot g\_\{\\tau\}\.\(52\)This includes them˙τ\\dot\{m\}\_\{\\tau\}term whenmmvaries\. Meanwhile, rearranging the scalar coefficient gives

α𝝆=αℛ−\(ντA\)⊤g\+sτ2∥g∥2−sτu⊤∇logγτ\+O\(h\),\\alpha\_\{\\boldsymbol\{\\rho\}\}=\\alpha\_\{\\mathcal\{R\}\}\-\(\\nu\_\{\\tau\}^\{A\}\)^\{\\top\}g\+\\frac\{s\_\{\\tau\}\}\{2\}\\\|g\\\|^\{2\}\-s\_\{\\tau\}u^\{\\top\}\\nabla\\log\\gamma\_\{\\tau\}\+O\(h\),\(53\)andsτtrΓ𝝆=−sτ∇⋅u−\(sτχ/2\)∥d∥2s\_\{\\tau\}\\operatorname\{tr\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}=\-s\_\{\\tau\}\\nabla\\cdot u\-\(s\_\{\\tau\}\\chi/2\)\\\|d\\\|^\{2\}\. Combining these expressions proves

α𝝆\+sτ​tr⁡Γ𝝆=A𝝆​\(τ,y\)\+O⁡\(h\)\.\\alpha\_\{\\boldsymbol\{\\rho\}\}\+s\_\{\\tau\}\\operatorname\{tr\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}=A\_\{\\boldsymbol\{\\rho\}\}\(\\tau,y\)\+O\(h\)\.\(54\)TheO⁡\(h\)O\(h\)term accounts for retainingμtA\\mu\_\{t\}^\{A\}andsts\_\{t\}in the finite scalar coefficient; it vanishes if those coefficients are evaluated atτ\\tauinstead\.

Finally,Δ​y⊤​Γ𝝆​Δ​y=h​sτ​ξ⊤​Γ𝝆​ξ\+OL2​\(h3/2\)\\Delta y^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\Delta y=hs\_\{\\tau\}\\xi^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\xi\+O\_\{L^\{2\}\}\(h^\{3/2\}\)\. Together with \([50](https://arxiv.org/html/2610.00395#A2.E50)\), this proves \([20](https://arxiv.org/html/2610.00395#S4.E20)\)\. The quadratic correction is conditionally centered for a fixed candidate\. This establishes the continuous conditional\-mean interpretation, not pathwise equality or unbiasedness after same\-noise selection\.

### B\.4Connection to Previous Variance\-Minimizing Methods

#### B\.4\.1Comparison with DAS

For this comparison only, consider the ideal normalized weight

ω𝝆RN​\(x,y\):=qt​\(x\)​Ktb𝝆​\(y∣x\)qτ​\(y\)​Lτa​\(x∣y\),\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{RN\}\}\(x,y\):=\\frac\{q\_\{t\}\(x\)K\_\{t\}^\{b^\{\\boldsymbol\{\\rho\}\}\}\(y\\mid x\)\}\{q\_\{\\tau\}\(y\)L\_\{\\tau\}^\{a\}\(x\\mid y\)\},\(55\)whereY∼qτY\\sim q\_\{\\tau\}andX\|Y∼LτaX\\mid Y\\sim L\_\{\\tau\}^\{a\}\. Assume the required support conditions and finite second moments\. Direct integration gives

𝔼⁡\[ω𝝆RN∣Y=y\]\\displaystyle\\mathbb\{E\}\[\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{RN\}\}\\mid Y=y\]=q¯τ𝝆​\(y\)qτ​\(y\),\\displaystyle=\\frac\{\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\(y\)\}\{q\_\{\\tau\}\(y\)\},\(56\)VarY⁡\(𝔼⁡\[ω𝝆RN∣Y\]\)\\displaystyle\\operatorname\{Var\}\_\{Y\}\\\!\\left\(\\mathbb\{E\}\[\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{RN\}\}\\mid Y\]\\right\)=χ2\(q¯τ𝝆∥qτ\)\.\\displaystyle=\\chi^\{2\}\(\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\\\|q\_\{\\tau\}\)\.For fixedKtb𝝆K\_\{t\}^\{b^\{\\boldsymbol\{\\rho\}\}\}, the variance\-minimizing backward conditional is

Lτopt​\(x∣y\):=qt​\(x\)​Ktb𝝆​\(y∣x\)q¯τ𝝆​\(y\)\.L\_\{\\tau\}^\{\\mathrm\{opt\}\}\(x\\mid y\):=\\frac\{q\_\{t\}\(x\)K\_\{t\}^\{b^\{\\boldsymbol\{\\rho\}\}\}\(y\\mid x\)\}\{\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\(y\)\}\.\(57\)It makesωRN\\omega^\{\\mathrm\{RN\}\}constant inxxconditional onyy, hence removes the conditional\-variance term\. The forward\-marginal term in \([56](https://arxiv.org/html/2610.00395#A2.E56)\) is unchanged by the choice ofLL\. This is the ideal conditional\-variance criterion underlying the proposal component of DAS\[[26](https://arxiv.org/html/2610.00395#bib.bib27)\]; its Gaussian proposal uses a local reward approximation\. DAS also uses tempering to modify intermediate targets\.

Our coefficients change the auxiliary forward kernel as well as the matched backward drift\. The Gaussian\-surrogate comparison is therefore through \([46](https://arxiv.org/html/2610.00395#A2.E46)\) and \([48](https://arxiv.org/html/2610.00395#A2.E48)\)\. It does not assert exact conditional\-variance elimination or exact RN correction for the implemented adaptive weights\. For clarity, even for the ideal RN weight,

𝔼L\[logω𝝆RN∣Y=y\]=logq¯τ𝝆​\(y\)qτ​\(y\)−KL\(Lτa\(⋅∣y\)∥Lτopt\(⋅∣y\)\)\.\\mathbb\{E\}\_\{L\}\[\\log\\omega\_\{\\boldsymbol\{\\rho\}\}^\{\\mathrm\{RN\}\}\\mid Y=y\]=\\log\\frac\{\\bar\{q\}\_\{\\tau\}^\{\\,\\boldsymbol\{\\rho\}\}\(y\)\}\{q\_\{\\tau\}\(y\)\}\-\\mathrm\{KL\}\\\!\\left\(L\_\{\\tau\}^\{a\}\(\\cdot\\mid y\)\\\|L\_\{\\tau\}^\{\\mathrm\{opt\}\}\(\\cdot\\mid y\)\\right\)\.\(58\)Thus the exact RN identity cannot be transferred unchanged to log weights; our forward\-KL relation forlog⁡ω^\\log\\widehat\{\\omega\}is a leading\-order result\.

#### B\.4\.2Comparison with DriftLite

DriftLite\-VCG\[[34](https://arxiv.org/html/2610.00395#bib.bib9)\]uses the FK compensation identity

Hr\(z;𝔟\):=∇⋅𝔟\(z\)\+𝔟\(z\)⊤∇logqr\(z\),−∇⋅\(𝔟qr\)\+qrHr\(⋅;𝔟\)=0\.H\_\{r\}\(z;\\mathfrak\{b\}\):=\\nabla\\cdot\\mathfrak\{b\}\(z\)\+\\mathfrak\{b\}\(z\)^\{\\top\}\\nabla\\log q\_\{r\}\(z\),\\qquad\-\\nabla\\cdot\(\\mathfrak\{b\}q\_\{r\}\)\+q\_\{r\}H\_\{r\}\(\\cdot;\\mathfrak\{b\}\)=0\.\(59\)Adding𝔟\\mathfrak\{b\}to the sampling drift andHrH\_\{r\}to its potential preserves the marginal path\. VCG selects a finite\-dimensional control by minimizing the residual\-potential variance\.

In our increasing reverse\-time convention,

−a𝝆⋆=−νA\+s​g−s​u𝝆\.\-a\_\{\\boldsymbol\{\\rho\}\}^\{\\star\}=\-\\nu^\{A\}\+sg\-su^\{\\boldsymbol\{\\rho\}\}\.Relative to𝝆=𝟎\\boldsymbol\{\\rho\}=\\boldsymbol\{0\}, the added sampling control is therefore𝔟𝝆=−s​u𝝆\\mathfrak\{b\}^\{\\boldsymbol\{\\rho\}\}=\-su^\{\\boldsymbol\{\\rho\}\}\. Its compensated potential, under the same unnormalized convention, is

Φ𝝆,r=A𝟎\(r,⋅\)−sr\[∇⋅ur𝝆\+\(ur𝝆\)⊤∇logqr\]=A𝝆\(r,⋅\)\.\\Phi\_\{\\boldsymbol\{\\rho\},r\}=A\_\{\\boldsymbol\{0\}\}\(r,\\cdot\)\-s\_\{r\}\\left\[\\nabla\\cdot u\_\{r\}^\{\\boldsymbol\{\\rho\}\}\+\(u\_\{r\}^\{\\boldsymbol\{\\rho\}\}\)^\{\\top\}\\nabla\\log q\_\{r\}\\right\]=A\_\{\\boldsymbol\{\\rho\}\}\(r,\\cdot\)\.\(60\)Thus the continuous control families agree after matching the target, fields, and time convention\. Fixingρ1\\rho\_\{1\}leaves an affine one\-dimensional family inρ2\\rho\_\{2\}\.

The population VCG criterion concernsVarqτ⁡\(A𝝆\)\\operatorname\{Var\}\_\{q\_\{\\tau\}\}\(A\_\{\\boldsymbol\{\\rho\}\}\)\. Our finite Gaussian log\-weight variance has this conditional\-mean component and the additional transition component in \([46](https://arxiv.org/html/2610.00395#A2.E46)\)\. The two population objectives can consequently have different minimizers\. The implemented rule instead maximizes predicted cumulative ESS using common\-noise candidate evaluations\.

The full reference displacement also retains selected finite\-step terms\. Writingz=h​sτ​ξz=\\sqrt\{hs\_\{\\tau\}\}\\xiandΔ​y0=z−h​a0⋆\\Delta y\_\{0\}=z\-ha\_\{0\}^\{\\star\}, for symmetricΓ𝝆\\Gamma\_\{\\boldsymbol\{\\rho\}\}we have

Δ​y0⊤​Γ𝝆​Δ​y0=z⊤​Γ𝝆​z−2​h​z⊤​Γ𝝆​a0⋆\+h2​\(a0⋆\)⊤​Γ𝝆​a0⋆\.\\Delta y\_\{0\}^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}\\Delta y\_\{0\}=z^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}z\-2hz^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}a\_\{0\}^\{\\star\}\+h^\{2\}\(a\_\{0\}^\{\\star\}\)^\{\\top\}\\Gamma\_\{\\boldsymbol\{\\rho\}\}a\_\{0\}^\{\\star\}\.\(61\)The first term estimates the trace contribution; the other terms areOp​\(h3/2\)O\_\{p\}\(h^\{3/2\}\)andO⁡\(h2\)O\(h^\{2\}\)\. They are available from the same fixed\-displacement JVP, but do not by themselves establish a higher\-order scheme because other terms of those orders are omitted\. The proved accuracy remainsOL2​\(h3/2\)O\_\{L^\{2\}\}\(h^\{3/2\}\)for the defined Gaussian log\-weight surrogate, with the same\-noise and stabilization distinctions stated above\.

## Appendix CClass\-Removal Experimental Details

This appendix documents the class\-removal experiments of §[5\.2\.1](https://arxiv.org/html/2610.00395#S5.SS2.SSS1)\. Source\-of\-truth reports with full provenance \(scripts, logs, per\-row CSVs\) are included in the supplementary materials asclass\_removal\_baseline\.mdandclass\_removal\_smc\.md\.

### C\.1Models, dataset, and metrics

##### Unconditional generator \(pAp^\{A\}\)\.

A standard DDPM UNet trained from scratch on CIFAR\-10 \(cifar10\_unconditional\_v2/best\_checkpoint\.pt, EMA state loaded\)\. Configuration:nf=128\\mathrm\{nf\}=128, channel multipliers\(1,2,2,2\)\(1,2,2,2\),22residual blocks per resolution, attention at×1616\\\!\\times\\\!16, GroupNorm and Swish,∼35\.7\\sim\\\!35\.7M parameters\. Sampling uses ancestral DDPM withT=1000T=1000steps and a linearβ\\betaschedule from10−410^\{\-4\}to2×10−22\{\\times\}10^\{\-2\}\. All methods \(baselines and ours\) share this backbone\.

##### Forget\-class generator \(pBp^\{B\}\)\.

For each forget classc∈\{0,1\}c\\in\\\{0,1\\\}, a UNet of identical architecture trained on the single\-class subset of CIFAR\-10 \(planes for c0, cars for c1\)\. Checkpoints come from the public release accompanying[Koulischer et al\. \[10\]](https://arxiv.org/html/2610.00395#bib.bib13); the network state is loaded as the rawnetdict\.

##### Evaluation\.

For class removal tasks, we used the same metrics as[Koulischer et al\. \[10\]](https://arxiv.org/html/2610.00395#bib.bib13)\. For each generated batch ofN=10,240N=10\{,\}240images we compute \(i\) Wrong%, the fraction classified as the forget class by theaaraki/vit\-base\-patch16\-224\-in21k\-finetuned\-cifar10ViT \(HuggingFace pipeline, one forward per image\); \(ii\) KL of the empirical nine\-keep\-class histogram to the uniform distributionpu=\(1/9,…,1/9\)p\_\{\\mathrm\{u\}\}=\(1/9,\\ldots,1/9\), defined as∑i∈keeppu​log⁡\(pu/qi\)\\sum\_\{i\\in\\text\{keep\}\}p\_\{\\mathrm\{u\}\}\\log\(p\_\{\\mathrm\{u\}\}/q\_\{i\}\)with natural log and no renormalization of the keep mass; and \(iii\) FID against the CIFAR\-10 train split filtered to drop the forget class \(≈45,000\\approx\\\!45\{,\}000reference images\), using InceptionV3 pool3 features\.

##### Pretrained\-model FID floor\.

Sampling each underlying UNet*unconditionally*\(no guidance\) atN=10,240N=10\{,\}240and computing FID against the natural reference set yields:

These FIDs use single\-class reference sets and are not directly comparable to the FID column reported in §[5\.2\.1](https://arxiv.org/html/2610.00395#S5.SS2.SSS1); they are reported as a sanity check on the underlying models\. The unconditional value5\.435\.43is, however, a lower bound for the FID of any class\-removal method, since at zero guidance every method reduces to unconditional sampling\. The asymmetric self\-FID of the two forget\-class models \(cars sharper than planes\) is consistent with the asymmetric difficulty observed in §[5\.2\.1](https://arxiv.org/html/2610.00395#S5.SS2.SSS1)between c0 and c1\.

### C\.2Hyperparameters and per\-method scale grids

All methods useN=10,240N=10\{,\}240samples per configuration and seed00\(per\-batch seeds are\{0,1,…\}\\\{0,1,\\ldots\\\}for resumable sampling\)\. Batch sizes are256256\(baselines\) and512512\(c0 SMC\) /10241024\(c1 SMC\)\.

##### Composition\.

εguided=εu−λ⁡\(εf−εu\)\\varepsilon\_\{\\text\{guided\}\}=\\varepsilon\_\{u\}\-\\lambda\(\\varepsilon\_\{f\}\-\\varepsilon\_\{u\}\)\. Scale gridλ∈\{0\.05,0\.10,0\.20,0\.30,0\.40,0\.50,0\.70,1\.00\}\\lambda\\in\\\{0\.05,0\.10,0\.20,0\.30,0\.40,0\.50,0\.70,1\.00\\\}on both classes\.

##### Safe Latent Diffusion\.

Safe Latent Diffusion\[[8](https://arxiv.org/html/2610.00395#bib.bib15)\]with the hyperparameter set used in the public DNG repository\[[10](https://arxiv.org/html/2610.00395#bib.bib13)\]: element\-wise gated correction with EMA momentum, threshold=0\.04=0\.04,ss=100\.0s\_\{s\}=100\.0, momentum decayβm=0\.1\\beta\_\{m\}=0\.1, momentum rescalesm=0\.2s\_\{m\}=0\.2\. Same scale grid as Composition\.

##### Dynamic Negative Guidance \(using defaults from[Koulischer et al\. \[10\]](https://arxiv.org/html/2610.00395#bib.bib13)\)\.

Per\-particle scaleλt=λ⋅pt/\(1−pt\)\\lambda\_\{t\}=\\lambda\\cdot p\_\{t\}/\(1\-p\_\{t\}\)with running posteriorptp\_\{t\}\. Hyperparameters: prior0\.010\.01, temperature0\.20\.2,pmin=10−6p\_\{\\min\}=10^\{\-6\},pmax=0\.8p\_\{\\max\}=0\.8, offset2×10−42\{\\times\}10^\{\-4\}\. Scale grids: eight points for class 0λ∈\{0\.10,0\.25,0\.50,1\.00,2\.00,5\.00,8\.00,12\.00\}\\lambda\\in\\\{0\.10,0\.25,0\.50,1\.00,2\.00,5\.00,8\.00,12\.00\\\}; eight points for class 1λ∈\{0\.10,0\.25,0\.50,1\.00,3\.50,5\.00,8\.00,12\.00\}\\lambda\\in\\\{0\.10,0\.25,0\.50,1\.00,3\.50,5\.00,8\.00,12\.00\\\}\.

##### Safe Denoiser\[[25](https://arxiv.org/html/2610.00395#bib.bib14)\]\.

Kernel\-fast repellency in pixel space with500500unsafe reference images \(selection seed4242\) andσ\\sigmaauto\-calibrated as the median pairwise reference distance in kernel space\. Time window\[0\.78,1\.00\]⋅T\[0\.78,1\.00\]\\cdot Tof the diffusion schedule\. Auto window\-quantile threshold calibration withq=0\.80q=0\.80, margin1\.61\.6,4040β\\betasample points,β\\beta\-quantile0\.250\.25\. Scale grids: sixteen points\{0\.05,0\.08,0\.125,0\.20,0\.33,0\.50,0\.75,1\.00,1\.50,2\.00,3\.00,4\.00,6\.00,8\.00,12\.00,16\.00\}\\\{0\.05,0\.08,0\.125,0\.20,0\.33,0\.50,0\.75,1\.00,1\.50,2\.00,3\.00,4\.00,6\.00,8\.00,12\.00,16\.00\\\}on c0 \(full operating curve\) and the upper eight\{2\.00,3\.00,4\.00,5\.00,6\.50,8\.00,9\.00,10\.00\}\\\{2\.00,3\.00,4\.00,5\.00,6\.50,8\.00,9\.00,10\.00\\\}on c1 \(where SD shows non\-trivial repellency\)\.

##### Proposed sampler\.

Sigmoid gateθ=σ⁡\(log⁡c\+βs​ℓ^\)\\theta=\\sigma\(\\log c\+\\beta\_\{s\}\\hat\{\\ell\}\)with bothc⁡\(t\)c\(t\)andβs​\(t\)\\beta\_\{s\}\(t\)on exp schedules\. Shared:c1=0\.1c\_\{1\}=0\.1,βs,1=0\.2\\beta\_\{s,1\}=0\.2;ℓ^\\hat\{\\ell\}\-clip2020,ℓ^\\hat\{\\ell\}\-temperatureτ=0\.5\\tau=0\.5,ℓ^\\hat\{\\ell\}\-offset2×10−42\{\\times\}10^\{\-4\}; ESS resampling threshold0\.50\.5;θ\\theta\-clamp\[0\.02,0\.98\]\[0\.02,0\.98\]\. Variance\-reduction \(RK\) inner loop withzz\-cap0\.250\.25,tmin=0\.10t\_\{\\min\}=0\.10, ramp0\.200\.20,1010gradient steps with backtracking\. Class\-specific schedule constants:c0=1c\_\{0\}=1,γbs=2\\gamma\_\{\\text\{bs\}\}=2,γc=1\\gamma\_\{c\}=1for c0;c0=3c\_\{0\}=3,γbs=5\\gamma\_\{\\text\{bs\}\}=5,γc=5\\gamma\_\{c\}=5for c1 \(a more aggressive ramp, selected from a pilotN=512N=512sweep to match DNG performance on the harder c1 task; see discussion in §[5\.2\.1](https://arxiv.org/html/2610.00395#S5.SS2.SSS1)\)\. Scale \(=βs,0=\\beta\_\{s,0\}\) grid: twelve points\{0\.5,0\.6,0\.8,1,1\.5,2,3,4,5,6,7,8\}\\\{0\.5,0\.6,0\.8,1,1\.5,2,3,4,5,6,7,8\\\}on c0; eight points\{1,1\.5,2,3,4,5,6,7\}\\\{1,1\.5,2,3,4,5,6,7\\\}on c1\.

### C\.3Compute and provenance

Generation was distributed across eight NVIDIA A6000\-class GPUs in waves; per\-configuration wall\-clock atN=10,240N=10\{,\}240is roughly280280–560560minutes for baselines and195195–285285minutes for the proposed sampler \(longer batches but only1010batches atbatch=1024\\text\{batch\}=1024for c1 vs2020batches atbatch=512\\text\{batch\}=512for c0\)\. Total compute for the class\-removal experiments is approximately∼850\\sim 850GPU\-hours\.

## Appendix DMulticlass\-Selection Experimental Details

### D\.1Models, dataset, and metrics

##### Five\-class generators \(pAp^\{A\}andpBp^\{B\}\)\.

Both are DDPM UNets of the same architecture as the unconditional generator of App\.[C](https://arxiv.org/html/2610.00395#A3), trained from scratch on disjoint five\-class subsets of CIFAR\-10 \(25,00025\{,\}000images each, EMA state loaded\)\. Forward task:pAp^\{A\}iscifar10\_classes\_01234\_v2/best\_checkpoint\.pt\(classes\{0,1,2,3,4\}\\\{0,1,2,3,4\\\}\) andpBp^\{B\}iscifar10\_classes\_34567\_v2/best\_checkpoint\.pt\(classes\{3,4,5,6,7\}\\\{3,4,5,6,7\\\}\); the two shared classes are\{3,4\}\\\{3,4\\\}\(cat, deer\) and the keep set isA∖B=\{0,1,2\}A\\setminus B=\\\{0,1,2\\\}\(airplane, automobile, bird\)\. Reverse task: roles are swapped and the keep set is\{5,6,7\}\\\{5,6,7\\\}\(dog, frog, horse\)\. Sampling protocol andβ\\beta\-schedule match App\.[C](https://arxiv.org/html/2610.00395#A3)\. The DNGuncond\{\}\_\{\\text\{uncond\}\}baseline additionally uses the unconditional ten\-class generator of App\.[C](https://arxiv.org/html/2610.00395#A3)as its base denoiser; the proposed sampler does not use the unconditional model\.

##### Evaluation\.

For each configuration we drawN=10,240N=10\{,\}240samples and report: \(i\) Wrong%, the fraction classified into the unwanted\-only classesB∖AB\\setminus Aby the same ViT classifier as App\.[C](https://arxiv.org/html/2610.00395#A3); \(ii\) FID against the CIFAR\-10 train split filtered to the keep classes \(15,00015\{,\}000reference images for each direction\); \(iii\) Recall\[[41](https://arxiv.org/html/2610.00395#bib.bib32)\]against the same keep\-class reference set, computed by the standardkk\-nearest\-neighbour manifold construction withk=3k=3on InceptionV3 pool3 features; and \(iv\) KL of the empirical\|𝒦\|=3\|\\mathcal\{K\}\|=3\-bin keep\-class histogram to uniform, defined as∑i∈𝒦pu​log⁡\(pu/qi\)\\sum\_\{i\\in\\mathcal\{K\}\}p\_\{\\mathrm\{u\}\}\\log\(p\_\{\\mathrm\{u\}\}/q\_\{i\}\)withpu=\(1/3,1/3,1/3\)p\_\{\\mathrm\{u\}\}=\(1/3,1/3,1/3\)\. The choice of Recall over KL as the main\-text plotting axis is discussed below\.

These FIDs are computed on five\-class reference sets and are not directly comparable to the FID column reported in §[5\.2\.2](https://arxiv.org/html/2610.00395#S5.SS2.SSS2), which uses three\-class keep references\. They serve as a floor on the FID achievable by any specificity method that retains the fullAA\-modes, since at zero guidance every method reduces to samplingpAp^\{A\}\. The mild asymmetry between forward and reverse \(6\.926\.92vs\.6\.766\.76\) is consistent with the asymmetric difficulty observed in §[5\.2\.2](https://arxiv.org/html/2610.00395#S5.SS2.SSS2)\.

##### Choice of metric: Recall over KL\.

Fig\.[6](https://arxiv.org/html/2610.00395#A4.F6)reports the Wrong %–KL Pareto for the forward direction for completeness; the method ordering is consistent with the Recall plot in the main text\. We prefer Recall over KL in multiclass selection for two reasons\. First, KL to uniform over\|𝒦\|=3\|\\mathcal\{K\}\|=3keep classes has only three histogram bins, so it measures class\-level balance but is blind to intra\-class diversity: a method can attainKL≈0\\mathrm\{KL\}\\approx 0while collapsing each kept class to a single stereotypical mode \(e\.g\. the full variety of airplane views to a single canonical pose\), and this mode shaving is invisible\. Recall instead uses15,00015\{,\}000reference points in20482048\-dimensional InceptionV3 feature space with the standardkk\-NN manifold construction, so each real image votes independently on whether its local neighbourhood is covered, and boundary\-adjacent modes lost under blanket guidance \(e\.g\. unusual horse poses, dark\-background frogs close in score space to the suppressed cat class\) show up directly as a Recall drop\. The nine\-bin histogram of class removal has enough resolution to compensate; at\|𝒦\|=3\|\\mathcal\{K\}\|=3it does not\. Second, theevaluate\.pyKL computation uses unnormalised keep\-class fractions \(it divides by total sample count rather than renormalising over keep classes\), so the raw KL inherits alog⁡\(1/\(1−W\)\)\\log\\\!\\bigl\(1/\(1\-W\)\\bigr\)artefact that is monotone in Wrong %; the number can be corrected post hoc, but Recall has no such ambiguity\.

![Refer to caption](https://arxiv.org/html/2610.00395v1/full_pareto_kl_012.png)Figure 6:Wrong %–KL Pareto for the forward multiclass\-selection task \(keep\{0,1,2\}\\\{0,1,2\\\}\)\. The method ordering is consistent with the Recall plot in Fig\.[3](https://arxiv.org/html/2610.00395#S5.F3)\. The reverse direction exhibits the same method ordering on FID and Recall but not on KL: with only three keep classes the histogram is too coarse to discriminate adaptive methods \(DNG, SMC\) from uniform\-strength baselines \(Composition, SLD\) on class balance alone\.

### D\.2Hyperparameters and per\-method scale grids

All methods useN=10,240N=10\{,\}240samples per configuration and seed00\(per\-batch incremental seeding\)\. Baselines run at batch256256; the proposed sampler runs at batch512512\(forward\) and10241024\(reverse\)\.

##### Composition\.

εguided=εA−λ⁡\(εB−εA\)\\varepsilon\_\{\\text\{guided\}\}=\\varepsilon\_\{A\}\-\\lambda\(\\varepsilon\_\{B\}\-\\varepsilon\_\{A\}\)withεA,εB\\varepsilon\_\{A\},\\varepsilon\_\{B\}the score networks ofpAp^\{A\}andpBp^\{B\}\. Scale grid \(both directions\):λ∈\{0\.1,0\.3,0\.5,0\.8,1\.2,1\.8\}\\lambda\\in\\\{0\.1,0\.3,0\.5,0\.8,1\.2,1\.8\\\}\.

##### Safe Latent Diffusion\.

SLD with the hyperparameter set used in the public DNG repository\[[10](https://arxiv.org/html/2610.00395#bib.bib13)\]: threshold=0\.04=0\.04,ss=100\.0s\_\{s\}=100\.0, momentum decayβm=0\.2\\beta\_\{m\}=0\.2, momentum rescalesm=0\.1s\_\{m\}=0\.1, no warmup\. Scale grid \(both directions\):\{0\.4,0\.6,0\.8,1\.0,1\.2,1\.6\}\\\{0\.4,0\.6,0\.8,1\.0,1\.2,1\.6\\\}\.

##### Dynamic Negative Guidance\.

DNG: per\-particle adaptive scaleλt=λ⋅pt/\(1−pt\)\\lambda\_\{t\}=\\lambda\\cdot p\_\{t\}/\(1\-p\_\{t\}\)with running posteriorptp\_\{t\}, using the unconditional generator as the base denoiser, as being faithful to the original paper setting\. Hyperparameters: prior0\.010\.01, temperature0\.20\.2,pmin=10−6p\_\{\\min\}=10^\{\-6\},pmax=0\.8p\_\{\\max\}=0\.8, offset2×10−42\{\\times\}10^\{\-4\}\.

##### Safe Denoiser\.

Kernel\-fast repellency in pixel space, base=pA=p^\{A\}\(the wanted five\-class model\)\. Reference set:2,5002\{,\}500unsafe images sampled from the unwanted\-class CIFAR\-10 train subset;σ=4\.00\\sigma=4\.00\(fixed, not auto\-calibrated\); time window\[0\.70,1\.00\]⋅T\[0\.70,1\.00\]\\cdot T; auto window\-quantile threshold calibration with margin2\.02\.0,β\\beta\-quantile0\.250\.25,5050sample points, calibration windowq=0\.80q=0\.80\. Scale grid \(forward,N=10,240N=10\{,\}240\):\{1\.5,2\.5,3\.5,4\.5,6\.0,8\.0,11\.0,15\.0\}\\\{1\.5,2\.5,3\.5,4\.5,6\.0,8\.0,11\.0,15\.0\\\}\. Safe Denoiser is not reported on the reverse task; see §[D\.3](https://arxiv.org/html/2610.00395#A4.SS3)for the rationale\.

##### Proposed sampler\.

Sigmoid gateθ=σ⁡\(log⁡c\+βs​ℓ^\)\\theta=\\sigma\(\\log c\+\\beta\_\{s\}\\hat\{\\ell\}\)with bothc⁡\(t\)c\(t\)andβs​\(t\)\\beta\_\{s\}\(t\)on exp schedules\. Shared:c0=1\.0c\_\{0\}=1\.0,c1=0\.1c\_\{1\}=0\.1,βs,1=0\.2\\beta\_\{s,1\}=0\.2,γbs=2\.0\\gamma\_\{\\text\{bs\}\}=2\.0,γc=1\.0\\gamma\_\{c\}=1\.0,ℓ^\\hat\{\\ell\}\-temperatureτ=0\.2\\tau=0\.2,θ\\theta\-clamp\[0\.02,0\.98\]\[0\.02,0\.98\],log⁡w\\log w\-clip8080, ESS resampling threshold0\.50\.5\. Variance\-reduction \(RK\) inner loop:zz\-cap0\.250\.25,tmin=0\.10t\_\{\\min\}=0\.10, ramp0\.200\.20,1010gradient steps with backtracking, learning rate0\.10\.1\.ρ1\\rho\_\{1\}fixed as 0\. Direction\-specific: forward usesℓ^\\hat\{\\ell\}\-clip2020andℓ^\\hat\{\\ell\}\-offset2×10−42\{\\times\}10^\{\-4\}; reverse usesℓ^\\hat\{\\ell\}\-clip1010andℓ^\\hat\{\\ell\}\-offset6×10−56\{\\times\}10^\{\-5\}\.

### D\.3Reverse Safe Denoiser exclusion

We do not report Safe Denoiser on the reverse task because every configuration we tried sits outside the plotted region of Fig\.[3](https://arxiv.org/html/2610.00395#S5.F3)\. Across anN=512N=512sweep overσ∈\{4,20,40,80,auto≈36\}\\sigma\\in\\\{4,20,40,80,\\text\{auto\}\\\!\\approx\\\!36\\\}, scalesλ∈\[0\.5,20\]\\lambda\\in\[0\.5,20\], and three choices of unsafe reference set \(full\{0,1,2,3,4\}\\\{0,1,2,3,4\\\}, overlap\-only\{3,4\}\\\{3,4\\\}, and far\-only\{0,1,2,8,9\}\\\{0,1,2,8,9\\\}\), Wrong%\\,\\%stayed in\[32,41\]%\[32,41\]\\,\\%and FID in\[60,121\]\[60,121\]; the next\-worst baseline, Composition atλ=1\.8\\lambda=1\.8, already reaches Wrong=2\.9%=2\.9\\,\\%at FID54\.6454\.64\. The failure is not a gate issue: the apply rate is saturated at1\.01\.0in every run and the auto\-calibratedβ\\beta\-threshold is essentially unchanged from the forward task\.

The cause is a pixel\-space discriminability issue specific to the reverse direction\. SD builds its repellent direction asx^0−u^​\(x^0\)\\widehat\{x\}\_\{0\}\-\\widehat\{u\}\(\\widehat\{x\}\_\{0\}\), whereu^​\(x^0\)\\widehat\{u\}\(\\widehat\{x\}\_\{0\}\)is a Nadaraya–Watson average of the unsafe references weighted by an RBF kernel of bandwidthσ\\sigmain pixel space\. For a sample on the reverse keep manifold\{5,6,7\}=\{dog,frog,horse\}\\\{5,6,7\\\}=\\\{\\text\{dog\},\\text\{frog\},\\text\{horse\}\\\}, the nearest unsafe references are the cat and deer exemplars \(the overlap classes\{3,4\}\\\{3,4\\\}\), not the vehicles or bird \(\{0,1,2\}\\\{0,1,2\\\}\), because all animal classes are visually similar at×3232\\\!\\times\\\!32resolution\. The kernel\-weighted unsafe centroidu^​\(x^0\)\\widehat\{u\}\(\\widehat\{x\}\_\{0\}\)therefore lies inside the keep manifold, and the residual repellent direction damages the keep classes rather than separating them from the unwanted modes\. Smallσ\\sigmamakes the cat/deer dominance more local but does not remove it; largeσ\\sigmacollapsesu^\\widehat\{u\}to a uniform reference mean, which is also non\-discriminative\. Restricting the unsafe set to the overlap classes\{3,4\}\\\{3,4\\\}only, or to the far\-only classes\{0,1,2,8,9\}\\\{0,1,2,8,9\\\}, does not break the Wrong floor; the overlap\-only run withσ\\sigmaauto and scale88collapsed to Wrong=56\.4%=56\.4\\,\\%at FID=121\.0=121\.0\. The floor itself is approximately the wanted\-only prior:pAp^\{A\}already places roughly30%30\\%of its mass on the overlap classes by construction, and SD cannot reduce that mass without distorting\{5,6,7\}\\\{5,6,7\\\}\.

The forward task does not exhibit this failure because the forward keep set\{0,1,2\}=\{plane,automobile,bird\}\\\{0,1,2\\\}=\\\{\\text\{plane\},\\text\{automobile\},\\text\{bird\}\\\}is pixel\-disjoint from the animal\-only unsafe set: the kernel cleanly separates the keep manifold from the unwanted modes, and SD\-forward traces a usable Wrong/FID curve\. The class\-removal task of §[5\.2\.1](https://arxiv.org/html/2610.00395#S5.SS2.SSS1)avoids the failure by construction, since the single forget class is disjoint from the nine keep classes\.

### D\.4Compute and provenance

Generation was distributed across eight NVIDIA A6000\-class GPUs in waves; per\-configuration wall\-clock atN=10,240N=10\{,\}240is roughly200200–360360minutes for Composition, SLD, and DNG,240240–360360minutes for SD \(the kernel evaluation against2,5002\{,\}500reference images adds a fixed per\-step cost\), and140140–200200minutes for the proposed sampler \(1010–2020batches at batch512512or10241024\)\. Total compute for the multiclass\-selection experiments is approximately345345GPU\-hours \(≈275\\approx 275baselines,≈70\\approx 70proposed sampler\)\. All configuration scripts, per\-configuration logs, and aggregation code are included in the supplementary materials\.

## Appendix EText\-to\-Image Experimental Details

This appendix describes the text\-to\-image experiments in §[5\.3](https://arxiv.org/html/2610.00395#S5.SS3)\. We use the same local SMC construction as in the main method, together with fixed numerical stabilizers for high\-dimensional latent diffusion\. Unless stated otherwise, one configuration, denoted byc​48c48, is used for all five prompts and both negative\-prompt types\. Thus, the same hyperparameters are used in all ten experimental settings\.

### E\.1Experimental Setting

##### Diffusion model\.

We use Stable Diffusion v1\.4\[[42](https://arxiv.org/html/2610.00395#bib.bib34)\]from the publicCompVis/stable\-diffusion\-v1\-4checkpoint\. Each experiment usesK=100K=100denoising steps andN=32N=32particles\.

### E\.2Hyperparameters

At stepkk, let

λk:=kK−1,λk∈\[0,1\],\\lambda\_\{k\}:=\\frac\{k\}\{K\-1\},\\qquad\\lambda\_\{k\}\\in\[0,1\],\(62\)denote the progress from the initial noise distribution to the generated image\. The intermediate target is

qk​\(x\)∝pkA​\(x\)​\(1−θk​\(x\)\)mk\.q\_\{k\}\(x\)\\propto p\_\{k\}^\{A\}\(x\)\\bigl\(1\-\\theta\_\{k\}\(x\)\\bigr\)^\{m\_\{k\}\}\.\(63\)The gate power increases linearly,

mk=26​λk,m\_\{k\}=26\\lambda\_\{k\},\(64\)so that the specificity constraint is introduced gradually and the terminal target usesmK−1=26m\_\{K\-1\}=26\.

We use

ηk=0\.75,τℓ=0\.75,\\eta\_\{k\}=0\.75,\\qquad\\tau\_\{\\ell\}=0\.75,\(65\)whereτℓ\\tau\_\{\\ell\}scales the estimated log\-density\-ratio increments used by the gate\. The gate offset is scheduled through

log⁡ck=log⁡c⋆−7​w​\(λk\)−0\.05​sin2⁡\(π​λk\),log⁡c⋆=61\.65\.\\log c\_\{k\}=\\log c\_\{\\star\}\-7w\(\\lambda\_\{k\}\)\-0\.05\\sin^\{2\}\(\\pi\\lambda\_\{k\}\),\\qquad\\log c\_\{\\star\}=61\.65\.\(66\)Here,w⁡\(λ\)w\(\\lambda\)increases smoothly from00to11over the first quarter of the trajectory, remains equal to11over the middle half, and returns smoothly to00over the final quarter\. We use cosine transitions at both boundaries\. Althoughc⋆c\_\{\\star\}is numerically large, the gate depends only onlog⁡ck\\log c\_\{k\}; its value compensates for the scale and offset of the estimated latent\-space log\-density ratio\.

The forward correction is

uk\(ρ1,ρ2\)​\(x\)=ρ1​dk​\(x\)\+ρ2,k​gk​\(x\),u\_\{k\}^\{\(\\rho\_\{1\},\\rho\_\{2\}\)\}\(x\)=\\rho\_\{1\}d\_\{k\}\(x\)\+\\rho\_\{2,k\}g\_\{k\}\(x\),\(67\)where

dk=∇log⁡pkA−∇log⁡pkB,gk=∇ℛk=ζk​dk,d\_\{k\}=\\nabla\\log p\_\{k\}^\{A\}\-\\nabla\\log p\_\{k\}^\{B\},\\qquad g\_\{k\}=\\nabla\\mathcal\{R\}\_\{k\}=\\zeta\_\{k\}d\_\{k\},\(68\)andℛk=mk​log⁡\(1−θk\)\\mathcal\{R\}\_\{k\}=m\_\{k\}\\log\(1\-\\theta\_\{k\}\)\. We fix

throughout sampling\. At each active optimization step,ρ2,k\\rho\_\{2,k\}is selected from101101uniformly spaced candidates in

ρ2,k∈\[0,1\.2\]\.\\rho\_\{2,k\}\\in\[0,1\.2\]\.\(70\)The valueρ2,k=0\\rho\_\{2,k\}=0is always included as a guard candidate\. Each step evaluates the full grid independently, without warm\-starting or fixed\-point refinement\. Optimization is disabled during the final denoising step\.

For every candidate, we add the stabilized increment to the cumulative log weight carried from the preceding step and compute the normalized ESS of the result\. We select the candidate with the largest predicted full cumulative ESS\. Thus, the optimization criterion is neither the incremental ESS alone nor an ESS computed from clipped ancestor\-selection weights\.

##### Mapping to Proposition[2](https://arxiv.org/html/2610.00395#Thmproposition2)\.

The latent\-space contrast estimator is rescaled byτℓ\\tau\_\{\\ell\}before entering the gate\. Equivalently, the implementation uses the effective gate slope

ηkeff:=τℓ​ηk\.\\eta\_\{k\}^\{\\mathrm\{eff\}\}:=\\tau\_\{\\ell\}\\eta\_\{k\}\.\(71\)Accordingly, the gate\-dependent coefficients in Proposition[2](https://arxiv.org/html/2610.00395#Thmproposition2)are evaluated as

ζk\\displaystyle\\zeta\_\{k\}=mk​ηkeff​θk,\\displaystyle=m\_\{k\}\\eta\_\{k\}^\{\\mathrm\{eff\}\}\\theta\_\{k\},\(72\)χk\\displaystyle\\chi\_\{k\}=mk​\(ηkeff\)2​θk​\(1−θk\)\.\\displaystyle=m\_\{k\}\(\\eta\_\{k\}^\{\\mathrm\{eff\}\}\)^\{2\}\\theta\_\{k\}\(1\-\\theta\_\{k\}\)\.

##### Time\-varying target power\.

Becausemkm\_\{k\}varies over the trajectory, the one\-step gate contribution includes the discrete power\-schedule term

\(mk\+1−mk\)​log⁡\(1−θk\)\.\(m\_\{k\+1\}\-m\_\{k\}\)\\log\(1\-\\theta\_\{k\}\)\.\(73\)We use this discrete correction once in the implemented Proposition[2](https://arxiv.org/html/2610.00395#Thmproposition2)increment; it is the finite\-step counterpart of them˙\\dot\{m\}term in Appendix[B\.1](https://arxiv.org/html/2610.00395#A2.SS1)\.

### E\.3Finite\-Step Stabilization and Resampling

The Proposition\-2 expansion contains several particle\-wise contributions that can be heavy\-tailed at finite step size\. With only eight particles, a single extreme value can dominate the ESS objective\. We therefore stabilize three groups of terms while using exactly the same stabilized expansion for coefficient selection and propagated weights\.

Letv=\(v1,…,vN\)v=\(v\_\{1\},\\ldots,v\_\{N\}\)be a particle\-wise contribution and define

v~:=mediani⁡\(vi\),δi:=\|vi−v~\|\.\\widetilde\{v\}:=\\operatorname\{median\}\_\{i\}\(v\_\{i\}\),\\qquad\\delta\_\{i\}:=\|v\_\{i\}\-\\widetilde\{v\}\|\.\(74\)Letδ\(r\+1\)\\delta\_\{\(r\+1\)\}be the\(r\+1\)\(r\+1\)\-st largest centered deviation\. For an optional absolute boundBB, define

Tv\(r,B\):=min⁡\{δ\(r\+1\),B\},T\_\{v\}^\{\(r,B\)\}:=\\min\\\{\\delta\_\{\(r\+1\)\},B\\\},\(75\)whereB=∞B=\\inftydenotes no absolute bound\. We use the median\-centered winsorization operator

𝒲r,B​\(vi\):=v~\+sign⁡\(vi−v~\)​min​\{\|vi−v~\|,Tv\(r,B\)\}\.\\mathcal\{W\}\_\{r,B\}\(v\_\{i\}\):=\\widetilde\{v\}\+\\operatorname\{sign\}\(v\_\{i\}\-\\widetilde\{v\}\)\\min\\left\\\{\|v\_\{i\}\-\\widetilde\{v\}\|,T\_\{v\}^\{\(r,B\)\}\\right\\\}\.\(76\)Ties at the threshold are retained\. ForN=32N=32, we user=1r=1, so the rank rule controls only the most extreme centered deviation\.

We decompose the raw expansion as

ℓ^k,iP2=Gk,i\+Hk,i0\+Jk,iraw,\\widehat\{\\ell\}\_\{k,i\}^\{\\mathrm\{P2\}\}=G\_\{k,i\}\+H\_\{k,i\}^\{0\}\+J\_\{k,i\}^\{\\mathrm\{raw\}\},\(77\)whereGk,iG\_\{k,i\}contains the gate\-time and target\-curvature terms,Jk,irawJ\_\{k,i\}^\{\\mathrm\{raw\}\}is the Jacobian–vector\-product contribution, andHk,i0H\_\{k,i\}^\{0\}contains the forward, reverse, and proposal\-curvature terms\. The gate group is stabilized as

Gk,istab=𝒲1,0\.10​\(Gk,i\)\.G\_\{k,i\}^\{\\mathrm\{stab\}\}=\\mathcal\{W\}\_\{1,0\.10\}\(G\_\{k,i\}\)\.\(78\)For the JVP contribution, we first apply the fixed shrinkage

Jk,ishrunk=0\.75​Jk,iraw,J\_\{k,i\}^\{\\mathrm\{shrunk\}\}=0\.75J\_\{k,i\}^\{\\mathrm\{raw\}\},\(79\)and then use

Jk,istab=𝒲1,0\.25​\(Jk,ishrunk\)\.J\_\{k,i\}^\{\\mathrm\{stab\}\}=\\mathcal\{W\}\_\{1,0\.25\}\\left\(J\_\{k,i\}^\{\\mathrm\{shrunk\}\}\\right\)\.\(80\)Finally, the combined kernel contribution is stabilized after inserting the JVP term,

Hk,istab=𝒲1,0\.10​\(Hk,i0\+Jk,istab\),H\_\{k,i\}^\{\\mathrm\{stab\}\}=\\mathcal\{W\}\_\{1,0\.10\}\\left\(H\_\{k,i\}^\{0\}\+J\_\{k,i\}^\{\\mathrm\{stab\}\}\\right\),\(81\)and the propagated increment is

ℓ^k,istab=Gk,istab\+Hk,istab\.\\widehat\{\\ell\}\_\{k,i\}^\{\\mathrm\{stab\}\}=G\_\{k,i\}^\{\\mathrm\{stab\}\}\+H\_\{k,i\}^\{\\mathrm\{stab\}\}\.\(82\)The same operations, in the same order, are applied to every candidate in theρ2,k\\rho\_\{2,k\}grid\.

##### Resampling\.

LetLk,iL\_\{k,i\}denote the full cumulative log weight after adding \([82](https://arxiv.org/html/2610.00395#A5.E82)\)\. Resampling is triggered when the normalized ESS computed directly fromLkL\_\{k\}falls below0\.300\.30\. We require at least1010denoising steps between successive events and permit resampling over the entire trajectory\. Hence, a100100\-step trajectory can contain at most ten resampling events\.

The resampling trigger always uses the unclipped full cumulative weights\. To avoid a single particle dominating ancestor selection, we draw ancestors using the median\-centered weights

L~k,i=medianj⁡\(Lk,j\)\+clip⁡\(Lk,i−medianj⁡\(Lk,j\),−1\.5,1\.5\)\.\\widetilde\{L\}\_\{k,i\}=\\operatorname\{median\}\_\{j\}\(L\_\{k,j\}\)\+\\operatorname\{clip\}\\left\(L\_\{k,i\}\-\\operatorname\{median\}\_\{j\}\(L\_\{k,j\}\),\-1\.5,1\.5\\right\)\.\(83\)The residualLk,i−L~k,iL\_\{k,i\}\-\\widetilde\{L\}\_\{k,i\}is transferred to every selected child and carried into subsequent steps\. Thus, clipping changes only the ancestor proposal probabilities; it does not discard the corresponding importance\-weight correction\.

These finite\-step operations leave the analytic target in \([63](https://arxiv.org/html/2610.00395#A5.E63)\) unchanged\. The propagated weight is the stabilized retained expansion in Proposition[2](https://arxiv.org/html/2610.00395#Thmproposition2), including its scheduled\-power correction\. It is not the exact finite\-step Gaussian transition\-density ratio\.

### E\.4Prompts

We use the five scenes from[Koulischer et al\. \[10\]](https://arxiv.org/html/2610.00395#bib.bib13)\. Each scene is paired with one semantically related and one unrelated negative prompt, resulting in ten experimental settings\.

Table 3:Text\-to\-image prompts\.
### E\.5Evaluation Metrics

For each method and condition, we generate3232images\. We compute CLIP similarity using theopenai/clip\-vit\-large\-patch14model\[[44](https://arxiv.org/html/2610.00395#bib.bib33)\]\. LetCLIP\+\\mathrm\{CLIP\}\_\{\+\}denote similarity to the positive prompt andCLIP−\\mathrm\{CLIP\}\_\{\-\}similarity to the negative prompt used during generation\. We report both quantities and the specificity margin

Margin=CLIP\+−CLIP−\.\\mathrm\{Margin\}=\\mathrm\{CLIP\}\_\{\+\}\-\\mathrm\{CLIP\}\_\{\-\}\.\(84\)
To measure particle diversity after resampling, letfjf\_\{j\}be the fraction of final particles descended from initial particlejj\. We report the number of unique surviving roots, the largest family fraction

Fmax=maxj⁡fj,F\_\{\\max\}=\\max\_\{j\}f\_\{j\},\(85\)and the effective number of roots

Reff=1∑jfj2\.R\_\{\\mathrm\{eff\}\}=\\frac\{1\}\{\\sum\_\{j\}f\_\{j\}^\{2\}\}\.\(86\)A largerReffR\_\{\\mathrm\{eff\}\}, a larger number of unique roots, and a smallerFmaxF\_\{\\max\}indicate less genealogical collapse\.

### E\.6Ablations

#### E\.6\.1JVP Contribution

We evaluate the JVP contribution on Prompt 1 with its related negative prompt\. On the same selected trajectory, we reconstruct the raw Proposition 2 increment before winsorization both with and without the JVP term and compare each surrogate with the exact finite one\-step Gaussian log\-weight\. All metrics are computed after centering across particles within each non\-degenerate step, since a common log\-weight offset does not affect the normalized particle weights\. The exact finite Gaussian quantity requires an additional selected\-endpoint denoiser evaluation and is used only as an evaluation diagnostic; both propagation and proposal optimization continue to use the stabilized Proposition 2 expansion\.

Table[4](https://arxiv.org/html/2610.00395#A5.T4)shows that including the JVP term increases the Pearson correlation from0\.6890\.689to0\.8040\.804, reduces centered MAE by22\.0%22\.0\\%, and reduces centered relative RMSE by13\.0%13\.0\\%\. It also reduces the mean absolute normalized\-ESS error by7\.4%7\.4\\%\. The no\-JVP result is obtained by removing the JVP term from the same raw expansion, so the comparison holds the states, noise realizations, and all remaining terms fixed\.

Table 4:Agreement of the raw Proposition 2 surrogate with the exact finite one\-step Gaussian log\-weight on seed10,00010\{,\}000\. Metrics are computed after particle\-wise centering within each non\-degenerate step\.
#### E\.6\.2Variance Proposal Optimization

We evaluate variance\-proposal optimization on Prompt 1 with its related negative prompt using five paired seeds\. Each run uses eight particles and100100diffusion steps\. We fixρ1=−2\\rho\_\{1\}=\-2and compare a proposal selected by maximizing the full cumulative ESS over101101values ofρ2∈\[0,1\.2\]\\rho\_\{2\}\\in\[0,1\.2\]with the fixed proposalρ2=0\\rho\_\{2\}=0\. The two conditions use identical random seeds and share all other hyperparameters, including the stabilized Proposition 2 weights, the ancestor\-weight cap, and the minimum separation of ten steps between resampling events\.

Table 5:Effect of full\-ESS optimization ofρ2\\rho\_\{2\}on CLIP specificity and particle genealogy\. Values are mean±\\pmstandard deviation over five paired seeds when 8 images generated\.As shown in Table[5](https://arxiv.org/html/2610.00395#A5.T5), optimizingρ2\\rho\_\{2\}increases the effective and unique numbers of roots by45\.0%45\.0\\%and38\.1%38\.1\\%, respectively\. It reduces the largest family fraction by29\.4%29\.4\\%and the number of resampling events by55\.6%55\.6\\%\. The mean full cumulative ESS also increases from0\.579±0\.0790\.579\\pm 0\.079to0\.685±0\.0240\.685\\pm 0\.024, with an improvement in all five paired seeds\. The mean CLIP margin increases by0\.00690\.0069, from0\.12940\.1294to0\.13630\.1363\. Its minimum, mean, median, and maximum are0\.1256/0\.1363/0\.1368/0\.14770\.1256/0\.1363/0\.1368/0\.1477, compared with0\.1135/0\.1294/0\.1307/0\.14560\.1135/0\.1294/0\.1307/0\.1456forρ2=0\\rho\_\{2\}=0\.

The margin improves in three of the five paired seeds, whereas the unique and effective root counts improve in four\. These results therefore support a positive average effect on both specificity and genealogy, together with a consistent improvement in full ESS, rather than uniform per\-seed dominance\.

#### E\.6\.3Compute

All experiments use NVIDIA A100 40GB GPUs\. At each active step, two forward\-mode JVPs are computed, one for each conditional branch\. Related and unrelated conditions are run concurrently on separate four\-GPU groups, while their particle weights, resampling operations, and genealogies remain independent\.

### E\.7Additional Visualizations

![Refer to caption](https://arxiv.org/html/2610.00395v1/prompt01_medieval_feast_related_curated_2x4.png)Figure 7:Qualitative samples from our method for the medieval\-feast prompt with the related negative prompt\.![Refer to caption](https://arxiv.org/html/2610.00395v1/prompt02_english_breakfast_related_curated_2x4.png)Figure 8:Qualitative samples from our method for the English\-breakfast prompt with the related negative prompt\.![Refer to caption](https://arxiv.org/html/2610.00395v1/prompt03_dinner_table_related_curated_2x4.png)Figure 9:Qualitative samples from our method for the dinner\-table prompt with the related negative prompt\.![Refer to caption](https://arxiv.org/html/2610.00395v1/prompt04_art_workshop_related_curated_2x4.png)Figure 10:Qualitative samples from our method for the art\-workshop prompt with the related negative prompt\.![Refer to caption](https://arxiv.org/html/2610.00395v1/prompt05_antique_store_related_curated_2x4.png)Figure 11:Qualitative samples from our method for the antique\-store prompt with the related negative prompt\.

## Appendix FSpecificity\-Aware pMHC Binder Design

### F\.1Task Setting

We study specificity\-aware protein\-binder design for two closely related peptide–MHC class I complexes\. The desired target is HLA\-A\*02:01–NLVPMVATV, whereas the undesired target is HLA\-A\*02:01–NLVFMVATV\. The two peptides differ only by a Pro\-to\-Phe substitution at peptide position 4\. Their normalized T\-cell activation values in the BATCAVE mutational\-scan dataset are1\.0001\.000and0\.0310\.031, respectively\. These measurements are used only to select a biologically relevant specificity pair and are not treated as affinity labels for the generated binders\.

Both targets contain the same HLA\-A\*02:01 heavy chain andβ2\\beta\_\{2\}\-microglobulin\. The generated object is an8080\-residue protein binder, denoted by chainXX\. The conditioning target contains the HLA heavy chain,β2\\beta\_\{2\}\-microglobulin, and the nine\-residue peptide; no T\-cell receptor is included\. Peptide residues 3–6, which include the mutated position, define the intended binding region, whileβ2\\beta\_\{2\}\-microglobulin is marked as non\-binding\.

### F\.2Construction of the Paired Targets

The desired and undesired pMHC structures are initially predicted with Boltz\-2\. The undesired complex is aligned to the desired complex using a Kabsch fit on the HLA platform, defined by the Cα\\alphaatoms of HLA residues 1–180\. We then construct an exact shared molecular context by copying the desired HLA andβ2\\beta\_\{2\}\-microglobulin coordinates into both conditions, while retaining the corresponding desired or mutated peptide\.

Consequently, the two inputs share the same presentation scaffold and differ only in the peptide sequence and peptide conformation\. During generation, a Kabsch fit on the same HLA platform removes the small numerical frame drift between the two conditional denoiser evaluations\. Across the reported Proposition 2 generation, the median fitted rotation over the final low\-noise quarter of sampling is0\.029∘0\.029^\{\\circ\}, and the maximum across backbone particles is0\.179∘0\.179^\{\\circ\}\. Thus, the contrastive field is not dominated by an arbitrary rigid\-body mismatch\.

### F\.3Binder\-Only Contrastive Guidance

At each reverse\-diffusion step, the same noisy molecular state is evaluated under the desired conditionAAand the undesired conditionBB\. LetvAv\_\{A\}andvBv\_\{B\}denote the corresponding denoising velocities\. After rotatingvBv\_\{B\}into the frame of conditionAA, the target binder update is

vguided,X=vA,X\+st​\(vA,X−vB,X​RB→A\),v\_\{\\mathrm\{guided\},X\}=v\_\{A,X\}\+s\_\{t\}\\left\(v\_\{A,X\}\-v\_\{B,X\}R\_\{B\\rightarrow A\}\\right\),\(87\)whereRB→AR\_\{B\\rightarrow A\}is the HLA\-derived frame correction andsts\_\{t\}is the likelihood\-ratio\-dependent guidance coefficient\. Only the generated binder is modified\. The HLA,β2\\beta\_\{2\}\-microglobulin, and peptide atoms retain the desired\-condition update\.

The two conditional evaluations use the same noisy coordinates, feature seed, noise realization, sampling step, and rigid\-body augmentation\. Therefore, the difference in \([87](https://arxiv.org/html/2610.00395#A6.E87)\) isolates the conditional contrast between the desired and undesired targets\. No unconditional model is evaluated; this is two\-condition contrastive guidance rather than conventional classifier\-free guidance\.

Letdtd\_\{t\}denote the one\-unit binder\-only contrastive field and letgt=st​dtg\_\{t\}=s\_\{t\}d\_\{t\}denote the target guided field\. The proposal family is

ut​\(ρ2\)=ρ1​dt\+ρ2​gt,qt​\(ρ2\)=ut​\(ρ2\)−gt\.u\_\{t\}\(\\rho\_\{2\}\)=\\rho\_\{1\}d\_\{t\}\+\\rho\_\{2\}g\_\{t\},\\qquad q\_\{t\}\(\\rho\_\{2\}\)=u\_\{t\}\(\\rho\_\{2\}\)\-g\_\{t\}\.\(88\)We fixρ1=−1\\rho\_\{1\}=\-1and chooseρ2\\rho\_\{2\}adaptively from2525uniformly spaced values in\[0,1\.2\]\[0,1\.2\]\. At each active step, the selected value maximizes the full cumulative ESS after adding the candidate Proposition 2 increment:

ρ2,t=arg⁡maxρ2∈ℛ​ESS⁡\(log⁡Wt−1\+Δ​log⁡w^tP2​\(ρ2\)\)\.\\rho\_\{2,t\}=\\arg\\max\_\{\\rho\_\{2\}\\in\\mathcal\{R\}\}\\operatorname\{ESS\}\\left\(\\log W\_\{t\-1\}\+\\widehat\{\\Delta\\log w\}^\{\\,\\mathrm\{P2\}\}\_\{t\}\(\\rho\_\{2\}\)\\right\)\.\(89\)The same retained Proposition 2 expansion is used for both selectingρ2\\rho\_\{2\}and propagating the particle weights\. It includes the forward\- and reverse\-α\\alphaterms, the target\- and proposal\-curvature terms, and the fixed\-displacement finite\-difference JVP term\. It is not replaced by an exact finite Gaussian transition ratio\.

### F\.4Compared Methods

We compare Ours with the following methods:

1. 1\.BoltzGen \(target A only\):the original BoltzGen sampler conditioned only on the desired pMHC, with the undesired target absent during generation and no negative guidance;
2. 2\.Fixed guidance:two\-condition contrastive guidance with coefficientλ=0\.25\\lambda=0\.25;
3. 3\.DNG:Dynamic Negative Guidance with base gain1818; and
4. 4\.Ours:likelihood\-ratio gating with the adaptive two\-field proposal and retained Proposition 2 weights\.

For Ours, letℓ^t\\widehat\{\\ell\}\_\{t\}denote the accumulated local reverse\-kernel log\-likelihood contrast\. We use

θt=σ⁡\(log⁡c\+η​ℓ^t\),st=m​η​τ​θt\.\\theta\_\{t\}=\\sigma\\left\(\\log c\+\\eta\\widehat\{\\ell\}\_\{t\}\\right\),\\qquad s\_\{t\}=m\\eta\\tau\\theta\_\{t\}\.\(90\)The reported configuration is

c=6,η=0\.005,τ=0\.75,m=2022\.22,ρ1=−1\.c=6,\\qquad\\eta=0\.005,\\qquad\\tau=0\.75,\\qquad m=2022\.22,\\qquad\\rho\_\{1\}=\-1\.\(91\)Sinceℓ^0=0\\widehat\{\\ell\}\_\{0\}=0, the initial gate and guidance coefficient are

θ0=c1\+c=67,s0=m​η​τ​c1\+c=6\.5\.\\theta\_\{0\}=\\frac\{c\}\{1\+c\}=\\frac\{6\}\{7\},\\qquad s\_\{0\}=m\\eta\\tau\\frac\{c\}\{1\+c\}=6\.5\.\(92\)We clipℓ^t\\widehat\{\\ell\}\_\{t\}to\[−100,100\]\[\-100,100\]\. The proposal variance scale is0\.100\.10\. The finite\-difference JVP uses step size0\.010\.01and shrinkage coefficient0\.750\.75\.

For numerical robustness, the gate, JVP, and remaining kernel contributions to the Proposition 2 log\-weight are median\-centered and winsorized separately, with maximum centered magnitudes0\.100\.10,0\.250\.25, and0\.100\.10, respectively\. No additional clip is applied to their summed increment\. Resampling is permitted when the full cumulative ESS fraction falls below0\.300\.30, subject to a ten\-step cooldown, at most ten resampling events, and no resampling in the final ten steps\. If resampling occurs, ancestor\-selection log\-weights are median\-centered and capped at1\.51\.5, and the residual weight correction is carried forward\. These operations robustify the Proposition 2 estimator without changing the underlying target expansion\. We do not apply a separate cap to the contrastive displacement\.

### F\.5Generation and Structural Verification

We use eight matched seed blocks\. Each method generates eight binder backbones per block with200200reverse\-diffusion steps and retains one inverse\-folded sequence per backbone\. The reported evaluation therefore contains

8​seed blocks×8​backbones per block=64​binders per method\.8\\ \\text\{seed blocks\}\\times 8\\ \\text\{backbones per block\}=64\\ \\text\{binders per method\}\.\(93\)For Ours, the eight backbones within each block form an interacting particle batch\. The other methods use the same seed blocks and output budget\.

Although the sampler permits SMC resampling, no resampling event occurs in any of the eight particle batches for Ours\. All eight initial roots survive within every block, and the ancestor\-weight clip is never activated\.

Each binder sequence is independently modeled with both the desired and undesired pMHC using Boltz\-2\. We generate three predictions per target condition using200200sampling steps and three recycling steps:

64​binders×2​conditions×3​predictions=384​verifier structures per method\.64\\ \\text\{binders\}\\times 2\\ \\text\{conditions\}\\times 3\\ \\text\{predictions\}=384\\ \\text\{verifier structures per method\}\.\(94\)
All four methods use the same eight seed blocks, binder count, inverse\-folding budget, target pair, verifier settings, and specificity metric\. Thus, differences in Table[6](https://arxiv.org/html/2610.00395#A6.T6)are not caused by unequal sequence or verification budgets\.

### F\.6Specificity Metric

Boltz\-2 is used as a structural verifier rather than as a calibrated protein–protein affinity predictor\. LetM\(i,c,k\)M^\{\(i,c,k\)\}denote the Boltz\-2pair\_chains\_iptmmatrix for binderii, conditioncc, and verifier predictionkk\. For each prediction, we symmetrize the binder–peptide interface score:

ai,c,k=12​\(MX,P\(i,c,k\)\+MP,X\(i,c,k\)\)\.a\_\{i,c,k\}=\\frac\{1\}\{2\}\\left\(M^\{\(i,c,k\)\}\_\{X,P\}\+M^\{\(i,c,k\)\}\_\{P,X\}\\right\)\.\(95\)The condition\-level score is the mean over theKKverifier predictions:

Ai,c=1K​∑k=1Kai,c,k\.A\_\{i,c\}=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}a\_\{i,c,k\}\.\(96\)The specificity margin of binderiiis

Δi=Ai,desired−Ai,undesired\.\\Delta\_\{i\}=A\_\{i,\\mathrm\{desired\}\}\-A\_\{i,\\mathrm\{undesired\}\}\.\(97\)A positive margin indicates greater predicted binder–peptide interface confidence for the desired pMHC\. We report the mean, median, and7575th percentile \(Q75\) of the margin, together with the fraction of binders for whichΔi\>0\\Delta\_\{i\}\>0\. No whole\-complex ipTM, contact term, HLA\-interface term, or manually weighted composite score enters this metric\.

### F\.7Results

Table 6:Specificity on the HLA\-A\*02:01–NLVPMVATVversus HLA\-A\*02:01–NLVFMVATVbinder\-design task\. Here,NNis the number of evaluated binders andKKis the number of Boltz\-2 predictions per binder and target condition\. All methods useK=3K=3; higher values are better\.Ours achieves the largest mean, median, and Q75 specificity margins\. Relative to target\-A\-only BoltzGen, the strongest baseline on these statistics, Ours improves the mean by3\.9%3\.9\\%, the median by14\.8%14\.8\\%, and Q75 by16\.6%16\.6\\%\. Fixed guidance produces the largest fraction of positive margins\. Thus, Ours improves the center and upper quartile of the specificity distribution, although it does not maximize the fraction of positive designs\.

##### Representative binder\.

The binder shown in Figure[5](https://arxiv.org/html/2610.00395#S5.F5)has the sequence

VTLTVEFTYD PSKVGAVIHS PLVLTEDGYK VKVTLTISED GETITVKVRL YDSDNRLISD YKAEVTVEGD TITVTVTFDP

where spaces are inserted only for readability\. Across the three verifier predictions per condition, its mean desired and undesired binder–peptide iPTM values are0\.86730\.8673and0\.51740\.5174, respectively, giving a specificity margin of0\.34990\.3499\. The binder contacts the desired peptide in two of the three predictions and the undesired peptide in none\. All six verifier structures are complete, and no binder–target heavy\-atom clashes below1\.51\.5Å are detected\.

##### Sensitivity\.

The reported configuration was selected in preliminary hyperparameter screens\. Table[6](https://arxiv.org/html/2610.00395#A6.T6)reports the complete matched6464\-binder evaluation rather than a smaller tuning subset\. Its mean, median, and Q75 margins are0\.18560\.1856,0\.18220\.1822, and0\.33370\.3337, respectively\.

##### Interpretation\.

These results show that likelihood\-ratio gating with an adaptive two\-field proposal and Proposition 2 weight propagation can shift the center of the generated\-binder distribution toward stronger predicted recognition of the desired peptide relative to a single\-residue off\-target\. The reported ipTM margin is a structural interface\-confidence surrogate and should not be interpreted as a calibrated binding affinity\. Experimental binding and off\-target measurements remain necessary to validate the generated specificity\.