Why DDIM Hallucinates More than DDPM: A Theoretical Analysis of Reverse Dynamics

# Why DDIM Hallucinates More than DDPM: A Theoretical Analysis of Reverse Dynamics
Source: [https://arxiv.org/html/2605.06831](https://arxiv.org/html/2605.06831)
Samanyu AroraAbhinav N\. HarishIshaan KharbandaHung Yun TsengGrigorios G\. Chrysos

###### Abstract

We theoretically study the hallucination phenomena in two canonical diffusion samplers: the stochastic Denoising Diffusion Probabilistic Model \(DDPM\) and the deterministic Denoising Diffusion Implicit Model \(DDIM\)\. We analyze the reverse ODE \(DDIM\) and SDE \(DDPM\) for a Gaussian mixture target, proving that after a critical timeτ\\tau, \(a\) DDIM can become stuck on the segment connecting the two nearest modes and \(b\) DDPMstochasticityhelps it become unstuck from this region, thus avoiding hallucination\. Our empirical validation verifies that DDPM has a significantly lower hallucination rate than DDIM when this region is entered\. Building on our observations, we exhibit how using additional stochastic steps can help DDIM avoid hallucinations and offer new insights on how to design improved samplers\.

Machine Learning, ICML

## 1Introduction

Despite significant progress, state\-of\-the\-art diffusion models still exhibit hallucinations which do not abide by the structural, semantic constraints present in their training distribution\. This includes distorted hands, implausible object boundaries, and temporal inconsistencies in video\. Concretely, hallucinations can be characterized by generations which arise when the reverse diffusion trajectory is guided toward regions outside the set of constraints implied by the target distribution\.

To address this issue, prior work has focused on a variety of different strategies\. In the domain of text generation,Lu et al\. \([2025](https://arxiv.org/html/2605.06831#bib.bib21)\)identify a local generation bias in image diffusion models by highlighting their inability to model global context in MNIST digits\.Aithal et al\. \([2024](https://arxiv.org/html/2605.06831#bib.bib1)\)characterize hallucinations as mode interpolation in Gaussian mixture models and propose a metric for detecting interpolations\.Triaridis et al\. \([2025](https://arxiv.org/html/2605.06831#bib.bib32)\)explore dynamic guidance as a means of mitigating hallucination\. Although these works provide strategies to detect hallucination, none of them explore*why*hallucinations arise by systematically studying the design and mechanics of the reverse diffusion process\.

We focus on the distinction between two popular samplers: the stochastic Denoising Diffusion Probabilistic Models \(DDPM\)\(Ho et al\.,[2020](https://arxiv.org/html/2605.06831#bib.bib16)\)and the deterministic Denoising Diffusion Implicit Models \(DDIM\)\(Song et al\.,[2021a](https://arxiv.org/html/2605.06831#bib.bib29)\)\. These samplers are trained identically; however, they differ in the sampling strategy they employ during the reverse process\. DDPM conducts an equal number of forward and reverse steps, matching the introduction of noise in the forward diffusion SDE\. DDIM, on the other hand, doesnotinclude this noise in the reverse process\. DDIM introduces step skipping through a non\-Markovian reverse process, offering faster inference\. We identify that removing the noise in the reverse process contributes significantly to hallucinations\. This observation has been made empirically by prior works in the context of counting hallucinations\(Fu et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib15)\)\. However, to the best of our knowledge, a solid theoretical characterization of this phenomenon remains absent\. We fill this gap and address hallucination as astructural artifactof the reverse process\. Concretely, we ask the following question:

Can we formally characterize why DDIM hallucinates more than DDPM?

To answer this question, we study the reverse Probability Flow \(PF\) ODE \(DDIM\) and the reverse Variance Preserving \(VP\) SDE \(DDPM\) when the data distribution is anNN\-mode Gaussian mixture111Under mild regularity conditions, the marginal densities are identical for DDIM and DDPM\.\. This setup lends itself to rigorous theoretical analysis\. Additionally, we study hallucinations formulated as mode interpolations: a phenomenon where a sampled diffusion trajectory ends up at the line connecting modes of the data distribution\(Aithal et al\.,[2024](https://arxiv.org/html/2605.06831#bib.bib1)\)\.

As demonstrated in[Figure1\(b\)](https://arxiv.org/html/2605.06831#S1.F1.sf2), our analysis reveals that DDIM/DDPM convergence occurs in 2 phases: \(i\) an early convergence phase dominated by attraction to the nearest line segment between two data modes and \(ii\) a late convergence phase governing motion along the line connecting these two modes\. We demonstrate that the final steps of a trajectory under the late convergence phase determine the outcome of sampling and are a crucial difference between these two samplers\. Specifically, in this final phase, the noise component of DDPM helps it escape regions on this line segment where DDIM gets stuck, thus avoiding hallucinations\. Our theory demonstrates that this is the foundational reason that DDPM has an edge over DDIM,regardless of the number of reverse steps employed\.

These findings rule out several previously held beliefs on DDIM hallucinations\. For example, although practitioners often skip DDIM steps and thus incur numerical error when sampling, we demonstrate that this does not explain the hallucination rate gap\. We find empirically that the DDIM and DDPM hallucination rate gap persists under the same discretization, and our theoretical results are derived under zero numerical error\. Additionally,Aithal et al\. \([2024](https://arxiv.org/html/2605.06831#bib.bib1)\)attribute mode interpolation to the smoothing of the score function due to approximation with a neural network\. Our results, however, are derived under theexact score function, i\.e, the expected score on using infinite samples from the distribution\. We find that, with the exact score, the mechanism causing mode interpolation can be cleanly characterized\.

In summary, our contributions are as follows:

1. 1\.We rigorously study the source of DDIM hallucinations as observed in anNN\-mode Gaussian mixture, demonstrating that after a critical timeτ\\tau, DDIM trajectories converge to the nearest line segment joining two modes and then can become stuck near the midpoint, thus hallucinating\. This is illustrated in[Figure1](https://arxiv.org/html/2605.06831#S1.F1)\.
2. 2\.We leverage this to provide a theoretical justification for why DDIM hallucinates more than DDPM: the noise of DDPM can help it become unstuck from this hallucination region around the midpoint\. This is illustrated in[Figure1\(c\)](https://arxiv.org/html/2605.06831#S1.F1.sf3)\.
3. 3\.Empirically, we invalidate that the DDIM hallucination rate gap can be explained by step skipping and demonstrate that adding a few DDPM steps after DDIM converges near the midpoint neighborhood can help the trajectory escape, lowering hallucination rate\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x1.png)![Refer to caption](https://arxiv.org/html/2605.06831v1/x2.png)

\(a\)
![Refer to caption](https://arxiv.org/html/2605.06831v1/x3.png)

\(b\)
![Refer to caption](https://arxiv.org/html/2605.06831v1/x4.png)

\(c\)

Figure 1:\(a\) In 100,000 generated samples for a 25\-mode Gaussian mixture target, despite using thesame pretrained model, DDPM \(left\) hallucinates significantly less than DDIM \(right\)\. \(b\) Towards thebeginningof the reverse process, the trajectory selects a line segment to converge to\. After that, the trajectoryconverges rapidlyto the nearest line segment: either thetrue modeor themidpoint neighborhood\. \(c\) Within the midpoint neighborhood, DDIM canget stuckand hallucinate, while DDPM canescapewith added noise and avoid hallucination\.
## 2Problem Setting

Notation: Bolded lowercase letters𝒙∈ℝϖ\\bm\{x\}\\in\\mathbb\{R\}^\{\\varpi\}denote vectors of dimensionϖ\\varpi\.\[N\]\[N\], for scalarN∈ℕN\\in\\mathbb\{N\}, denotes the integer subset\{1,2,…,N\}\\\{1,2,\.\.\.,N\\\}\. For a functionff, we use∇𝒙f\\nabla\_\{\\bm\{x\}\}fto denote its gradient iff:ℝϖ→ℝf:\\mathbb\{R\}^\{\\varpi\}\\to\\mathbb\{R\}and its Jacobian iff:ℝϖ→ℝof:\\mathbb\{R\}^\{\\varpi\}\\to\\mathbb\{R\}^\{o\}\. We denote byxt:\[0,T\]→ℝx\_\{t\}:\[0,T\]\\to\\mathbb\{R\}and𝒙t:\[0,T\]→ℝϖ\\bm\{x\}\_\{t\}:\[0,T\]\\to\\mathbb\{R\}^\{\\varpi\}time\-dependent scalar\-valued and vector\-valued functions, respectively, from scalar timeTTto0unless otherwise specified\. We usex˙t\\dot\{x\}\_\{t\}to denote the time derivative ofxtx\_\{t\}and similarly use𝒙˙t\\dot\{\\bm\{x\}\}\_\{t\}for𝒙t\\bm\{x\}\_\{t\}; for all other derivatives, we use the notationdd​x​f​\(x\)\\frac\{d\}\{dx\}f\(x\)\.𝒩​\(𝒙;𝝁,σ2​𝑰\)\\mathcal\{N\}\(\\bm\{x\};\\bm\{\\mu\},\\sigma^\{2\}\\bm\{I\}\)is used to denote the probability density function of aϖ\\varpi\-dimensional Gaussian centered at𝝁\\bm\{\\mu\}with varianceσ2​𝑰\\sigma^\{2\}\\bm\{I\}\. We write stochastic differential equations \(SDE\) asd​𝒙t=b​\(t,𝒙t\)​d​t\+σ​\(t,𝒙t\)​d​𝑾td\\bm\{x\}\_\{t\}=b\(t,\\bm\{x\}\_\{t\}\)dt\+\\sigma\(t,\\bm\{x\}\_\{t\}\)d\\bm\{W\}\_\{t\}where𝑾t∈ℝϖ×1\\bm\{W\}\_\{t\}\\in\\mathbb\{R\}^\{\{\\varpi\}\\times 1\}is the standard Brownian motion\. We provide a full list of notation in[AppendixA](https://arxiv.org/html/2605.06831#A1), and a symbol table in[AppendixJ](https://arxiv.org/html/2605.06831#A10)\.

Diffusion Models: The goal of diffusion models is to learn to transport a simple base distributionpbasep\_\{\\text\{base\}\}to a more complicated target distributionpdatap\_\{\\text\{data\}\}\. Throughout, we considerpbasep\_\{\\text\{base\}\}to be the standard Gaussian𝒩​\(0,𝑰\)\\mathcal\{N\}\(0,\\bm\{I\}\)\(Sohl\-Dickstein et al\.,[2015](https://arxiv.org/html/2605.06831#bib.bib27)\)\. We first define the DDPM Variance Preserving \(VP\) forward SDE\(Song et al\.,[2021b](https://arxiv.org/html/2605.06831#bib.bib30)\):

d​𝒙t=−12​βt​𝒙t​d​t\+βt​d​𝑾t,d\\bm\{x\}\_\{t\}\\;=\\;\-\\tfrac\{1\}\{2\}\\beta\_\{t\}\\,\\bm\{x\}\_\{t\}\\,dt\\;\+\\;\\sqrt\{\\beta\_\{t\}\}\\,d\\bm\{W\}\_\{t\},\(1\)
whereβt\\beta\_\{t\}is a noise schedule and𝑾t\\bm\{W\}\_\{t\}is the standard Brownian motion\. We then denoteα¯t:=exp⁡\(−∫0tβ​\(s\)​𝑑s\)\\bar\{\\alpha\}\_\{t\}:=\\exp\(\-\\int\_\{0\}^\{t\}\\beta\(s\)ds\)\. We consider the standard setting whereα¯t∈C2\\bar\{\\alpha\}\_\{t\}\\in C^\{2\},α¯0=1\\bar\{\\alpha\}\_\{0\}=1, andα¯T=γ\\bar\{\\alpha\}\_\{T\}=\\gammafor someγ≈0\\gamma\\approx 0\. By Anderson’s theorem\(Anderson,[1982](https://arxiv.org/html/2605.06831#bib.bib2)\),[Eq\.1](https://arxiv.org/html/2605.06831#S2.E1)can be reversed as:

d​𝒙t=−12​βt​\(𝒙t\+2​∇𝒙tlog⁡pt​\(𝒙t\)\)​d​t\+βt​d​𝑾¯t,d\\bm\{x\}\_\{t\}\\;=\\;\-\\frac\{1\}\{2\}\\beta\_\{t\}\\left\(\\bm\{x\}\_\{t\}\+2\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)\\right\)dt\\;\+\\;\\sqrt\{\\beta\_\{t\}\}\\,d\\bar\{\\bm\{W\}\}\_\{t\},\(2\)for reversed Brownian motion𝑾¯t\\bar\{\\bm\{W\}\}\_\{t\}\. Here, the marginalsptp\_\{t\}are given by convolving the target distributionpdatap\_\{\\text\{data\}\}with a standard Gaussian:pt​\(𝒙\)=∫pdata​\(𝒚\)​𝒩​\(𝒙;α¯t​𝒚,\(1−α¯t\)​𝑰\)​𝑑𝒚p\_\{t\}\(\\bm\{x\}\)=\\int p\_\{\\text\{data\}\}\(\\bm\{y\}\)\\mathcal\{N\}\(\\bm\{x\};\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\bm\{y\},\(1\-\\bar\{\\alpha\}\_\{t\}\)\\bm\{I\}\)d\\bm\{y\}\. Since access to the true score∇𝒙tlog⁡pt​\(𝒙t\)\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)is usually unavailable in practice, one instead learns the score fields𝜽​\(𝒙t\)s\_\{\\bm\{\\theta\}\}\(\\bm\{x\}\_\{t\}\)\. Then, discretizing[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2)withsθs\_\{\\theta\}, using Euler\-Maruyama’s method, provides a stochastic algorithm to sample from target distributionpdatap\_\{\\text\{data\}\}, beginning with a base instance𝒙T∼𝒩​\(0,𝑰\)\\bm\{x\}\_\{T\}\\sim\\mathcal\{N\}\(0,\\bm\{I\}\)\.

For DDIM, we define the \(reverse\-time\) probability flow ordinary differential equation \(PF ODE\):

𝒙t˙=−12​βt​\(𝒙t\+∇𝒙tlog⁡pt​\(𝒙t\)\),\\dot\{\\bm\{x\}\_\{t\}\}=\-\\frac\{1\}\{2\}\\beta\_\{t\}\(\\bm\{x\}\_\{t\}\+\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)\),\(3\)
where the marginalsptp\_\{t\}are the same as those in[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2)\. Notice that[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)is exactly[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2)without the noise term, up to a constant multiplication of the score\. Similarly, one estimates the score fields𝜽​\(𝒙t\)s\_\{\\bm\{\\theta\}\}\(\\bm\{x\}\_\{t\}\)and discretizes[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)with a first\-order exponential integrator\(Lu et al\.,[2022](https://arxiv.org/html/2605.06831#bib.bib20)\)to obtain adeterministicsampling method forpdatap\_\{\\text\{data\}\}\(Song et al\.,[2021a](https://arxiv.org/html/2605.06831#bib.bib29)\)\. In the main paper, we theoretically study the dynamics of[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)and[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2)assuming access to the exact score function\. We discuss this regularity condition in[AppendixI](https://arxiv.org/html/2605.06831#A9)\. We also provide an empirical study under the inexact score function in the main paper experiments \([Sec\.6](https://arxiv.org/html/2605.06831#S6)\) and theoretically characterize the inexact score function dynamics in[AppendixC](https://arxiv.org/html/2605.06831#A3)\.

Finally, we consider a Gaussian mixture target distribution ofNNcomponents,pdata​\(𝒙\):=∑i=1Nπi​𝒩​\(𝒙;𝝁\(i\),σ2​𝑰\)p\_\{\\text\{data\}\}\(\\bm\{x\}\):=\\sum\_\{i=1\}^\{N\}\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{x\};\\bm\{\\mu\}^\{\(i\)\},\\sigma^\{2\}\\bm\{I\}\)\. Applying linearity of convolution yields thatpt​\(𝒙\)=∑i=1Nπi​𝒩​\(𝒙;𝝁t\(i\),σt2​𝑰\)p\_\{t\}\(\\bm\{x\}\)=\\sum\_\{i=1\}^\{N\}\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{x\};\\bm\{\\mu\}\_\{t\}^\{\(i\)\},\\sigma\_\{t\}^\{2\}\\bm\{I\}\), where𝝁t\(i\):=α¯​𝝁\(i\)\\bm\{\\mu\}\_\{t\}^\{\(i\)\}:=\\sqrt\{\\bar\{\\alpha\}\}\\bm\{\\mu\}^\{\(i\)\}andσt2=σ2​α¯t\+\(1−α¯t\)\\sigma\_\{t\}^\{2\}=\\sigma^\{2\}\\bar\{\\alpha\}\_\{t\}\+\(1\-\\bar\{\\alpha\}\_\{t\}\)\. We then denoteσ~t2:=σt2α¯t\\tilde\{\\sigma\}\_\{t\}^\{2\}:=\\frac\{\\sigma\_\{t\}^\{2\}\}\{\\bar\{\\alpha\}\_\{t\}\}as the effective variance; note that this starts large at the beginning of the reverse process, but becomesσ2\\sigma^\{2\}at the end, i\.e, the variance of an individual Gaussian mode\. In the main paper, we consider the equal weights case whereπi=πj​∀i,j∈\[N\]\\pi\_\{i\}=\\pi\_\{j\}\\;\\forall i,j\\in\[N\]; we extend this to the unequal weights case in[AppendixB](https://arxiv.org/html/2605.06831#A2)of the appendix\.

## 3Related Work

Diffusion Models: Diffusion models generate samples by reversing a noise corruption process, with DDPM\(Ho et al\.,[2020](https://arxiv.org/html/2605.06831#bib.bib16)\)and its deterministic fewer\-step counterpart DDIM\(Song et al\.,[2021a](https://arxiv.org/html/2605.06831#bib.bib29)\)\. Score\-based generative modeling unifies DDPM\-style reverse SDEs and DDIM\-style PF ODEs\(Song et al\.,[2021b](https://arxiv.org/html/2605.06831#bib.bib30)\); recent theory studies the gap between ODE\- and SDE\- sampling distributions\(Deveney et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib12)\), convergence guarantees for DDIM vs\. DDPM\(Beyler & Bach,[2025](https://arxiv.org/html/2605.06831#bib.bib4)\), and statistical guarantees on the PF ODE\(Chen et al\.,[2023](https://arxiv.org/html/2605.06831#bib.bib11); Cai & Li,[2025](https://arxiv.org/html/2605.06831#bib.bib8)\)\. Recent work also explores differences between the ODE and SDE in terms of performance\(Cao et al\.,[2023](https://arxiv.org/html/2605.06831#bib.bib9)\)and proposes hybrid samplers\(Xu et al\.,[2023](https://arxiv.org/html/2605.06831#bib.bib34)\)\. However, these works do not explicitly study hallucinations\. Additionally, instead of studying how ODE/SDE trajectories evolve, they compare statistical distances \(e\.g\., Wasserstein distance\(Deveney et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib12)\)\)\. These analyses do not provide insight intowhereandwhenhallucinations arise\. To our knowledge, our work is the first to mathematically study the differences in hallucination mechanisms between DDIM and DDPM from the lens of the reverse ODE and SDE\.

In our work, we also isolate key times in the reverse process trajectories for DDIM and DDPM\. Several recent works isolate key times in the reverse trajectory as well: semantic features emerge in narrow time windows\(Li & Chen,[2024](https://arxiv.org/html/2605.06831#bib.bib18)\)and, given the memorizing score function, there are distinct regimes of feature emergence and memorization\(Biroli et al\.,[2024](https://arxiv.org/html/2605.06831#bib.bib5)\)\. Notably, these times and reverse process regimes are obtained under different regularity conditions and differ from ours\.Bonnaire et al\. \([2025](https://arxiv.org/html/2605.06831#bib.bib6)\)study critical timesteps during diffusiontraining\(e\.g\., the relevant steps of SGD\), but this differs from our study of relevant steps in the reverse process\.

Aside from quantifying key timesteps in the reverse process,NN\-mode Gaussian mixtures remain a central controlled testbed for understanding diffusion sampler behavior, with recent theory directly analyzing diffusion guidance\(Wu et al\.,[2024](https://arxiv.org/html/2605.06831#bib.bib33)\), smoothness\(Liang et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib19)\), convergence\(Li et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib17)\), memorization\(Buchanan et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib7)\), and generalization\(Zhang et al\.,[2026](https://arxiv.org/html/2605.06831#bib.bib36)\)properties inNN\-Gaussian\-mixture regimes\. Our work adds to this thread of research, filling a gap in the mathematical understanding of hallucinations across different types of diffusion inference processes\.

Hallucinations in Diffusion Models: A growing line of work defines diffusion hallucinations as “mode interpolations”, or sampled instances that are interpolations between modes of the target distribution\(Aithal et al\.,[2024](https://arxiv.org/html/2605.06831#bib.bib1)\)\. For example, in our Gaussian mixture setting, modes are the means of the Gaussian components and mode interpolations are instances which lie on the line segment joining any two modes\. This is distinct from“mode collapse”\(Zhang et al\.,[2018](https://arxiv.org/html/2605.06831#bib.bib35); Thanh\-Tung & Tran,[2020](https://arxiv.org/html/2605.06831#bib.bib31)\), which describes a loss of diversity due to collapsing onto a subset of the true modes\. In line with these recent works, we use mode interpolation as our working definition of hallucination, finding both empirically and theoretically that DDIM interpolates more often than DDPM\. In this work, we assume awell\-separated Gaussian mixture\(Shah et al\.,[2023](https://arxiv.org/html/2605.06831#bib.bib25)\)\. This assumption is natural for any rigorous analysis of mode interpolation hallucinations; if regions of high probability mass around the modes overlap, then there are no mode interpolations since all samples are in these valid regions\. Furthermore, whileAithal et al\. \([2024](https://arxiv.org/html/2605.06831#bib.bib1)\)attribute mode interpolation solely to the smoothing of the learned diffusion score field, we find that the mechanism underlying mode interpolation can arise even with the exact, non\-smoothed score field\.

Furthermore, many papers study detection and mitigation of hallucinations\(Betti et al\.,[2026](https://arxiv.org/html/2605.06831#bib.bib3); Chandran\.C et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib10); Triaridis et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib32); Somepalli et al\.,[2023](https://arxiv.org/html/2605.06831#bib.bib28)\); in contrast, we study how hallucinations arise specifically due to the choice of inference procedure\. This emphasis is further supported by concurrent work showing higher counting hallucination rates for DDIM than DDPM\(Fu et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib15)\), distributional differences in ODE\- and SDE\-based samplers\(Deveney et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib12); Beyler & Bach,[2025](https://arxiv.org/html/2605.06831#bib.bib4)\), and empirical evidence that stochastic samplers outperform deterministic ones in scientific generation tasks\(Shehata et al\.,[2025](https://arxiv.org/html/2605.06831#bib.bib26)\)\. Additionally,Triaridis et al\. \([2025](https://arxiv.org/html/2605.06831#bib.bib32)\)highlight the need to understand the difference in hallucination mechanisms between DDIM and DDPM for future mitigation methods, rather than treating mitigation purely as a post\-hoc problem\.

## 4DDIM Hallucinations

In what follows, we aim to study how DDIM hallucinates\. To do so, we first demonstrate that after the variance ofptp\_\{t\}becomes sufficiently small, DDIM in[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)converges to theε\\varepsilon\-tube around the nearesti,ji,j\-mode segment, which is the line segment joining the two nearest modes𝝁t\(i\)\\bm\{\\mu\}\_\{t\}^\{\(i\)\}and𝝁t\(j\)\\bm\{\\mu\}\_\{t\}^\{\(j\)\}illustrated in[Figure2](https://arxiv.org/html/2605.06831#S4.F2)\. Then, the rest of the DDIM dynamics remain within this tube, parallel to the nearesti,ji,j\-mode segment\. Thus, if DDIM does not converge directly to one of the modes, then mode interpolation can happen on thei,ji,j\-mode segment\. Next, we demonstrate that there exists a radius around the midpoint of the nearesti,ji,j\-mode segment such that DDIM cannot escape\. In[Sec\.5](https://arxiv.org/html/2605.06831#S5), we move to studying DDPM, demonstrating that the noise term in[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2)leads to DDPM escaping the midpoint neighborhood, thus avoiding hallucination\.

To begin, we prove that DDIM with[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)converges to the tube around the nearesti,ji,j\-mode segment after sufficient time\. Firstly, we define anε\\varepsilon\-extendeda,ba,b\-mode segment fora,b∈\[N\]a,b\\in\[N\]asLt,ε\(a,b\):=\{𝝁t\(b\)\+v​\(𝝁t\(a\)−𝝁t\(b\)\):v∈\[−ε,1\+ε\]\}L\_\{t,\\varepsilon\}^\{\(a,b\)\}:=\\\{\\bm\{\\mu\}\_\{t\}^\{\(b\)\}\+v\(\\bm\{\\mu\}\_\{t\}^\{\(a\)\}\-\\bm\{\\mu\}\_\{t\}^\{\(b\)\}\):v\\in\[\-\\varepsilon,1\+\\varepsilon\]\\\}\. Whenε=0\\varepsilon=0, this refers to just the line segment joining the two modes; we denote this byLt\(a,b\)L\_\{t\}^\{\(a,b\)\}\. Next, we define the orthogonal distance of a trajectory to anε\\varepsilon\-extendeda,ba,b\-mode segment asd⟂​\(𝒙t,Lt,ε\(a,b\)\):=inf𝒚∈Lt,ε\(a,b\)‖𝒚−𝒙t‖2d\_\{\\perp\}\(\\bm\{x\}\_\{t\},L\_\{t,\\varepsilon\}^\{\(a,b\)\}\):=\\inf\_\{\\bm\{y\}\\in L\_\{t,\\varepsilon\}^\{\(a,b\)\}\}\|\|\\bm\{y\}\-\\bm\{x\}\_\{t\}\|\|\_\{2\}and aζ\\zeta\-tube aroundLt\(a,b\)L\_\{t\}^\{\(a,b\)\}asTubet,ζ\(a,b\):=\{𝒚∈ℝϖ:∃𝒌∈Lt\(a,b\)​s\.t\.​‖𝒚−𝒌‖2≤ζ\}\.\\mathrm\{Tube\}\_\{t,\\zeta\}^\{\(a,b\)\}:=\\\{\\bm\{y\}\\in\\mathbb\{R\}^\{\\varpi\}:\\exists\\bm\{k\}\\in L\_\{t\}^\{\(a,b\)\}\\;\\text\{s\.t\.\}\\;\|\|\\bm\{y\}\-\\bm\{k\}\|\|\_\{2\}\\leq\\zeta\\\}\.We similarly define theε\\varepsilon\-ball around a mode\. These key objects are illustrated in[Figure2](https://arxiv.org/html/2605.06831#S4.F2)\. We then state our primary assumption\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x5.png)Figure 2:\(1\)In black, we have the line segmentLt\(i,j\)L\_\{t\}^\{\(i,j\)\}joining two modes\.\(2\)Together with the red portion, this formsLt,ε\(i,j\)L\_\{t,\\text\{\{\\color\[rgb\]\{0\.72,0,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.72,0,0\}\\pgfsys@color@cmyk@stroke\{0\}\{0\.89\}\{0\.94\}\{0\.28\}\\pgfsys@color@cmyk@fill\{0\}\{0\.89\}\{0\.94\}\{0\.28\}$\\varepsilon$\}\}\}^\{\(i,j\)\}\.\(3\)We then have theε\\varepsilon\-ball surrounding modesiiandjj\.\(4\)Next, we haveTube\(i,j\)t,ε\{\}\_\{t,\\text\{\{\\color\[rgb\]\{0,0\.88,0\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.88,0\}\\pgfsys@color@cmyk@stroke\{0\.91\}\{0\}\{0\.88\}\{0\.12\}\\pgfsys@color@cmyk@fill\{0\.91\}\{0\}\{0\.88\}\{0\.12\}$\\varepsilon$\}\}\}^\{\(i,j\)\}\.\(5\)We also illustrate the midpoint of the line segment𝒚t∗\\bm\{y\}\_\{t\}^\{\*\}\(where𝒘t=0\\bm\{w\}\_\{t\}=0\), discussed in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. This provides a high\-level description of key objects used throughout[Sec\.4](https://arxiv.org/html/2605.06831#S4), and is not intended to describe the boundary between true and interpolated samples\.###### Assumption 4\.1\(Two\-Mode Dominance\)\.

Consider a reverse diffusion process governed by[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)with Gaussian mixture target distributionpdatap\_\{\\text\{data\}\}\. Assume that there exists a timeτ1∈\(0,T\)\\tau\_\{1\}\\in\(0,T\)such that fort≤τ1t\\leq\\tau\_\{1\}there existsi,j∈\[N\]i,j\\in\[N\],i≠ji\\neq j, where, for anyk∉\{i,j\}k\\notin\\\{i,j\\\}:

‖𝒙t−𝝁t\(k\)‖22≥min⁡\(‖𝒙t−𝝁t\(i\)‖22,‖𝒙t−𝝁𝒕\(j\)‖22\)\+Δt,\|\|\\bm\{x\}\_\{t\}\-\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\|\|\_\{2\}^\{2\}\\geq\\min\\left\(\|\|\\bm\{x\}\_\{t\}\-\\bm\{\\mu\}\_\{t\}^\{\(i\)\}\|\|\_\{2\}^\{2\},\|\|\\bm\{x\}\_\{t\}\-\\bm\{\\mu\_\{t\}\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\\right\)\+\\Delta\_\{t\},
for some marginΔt≥2​σt2​κ\\Delta\_\{t\}\\geq 2\\sigma\_\{t\}^\{2\}\\kappawhereκ\>1\\kappa\>1\.

Remark: In the reverse process, the variance ofptp\_\{t\}begins large but decays toσ2\\sigma^\{2\}byt=0t=0\. Thus, we assume a critical time such that theσt\\sigma\_\{t\}is sufficiently small with respect to the distances from the modes\. Note that in the case that the trajectory is very close to one particular mode,[Sec\.4](https://arxiv.org/html/2605.06831#S4)can hold for several pairs of modes, for example modesi,ji,jand modesi,ki,k\. In this case, as we demonstrate later, the instance will simply converge to modeii, thus converging to the line segment betweeniiand any associated mode, includingi,ji,j\. Otherwise,[Sec\.4](https://arxiv.org/html/2605.06831#S4)says that, after sufficient time in the reverse process, the trajectory is nearer to a pair of modes than any other modes\. Next, the bound on the margin is saying that this distance is sufficiently large with respect to the variance of the mixture\. This well\-separation assumption is critical for any analysis of mode interpolations, ensuring that the restN−2N\-2modes do not have regions of high probability mass which overlap with modesiiandjj\. If this assumption does not hold, a mode interpolation between modesiiandjjmay not be a hallucination and may instead be a high probability instance underpdatap\_\{\\mathrm\{data\}\}\. Such well\-separation conditions are common in diffusion analyses\(Shah et al\.,[2023](https://arxiv.org/html/2605.06831#bib.bib25)\)\. We empirically verify this assumption in[FigureE\.6](https://arxiv.org/html/2605.06831#A5.F6), finding that it holds for largeκ\>\>1\\kappa\>\>1\.

Given this assumption, we prove our first core result, demonstrating that DDIM converges fast to theε\\varepsilon\-tube around thei,ji,j\-mode segment aftert<τ1t<\\tau\_\{1\}in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\.

###### Theorem 4\.2\(Convergence toTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}\)\.

Suppose[Sec\.4](https://arxiv.org/html/2605.06831#S4)holds\. Then, there exists a timeτ≤τ1\\tau\\leq\\tau\_\{1\}such that for allt∈\[0,τ\)t\\in\[0,\\tau\), we have exponential decay of the orthogonal distance to theε\\varepsilon\-tube of the nearesti,ji,j\-mode segment, i\.e\.:

d⟂​\(𝒙t,Lt,ε\(i,j\)\)≤𝒪​\(d⟂​\(𝒙T,LT,ε\(i,j\)\)​exp⁡\(−u​\(t\)\)\+ε\),\\displaystyle\\text\{d\}\_\{\\perp\}\(\\bm\{x\}\_\{t\},L\_\{t,\\varepsilon\}^\{\(i,j\)\}\)\\leq\\mathcal\{O\}\\left\(\\text\{d\}\_\{\\perp\}\(\\bm\{x\}\_\{T\},L\_\{T,\\varepsilon\}^\{\(i,j\)\}\)\\exp\(\-u\(t\)\)\+\\varepsilon\\right\),
whereε∈𝒪​\(N​exp⁡\(−κ\)\)\\varepsilon\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)andu​\(t\):=∫tTβs2​σs2​𝑑su\(t\):=\\int\_\{t\}^\{T\}\\frac\{\\beta\_\{s\}\}\{2\\sigma\_\{s\}^\{2\}\}\\;ds, which is monotonically increasing ast→0t\\to 0\. This yields that, for anyϕ∈\(0,1\)\\phi\\in\(0,1\), with probability at least1−2​exp⁡\(−\(ϖ\)​ϕ2/8\)1\-2\\exp\(\-\(\\varpi\)\\phi^\{2\}/8\), we have that:

d⟂​\(𝒙t,Lt,ε\(i,j\)\)≤𝒪​\(\(ϖ\)​\(1\+ϕ\)​exp⁡\(−u​\(t\)\)\+ε\)\.\\displaystyle\\text\{d\}\_\{\\perp\}\(\\bm\{x\}\_\{t\},L\_\{t,\\varepsilon\}^\{\(i,j\)\}\)\\leq\\mathcal\{O\}\\left\(\\sqrt\{\(\\varpi\)\(1\+\\phi\)\}\\exp\(\-u\(t\)\)\+\\varepsilon\\right\)\.

Proof Sketch: Rescale[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)and then compute the dynamics around the nearesti,ji,j\-mode segment\. Then, decouple the dynamics into parallel and orthogonal components\. Finally, solve a first\-order linear ODE for the orthogonal component and show that the parallel component remains restricted to only theε\\varepsilon\-extended nearesti,ji,j\-mode segment\. A full proof is provided in[Sec\.H\.1](https://arxiv.org/html/2605.06831#A8.SS1)\.

Note that the rate of convergence toTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}is damped by the initial distance from the segment, which scales withϖ\\sqrt\{\\varpi\}with high probability\. Next, note that afterτ\\tau, if the trajectory entersTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}, the trajectory stays there\.

###### Corollary 4\.3\(Trapping inTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}\)\.

For anyt∈\[0,τ\]t\\in\[0,\\tau\], whereτ\\tauis defined in[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2), if𝐱t\\bm\{x\}\_\{t\}governed by the dynamics of[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)entersTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}, whereε∈𝒪​\(N​exp⁡\(−κ\)\)\\varepsilon\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\), it stays there\.

As such, once the instance is inTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}, the only nontrivial component of the dynamics are the parallel dynamics to the nearesti,ji,j\-mode segmentLt,ε\(i,j\)L\_\{t,\\varepsilon\}^\{\(i,j\)\}withinTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}\. We thus study these parallel dynamics; however, before doing so, we introduce a new assumption\.

###### Assumption 4\.4\(Modes Sufficiently Far Apart\)\.

Consider a reverse diffusion process governed by[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)with Gaussian mixture target distributionpdatap\_\{\\text\{data\}\}\. Then, consider timeτ2∈\(0,T\)\\tau\_\{2\}\\in\(0,T\), such that fort≤τ2t\\leq\\tau\_\{2\}:

ℓ2:=‖𝝁\(i\)−𝝁\(j\)‖22≥4​κ​σ~t2​ϖ,\\ell^\{2\}:=\|\|\\bm\{\\mu\}^\{\(i\)\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\\geq 4\\kappa\\tilde\{\\sigma\}\_\{t\}^\{2\}\\varpi,
whereκ\>1\\kappa\>1from[Sec\.4](https://arxiv.org/html/2605.06831#S4)andσ~t2=σt2α¯t\\tilde\{\\sigma\}\_\{t\}^\{2\}=\\frac\{\\sigma\_\{t\}^\{2\}\}\{\\bar\{\\alpha\}\_\{t\}\}\.

Remark: Here, we assume that the effective variance is small enough such that thedistance between the two modes of interest,𝝁\(i\)\\bm\{\\mu\}^\{\(i\)\}and𝝁\(j\)\\bm\{\\mu\}^\{\(j\)\}, is larger\. Furthermore,ℓ\\ellisnottime\-dependent\. Whileκ\\kappain[Sec\.4](https://arxiv.org/html/2605.06831#S4)could be different, we take it to be the same as[Sec\.4](https://arxiv.org/html/2605.06831#S4)for simplicity\.[Sec\.4](https://arxiv.org/html/2605.06831#S4)scales with dimensionϖ\\varpiin order to ensure regions of high probability mass do not overlap between modesiiandjjin high dimensions\(Saremi & Hyvärinen,[2019](https://arxiv.org/html/2605.06831#bib.bib24)\), so that mode interpolations remain a valid definition of hallucination\. We empirically verify this assumption in[FigureE\.6](https://arxiv.org/html/2605.06831#A5.F6), finding that it holds for largeκ\>\>1\\kappa\>\>1\.

Next, we find that there are stable equilibria near the true modes:

###### Proposition 4\.5\(Mode Stability\)\.

Suppose[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.4](https://arxiv.org/html/2605.06831#S4)hold\. Considert≤τ^t\\leq\\hat\{\\tau\}whereτ^=min⁡\{τ,τ2\}\\hat\{\\tau\}=\\min\\\{\\tau,\\tau\_\{2\}\\\}\. Then, with respect to the parallel dynamics toLt,ε\(i,j\)L\_\{t,\\varepsilon\}^\{\(i,j\)\}while the trajectory is inTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}, there exist instantaneous stable equilibria in aε\\varepsilon\-ball around modes𝛍t\(i\)\\bm\{\\mu\}\_\{t\}^\{\(i\)\}and𝛍t\(j\)\\bm\{\\mu\}\_\{t\}^\{\(j\)\}, whereε∈𝒪​\(N​exp⁡\(−κ\)\)\\varepsilon\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\), forκ≥2​log⁡N\+Θ​\(1\)\\kappa\\geq 2\\log N\+\\Theta\(1\)\.

Proof Sketch: Apply the intermediate value theorem to show existence, then bound the drift derivative to check stability\. See[Sec\.H\.3](https://arxiv.org/html/2605.06831#A8.SS3)for a full proof\.

Remark: Note that these parallel dynamics are theexactparallel dynamics in[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50), incorporating an additional term due to the influence of theN−2N\-2other modes, which is bounded byε\\varepsilon\.

Furthermore, trajectories in a neighborhood of these instantaneous equilibria converge to astable trajectoryaround the equilibria\.

###### Corollary 4\.6\(Tracking Stability\)\.

With the same conditions as[Sec\.4](https://arxiv.org/html/2605.06831#S4), there exists a stable trajectory in a neighborhood aroundξt∗,i\\xi\_\{t\}^\{\*,i\}such that any two trajectories, in that same neighborhood, converge to the stable trajectory\. This holds similarly forξt∗,j\\xi\_\{t\}^\{\*,j\}\.

Remark: In particular, this means that trajectories which end up nearξt∗,i\\xi\_\{t\}^\{\*,i\}stay nearξt∗,i\\xi\_\{t\}^\{\*,i\}, and same forξt∗,j\\xi\_\{t\}^\{\*,j\}\.

Next, we find that there is a neighborhood around the midpoint for which the parallel dynamics aretrapped\. Effectively, if an instance is trapped around the midpoint, this yields mode interpolation\.

###### Proposition 4\.7\(Trapping in Midpoint Neighborhood\)\.

Suppose[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.4](https://arxiv.org/html/2605.06831#S4)hold\. Letτ^=min⁡\{τ,τ2\}\\hat\{\\tau\}=\\min\\\{\\tau,\\tau\_\{2\}\\\}\. Furthermore, supposeπi=πj\\pi\_\{i\}=\\pi\_\{j\}\. Then:

1. 1\.Fort≤τ^t\\leq\\hat\{\\tau\}, consider the approximate parallel dynamics inTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}, and denote these dynamics byξt\\xi\_\{t\}\. Then, the point𝒚t∗=𝝁\(i\)\+𝝁\(j\)2\+𝒘t\\bm\{y\}^\{\*\}\_\{t\}=\\frac\{\\bm\{\\mu\}^\{\(i\)\}\+\\bm\{\\mu\}^\{\(j\)\}\}\{2\}\+\\bm\{w\}\_\{t\}, where𝒘t\\bm\{w\}\_\{t\}is the component of[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)orthogonal toLt,ε\(i,j\)L\_\{t,\\varepsilon\}^\{\(i,j\)\}, is a hyperbolic \(instantaneously\) unstable equilibrium with respect to these dynamics, with unstable eigenvalueλt:=ℓ24​σ~t2−1\\lambda\_\{t\}:=\\frac\{\\ell^\{2\}\}\{4\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\-1\. Note that𝒚t∗\\bm\{y\}^\{\*\}\_\{t\}is exactly whereξt=12\\xi\_\{t\}=\\frac\{1\}\{2\}\.
2. 2\.Supposeτ3≤τ^\\tau\_\{3\}\\leq\\hat\{\\tau\}\. Considerξ​\(⋅\)\\xi\(\\cdot\)as the approximate parallel dynamics inTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}\. Choose a \(nondimensionalized\) radiusϑ∈\(0,12\]\\vartheta\\in\(0,\\frac\{1\}\{2\}\]satisfying0<ϑ<λ¯2​\(1\+λ¯\)20<\\vartheta<\\frac\{\\underline\{\\lambda\}\}\{2\(1\+\\bar\{\\lambda\}\)^\{2\}\}, whereλ¯:=min\{t∈\[0,τ3\]\}⁡λt\\underline\{\\lambda\}:=\\min\_\{\\\{t\\in\[0,\\tau\_\{3\}\]\\\}\}\\lambda\_\{t\}andλ¯:=max\{t∈\[0,τ3\]\}⁡λt\\bar\{\\lambda\}:=\\max\_\{\\\{t\\in\[0,\\tau\_\{3\}\]\\\}\}\\lambda\_\{t\}\. LetΛ\+:=λ¯\+2​\(1\+λ¯\)2​ϑ\\Lambda\_\{\+\}:=\\bar\{\\lambda\}\+2\(1\+\\bar\{\\lambda\}\)^\{2\}\\vartheta\. Then, if0<\|ξτ3−12\|<ϑ​exp⁡\(−Λ\+​τ3\)0<\|\\xi\_\{\\tau\_\{3\}\}\-\\frac\{1\}\{2\}\|<\\vartheta\\exp\(\-\\Lambda\_\{\+\}\\tau\_\{3\}\), then: \|ξt−12\|<ϑ​for every​t∈\[0,τ3\]\.\|\\xi\_\{t\}\-\\frac\{1\}\{2\}\|<\\vartheta\\;\\text\{for every\}\\;t\\in\[0,\\tau\_\{3\}\]\. Equivalently, to possibly exit the midpoint neighborhood of radiusϑ\\vartheta, a necessary condition is: τ3≥1Λ\+​log⁡\(ϑ\|ξτ3−12\|\)\.\\tau\_\{3\}\\geq\\frac\{1\}\{\\Lambda\_\{\+\}\}\\log\(\\frac\{\\vartheta\}\{\|\\xi\_\{\\tau\_\{3\}\}\-\\frac\{1\}\{2\}\|\}\)\.

Proof Sketch: Compute the root of the approximate parallel dynamics and check its derivative’s positivity\. Compute its unstable Jacobian eigenvalueλt\\lambda\_\{t\}\. Bound the second derivative of the parallel dynamics and use this to bound the derivative of\|ξt−12\|\|\\xi\_\{t\}\-\\frac\{1\}\{2\}\|; applying Grönwall’s and leveraging our choice ofϑ\\varthetayields the result\. A full proof is provided in[Sec\.H\.5](https://arxiv.org/html/2605.06831#A8.SS5)\.

Discussion: This proposition demonstrates that, for a sufficiently small time budgetτ3\\tau\_\{3\}, DDIM trajectories cannot leave aϑ\\vartheta\-neighborhood of the midpoint\. Even though the midpoint repels the instance along the line as an unstable equilibria, it does not do so sufficiently fast enough to reach the true modes\. Additionally, note that if𝒘t\\bm\{w\}\_\{t\}is nonzero, this becomes a hyperbolic saddle: the negative eigenvalues represent attraction toLt\(i,j\)L\_\{t\}^\{\(i,j\)\}\. This corresponds with the fact that the component of[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)orthogonal toLt\(i,j\)L\_\{t\}^\{\(i,j\)\}vanishes in sufficient time per[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)\. We empirically demonstrate this in[FigureE\.12](https://arxiv.org/html/2605.06831#A5.F12)\.

Note that we use the “approximate” parallel dynamics given in[Eq\.H\.234](https://arxiv.org/html/2605.06831#A8.E234)to prove this result\. The approximate dynamics ignore theε\\varepsilonterm induced by theN−2N\-2modes\. In[AppendixB](https://arxiv.org/html/2605.06831#A2), we prove that using the exact parallel dynamics in[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50)only shifts the location of the point which trajectories get stuck near\. Empirically, as demonstrated in[FigureE\.12](https://arxiv.org/html/2605.06831#A5.F12), we find that the location shift is negligible\. We provide a comprehensive theoretical study of this in[AppendixB](https://arxiv.org/html/2605.06831#A2)\. Additionally, the saddle still exists under score estimation error, as demonstrated in[AppendixC](https://arxiv.org/html/2605.06831#A3)\.

In particular, this defines two regimes of convergence in the diffusion reverse process for theNNcomponent Gaussian mixture, which we illustrate in[Figure1](https://arxiv.org/html/2605.06831#S1.F1): \(a\) Afterτ\\tau,𝒙t\\bm\{x\}\_\{t\}moves fast towardsTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}\(b\) If𝒙t\\bm\{x\}\_\{t\}converges outside aϑ\\vartheta\-neighborhood of the midpoint, near one of the true modes𝝁t\(i\)\\bm\{\\mu\}\_\{t\}^\{\(i\)\}or𝝁t\(j\)\\bm\{\\mu\}\_\{t\}^\{\(j\)\}, it converges fast to this true mode\. Otherwise, if it converges near the midpoint𝒚t∗\\bm\{y\}\_\{t\}^\{\*\}, the trajectory gets “stuck” and requires large remaining timeτ3\\tau\_\{3\}to leave\.

## 5DDPM Hallucinations

Recall that DDPM follows the reverse SDE[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2), which shares the same drift as the PF ODE[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)\(up to a constant factor\) but includes an additional Brownian forcing term\. Consequently, DDPM exhibits the same two qualitative convergence stages as DDIM, described in[Sec\.4](https://arxiv.org/html/2605.06831#S4), which we also verify empirically in[Sec\.6](https://arxiv.org/html/2605.06831#S6)\. Here we focus on the final stage, when the trajectory is governed by the parallel dynamics withinTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}\. In this regime, we show that DDPM is substantially less likely than DDIM to terminate in the midpoint neighborhood responsible for mode interpolation hallucinations\.

For modes𝝁\(i\)\\bm\{\\mu\}^\{\(i\)\}and𝝁\(j\)\\bm\{\\mu\}^\{\(j\)\}, we define their bisector hyperplane asH\(i,j\):=\{𝒚∈ℝϖ:‖𝒚−𝝁\(i\)‖22=‖𝒚−𝝁\(j\)‖22\}H^\{\(i,j\)\}:=\\Big\\\{\\bm\{y\}\\in\\mathbb\{R\}^\{\\varpi\}:\\ \\\|\\bm\{y\}\-\\bm\{\\mu\}^\{\(i\)\}\\\|\_\{2\}^\{2\}=\\\|\\bm\{y\}\-\\bm\{\\mu\}^\{\(j\)\}\\\|\_\{2\}^\{2\}\\Big\\\}\. InsideTubet,ε\(i,j\)\\mathrm\{Tube\}^\{\(i,j\)\}\_\{t,\\varepsilon\},[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)implies that afterτ\\tau, the relevant motion reduces to a one\-dimensional displacement fromH\(i,j\)H^\{\(i,j\)\}along its unit normal\. In what follows, we make this reduction explicit and then bound the probability of ending near the midpoint\.

Formally, let𝒖:=𝝁\(j\)−𝝁\(i\)‖𝝁\(j\)−𝝁\(i\)‖2\\bm\{u\}:=\\frac\{\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\}\{\\\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\\\|\_\{2\}\}and define the signed bisector coordinateAt:=⟨𝒙t−12​\(𝝁\(i\)\+𝝁\(j\)\),𝒖⟩A\_\{t\}:=\\Big\\langle\\bm\{x\}\_\{t\}\-\\tfrac\{1\}\{2\}\(\\bm\{\\mu\}^\{\(i\)\}\+\\bm\{\\mu\}^\{\(j\)\}\),\\,\\bm\{u\}\\Big\\rangleand note thatAt=0A\_\{t\}=0if and only if𝒙t∈H\(i,j\)\\bm\{x\}\_\{t\}\\in H^\{\(i,j\)\}, ranging from−\(ℓ\+ε\)2\-\\frac\{\(\\ell\+\\varepsilon\)\}\{2\}andℓ\+ε2\\frac\{\\ell\+\\varepsilon\}\{2\}along theϵ\\epsilon\-extendedi,ji,j\-mode segment\. Applying Itô’s formula toAtA\_\{t\}and defining the one\-dimensional Brownian motiond​Bt:=⟨𝒖,d​𝑾¯t⟩dB\_\{t\}:=\\langle\\bm\{u\},d\\bar\{\\bm\{W\}\}\_\{t\}\\rangleyields:

d​At=b​\(At,t\)​d​t\+2​η​\(t\)​d​Bt,AT=0,dA\_\{t\}\\;=\\;b\(A\_\{t\},t\)\\,dt\\;\+\\;\\sqrt\{2\\eta\(t\)\}\\,dB\_\{t\},\\qquad A\_\{T\}=0,\(4\)whereη​\(t\)=12​βt\\eta\(t\)=\\tfrac\{1\}\{2\}\\beta\_\{t\}andb​\(At,t\)=⟨−12​βt​𝒙t−βt​∇𝒙tlog⁡pt​\(𝒙t\),𝒖⟩b\(A\_\{t\},t\)=\\big\\langle\-\\tfrac\{1\}\{2\}\\beta\_\{t\}\\,\\bm\{x\}\_\{t\}\-\\beta\_\{t\}\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\),\\,\\bm\{u\}\\big\\rangle\. We then leverage this SDE to bound the probability of DDPM terminating in aϑ\\vartheta\-neighborhood of𝒚t∗\\bm\{y\}\_\{t\}^\{\*\}\.

###### Proposition 5\.1\(DDPM Terminal Midpoint Bound\)\.

LetAtA\_\{t\}satisfy[Eq\.4](https://arxiv.org/html/2605.06831#S5.E4)ont∈\[0,τ3\]t\\in\[0,\\tau\_\{3\}\], withη∈L1​\(\[0,τ3\]\)\\eta\\in L^\{1\}\(\[0,\\tau\_\{3\}\]\)andη​\(t\)≤ηmax\\eta\(t\)\\leq\\eta\_\{\\max\}for allt∈\[0,τ3\]t\\in\[0,\\tau\_\{3\}\]\. Fixϑ\>0\\vartheta\>0\. Assume there exists a measurable functionK:\[0,τ3\]→\[0,∞\)K:\[0,\\tau\_\{3\}\]\\to\[0,\\infty\)such that for everyt∈\[0,τ3\]t\\in\[0,\\tau\_\{3\}\]and everya∈\[−2​ϑ,2​ϑ\]a\\in\[\-2\\vartheta,2\\vartheta\],

\|b​\(a,t\)\|≤K​\(t\)​\|a\|\.\|b\(a,t\)\|\\leq K\(t\)\\,\|a\|\.\(5\)Assume further there existsλrep\>0\\lambda\_\{\\mathrm\{rep\}\}\>0such that for allt∈\[0,τ3\]t\\in\[0,\\tau\_\{3\}\],

b​\(a,t\)\\displaystyle b\(a,t\)≤−λrep​a∀a≥ϑ,\\displaystyle\\leq\-\\lambda\_\{\\mathrm\{rep\}\}\\,a\\qquad\\forall\\,a\\geq\\vartheta,\(6\)b​\(a,t\)\\displaystyle b\(a,t\)≥−λrep​a∀a≤−ϑ\.\\displaystyle\\geq\-\\lambda\_\{\\mathrm\{rep\}\}\\,a\\qquad\\forall\\,a\\leq\-\\vartheta\.Then the terminal midpoint event satisfies

ℙ​\(\|A0\|≤ϑ\)≤\\displaystyle\\mathbb\{P\}\\big\(\|A\_\{0\}\|\\leq\\vartheta\\big\)\\leq4π​exp⁡\(−π2​∫0τ3η​\(s\)​𝑑s16​\(1\+∫0τ3K​\(s\)​𝑑s\)2​ϑ2\)\\displaystyle\\frac\{4\}\{\\pi\}\\exp\\\!\\left\(\-\\frac\{\\pi^\{2\}\\int\_\{0\}^\{\\tau\_\{3\}\}\\eta\(s\)\\,ds\}\{16\\Big\(1\+\\int\_\{0\}^\{\\tau\_\{3\}\}K\(s\)\\,ds\\Big\)^\{2\}\\,\\vartheta^\{2\}\}\\right\)\(7\)\+2​exp⁡\(−λrep​ϑ22​ηmax\)\.\\displaystyle\+2\\exp\\\!\\Big\(\-\\frac\{\\lambda\_\{\\mathrm\{rep\}\}\\,\\vartheta^\{2\}\}\{2\\eta\_\{\\max\}\}\\Big\)\.

Proof Sketch: We upper bound the probability of terminating in the midpoint neighborhood by splitting trajectories into two cases and combining their bounds: \(i\) those that remain within a2​ϑ2\\vartheta\-slab around the bisector for the entire reverse\-time interval, and \(ii\) those that exit this slab\. On theconfinement event\(i\),[Eq\.5](https://arxiv.org/html/2605.06831#S5.E5)implies that staying trapped forces the stochastic martingale term to also remain trapped\. UsingDubins & Schwarz \([1965](https://arxiv.org/html/2605.06831#bib.bib13)\), we view this martingale as a time\-changed Brownian motion with time horizon given by its quadratic variation2​∫0Tη​\(s\)​𝑑s2\\int\_\{0\}^\{T\}\\eta\(s\)\\,ds, and a standard Brownian confinement estimate yields the first exponential term\. On theexit event\(ii\), to still terminate near the midpoint, the path must leave to distance2​ϑ2\\varthetaand later return to withinϑ\\vartheta\. The strong Markov property at the exit time reduces this to a return probability from±2​ϑ\\pm 2\\vartheta\. Under[Eq\.6](https://arxiv.org/html/2605.06831#S5.E6), an exponential supermartingale and optional stopping bound this return, giving the second exponential term\. A full proof can be found in[Sec\.H\.6](https://arxiv.org/html/2605.06831#A8.SS6)\.

Discussion\.[Sec\.5](https://arxiv.org/html/2605.06831#S5)is the DDPM analogue of the DDIM midpoint persistence result in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. After trajectories enterTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}, DDIM’s remaining failure mode is to terminate near the bisector, producing mode interpolation samples\. This proposition shows that DDPM makes this terminal midpoint event exponentially unlikely, so trajectories in the stage where they are withinTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}overwhelmingly tend to end near one of the two true modes instead\. We validate this empirically in[Sec\.6](https://arxiv.org/html/2605.06831#S6)\. The assumptions of[Sec\.5](https://arxiv.org/html/2605.06831#S5)are further detailed in[AppendixF](https://arxiv.org/html/2605.06831#A6)and are verified in[Sec\.E\.7](https://arxiv.org/html/2605.06831#A5.SS7)\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x6.png)Figure 3:Hallucination rate for varying number of DDIM steps used in the reverse process\. Notice that the number of DDIM interpolated samples is consistently larger than that of DDPM\. Thus, this invalidates the idea that the gap between DDIM and DDPM hallucination rates arises due to skipping steps\.![Refer to caption](https://arxiv.org/html/2605.06831v1/x7.png)\(a\)DDIM convergence to thei,ji,j\-mode segment\.
![Refer to caption](https://arxiv.org/html/2605.06831v1/x8.png)\(b\)DDPM convergence to thei,ji,j\-mode segment\.
![Refer to caption](https://arxiv.org/html/2605.06831v1/x9.png)

Figure 4:For both DDIM \([Figure4\(a\)](https://arxiv.org/html/2605.06831#S5.F4.sf1)\) and DDPM \([Figure4\(b\)](https://arxiv.org/html/2605.06831#S5.F4.sf2)\), we plot the convergence rate to the nearesti,ji,j\-mode segment across 100,000 trajectories, finding that convergence occurs afterτ1\\tau\_\{1\}and thus validating[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)\. Note thati,ji,jchange across time in these figures; however, as expected, afterτ1\\tau\_\{1\}they become fixed\. We plotε/ϖ\\varepsilon/\\varpias a dotted black line, finding that convergence toTubet,ε\(i,j\)\\mathrm\{Tube\}\_\{t,\\varepsilon\}^\{\(i,j\)\}is afterτ2\\tau\_\{2\}; thus, only the parallel dynamics become relevant at this stage\. We also find that DDPM ends up closer to the nearesti,ji,j\-mode segment \(dotted red line\)\. Note that DDIM has 50 steps due to quadratic skipping\. Uncertainty is quantified in the blue region as the95%95\\%confidence interval across trajectories\.
## 6Experiments

We provide experiments that justify our theoretical assumptions; demonstrate our theoretical results; and validate our subsequent interpretations\. We primarily experiment with a Gaussian mixture consisting of 25 modes arranged on a uniform grid in 2 dimensions; we study this dataset with higher dimensions in[Sec\.E\.2](https://arxiv.org/html/2605.06831#A5.SS2)\. Unless otherwise stated, we use 1000 DDPM steps \(T=1000T=1000\) and 50 DDIM sampling steps with quadratic skipping\. Firstly, we show that mode interpolation is a primary source of hallucinations during sampling\. We also demonstrate that the high hallucination rate gap between DDIM and DDPM isnotdue to DDIM skipping steps \(i\.e\., PF ODE discretization\) in the reverse process\. Finally, we demonstrate that DDPM noise helps escape theϑ\\vartheta\-neighborhood around the midpoint, thus avoiding hallucinations, as predicted by our theoretical results\. Further experimental details are provided in[AppendixF](https://arxiv.org/html/2605.06831#A6); we also provide additional experiments on images as well as ablations in[AppendixE](https://arxiv.org/html/2605.06831#A5)\.

DDIM Hallucinates More than DDPM: In[Figure1\(a\)](https://arxiv.org/html/2605.06831#S1.F1.sf1), we find that DDIM hallucinates more than DDPM\.

We classify a sample as amode interpolationif it lies more than 5σ\\sigmafrom every true mode but within 5σ\\sigmaof a line segment joining two modes\. Note that we omit<0\.01%<0\.01\\%of instances which land outside5​σ5\\sigmaof any line segment; these instances arise due to score error, and we do not focus on them for the purposes of this work\. We expand on our choice of threshold and characterization of these samples in[AppendixF](https://arxiv.org/html/2605.06831#A6)\. Additionally, we reproduce the recent results ofFu et al\. \([2025](https://arxiv.org/html/2605.06831#bib.bib15)\)in[Sec\.E\.3](https://arxiv.org/html/2605.06831#A5.SS3), demonstrating that DDIM has higher counting hallucinations than DDPM as well for an image dataset\.

Crucially, in[Figure3](https://arxiv.org/html/2605.06831#S5.F3), we demonstrate that the significantly higher hallucination rate of DDIM isnot due to skipping steps\. Specifically, from coarse ODE discretizations to fine ODE discretizations, the hallucination rate of DDIM remains larger than that of DDPM\.

Finally, we observe that for an axis\-aligned grid, hallucinations rarely happen on the diagonals joining two modes\. This is also observed byAithal et al\. \([2024](https://arxiv.org/html/2605.06831#bib.bib1)\)\. We provide theoretical justification for this in[AppendixD](https://arxiv.org/html/2605.06831#A4)\.

Convergence to Nearesti,ji,j\-Mode Segment: In[Figure4](https://arxiv.org/html/2605.06831#S5.F4), we validate[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)empirically, firstly finding that[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[4](https://arxiv.org/html/2605.06831#S4)hold for largeκ=7\\kappa=7\. In[Sec\.E\.1](https://arxiv.org/html/2605.06831#A5.SS1), we find that these assumptions hold for a range of other largeκ\\kappaas well\. We then find that DDIM and DDPM converge to theε\\varepsilon\-extended nearesti,ji,j\-mode segment afterτ1\\tau\_\{1\}, whereε=N​exp⁡\(−κ\)\\varepsilon=N\\exp\(\-\\kappa\)\. Note that this line segment varies with time, andi,ji,jcan change over the reverse process\. Importantly, to verify[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2), we compute the distance \(andε\\varepsilon\) rescaled by dividingϖ\\sqrt\{\\varpi\}\. We also find that the trajectories pass into theε\\varepsilon\-tube afterτ2\\tau\_\{2\}, empirically verifying that the phenomena described in[Sec\.4](https://arxiv.org/html/2605.06831#S4)are relevant as soon as the trajectory enters theε\\varepsilon\-tube\. These results are forϖ=2\\varpi=2; results for higher dimensions are in[Sec\.E\.2](https://arxiv.org/html/2605.06831#A5.SS2)\.

DDPM Noise Helps Avoid Hallucinations: In what follows, we aim to demonstrate that[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.5](https://arxiv.org/html/2605.06831#S5)hold for a nontrivial radiusϑ\\vartheta\. That is, forτ3=9\\tau\_\{3\}=9\(sinceτ2=9\\tau\_\{2\}=9in[Figure4](https://arxiv.org/html/2605.06831#S5.F4)\) remaining steps, DDIM trajectories get stuck within aϑ\\vartheta\-neighborhood of the midpoint𝒚t∗\\bm\{y\}\_\{t\}^\{\*\}while DDPM trajectories escape\. Specifically, we start DDIM and DDPM \(with appropriately adjustedβt\\beta\_\{t\}\) alongLt\(a,b\)L\_\{t\}^\{\(a,b\)\}at timeτ3\\tau\_\{3\}and evaluate the hallucination rate after starting at this point\. We do so for 100,000 trajectories across alla,b∈\[N\]a,b\\in\[N\]\. Critically, in[Figure5](https://arxiv.org/html/2605.06831#S6.F5), we find that there exists aϑ​\(τ3\)\>0\\vartheta\(\\tau\_\{3\}\)\>0such that if a trajectory starts withinϑ​\(τ3\)\\vartheta\(\\tau\_\{3\}\)of the midpoint, it hallucinates\. Next, the figure demonstrates that within this radius, DDPM has a significantly lower hallucination rate than DDIM\. We thus conclude that the noise term in DDPM helps it escape theϑ\\vartheta\-neighborhood around the midpoint, thus avoiding hallucinations\. In[Figure5](https://arxiv.org/html/2605.06831#S6.F5), we leverage this insight to evaluate the following DDIM\-DDPM hybrid procedure: we addzzDDPM steps to DDIM afterτ3\\tau\_\{3\}, and find that this helps avoid hallucinations as well\. In[Sec\.E\.4](https://arxiv.org/html/2605.06831#A5.SS4), we demonstrate that our results hold across various choices ofτ3\\tau\_\{3\}\. Note thatτ3\\tau\_\{3\}is taken with respect to DDIM steps throughout our experiments\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x10.png)Figure 5:Starting DDIM atτ3=9\\tau\_\{3\}=9, we find that forϑ=0\.15​ℓt\\vartheta=0\.15\\ell\_\{t\}, DDIM gets stuck before it can reach the true modes, i\.e\., it hallucinates, as predicted by[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. Furthermore, DDPM has a lower hallucination rate within this sameϑ\\vartheta\. Thus, we conclude that DDPM noise helps escape theϑ\\vartheta\-neighborhood around the midpoint, as predicted by[Sec\.5](https://arxiv.org/html/2605.06831#S5)\. Given this, we find that addingzzDDPM steps after starting DDIM atτ3\\tau\_\{3\}helps mitigate hallucination\. Uncertainty is quantified by the standard error across mode pairs\.
## 7Discussion

We present a geometric analysis of hallucinations in diffusion models through the lens of Gaussian mixtures\. In particular, after sufficient time, diffusion trajectories undergo two key convergence phases: rapid collapse onto a nearby line segment connecting two modes, followed by dynamics along this line\. Next, if a trajectory does not converge to one of the modes, it can get stuck near a point; in the case that Gaussian mixture weights are equal, this is at the midpoint\. This provides insight into the core mechanisms which differ between DDIM and DDPM: stochastic noise can help DDPM escape the neighborhood of the midpoint to one of the true modes and avoid hallucination\.

Our results thus imply that the noise term is beneficial for reducing mode interpolation hallucinations\. Empirically, we find this to be the case as well: adding only a few diffusion steps at the end of the reverse process significantly lowers the hallucination rate\. We hope that these results highlight to the community the pitfalls of preferring DDIM over DDPM and inspire future work on designing flow\-based generative models which avoid the midpoint neighborhood entirely, thus mitigating hallucinations\.

Limitations and Future Work: Our primary analysis assumes a Gaussian mixture target distribution with well\-separated modes\. While we prove some results under an inexact score in[AppendixC](https://arxiv.org/html/2605.06831#A3), a full characterization of what changes under score error is left to future work\. Additionally,Aithal et al\. \([2024](https://arxiv.org/html/2605.06831#bib.bib1)\)highlight that mode interpolation can also happen in image latent space\. Thus, a natural extension of our theoretical analysis to images is studying latent diffusion\(Rombach et al\.,[2022](https://arxiv.org/html/2605.06831#bib.bib23)\)with Gaussian mixture latents\. Furthermore, we believe that our work provides a first step towards designing adaptive samplers or models which detect when trajectories are stuck and inject DDPM noise accordingly, thus mitigating hallucinations while keeping the computational benefits of DDIM\.

## Broader Impacts

Hallucinations, generations that violate structural or semantic constraints of the training distribution, are a key obstacle in the adoption of state\-of\-the\-art generative models, like diffusion, in safety critical contexts\. Our work contributes to removing this obstacle by giving a mechanistic account of one category of hallucination: mode interpolation, where trajectories terminate between modes of the target distribution\. We identify why DDIM, a commonly used diffusion inference procedure, has a higher hallucination rate than its stochastic counterpart, DDPM\. Concretely, the noise in the DDPM reverse process can help it get unstuck from a region around the midpoint, where DDIM would otherwise hallucinate\. Empirically, we highlight that even mild noise injection can be beneficial, guiding practitioners to safer generative modeling\. Altogether, we believe our work has positive societal impact\.

## References

- Aithal et al\. \(2024\)Aithal, S\. K\., Maini, P\., Lipton, Z\. C\., and Kolter, J\. Z\.Understanding hallucinations in diffusion models through mode interpolation\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2024\.
- Anderson \(1982\)Anderson, B\. D\.Reverse\-time diffusion equation models\.*Stochastic Processes and their Applications*, 1982\.
- Betti et al\. \(2026\)Betti, F\., Baraldi, L\., Baraldi, L\., Cucchiara, R\., and Sebe, N\.Hallucination early detection in diffusion models\.*International Journal of Computer Vision \(IJCV\)*, 2026\.
- Beyler & Bach \(2025\)Beyler, E\. and Bach, F\.Convergence of deterministic and stochastic diffusion\-model samplers: A simple analysis in wasserstein distance\.*arXiv 2508\.03210*, 2025\.
- Biroli et al\. \(2024\)Biroli, G\., Bonnaire, T\., de Bortoli, V\., and Mézard, M\.Dynamical regimes of diffusion models\.In*Nature Communications*, 2024\.
- Bonnaire et al\. \(2025\)Bonnaire, T\., Urfin, R\., Biroli, G\., and Mezard, M\.Why diffusion models don’t memorize: The role of implicit dynamical regularization in training\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2025\.
- Buchanan et al\. \(2025\)Buchanan, S\., Pai, D\., Ma, Y\., and Bortoli, V\. D\.On the edge of memorization in diffusion models\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2025\.
- Cai & Li \(2025\)Cai, C\. and Li, G\.Minimax optimality of the probability flow ode for diffusion models\.*arXiv 2503\.09583*, 2025\.
- Cao et al\. \(2023\)Cao, Y\., Chen, J\., Luo, Y\., and Zhou, X\.Exploring the optimal choice for generative processes in diffusion models: Ordinary vs stochastic differential equations\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2023\.
- Chandran\.C et al\. \(2025\)Chandran\.C, B\., Anumasa, S\., and Liu, D\.Laplacian score sharpening for mitigating hallucination in diffusion models\.*arXiv 2511\.07496*, 2025\.
- Chen et al\. \(2023\)Chen, S\., Chewi, S\., Lee, H\., Li, Y\., Lu, J\., and Salim, A\.The probability flow ode is provably fast\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2023\.
- Deveney et al\. \(2025\)Deveney, T\., Stanczuk, J\., Kreusser, L\., Budd, C\., and Schönlieb, C\.\-B\.Closing the ode\-sde gap in score\-based diffusion models through the fokker\-planck equation\.*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, 2025\.
- Dubins & Schwarz \(1965\)Dubins, L\. E\. and Schwarz, G\.On continuous martingales\.*Proceedings of the National Academy of Sciences of the United States of America*, 1965\.
- Fernandez\-Granda \(2017\)Fernandez\-Granda, C\.Lecture 3: Nyu optimization\-based data analysis lecture notes, 2017\.
- Fu et al\. \(2025\)Fu, S\., Zhou, J\., Chen, Q\., Jing, H\., Nguyen, H\. A\., Liu, X\., Zeng, Z\., Ma, L\., Zhang, Q\., and Wu, Q\.Counting hallucinations in diffusion models\.*arXiv 2510\.13080*, 2025\.
- Ho et al\. \(2020\)Ho, J\., Jain, A\., and Abbeel, P\.Denoising diffusion probabilistic models\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2020\.
- Li et al\. \(2025\)Li, G\., Cai, C\., and Wei, Y\.Dimension\-free convergence of diffusion models for approximate gaussian mixtures\.*arXiv 2504\.05300*, 2025\.
- Li & Chen \(2024\)Li, M\. and Chen, S\.Critical windows: Non\-asymptotic theory for feature emergence in diffusion models\.In*International Conference on Machine Learning \(ICML\)*, 2024\.
- Liang et al\. \(2025\)Liang, Y\., Shi, Z\., Song, Z\., and Zhou, Y\.Unraveling the smoothness properties of diffusion models: A gaussian mixture perspective\.*International Conference on Computer Vision \(ICCV\)*, 2025\.
- Lu et al\. \(2022\)Lu, C\., Zhou, Y\., Bao, F\., Chen, J\., Li, C\., and Zhu, J\.DPM\-solver: A fast ODE solver for diffusion probabilistic model sampling in around 10 steps\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2022\.
- Lu et al\. \(2025\)Lu, R\., Wang, R\., Lyu, K\., Jiang, X\., Huang, G\., and Wang, M\.Towards understanding text hallucination of diffusion models via local generation bias\.In*International Conference on Learning Representations \(ICLR\)*, 2025\.
- Mörters & Peres \(2010\)Mörters, P\. and Peres, Y\.*Brownian Motion*\.Cambridge Series in Statistical and Probabilistic Mathematics\. Cambridge University Press, 2010\.
- Rombach et al\. \(2022\)Rombach, R\., Blattmann, A\., Lorenz, D\., Esser, P\., and Ommer, B\.High\-resolution image synthesis with latent diffusion models\.In*Conference on Computer Vision and Pattern Recognition \(CVPR\)*, 2022\.
- Saremi & Hyvärinen \(2019\)Saremi, S\. and Hyvärinen, A\.Neural empirical bayes\.*Journal of Machine Learning Research*, 2019\.
- Shah et al\. \(2023\)Shah, K\., Chen, S\., and Klivans, A\.Learning mixtures of gaussians using the DDPM objective\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2023\.
- Shehata et al\. \(2025\)Shehata, Y\., Holzschuh, B\., and Thuerey, N\.Improved sampling of diffusion models in fluid dynamics with tweedie’s formula\.In*International Conference on Learning Representations \(ICLR\)*, 2025\.
- Sohl\-Dickstein et al\. \(2015\)Sohl\-Dickstein, J\., Weiss, E\. A\., Maheswaranathan, N\., and Ganguli, S\.Deep unsupervised learning using nonequilibrium thermodynamics\.In*International Conference on Machine Learning \(ICML\)*, 2015\.
- Somepalli et al\. \(2023\)Somepalli, G\., Singla, V\., Goldblum, M\., Geiping, J\., and Goldstein, T\.Understanding and mitigating copying in diffusion models\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2023\.
- Song et al\. \(2021a\)Song, J\., Meng, C\., and Ermon, S\.Denoising diffusion implicit models\.In*International Conference on Learning Representations \(ICLR\)*, 2021a\.
- Song et al\. \(2021b\)Song, Y\., Sohl\-Dickstein, J\., Kingma, D\. P\., Kumar, A\., Ermon, S\., and Poole, B\.Score\-based generative modeling through stochastic differential equations\.In*International Conference on Learning Representations \(ICLR\)*, 2021b\.
- Thanh\-Tung & Tran \(2020\)Thanh\-Tung, H\. and Tran, T\.Catastrophic forgetting and mode collapse in gans\.In*International Joint Conference on Neural Networks \(IJCNN\)*, 2020\.
- Triaridis et al\. \(2025\)Triaridis, K\., Graikos, A\., Chatziagapi, A\., Chrysos, G\. G\., and Samaras, D\.Mitigating diffusion model hallucinations with dynamic guidance\.*arXiv 2510\.05356*, 2025\.
- Wu et al\. \(2024\)Wu, Y\., Chen, M\., Li, Z\., Wang, M\., and Wei, Y\.Theoretical insights for diffusion guidance: A case study for gaussian mixture models\.In*International Conference on Machine Learning \(ICML\)*, 2024\.
- Xu et al\. \(2023\)Xu, Y\., Deng, M\., Cheng, X\., Tian, Y\., Liu, Z\., and Jaakkola, T\. S\.Restart sampling for improving generative processes\.In*Advances in Neural Information Processing Systems \(NeurIPS\)*, 2023\.
- Zhang et al\. \(2018\)Zhang, Z\., Li, M\., and Yu, J\.On the convergence and mode collapse of gan\.*SIGGRAPH Asia Technical Briefs*, 2018\.
- Zhang et al\. \(2026\)Zhang, Z\., Li, X\., Li, X\., Shi, L\., Wu, M\., Tao, M\., and Qu, Q\.Generalization of diffusion models arises with a balanced representation space\.In*International Conference on Learning Representations \(ICLR\)*, 2026\.

## Contents of the Appendix

- •In[AppendixA](https://arxiv.org/html/2605.06831#A1)we provide a full list of notation\.
- •In[AppendixB](https://arxiv.org/html/2605.06831#A2), we provide theoretical results characterizing how our analysis changes if we assume unequally weighted modes \(i\.e\.,πi≠πj\\pi\_\{i\}\\neq\\pi\_\{j\}\)\.
- •In[AppendixC](https://arxiv.org/html/2605.06831#A3), we provide theoretical results characterizing how our analysis changes if we assume the learned inexact score, rather than the exact score in our main paper\.
- •In[AppendixD](https://arxiv.org/html/2605.06831#A4), we provide a theoretical result characterizing why mode interpolations rarely happen on the diagonal lines joining two modes in the axis\-aligned grid we use throughout our experiments\.
- •In[AppendixE](https://arxiv.org/html/2605.06831#A5), we provide additional experiments, including verifying assumptions, empirically evaluating higher dimensional Gaussian mixtures and images, and running ablations on several hyperparameters in our experiments\.
- •In[AppendixF](https://arxiv.org/html/2605.06831#A6), we provide implementation details for our experiments\.
- •In[AppendixG](https://arxiv.org/html/2605.06831#A7)we provide helpful lemmas, which are then used in our proofs in[AppendixH](https://arxiv.org/html/2605.06831#A8)\.
- •In[AppendixI](https://arxiv.org/html/2605.06831#A9), we discuss the use of the exact score throughout our paper\.
- •In[AppendixJ](https://arxiv.org/html/2605.06831#A10), we provide a table of symbols used throughout our paper\.

## Appendix AFull Notation

We letx∈ℝx\\in\\mathbb\{R\}denote a scalar\. Bolded lowercase letters𝒙∈ℝϖ\\bm\{x\}\\in\\mathbb\{R\}^\{\\varpi\}denote vectors of dimensionϖ\\varpi\. Bolded upper case letters𝑿∈ℝϖ×ϖ\\bm\{X\}\\in\\mathbb\{R\}^\{\\varpi\\times\\varpi\}denote matrices withϖ\\varpirows and columns, unless dimensionalities are explicitly specified\.\[N\]\[N\], for scalarN∈ℝN\\in\\mathbb\{R\}, denotes the integer subset\{1,2,…,N\}\\\{1,2,\.\.\.,N\\\}\. We use‖𝒙‖2\|\|\\bm\{x\}\|\|\_\{2\}to denote theℓ2\\ell\_\{2\}norm of a vector𝒙\\bm\{x\}\. We use⟨𝒙1,𝒙2⟩\\langle\\bm\{x\}\_\{1\},\\bm\{x\}\_\{2\}\\rangleto denote the inner product between vectors𝒙1\\bm\{x\}\_\{1\},𝒙2\\bm\{x\}\_\{2\}, which is the dot product𝒙T​𝒙\\bm\{x\}^\{T\}\\bm\{x\}unless explicitly specified otherwise\. For a functionff, we use∇𝒙f\\nabla\_\{\\bm\{x\}\}fto denote its gradient iff:ℝϖ→ℝf:\\mathbb\{R\}^\{\\varpi\}\\to\\mathbb\{R\}and its Jacobian iff:ℝϖ→ℝof:\\mathbb\{R\}^\{\\varpi\}\\to\\mathbb\{R\}^\{o\}\. We denote byxt:\[0,T\]→ℝx\_\{t\}:\[0,T\]\\to\\mathbb\{R\}and𝒙t:\[0,T\]→ℝϖ\\bm\{x\}\_\{t\}:\[0,T\]\\to\\mathbb\{R\}^\{\\varpi\}time\-dependent scalar\-valued and vector\-valued functions, respectively, from time scalar timeTTto0unless otherwise specified\. We usex˙t\\dot\{x\}\_\{t\}to denote the time derivative ofxtx\_\{t\}and similarly use𝒙˙t\\dot\{\\bm\{x\}\}\_\{t\}for𝒙t\\bm\{x\}\_\{t\}; for all other derivatives, we use the notationdd​x​f​\(x\)\\frac\{d\}\{dx\}f\(x\)\.𝒩​\(𝒙;𝝁,σ2​𝑰\)\\mathcal\{N\}\(\\bm\{x\};\\bm\{\\mu\},\\sigma^\{2\}\\bm\{I\}\)is used to denote the probability density function of aϖ\\varpi\-dimensional Gaussian centered at𝝁\\bm\{\\mu\}with varianceσ2​𝑰\\sigma^\{2\}\\bm\{I\}, where𝑰\\bm\{I\}is the identity matrix\. We write stochastic differential equations \(SDE\) asd​𝒙t=b​\(t,𝒙t\)​d​t\+σ​\(t,𝒙t\)​d​𝑾td\\bm\{x\}\_\{t\}=b\(t,\\bm\{x\}\_\{t\}\)dt\+\\sigma\(t,\\bm\{x\}\_\{t\}\)d\\bm\{W\}\_\{t\}where𝑾t∈ℝϖ×1\\bm\{W\}\_\{t\}\\in\\mathbb\{R\}^\{\{\\varpi\}\\times 1\}is the standard Brownian motion, which is equivalent to the weak form𝒙t=𝒙0\+∫0tb​\(s,𝒙s\)​𝑑s\+∫0tσ​\(s,𝒙s\)​𝑑𝑾t\\bm\{x\}\_\{t\}=\\bm\{x\}\_\{0\}\+\\int\_\{0\}^\{t\}b\(s,\\bm\{x\}\_\{s\}\)ds\+\\int\_\{0\}^\{t\}\\sigma\(s,\\bm\{x\}\_\{s\}\)d\\bm\{W\}\_\{t\}\. We say that, forf,g:ℝϖ→ℝof,g:\\mathbb\{R\}^\{\\varpi\}\\to\\mathbb\{R\}^\{o\},f∈𝒪​\(g\)f\\in\\mathcal\{O\}\(g\)if∃c1\>0\\exists c\_\{1\}\>0s\.t\.f​\(𝒙\)≤c1​g​\(𝒙\)f\(\\bm\{x\}\)\\leq c\_\{1\}g\(\\bm\{x\}\)forxxsufficiently large;f∈Ω​\(g\)f\\in\\Omega\(g\)if∃c2\>0\\exists c\_\{2\}\>0s\.t\.f​\(x\)≥c2​g​\(x\)f\(x\)\\geq c\_\{2\}g\(x\)forxxsufficiently large; andf∈Θ​\(g\)f\\in\\Theta\(g\)iff∈𝒪​\(g\)f\\in\\mathcal\{O\}\(g\)andf∈Ω​\(g\)f\\in\\Omega\(g\)\.

## Appendix BSaddle Existence for Unequal Weights and Exact Parallel Dynamics

In[Sec\.4](https://arxiv.org/html/2605.06831#S4), we demonstrated that, after sufficient time in the reverse process, the midpoint was an equilibrium with respect to the approximate parallel dynamics, under equal weights of thei,ji,j\-mode segmentπi=πj\\pi\_\{i\}=\\pi\_\{j\}\. Here:

1. 1\.Under unequal weights, we demonstrate a sufficient condition for an equilibrium to exist\.
2. 2\.Under the exact dynamics, we demonstrate that there exists an equilibrium, and that it does not differ greatly from the midpoint\. This is also empirically justified by[FigureE\.12](https://arxiv.org/html/2605.06831#A5.F12)\.

###### Proposition B\.1\(Equilibria Location under Unequal Weights and Exact Dynamics\)\.

Suppose[Sec\.4](https://arxiv.org/html/2605.06831#S4),[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.4](https://arxiv.org/html/2605.06831#S4)hold and fixt≤τ^t\\leq\\hat\{\\tau\}\(withτ^\\hat\{\\tau\}same as in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\)\. Consider the induced 1D parallel dynamics inside the tube\.

1. 1\.\(Unequal weights\)\.Letℓ:=‖𝝁\(i\)−𝝁\(j\)‖2\\ell:=\\\|\\bm\{\\mu\}^\{\(i\)\}\-\\bm\{\\mu\}^\{\(j\)\}\\\|\_\{2\}\. For the approximate parallel dynamics in Tube\(i,j\)t,ε\{\}\_\{t,\\varepsilon\}^\{\(i,j\)\}, there exists a point parallel toLt,ε\(i,jL\_\{t,\\varepsilon\}^\{\(i,j\}, which we denote byξi​j⋆​\(t\)∈\(0,1\)\\xi^\{\\star\}\_\{ij\}\(t\)\\in\(0,1\), which satisfies: log⁡\(ξi​j⋆​\(t\)1−ξi​j⋆​\(t\)\)=log⁡\(πjπi\)\+ℓ2σ~t2​\(ξi​j⋆​\(t\)−12\)\.\\log\\\!\\Big\(\\frac\{\\xi^\{\\star\}\_\{ij\}\(t\)\}\{1\-\\xi^\{\\star\}\_\{ij\}\(t\)\}\\Big\)=\\log\\\!\\Big\(\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\Big\)\+\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\Big\(\\xi^\{\\star\}\_\{ij\}\(t\)\-\\frac\{1\}\{2\}\\Big\)\.\(B\.8\) In particular,πi=πj\\pi\_\{i\}=\\pi\_\{j\}impliesξi​j⋆​\(t\)=12\\xi^\{\\star\}\_\{ij\}\(t\)=\\tfrac\{1\}\{2\}, andπj<πi\\pi\_\{j\}<\\pi\_\{i\}impliesξi​j⋆​\(t\)\>12\\xi^\{\\star\}\_\{ij\}\(t\)\>\\tfrac\{1\}\{2\}\(and vice versa\)\. Moreover, Fi​j,t′​\(ξi​j⋆​\(t\)\)=ℓ2σ~t2​ξi​j⋆​\(t\)​\(1−ξi​j⋆​\(t\)\)−1,F^\{\\prime\}\_\{ij,t\}\\\!\\left\(\\xi^\{\\star\}\_\{ij\}\(t\)\\right\)=\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\,\\xi^\{\\star\}\_\{ij\}\(t\)\\bigl\(1\-\\xi^\{\\star\}\_\{ij\}\(t\)\\bigr\)\-1,\(B\.9\)soξi​j⋆​\(t\)\\xi^\{\\star\}\_\{ij\}\(t\)is hyperbolic unstable wheneverFi​j,t′​\(ξi​j⋆​\(t\)\)\>0F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\\star\}\_\{ij\}\(t\)\)\>0\.
2. 2\.\(Exact Parallel Dynamics\)\.Under the exact parallel dynamics characterized in[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50), between the two stable equilibria near modes𝝁\(i\)\\bm\{\\mu\}^\{\(i\)\}and𝝁\(j\)\\bm\{\\mu\}^\{\(j\)\}discussed in[Sec\.4](https://arxiv.org/html/2605.06831#S4), there exists an equilibriumξN⋆​\(t\)\\xi^\{\\star\}\_\{N\}\(t\)of the exact parallel dynamics Furthermore, forκ\\kappasufficiently large, assume there an intervalIIcontainingξi​j⋆​\(t\)\\xi^\{\\star\}\_\{ij\}\(t\)such that, for somem\>0m\>0: Fi​j,t′​\(ξ\)≥m\>0∀ξ∈I,F^\{\\prime\}\_\{ij,t\}\(\\xi\)\\geq m\>0\\qquad\\forall\\,\\xi\\in I,\(B\.10\) whereFi​j,tF\_\{ij,t\}is the drift of the approximate parallel dynamics\. Then,ξN⋆​\(t\)∈I\\xi^\{\\star\}\_\{N\}\(t\)\\in Isatisfies \|ξN⋆​\(t\)−ξi​j⋆​\(t\)\|≤ε/m\.\\big\|\\xi^\{\\star\}\_\{N\}\(t\)\-\\xi^\{\\star\}\_\{ij\}\(t\)\\big\|\\leq\\varepsilon/m\.\(B\.11\) whereε∈𝒪​\(N​exp⁡\(−κ\)\)\\varepsilon\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)\.

Proof sketch: Item \(1\) follows by analyzing the approximate parallel dynamics for a single pair of components: the equilibrium condition yields the stated log\-odds relation, and the location and slope follow from monotonicity and differentiation of the logistic responsibility\. For item \(2\), under two\-mode dominance the exact parallel dynamics are a uniformly small perturbation of the approximate dynamics\. Existence of an interior unstable equilibrium then follows by continuity and the intermediate value theorem, and a mean\-value argument yields the stated displacement bound\. A full proof is given in[Sec\.H\.7](https://arxiv.org/html/2605.06831#A8.SS7)\.

Discussion: The key contribution of this result is twofold\. First, we show that the exact parallel dynamics admit an interior equilibria rather than assuming its existence\. Second, we show that this equilibrium is robust: its location does not deviate significantly under unequal mixture weights or when passing from the approximate to the exact dynamics\. Item \(1\) characterizes how unequal weights bias the saddle along the\(i,j\)\(i,j\)\-line toward the lower\-weight component\. Item \(2\) shows that under two\-mode dominance, the exact parallel dynamics differ from the approximate dynamics by only a small perturbation, so the interior saddle persists and remains close to its pairwise location\. Finally, since an equilibrium exists between two stable \(attracting\) ones \(which exist and are stable for the exact parallel dynamics per[Sec\.4](https://arxiv.org/html/2605.06831#S4)\), and the exact parallel dynamics are one\-dimensional, intuitively this is an equilibrium which repels i\.e\. it is unstable\. As such, it can also admit the confinement in a radius discussed in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\.

## Appendix CResults under Inexact Score

The preceding analysis leverages the closed\-form score of Gaussian mixtures\. In practice, the score∇𝒙log⁡pt​\(𝒙\)\\nabla\_\{\\bm\{x\}\}\\log p\_\{t\}\(\\bm\{x\}\)is approximated by a neural networksθ​\(𝒙,t\)s\_\{\\theta\}\(\\bm\{x\},t\)\. In this section, we characterize how score estimation error affects the mechanism established in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\.

###### Definition C\.1\(Score Error Field\)\.

Define the score error field as

ψ​\(𝒙,t\):=sθ​\(𝒙,t\)−∇𝒙log⁡pt​\(𝒙\)\.\\psi\(\\bm\{x\},t\):=s\_\{\\theta\}\(\\bm\{x\},t\)\-\\nabla\_\{\\bm\{x\}\}\\log p\_\{t\}\(\\bm\{x\}\)\.\(C\.12\)The learned reverse dynamics become

𝒙˙t=−12​βt​\(𝒙t\+∇𝒙tlog⁡pt​\(𝒙t\)\+ψ​\(𝒙t,t\)\)\.\\dot\{\\bm\{x\}\}\_\{t\}=\-\\frac\{1\}\{2\}\\beta\_\{t\}\\left\(\\bm\{x\}\_\{t\}\+\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)\+\\psi\(\\bm\{x\}\_\{t\},t\)\\right\)\.\(C\.13\)

###### Assumption C\.2\(Bounded Score Error\)\.

There existsϱ:\[0,T\]→ℝ≥0\\varrho:\[0,T\]\\to\\mathbb\{R\}\_\{\\geq 0\}such that‖ψ​\(𝐱,t\)‖2≤ϱ​\(t\)\\\|\\psi\(\\bm\{x\},t\)\\\|\_\{2\}\\leq\\varrho\(t\)for all𝐱∈Tubet,ϵ\(i,j\)\\bm\{x\}\\in\\mathrm\{Tube\}^\{\(i,j\)\}\_\{t,\\epsilon\}\.

###### Assumption C\.3\(Lipschitz Score Error\)\.

There existsLψ:\[0,T\]→ℝ≥0L\_\{\\psi\}:\[0,T\]\\to\\mathbb\{R\}\_\{\\geq 0\}such that for all𝐱,𝐲∈Tubet,ϵ\(i,j\)\\bm\{x\},\\bm\{y\}\\in\\mathrm\{Tube\}^\{\(i,j\)\}\_\{t,\\epsilon\}:

‖ψ​\(𝒙,t\)−ψ​\(𝒚,t\)‖2≤Lψ​\(t\)​‖𝒙−𝒚‖2\.\\\|\\psi\(\\bm\{x\},t\)\-\\psi\(\\bm\{y\},t\)\\\|\_\{2\}\\leq L\_\{\\psi\}\(t\)\\\|\\bm\{x\}\-\\bm\{y\}\\\|\_\{2\}\.\(C\.14\)

Under these assumptions, we establish that \(1\) the saddle location shifts under score error, and \(2\) the unstable eigenvalue can decrease, amplifying the slowdown effect\.

###### Proposition C\.4\(Saddle Perturbation\)\.

Suppose[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.4](https://arxiv.org/html/2605.06831#S4)hold\. Letξi​j∗​\(t\)\\xi^\{\*\}\_\{ij\}\(t\)denote the unstable equilibrium of the exact parallel dynamics from[AppendixB](https://arxiv.org/html/2605.06831#A2), with unstable eigenvalueλt=Fi​j,t′​\(ξi​j∗​\(t\)\)\>0\\lambda\_\{t\}=F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)\>0\.

Then, forϱ​\(t\)\\varrho\(t\)sufficiently small, the perturbed parallel dynamics admit an equilibriumξθ∗​\(t\)\\xi^\{\*\}\_\{\\theta\}\(t\)satisfying

\|ξθ∗​\(t\)−ξi​j∗​\(t\)\|≤2​ϱ​\(t\)ℓ​λt\+O​\(ϱ​\(t\)2\),\|\\xi^\{\*\}\_\{\\theta\}\(t\)\-\\xi^\{\*\}\_\{ij\}\(t\)\|\\leq\\frac\{2\\varrho\(t\)\}\{\\ell\\lambda\_\{t\}\}\+O\(\\varrho\(t\)^\{2\}\),\(C\.15\)whereℓ=‖𝛍\(j\)−𝛍\(i\)‖2\\ell=\\\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\\\|\_\{2\}\.

Proof Sketch:Under score errorψ\\psi, the perturbed parallel drift becomesF~N,t​\(ξ\)=Fi​j,t​\(ξ\)\+et​\(ξ\)\\tilde\{F\}\_\{N,t\}\(\\xi\)=F\_\{ij,t\}\(\\xi\)\+e\_\{t\}\(\\xi\), where\|et​\(ξ\)\|≤ϱ​\(t\)/ℓ\|e\_\{t\}\(\\xi\)\|\\leq\\varrho\(t\)/\\ellby Cauchy\-Schwarz\. Applying the implicit function theorem at the unperturbed equilibrium and Taylor expanding yields the displacement bound\. A full proof is provided in[Sec\.H\.8](https://arxiv.org/html/2605.06831#A8.SS8)\.

###### Proposition C\.5\(Eigenvalue Perturbation\)\.

Suppose[Sec\.4](https://arxiv.org/html/2605.06831#S4),[Sec\.4](https://arxiv.org/html/2605.06831#S4),[AppendixC](https://arxiv.org/html/2605.06831#A3), and[AppendixC](https://arxiv.org/html/2605.06831#A3)hold\. The perturbed parallel dynamics have eigenvalue atξθ∗​\(t\)\\xi^\{\*\}\_\{\\theta\}\(t\):

λθ​\(t\)=λt\+𝒖⊤​∇𝒙ψ​\(𝒚​\(ξθ∗\),t\)​𝒖\+O​\(ϱ​\(t\)\)\.\\lambda\_\{\\theta\}\(t\)=\\lambda\_\{t\}\+\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\(\\bm\{y\}\(\\xi^\{\*\}\_\{\\theta\}\),t\)\\,\\bm\{u\}\+O\(\\varrho\(t\)\)\.\(C\.16\)In particular:

1. 1\.If𝒖⊤​∇𝒙ψ​𝒖<0\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\\,\\bm\{u\}<0\(error is “contractive” alongLt\(i,j\)L^\{\(i,j\)\}\_\{t\}\), thenλθ​\(t\)<λt\\lambda\_\{\\theta\}\(t\)<\\lambda\_\{t\}\.
2. 2\.If𝒖⊤​∇𝒙ψ​\(𝒚​\(ξθ∗\),t\)​𝒖≤−λt−C​ϱ​\(t\)\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\(\\bm\{y\}\(\\xi^\{\*\}\_\{\\theta\}\),t\)\\,\\bm\{u\}\\leq\-\\lambda\_\{t\}\-C\\varrho\(t\)for some constantC\>0C\>0, thenλθ​\(t\)≤0\\lambda\_\{\\theta\}\(t\)\\leq 0, and the perturbed saddle becomes instantaneously stable\.

Proof Sketch:Differentiating the perturbed driftF~N,t​\(ξ\)=Fi​j,t​\(ξ\)\+et​\(ξ\)\\tilde\{F\}\_\{N,t\}\(\\xi\)=F\_\{ij,t\}\(\\xi\)\+e\_\{t\}\(\\xi\)and usinget′​\(ξ\)=𝒖⊤​∇𝒙ψ​𝒖e^\{\\prime\}\_\{t\}\(\\xi\)=\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\\,\\bm\{u\}yields the eigenvalue perturbation\. A full proof is provided in[Sec\.H\.9](https://arxiv.org/html/2605.06831#A8.SS9)\.

We conclude by showing that DDPM’s noise mechanism provides robustness against eigenvalue perturbation\.

###### Proposition C\.6\(DDPM Robustness to Score Error\)\.

Consider the perturbed DDPM dynamics

d​At=bθ​\(At,t\)​d​t\+2​η​\(t\)​d​Bt,dA\_\{t\}=b\_\{\\theta\}\(A\_\{t\},t\)\\,dt\+\\sqrt\{2\\eta\(t\)\}\\,dB\_\{t\},\(C\.17\)wherebθ​\(At,t\)=b​\(At,t\)\+e~t​\(At\)b\_\{\\theta\}\(A\_\{t\},t\)=b\(A\_\{t\},t\)\+\\tilde\{e\}\_\{t\}\(A\_\{t\}\)with\|e~t​\(a\)\|≤ϱ​\(t\)/ℓ\|\\tilde\{e\}\_\{t\}\(a\)\|\\leq\\varrho\(t\)/\\ell\.

Defineϱmax:=supt∈\[0,τ^\]ϱ​\(t\)\\varrho\_\{\\max\}:=\\sup\_\{t\\in\[0,\\hat\{\\tau\}\]\}\\varrho\(t\)\. Suppose the exact drift satisfies the escape condition in[Sec\.5](https://arxiv.org/html/2605.06831#S5)with constantλrep\>0\\lambda\_\{\\mathrm\{rep\}\}\>0\. If

ϱmaxℓ<λrep​ϑ,\\frac\{\\varrho\_\{\\max\}\}\{\\ell\}<\\lambda\_\{\\mathrm\{rep\}\}\\vartheta,\(C\.18\)then the perturbed drift satisfies an escape condition with constantλ~rep:=λrep−ϱmaxℓ​ϑ\>0\\tilde\{\\lambda\}\_\{\\mathrm\{rep\}\}:=\\lambda\_\{\\mathrm\{rep\}\}\-\\frac\{\\varrho\_\{\\max\}\}\{\\ell\\vartheta\}\>0, and the escape bound holds withλrep\\lambda\_\{\\mathrm\{rep\}\}replaced byλ~rep\\tilde\{\\lambda\}\_\{\\mathrm\{rep\}\}\.

Proof:A full proof is provided in[Sec\.H\.10](https://arxiv.org/html/2605.06831#A8.SS10)\.

## Appendix DInterpolation on Mode Diagonals

In what follows, we provide a mathematical justification for why trajectories never mode interpolate on diagonals across grids:

###### Proposition D\.1\(Diagonal Avoidance on a Grid\)\.

Suppose[Sec\.4](https://arxiv.org/html/2605.06831#S4)holds with the same notation as above, and assume the true modes lie on an axis\-aligned rectangular grid\. Then, if the\(i,j\)\(i,j\)in[Sec\.4](https://arxiv.org/html/2605.06831#S4)corresponds to a diagonal pair of some grid cell, then for everyt≤τ1t\\leq\\tau\_\{1\}:

min⁡\{γ~i,t,γ~j,t\}≤exp⁡\(−κ\),\\min\\\{\\tilde\{\\gamma\}\_\{i,t\},\\tilde\{\\gamma\}\_\{j,t\}\\\}\\leq\\exp\(\-\\kappa\),\(D\.19\), whereγ~i,t\\tilde\{\\gamma\}\_\{i,t\},γ~j,t\\tilde\{\\gamma\}\_\{j,t\}denotes the rescaled responsibilities of modesi,ji,jper[AppendixG](https://arxiv.org/html/2605.06831#A7)\.

Proof Sketch\.We can reduce the problem to a single rectangular cell using the implications of[AppendixG](https://arxiv.org/html/2605.06831#A7)\. Suppose, for contradiction, that the dominant pair\(i,j\)\(i,j\)corresponds to a diagonal of this cell and that[Eq\.D\.19](https://arxiv.org/html/2605.06831#A4.E19)holds for this pair\. Then, using the geometric identity for a rectangle together with[Sec\.4](https://arxiv.org/html/2605.06831#S4), we obtain an upper bound onρ\\rhoof order𝒪​\(e−κ\)\\mathcal\{O\}\(e^\{\-\\kappa\}\)\. A full proof is given in[Sec\.H\.11](https://arxiv.org/html/2605.06831#A8.SS11)\.

Discussion: This proposition is purely geometric, and does not rely on the parallel dynamics\. The responsibility of a mode is its probability density proportional to that of all other modes; in particular, if the minimum ofγ~i,t\\tilde\{\\gamma\}\_\{i,t\},γ~j,t\\tilde\{\\gamma\}\_\{j,t\}is exponentially small, then the trajectory must be very close to the other mode\. In particular, by[Sec\.4](https://arxiv.org/html/2605.06831#S4), the trajectorymust converge to the other mode, since it is confined to Tube\(i,j\)t,ε\{\}\_\{t,\\varepsilon\}^\{\(i,j\)\}\. For example, if modeii’s responsibility is small, then the trajectory must converge to modejj\.

## Appendix EAdditional Experiments

### E\.1Verifying Assumptions:[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.4](https://arxiv.org/html/2605.06831#S4)

In[FigureE\.6](https://arxiv.org/html/2605.06831#A5.F6), for the Gaussian dataset withϖ=2\\varpi=2, we provide results demonstrating that forκ\>\>1\\kappa\>\>1,τ1\\tau\_\{1\}andτ2\\tau\_\{2\}exist and are valid\. In[Sec\.E\.2](https://arxiv.org/html/2605.06831#A5.SS2), we demonstrate that this holds similarly for higher dimensions\. Hence,[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.4](https://arxiv.org/html/2605.06831#S4)hold forκ\>\>1\\kappa\>\>1Note that, consistently,τ2<τ1\\tau\_\{2\}<\\tau\_\{1\}; in[Figure4](https://arxiv.org/html/2605.06831#S5.F4), we show thatτ2\\tau\_\{2\}is well before the time that the instance hits theε\\varepsilon\-tube, as predicted by[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)\. Additionally, inuu\-time per[AppendixG](https://arxiv.org/html/2605.06831#A7),[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.4](https://arxiv.org/html/2605.06831#S4)hold similarly\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x11.png)\(a\)DDIMτ1\\tau\_\{1\}andτ2\\tau\_\{2\}vs\.κ\\kappa\.
![Refer to caption](https://arxiv.org/html/2605.06831v1/x12.png)\(b\)DDPMτ1\\tau\_\{1\}andτ2\\tau\_\{2\}vs\.κ\\kappa\.

Figure E\.6:For both DDIM \([Figure6\(a\)](https://arxiv.org/html/2605.06831#A5.F6.sf1)\) and DDPM \([Figure6\(b\)](https://arxiv.org/html/2605.06831#A5.F6.sf2)\), we plot howτ1\\tau\_\{1\}andτ2\\tau\_\{2\}change vs\.κ\\kappaover 100,000 sampled trajectories, across 500 equidistantκ\\kappaincrements ranging from 1 to 20\. As expected, for largeκ\\kappa,[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[4](https://arxiv.org/html/2605.06831#S4)hold; this yields thatε\\varepsilonis small as well\. Furthermore, as expected, asκ\\kappaincreases, both of these decrease\. Note thatτ2\\tau\_\{2\}is the same across all trajectories, sinceℓ2\\ell^\{2\}is the same across all modei,ji,jpairs, so the standard deviation is 0\. Furthermore,𝔼​\[τ2\]\\mathbb\{E\}\[\\tau\_\{2\}\]is much smoother for DDPM than DDIM, since DDPM has many more sampled timesteps to evaluateτ2\\tau\_\{2\}at; however, since𝔼​\[τ1\]\\mathbb\{E\}\[\\tau\_\{1\}\]average 100,000 differentτ1\\tau\_\{1\}, it is smooth across DDPM and DDIM\.
### E\.2Higher Dimensional Gaussian Mixtures

Throughout, we use2525modes and furthest point sampling, keepingℓ\\ellroughly the same as in theϖ=2\\varpi=2case\. Notably, however, some modes are slightly more or slightly less apart thanℓ\\ell, hence adding uncertainty toτ2\\tau\_\{2\}across trajectories as well\. In[FiguresE\.7](https://arxiv.org/html/2605.06831#A5.F7)and[E\.8](https://arxiv.org/html/2605.06831#A5.F8), we validate[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[4](https://arxiv.org/html/2605.06831#S4)forϖ=2,4,8,16,32,64\\varpi=2,4,8,16,32,64\. We find that, as dimension increases,τ1\\tau\_\{1\}becomes larger \(i\.e\., nearer to the beginning of the reverse process\) andτ2\\tau\_\{2\}becomes smaller \(i\.e\., nearer to the end of the reverse process\) for both DDIM and DDPM, hence corresponding to differentκ\\kappa\(and thus differentε\\varepsilon\)\. Still, our two core assumptions still hold for largeκ\\kappa\. In[FigureE\.9](https://arxiv.org/html/2605.06831#A5.F9), we valuate[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)for both DDIM and DDPM across the sameϖ\\varpiincrements\. For both DDIM and DDPM, we find that exponential convergence still holds\. We also find thatε\\varepsilonincreases for DDIM as the dimension increases\. Furthermore,ε\\varepsilonincreases for DDPM as well, albeit at a much slower rate\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x13.png)\(a\)DDIM
![Refer to caption](https://arxiv.org/html/2605.06831v1/x14.png)\(b\)DDPM

Figure E\.7:τ1\\tau\_\{1\}vs\.κ\\kappafor varying dimensions\. We find thatτ1\\tau\_\{1\}increases as dimension increases across both DDIM and DDPM\.![Refer to caption](https://arxiv.org/html/2605.06831v1/x15.png)\(a\)DDIM
![Refer to caption](https://arxiv.org/html/2605.06831v1/x16.png)\(b\)DDPM

Figure E\.8:τ2\\tau\_\{2\}vs\.κ\\kappafor varying dimensions\. We find thatτ2\\tau\_\{2\}decreases as dimension increases across both DDIM and DDPM\.![Refer to caption](https://arxiv.org/html/2605.06831v1/x17.png)\(a\)DDIM
![Refer to caption](https://arxiv.org/html/2605.06831v1/x18.png)\(b\)DDPM

Figure E\.9:Orthogonal distance toε\\varepsilon\-extendedi,ji,j\-mode segment for DDIM\. We find that, as dimension increases,ε\\varepsilonincreases as well\. Furthermore, the difference in final distances for DDIM and DDPM become more exacerbated as dimension increases\.
### E\.3DDIM Hallucinates More than DDPM For Images

In[FigureE\.10](https://arxiv.org/html/2605.06831#A5.F10), we find that for a UNet denoiser pretrained on our 3 triangle dataset described in[AppendixF](https://arxiv.org/html/2605.06831#A6), across various choices of training epochs, DDIM consistently hallucinates more than DDPM\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x19.png)Figure E\.10:We compare the image hallucination rate of DDIM and DDPM for 1000 epochs of training using a UNet architecture on the 3 triangle dataset\. The evaluation is conducted on 1,000 samples uniformly sampled from𝒩​\(0,1\)\\mathcal\{N\}\(0,1\)Note that the hallucination rate of DDPM is consistently lower than DDIM in this case\.
### E\.4Ablation onτ3\\tau\_\{3\}

In[FigureE\.11](https://arxiv.org/html/2605.06831#A5.F11), we find that asτ3\\tau\_\{3\}increases across samplers, hallucination rate drops, as expected per[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. However, note that a largeτ3\\tau\_\{3\}is unrealistic; empirically, as demonstrated in[Figure4](https://arxiv.org/html/2605.06831#S5.F4),τ3\\tau\_\{3\}is less than 15\. Note that all results are in DDIM steps\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x20.png)\(a\)DDIM
![Refer to caption](https://arxiv.org/html/2605.06831v1/x21.png)\(b\)DDIM \+z=2z=2DDPM
![Refer to caption](https://arxiv.org/html/2605.06831v1/x22.png)\(c\)DDIM \+z=5z=5DDPM
![Refer to caption](https://arxiv.org/html/2605.06831v1/x23.png)\(d\)DDIM \+z=8z=8DDPM
![Refer to caption](https://arxiv.org/html/2605.06831v1/x24.png)\(e\)DDPM

Figure E\.11:Ablation onτ3\\tau\_\{3\}across various DDIM\-DDPM hybrid samplers, starting from the midpoint neighborhood and evaluating hallucination rate\. Uncertainty is quantified by the standard error across mode pairs\.
### E\.5Jacobian Eigenvalues

In[FigureE\.12](https://arxiv.org/html/2605.06831#A5.F12), we show that there exists a positive \(λt\\lambda\_\{t\}\) and negative eigenvalue\. These two eigenvalues are consistently positive \(negative\), demonstrating that the midpoint demonstrating that the midpoint is indeed a saddle\. Thus, we can conclude that score error and studying the approximate parallel dynamics \(rather than the exact parallel dynamics\) only has a small effect on the position of the slowdown point\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x25.png)Figure E\.12:Average DDIM Jacobian eigenvalues evaluated at the midpoint joining mode segments,λt\\lambda\_\{t\}, grouped into positive \(blue\) and negative \(green\) groups, during the reverse process\. A vertical dotted orange line representsτ2\\tau\_\{2\}\. We find that there exists both a positive and corresponding negative eigenvalue after timeτ2\\tau\_\{2\}\. Notably, there were no eigenvalues excluded from this plot; all midpoints had one positive eigenvalue and one negative eigenvalue\. In particular, the midpoint remains a hyperbolic saddle equilibria with appropriate eigenvalues\. This indicates that despite score error and our study of the approximate dynamics, the slowdown region does not greatly shift for a well\-trained diffusion model\. Finally, note that the Jacobian eigenvalues are identical \(up to numerical error\) for DDIM and DDPM, since[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)and[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2)have the same Jacobian\.
### E\.6Varying Noise Levelη\\etain DDIM

![Refer to caption](https://arxiv.org/html/2605.06831v1/x26.png)\(a\)Change in hallucination rate while varying DDIM noise levelη\\eta
![Refer to caption](https://arxiv.org/html/2605.06831v1/x27.png)\(b\)Change in midpoint neighborhood escape while varying DDIM noise levelη\\eta

Figure E\.13:Increasing DDIM noise levelη\\etadecreases mode interpolation hallucination rate \([Figure13\(a\)](https://arxiv.org/html/2605.06831#A5.F13.sf1)\) while helping DDIM trajectories escape the midpoint trapping region \([Figure13\(b\)](https://arxiv.org/html/2605.06831#A5.F13.sf2)\), demonstrating that SDE noise helps with reducing hallucinations, in line with our theoretical results\.In[FigureE\.13](https://arxiv.org/html/2605.06831#A5.F13), we provide a first characterization of how different noise levelsη\\eta\(Song et al\.,[2021a](https://arxiv.org/html/2605.06831#bib.bib29)\)applied to the DDIM steps affect hallucination rate and escape from the midpoint region \(contrasting to adding additional DDPM steps in[Figure5](https://arxiv.org/html/2605.06831#S6.F5)\)\. In line with our results in noise being beneficial for reducing hallucinations, we find that the highest level of noise results in the lowest amount of mode interpolations\. We also provide an analogue of[Figure5](https://arxiv.org/html/2605.06831#S6.F5)for this setting, demonstrating that adding noise to DDIM helps escape the midpoint neighborhood, in line with our result in[Sec\.5](https://arxiv.org/html/2605.06831#S5)\.

### E\.7Verifying Assumptions:[Sec\.5](https://arxiv.org/html/2605.06831#S5)

![Refer to caption](https://arxiv.org/html/2605.06831v1/x28.png)\(a\)K​\(t\)K\(t\)in[Sec\.5](https://arxiv.org/html/2605.06831#S5)
![Refer to caption](https://arxiv.org/html/2605.06831v1/x29.png)\(b\)λrep\\lambda\_\{\\mathrm\{rep\}\}in[Sec\.5](https://arxiv.org/html/2605.06831#S5)

Figure E\.14:K​\(t\)K\(t\)\([Figure14\(a\)](https://arxiv.org/html/2605.06831#A5.F14.sf1)\) andλrep\\lambda\_\{\\mathrm\{rep\}\}\([Figure14\(b\)](https://arxiv.org/html/2605.06831#A5.F14.sf2)\) computed across a total of 400k samples \(10k samples per line segment\) under the exact score\. We find thatK​\(t\)K\(t\)exists andλrep\>0\\lambda\_\{\\mathrm\{rep\}\}\>0as desired\. Details on how to compute these are included in[AppendixF](https://arxiv.org/html/2605.06831#A6)\.In[FigureE\.14](https://arxiv.org/html/2605.06831#A5.F14), we verify the assumptions onb​\(a,t\)b\(a,t\)in[Sec\.5](https://arxiv.org/html/2605.06831#S5)across 400k samples \(10k per each line segment\)\. We find that both assumptions hold, using the exact score presented in our[Sec\.5](https://arxiv.org/html/2605.06831#S5), across all 40 line segments for a total of 400,000 samples\. Details on howb​\(a,t\),K​\(t\),b\(a,t\),\\;K\(t\),andλrep\\lambda\_\{\\mathrm\{rep\}\}are computed are provided in[AppendixF](https://arxiv.org/html/2605.06831#A6)\. Note thatτ3=9\\tau\_\{3\}=9in DDIM steps corresponds toτ3=21\\tau\_\{3\}=21in DDPM steps\.

### E\.8Midpoint Trapping under the Exact Score

![Refer to caption](https://arxiv.org/html/2605.06831v1/x30.png)Figure E\.15:Exact score analogue of[Figure5](https://arxiv.org/html/2605.06831#S6.F5)\. We find that atϑ=0\.15​ℓt\\vartheta=0\.15\\ell\_\{t\}andτ3=3\\tau\_\{3\}=3, DDIM gets stuck in the midpoint neighborhood while DDPM escapes, confirming our theoretical results in[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.5](https://arxiv.org/html/2605.06831#S5)\.In[FigureE\.15](https://arxiv.org/html/2605.06831#A5.F15), we provide an analogue of[Figure5](https://arxiv.org/html/2605.06831#S6.F5)in the exact score case to demonstrate that the midpoint trapping mechanism we extract theoretically in[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.5](https://arxiv.org/html/2605.06831#S5)holds, as expected, under the exact score\. As discussed in[AppendixI](https://arxiv.org/html/2605.06831#A9), this then becomes useful and predictive in the learned score case as demonstrated by[Figure5](https://arxiv.org/html/2605.06831#S6.F5)and[Tab\.I\.1](https://arxiv.org/html/2605.06831#A9.T1)\.

### E\.9Hallucination Rate Ablations onTT,σ\\sigma, andℓ\\ell

![Refer to caption](https://arxiv.org/html/2605.06831v1/x31.png)Figure E\.16:Ablations on hallucination rate gap between DDIM and DDPM across various choices ofσ\\sigma,TT, andℓ\\ell\. We find that the hallucination rate gap between DDIM and DDPM persists across several different hyperparameter configurations\.In[FigureE\.16](https://arxiv.org/html/2605.06831#A5.F16), we demonstrate that the gap in mode interpolation hallucination rate persists across various choices ofTT,σ\\sigma, andℓ\\ell\. In general, hallucination rates remain roughly stable\. AsTTincreases, the gap increases: while discretization error does not explain the gap between DDIM and DDPM hallucinations, DDIM becomes a prohibitively coarse discretization of the PF ODE and thus has a higher hallucination rate\. Asσ\\sigmaincreases, the gap decreases\. This is because regions of high probability mass begin to overlap, so feasibly classifying samples as hallucinations becomes rare: this is precisely why[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[4](https://arxiv.org/html/2605.06831#S4)are required, since they ensure well\-separated modes where is a useful definition of hallucinations\. For similar reasons, asℓ\\elldecreases, the gap decreases\.

## Appendix FExperimental Details

### F\.1Gaussian Mixture

We train our experiments 25\-mode gaussian with varying dimensions\. Our setup contains 25 modes with an inter\-mode separation of22with each mode having a standard deviationσ\\sigmaof0\.020\.02\. Lengths and standard deviations are then normalized by2​ϖ2\\sqrt\{\\varpi\}during sampling\. We sample100,000100,000samples uniformly across all modes and train a 3\-layer MLP with a hidden size of 64 using a batch size of10,00010,000for10,00010,000epochs\. For higher dimensions, modes are sampled from a lattice of5φ5^\{\\varphi\}\. In higher dimensions \(φ\>2\\varphi\>2\) sampling of the 25 modes is done using farthest point sampling\. This is done on set of candidates given bymin⁡\(M,5φ\)\\operatorname\{min\}\(M,5^\{\\varphi\}\), with a value ofM=500,000M=500,000as the maximum number of candidate points\. Additionally, in higher dimensions, we use a stronger training pipeline of Fourier features, embedding the log signal\-to\-noise ratio, and varying depth to ensure approximation of the score\.

During sampling, we choose100,000100,000sampled freshly from a standard normal distribution𝒩​\(0,1\)\\mathcal\{N\}\(0,1\)and run inference on those samples in the reverse process\. Furthermore, in2​D2D, we eliminate invalid samples by removing those trajectories for which the final distance to the line joining two modes is greater than5​σ5\\sigma\. These are those trajectories for which the outcome is neither a mode interpolation nor a convergence to one of the true modes\. In the 2D Gaussian, this amounts to<0\.01%<0\.01\\%of the total number of samples used\. Mode interpolations in2​D2Dare thus samples which are within5​σ5\\sigmaof a nearby line but further than5​σ5\\sigmafrom any mode\. A lower threshold would result in false true samples with probability≈1\\approx 1across 100,000 samples\. For higher dimensional Gaussian datasets\(\>10\)\(\>10\), to handle concentration of measure, we use a threshold ofσ​\(ϖ\+4\)\\sigma\(\\sqrt\{\\varpi\}\+4\)instead\. Additionally, for the analysis of exponential convergence, we find that this threshold is hard to tune, since initial distances scale withϖ\\sqrt\{\\varpi\}; thus, any instance which is initially further thanϖ\\sqrt\{\\varpi\}at the beginning of the reverse process is removed\. This also amounts to a negligible amount of samples\.

### F\.2Counting Triangles

We create a dataset of 10,000 images of size64×6464\\times 64with 3 triangles placed randomly in the image\. The locations are chosen such that the images are non\-overlapping\. The goal is to faithfully generate images that maintain the number of triangles in the generated image with no constraint on the position of the triangles in the images generated\. Example images are provided in[FigureF\.17](https://arxiv.org/html/2605.06831#A6.F17)\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x32.png)
![Refer to caption](https://arxiv.org/html/2605.06831v1/x33.png)
![Refer to caption](https://arxiv.org/html/2605.06831v1/x34.png)

Figure F\.17:Samples chosen from the dataset of10,00010,000elements containing 3 triangles\. The positions are chosen such that the triangles are non\-overlapping\.
### F\.3Computing[Sec\.5](https://arxiv.org/html/2605.06831#S5)Objects

Note that we can write𝒙t=𝒎\+At​𝒖\+𝒛t\\bm\{x\}\_\{t\}=\\bm\{m\}\+A\_\{t\}\\bm\{u\}\+\\bm\{z\}\_\{t\}where⟨𝒛t,𝒖⟩=0\\langle\\bm\{z\}\_\{t\},\\bm\{u\}\\rangle=0and‖𝒛t‖2≤ε\|\|\\bm\{z\}\_\{t\}\|\|\_\{2\}\\leq\\varepsilon, since the trajectory is restricted to theε\\varepsilon\-tube\. This thus allows us to computeb​\(At,t\)b\(A\_\{t\},t\)with respect toAtA\_\{t\}\. Note that these requireb​\(a,t\)b\(a,t\)to be negative \(positive\) fora≥ϑa\\geq\\vartheta\(a≤−ϑa\\leq\-\\vartheta\), sinceb​\(a,t\)b\(a,t\)is moving in reverse time in the statement of[Sec\.5](https://arxiv.org/html/2605.06831#S5)\. Next, we check the two assumptions of[Sec\.5](https://arxiv.org/html/2605.06831#S5)across all line segments and across 50 intervals ofaaon each side of the line segment until the endpoints ofaa\. We also sample𝒛t\\bm\{z\}\_\{t\}uniformly in\[−ε,ε\]\[\-\\varepsilon,\\varepsilon\]at each evaluation ofaa\. For the assumption with respect toK​\(t\)K\(t\), the endpoints are\[−2​ϑ,2​ϑ\]\[\-2\\vartheta,2\\vartheta\]\. For the assumption with respect toλrep\\lambda\_\{\\mathrm\{rep\}\}, from the proof of[Sec\.5](https://arxiv.org/html/2605.06831#S5), note that the assumption withK​\(t\)K\(t\)bounds the probability that the trajectory stays in a2​ϑ2\\vartheta\-neighborhood and theλrep\\lambda\_\{\\mathrm\{rep\}\}assumption bounds the probability that the trajectory returns after exit\. Thus, in[Sec\.E\.7](https://arxiv.org/html/2605.06831#A5.SS7), we verify the assumption withλrep\\lambda\_\{\\mathrm\{rep\}\}for\|a\|∈\[ϑ,a∗\]\|a\|\\in\[\\vartheta,a^\{\*\}\], wherea∗a^\{\*\}is when return after exit happens0%0\\%of the time and DDPM trajectories interpolate with rate<0\.001<0\.001withτ3\\tau\_\{3\}steps remaining\.\[ϑ,a∗\]\[\\vartheta,a^\{\*\}\]is thus the region where return after exit becomes infeasible, which is precisely where the assumption onλrep\\lambda\_\{\\mathrm\{rep\}\}must hold\. In[Sec\.E\.7](https://arxiv.org/html/2605.06831#A5.SS7),a∗=0\.280a^\{\*\}=0\.280whereϑ=0\.15\\vartheta=0\.15\(nondimensionalized\)\.

### F\.4Additional Figures

In[FigureF\.18](https://arxiv.org/html/2605.06831#A6.F18), we illustrate the reduction to the DDPM dynamics along the bisector normal which is integral for our theory in[Sec\.5](https://arxiv.org/html/2605.06831#S5)\.

![Refer to caption](https://arxiv.org/html/2605.06831v1/x35.png)Figure F\.18:Reduction to 1D dynamics along the bisector normal\.Top: InsideTubet,ε\(i,j\)\\mathrm\{Tube\}^\{\(i,j\)\}\_\{t,\\varepsilon\}, the bisector hyperplaneH\(i,j\)H^\{\(i,j\)\}\(red\) is orthogonal toL\(i,j\)L^\{\(i,j\)\}\. The signed 1D coordinateAt=⟨xt−μ\(i\)\+μ\(j\)2,u⟩A\_\{t\}=\\langle x\_\{t\}\-\\frac\{\\mu^\{\(i\)\}\+\\mu^\{\(j\)\}\}\{2\},u\\ranglemeasures displacement fromH\(i,j\)H^\{\(i,j\)\}along the unit normaluu\.Bottom: The 1D reduction with modes at±ℓ2\\pm\\frac\{\\ell\}\{2\}, theϑ\\vartheta\-neighborhood around the midpoint, and unstable drift directions away from the bisector\.

## Appendix GHelpful Lemmas

###### Lemma G\.1\.

Given the diffusion marginals as defined in[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2), we can rewrite the score as:

∇𝒙tlog⁡pt​\(𝒙t\)=−1σt2​\(𝒙t−∑k=1Nγk​\(𝒙t\)​𝝁t\(k\)\)\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)=\-\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\left\(\\bm\{x\}\_\{t\}\-\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\\right\)\(G\.20\)
for the responsibilitiesγk​\(𝐱t\)=πk​𝒩​\(𝐱t;𝛍t\(k\),σt2​𝐈\)∑k=1Nπk​𝒩​\(𝐱t;𝛍t\(k\),σt2​𝐈\)\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)=\\frac\{\\pi\_\{k\}\\mathcal\{N\}\(\\bm\{x\}\_\{t\};\\bm\{\\mu\}\_\{t\}^\{\(k\)\},\\sigma\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\sum\_\{k=1\}^\{N\}\\pi\_\{k\}\\mathcal\{N\}\(\\bm\{x\}\_\{t\};\\bm\{\\mu\}\_\{t\}^\{\(k\)\},\\sigma\_\{t\}^\{2\}\\bm\{I\}\)\}, whereσt2=σ2​α¯t\+\(1−α¯t\)\\sigma\_\{t\}^\{2\}=\\sigma^\{2\}\\bar\{\\alpha\}\_\{t\}\+\(1\-\\bar\{\\alpha\}\_\{t\}\)\.

###### Proof\.

Firstly, we compute the gradient of the diffusion marginal as:

∇𝒙tpt​\(𝒙t\)\\displaystyle\\nabla\_\{\\bm\{x\}\_\{t\}\}p\_\{t\}\(\\bm\{x\}\_\{t\}\)=∑k=1Nπk​∇𝒙t𝒩​\(𝒙t;𝝁t\(k\),σt2​𝑰\)\\displaystyle=\\sum\_\{k=1\}^\{N\}\\pi\_\{k\}\\nabla\_\{\\bm\{x\}\_\{t\}\}\\mathcal\{N\}\(\\bm\{x\}\_\{t\};\\bm\{\\mu\}\_\{t\}^\{\(k\)\},\\sigma\_\{t\}^\{2\}\\bm\{I\}\)\(G\.21\)=∑k=1Nπk​∇𝒙t𝒩​\(𝒙t;𝝁t\(k\),σt2​𝑰\)​∇𝒙t\(−‖𝒙t−𝝁t\(k\)‖222​σt2\)\\displaystyle=\\sum\_\{k=1\}^\{N\}\\pi\_\{k\}\\nabla\_\{\\bm\{x\}\_\{t\}\}\\mathcal\{N\}\(\\bm\{x\}\_\{t\};\\bm\{\\mu\}\_\{t\}^\{\(k\)\},\\sigma\_\{t\}^\{2\}\\bm\{I\}\)\\nabla\_\{\\bm\{x\}\_\{t\}\}\\left\(\-\\frac\{\|\|\\bm\{x\}\_\{t\}\-\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\|\|\_\{2\}^\{2\}\}\{2\\sigma\_\{t\}^\{2\}\}\\right\)\(G\.22\)=∑k=1Nπk​∇𝒙t𝒩​\(𝒙t;𝝁t\(k\),σt2​𝑰\)​\(−𝒙t−𝝁t\(k\)σt2\)\\displaystyle=\\sum\_\{k=1\}^\{N\}\\pi\_\{k\}\\nabla\_\{\\bm\{x\}\_\{t\}\}\\mathcal\{N\}\(\\bm\{x\}\_\{t\};\\bm\{\\mu\}\_\{t\}^\{\(k\)\},\\sigma\_\{t\}^\{2\}\\bm\{I\}\)\\left\(\-\\frac\{\\bm\{x\}\_\{t\}\-\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\}\{\\sigma\_\{t\}^\{2\}\}\\right\)\(G\.23\)
Then, incorporating[Eq\.G\.23](https://arxiv.org/html/2605.06831#A7.E23)into the definition of the score yields:

∇𝒙tlog⁡pt​\(𝒙t\)\\displaystyle\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)=∇𝒙𝒕pt​\(𝒙t\)pt​\(𝒙t\)\\displaystyle=\\frac\{\\nabla\_\{\\bm\{x\_\{t\}\}\}p\_\{t\}\(\\bm\{x\}\_\{t\}\)\}\{p\_\{t\}\(\\bm\{x\}\_\{t\}\)\}\(G\.24\)=−1σt2​∑k=1Nπk​𝒩​\(𝒙t;𝝁t\(k\),σt2​𝑰\)pt​\(𝒙t\)​\(𝒙t−𝝁t\(k\)\)\\displaystyle=\-\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\sum\_\{k=1\}^\{N\}\\frac\{\\pi\_\{k\}\\mathcal\{N\}\(\\bm\{x\}\_\{t\};\\bm\{\\mu\}\_\{t\}^\{\(k\)\},\\sigma\_\{t\}^\{2\}\\bm\{I\}\)\}\{p\_\{t\}\(\\bm\{x\}\_\{t\}\)\}\(\\bm\{x\}\_\{t\}\-\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\)\(G\.25\)=−1σt2​∑k=1Nγk​\(𝒙t\)​\(𝒙t−𝝁t\(k\)\),definition ofγk\\displaystyle=\-\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\(\\bm\{x\}\_\{t\}\-\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\),\\text\{ definition of $\\gamma\_\{k\}$\}\(G\.26\)=−1σt2​\(∑k=1Nγk​\(𝒙t\)​\(𝒙t\)−∑k=1Nγk​\(𝒙t\)​𝝁t\(k\)\)\\displaystyle=\-\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\left\(\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\(\\bm\{x\}\_\{t\}\)\-\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\\right\)\(G\.27\)=−1σt2​\(𝒙t−∑k=1Nγk​\(𝒙t\)​𝝁t\(k\)\),since∑k=1Nγk​\(𝒙t\)=1\\displaystyle=\-\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\left\(\\bm\{x\}\_\{t\}\-\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\\right\),\\;\\text\{since $\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)=1$\}\(G\.28\)
as desired\. ∎

###### Lemma G\.2\.

Let𝐲t=𝐱tα¯t\\bm\{y\}\_\{t\}=\\frac\{\\bm\{x\}\_\{t\}\}\{\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}, where𝐱t\\bm\{x\}\_\{t\}is governed by[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3), and let𝛍^​\(𝐲t\)=∑i=1Nγ~i​\(𝐲t\)​𝛍\(i\)\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)=\\sum\_\{i=1\}^\{N\}\\tilde\{\\gamma\}\_\{i\}\(\\bm\{y\}\_\{t\}\)\\bm\{\\mu\}^\{\(i\)\}with new responsibilities:

γ~k​\(𝒚t\)=πk​𝒩​\(𝒚t;𝝁\(𝒌\),σ~t2​𝑰\)∑m=1Nπm​𝒩​\(𝒚t;𝝁\(𝒎\),σ~t2​𝑰\)\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)=\\frac\{\\pi\_\{k\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu^\{\(k\)\}\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\sum\_\{m=1\}^\{N\}\\pi\_\{m\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu^\{\(m\)\}\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\(G\.29\)
andσ~t2=σt2α¯t\\tilde\{\\sigma\}\_\{t\}^\{2\}=\\frac\{\\sigma\_\{t\}^\{2\}\}\{\\bar\{\\alpha\}\_\{t\}\}\.

Next, define the time\-rescalinguuas:

d​ud​t=βt2​σt2\\frac\{du\}\{dt\}=\\frac\{\\beta\_\{t\}\}\{2\\sigma\_\{t\}^\{2\}\}\(G\.30\)
Then, lettingt=ut=uWLOG, we have the rescaled dynamics:

𝒚t˙=𝝁^​\(𝒚t\)−𝒚t\\dot\{\\bm\{y\}\_\{t\}\}=\\bm\{\\hat\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\-\\bm\{y\}\_\{t\}\(G\.31\)

###### Proof\.

Firstly, letI​\(t\)=−∫0tβs​𝑑sI\(t\)=\-\\int\_\{0\}^\{t\}\\beta\_\{s\}ds, i\.e\.α¯t=eI​\(t\)\\bar\{\\alpha\}\_\{t\}=e^\{I\(t\)\}\. This yields that:

α¯˙t\\displaystyle\\dot\{\\bar\{\\alpha\}\}\_\{t\}=\(eI​\(t\)\)˙\\displaystyle=\\dot\{\(e^\{I\(t\)\}\)\}\(G\.32\)=eI​\(t\)​I​\(t\)˙\\displaystyle=e^\{I\(t\)\}\\dot\{I\(t\)\}\(G\.33\)=eI​\(t\)​\(−βt\)\\displaystyle=e^\{I\(t\)\}\(\-\\beta\_\{t\}\)\(G\.34\)=−βt​α¯t\\displaystyle=\-\\beta\_\{t\}\\bar\{\\alpha\}\_\{t\}\(G\.35\)
Next, by definition of𝒚t\\bm\{y\}\_\{t\}, we have that:

𝒚˙t\\displaystyle\\dot\{\\bm\{y\}\}\_\{t\}=𝒙t˙​α¯t−1/2\+𝒙t​dd​t​α¯t−1/2\\displaystyle=\\dot\{\\bm\{x\}\_\{t\}\}\\bar\{\\alpha\}\_\{t\}^\{\-1/2\}\+\\bm\{x\}\_\{t\}\\frac\{d\}\{dt\}\\bar\{\\alpha\}\_\{t\}^\{\-1/2\}\(G\.36\)=𝒙t˙​α¯t−1/2\+𝒙t​\(βt2​α¯t\)\\displaystyle=\\dot\{\\bm\{x\}\_\{t\}\}\\bar\{\\alpha\}\_\{t\}^\{\-1/2\}\+\\bm\{x\}\_\{t\}\\left\(\\frac\{\\beta\_\{t\}\}\{2\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\right\)\(G\.37\)=1α¯t​\(−βt2​\(𝒙t\+∇𝒙tlog⁡pt​\(𝒙t\)\)\)\+βt2​α¯t​𝒙t\\displaystyle=\\frac\{1\}\{\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\left\(\-\\frac\{\\beta\_\{t\}\}\{2\}\\left\(\\bm\{x\}\_\{t\}\+\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)\\right\)\\right\)\+\\frac\{\\beta\_\{t\}\}\{2\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\bm\{x\}\_\{t\}\(G\.38\)=−βt2​α¯t​𝒙t−βt2​α¯t​∇𝒙tlog⁡pt​\(𝒙t\)\+βt2​α¯t​𝒙t\\displaystyle=\\frac\{\-\\beta\_\{t\}\}\{2\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\bm\{x\}\_\{t\}\-\\frac\{\\beta\_\{t\}\}\{2\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)\+\\frac\{\\beta\_\{t\}\}\{2\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\bm\{x\}\_\{t\}\(G\.39\)=−βt2​α¯t​∇𝒙tlog⁡pt​\(𝒙t\)\\displaystyle=\-\\frac\{\\beta\_\{t\}\}\{2\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)\(G\.40\)=−βt2​α¯t​\(−1σt2​\(𝒙t−∑k=1Nγk​\(𝒙t\)​𝝁t\(k\)\)\),[AppendixG](https://arxiv.org/html/2605.06831#A7)\\displaystyle=\-\\frac\{\\beta\_\{t\}\}\{2\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\left\(\-\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\(\\bm\{x\}\_\{t\}\-\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\)\\right\),\\;\\text\{\\lx@cref\{creftype~refnum\}\{lemma:cvx\_comb\_of\_repsonsibilities\}\}\(G\.41\)=βt2​σt2​α¯t​\(∑k=1Nγk​\(𝒙t\)​𝝁t\(k\)−𝒙t\)\\displaystyle=\\frac\{\\beta\_\{t\}\}\{2\\sigma\_\{t\}^\{2\}\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\left\(\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\\bm\{\\mu\}\_\{t\}^\{\(k\)\}\-\\bm\{x\}\_\{t\}\\right\)\(G\.42\)=βt2​σt2​α¯t​\(α¯t​∑k=1Nγk​\(𝒙t\)​𝝁\(k\)−α¯t​𝒚t\)\\displaystyle=\\frac\{\\beta\_\{t\}\}\{2\\sigma\_\{t\}^\{2\}\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\\left\(\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\sum\_\{k=1\}^\{N\}\\gamma\_\{k\}\(\\bm\{x\}\_\{t\}\)\\bm\{\\mu\}^\{\(k\)\}\-\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\bm\{y\}\_\{t\}\\right\)\(G\.43\)=βt2​σt2​\(𝝁^​\(𝒚t\)−𝒚t\)\\displaystyle=\\frac\{\\beta\_\{t\}\}\{2\\sigma^\{2\}\_\{t\}\}\\left\(\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\-\\bm\{y\}\_\{t\}\\right\)\(G\.44\)
The chain rule yields the time rescaling:

d​𝒚td​u\\displaystyle\\frac\{d\\bm\{y\}\_\{t\}\}\{du\}=d​𝒚td​t​d​td​u\\displaystyle=\\frac\{d\\bm\{y\}\_\{t\}\}\{dt\}\\frac\{dt\}\{du\}\(G\.45\)=\(βt2​σt2​\(𝝁^​\(𝒚t\)−𝒚t\)\)​\(2​σt2βt\)\\displaystyle=\\left\(\\frac\{\\beta\_\{t\}\}\{2\\sigma\_\{t\}^\{2\}\}\\left\(\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\-\\bm\{y\}\_\{t\}\\right\)\\right\)\(\\frac\{2\\sigma\_\{t\}^\{2\}\}\{\\beta\_\{t\}\}\)\(G\.46\)=𝝁^​\(𝒚t\)−𝒚t\\displaystyle=\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\-\\bm\{y\}\_\{t\}\(G\.47\)
as desired\. ∎

Remark: This lemma takes the nonautonomous[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3), where the means evolve over time, to dynamics with respect to thestatic means\.

###### Lemma G\.3\.

Assume[Sec\.4](https://arxiv.org/html/2605.06831#S4)holds\. Considerτ1\\tau\_\{1\}from[Sec\.4](https://arxiv.org/html/2605.06831#S4)\.\. We decompose𝐲t\\bm\{y\}\_\{t\}into parallel and orthogonal components i\.e\.:

𝒚t=𝝁\(i\)\+ξt​ℓ​𝒖\+𝒘t,\\bm\{y\}\_\{t\}=\\bm\{\\mu\}^\{\(i\)\}\+\\xi\_\{t\}\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\},\(G\.48\)
whereℓ=‖𝛍\(j\)−𝛍\(i\)‖2,𝐮=𝛍\(j\)−𝛍\(i\)ℓ\\ell=\|\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\},\\bm\{u\}=\\frac\{\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\}\{\\ell\}, and𝐰t\\bm\{w\}\_\{t\}satisfies⟨𝐰t,𝐮⟩=0\\langle\\bm\{w\}\_\{t\},\\bm\{u\}\\rangle=0\.

Then, fort≤τ1t\\leq\\tau\_\{1\}, we have that:

𝒚˙=𝝁^​\(𝒚t\)−𝒚t\\dot\{\\bm\{y\}\}=\\bm\{\\hat\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\-\\bm\{y\}\_\{t\}\(G\.49\)
is equivalent to the dynamical system:

ξ˙t\\displaystyle\\dot\{\\xi\}\_\{t\}=γ~j​\(𝒚t\)−ξt\+δξt\\displaystyle=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\+\\delta\_\{\\xi\_\{t\}\}\(G\.50\)𝒘t˙\\displaystyle\\dot\{\\bm\{w\}\_\{t\}\}=−𝒘t\+𝜹𝒘t\\displaystyle=\-\\bm\{w\}\_\{t\}\+\\bm\{\\delta\}\_\{\\bm\{w\}\_\{t\}\}\(G\.51\)
where:

\|δξ\|,‖𝜹𝒘‖2≤𝒪​\(N​exp⁡\(−κ\)\)\|\\delta\_\{\\xi\}\|,\|\|\\bm\{\\delta\}\_\{\\bm\{w\}\}\|\|\_\{2\}\\leq\\mathcal\{O\}\\left\(N\\exp\(\-\\kappa\)\\right\)\(G\.52\)

###### Proof\.

Firstly, we have that:

𝒚˙t=∑k=1Nγ~k​\(𝒚t\)​𝝁\(k\)−𝒚t\\dot\{\\bm\{y\}\}\_\{t\}=\\sum\_\{k=1\}^\{N\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{y\}\_\{t\}\(G\.53\)
if and only if, expanding with[Eq\.G\.48](https://arxiv.org/html/2605.06831#A7.E48):

ℓ​ξ˙t​𝒖\+𝒘˙t=∑k=1Nγ~k​\(𝒚t\)​𝝁\(k\)−\(𝝁\(i\)\+ξt​ℓ​𝒖\+𝒘t\)\.\\ell\\dot\{\\xi\}\_\{t\}\\bm\{u\}\+\\dot\{\\bm\{w\}\}\_\{t\}=\\sum\_\{k=1\}^\{N\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\\bm\{\\mu\}^\{\(k\)\}\-\(\\bm\{\\mu\}^\{\(i\)\}\+\\xi\_\{t\}\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\)\.\(G\.54\)
Next, we have thatγ~i​\(𝒚t\)​\(𝝁\(i\)−𝝁\(i\)\)=0\\tilde\{\\gamma\}\_\{i\}\(\\bm\{y\}\_\{t\}\)\(\\bm\{\\mu\}^\{\(i\)\}\-\\bm\{\\mu\}^\{\(i\)\}\)=0;γ~j​\(𝒚t\)​\(𝝁\(j\)−𝝁\(i\)\)=γ~j​\(𝒚t\)​ℓ​𝒖\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\(\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\)=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\ell\\bm\{u\}; and𝑬t:=∑k∉\{i,j\}γ~k​\(𝒚t\)​\(𝝁\(k\)−𝝁\(i\)\)\\bm\{E\}\_\{t\}:=\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\), which when incorporated into[Eq\.G\.54](https://arxiv.org/html/2605.06831#A7.E54)yields:

ℓ​ξ˙t​𝒖\+𝒘˙t\\displaystyle\\ell\\dot\{\\xi\}\_\{t\}\\bm\{u\}\+\\dot\{\\bm\{w\}\}\_\{t\}=γ~j​\(𝒚t\)​ℓ​𝒖\+𝑬t−ξt​ℓ​𝒖−𝒘t\\displaystyle=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\ell\\bm\{u\}\+\\bm\{E\}\_\{t\}\-\\xi\_\{t\}\\ell\\bm\{u\}\-\\bm\{w\}\_\{t\}\(G\.55\)=\(γ~j​\(𝒚t\)−ξt\)​ℓ​𝒖−𝒘t\+𝑬t\.\\displaystyle=\(\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\)\\ell\\bm\{u\}\-\\bm\{w\}\_\{t\}\+\\bm\{E\}\_\{t\}\.\(G\.56\)
Next, noting that⟨𝒖,𝒖⟩=1\\langle\\bm\{u\},\\bm\{u\}\\rangle=1, it follows from[Eq\.G\.56](https://arxiv.org/html/2605.06831#A7.E56)that:

⟨ℓ​ξ˙t​𝒖,𝒖⟩=⟨\(γ~j​\(𝒚t\)−ξt\)​ℓ​𝒖−𝒘t\+𝑬t,𝒖⟩\\displaystyle\\langle\\ell\\dot\{\\xi\}\_\{t\}\\bm\{u\},\\bm\{u\}\\rangle=\\langle\(\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\)\\ell\\bm\{u\}\-\\bm\{w\}\_\{t\}\+\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle⇔\\displaystyle\\iff\(G\.57\)ℓ​ξ˙t=\(γ~j​\(𝒚t\)−ξt\)​ℓ\+⟨𝑬t,𝒖⟩\\displaystyle\\ell\\dot\{\\xi\}\_\{t\}=\(\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\)\\ell\+\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle⇔\\displaystyle\\iff\(G\.58\)ξ˙t=γ~j​\(𝒚t\)−ξt\+δξt,δξt:=1ℓ​⟨𝑬t,𝒖⟩\\displaystyle\\dot\{\\xi\}\_\{t\}=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\+\\delta\_\{\\xi\_\{t\}\},\\;\\delta\_\{\\xi\_\{t\}\}:=\\frac\{1\}\{\\ell\}\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle\(G\.59\)
yielding the first equation in[Eq\.G\.51](https://arxiv.org/html/2605.06831#A7.E51), the parallel dynamics alongLt,ε\(i,j\)L\_\{t,\\varepsilon\}^\{\(i,j\)\}\. Furthermore, by[Eq\.G\.58](https://arxiv.org/html/2605.06831#A7.E58), we have that:

ℓ​ξ˙t​𝒖=\(γ~j​\(𝒚t\)−ξt\)​ℓ​𝒖\+⟨𝑬t,𝒖⟩​𝒖\\ell\\dot\{\\xi\}\_\{t\}\\bm\{u\}=\(\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\)\\ell\\bm\{u\}\+\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle\\bm\{u\}\(G\.60\)
Thus, subtractingℓ​ξ˙t​𝒖\\ell\\dot\{\\xi\}\_\{t\}\\bm\{u\}from the RHS of[Eq\.G\.54](https://arxiv.org/html/2605.06831#A7.E54)and then plugging in[Eq\.G\.60](https://arxiv.org/html/2605.06831#A7.E60)yields:

𝒘˙t\\displaystyle\\dot\{\\bm\{w\}\}\_\{t\}=\(γ~j​\(𝒚t\)−ξt\)​ℓ​𝒖−𝒘t\+𝑬t−\(\(γ~j​\(𝒚t\)−ξt\)​ℓ​𝒖\+⟨𝑬t,𝒖⟩​𝒖\)\\displaystyle=\(\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\)\\ell\\bm\{u\}\-\\bm\{w\}\_\{t\}\+\\bm\{E\}\_\{t\}\-\\left\(\(\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\)\\ell\\bm\{u\}\+\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle\\bm\{u\}\\right\)\(G\.61\)=−𝒘t\+𝑬t−⟨𝑬t,𝒖⟩​𝒖\\displaystyle=\-\\bm\{w\}\_\{t\}\+\\bm\{E\}\_\{t\}\-\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle\\bm\{u\}\(G\.62\)
Note that⟨𝑬t,𝒖⟩𝒖=Proj\(𝒖\)\(𝑬t\)=:Proj∥\(𝑬t\)\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle\\bm\{u\}=\\text\{Proj\}\_\{\(\\bm\{u\}\)\}\(\\bm\{E\}\_\{t\}\)=:\\text\{Proj\}\_\{\\parallel\}\(\\bm\{E\}\_\{t\}\), yielding that𝑬t−⟨𝑬t,𝒖⟩𝒖=:Proj⟂\(𝑬t\)\\bm\{E\}\_\{t\}\-\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle\\bm\{u\}=:\\text\{Proj\}\_\{\\perp\}\(\\bm\{E\}\_\{t\}\)\. Incorporating this into[Eq\.G\.62](https://arxiv.org/html/2605.06831#A7.E62)yields:

𝒘˙t=−𝒘t\+𝜹𝒘t,𝜹𝒘t:=Proj⟂​\(𝑬t\)\.\\dot\{\\bm\{w\}\}\_\{t\}=\-\\bm\{w\}\_\{t\}\+\\bm\{\\delta\}\_\{\\bm\{w\}\_\{t\}\},\\;\\bm\{\\delta\}\_\{\\bm\{w\}\_\{t\}\}:=\\text\{Proj\}\_\{\\perp\}\(\\bm\{E\}\_\{t\}\)\.\(G\.63\)
This yields the second equation in[Eq\.G\.51](https://arxiv.org/html/2605.06831#A7.E51), the orthogonal dynamics with respect toLt,ε\(i,j\)L\_\{t,\\varepsilon\}^\{\(i,j\)\}\. To prove the lemma, it thus suffices to bound\|δξt\|\|\\delta\_\{\\xi\_\{t\}\}\|and‖𝜹𝒘t‖2\|\|\\bm\{\\delta\}\_\{\\bm\{w\}\_\{t\}\}\|\|\_\{2\}\. To do so, we begin by bounding the responsibilitiesγ~k​\(𝒚t\)\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)fork∉\{i,j\}k\\notin\\\{i,j\\\}:

γ~k​\(𝒚t\)\\displaystyle\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)=πk​𝒩​\(𝒚t;𝝁\(k\),σ~t2​𝑰\)∑m=1Nπm​𝒩​\(𝒚t;𝝁\(m\),σ~t2​𝑰\)\\displaystyle=\\frac\{\\pi\_\{k\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(k\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\sum\_\{m=1\}^\{N\}\\pi\_\{m\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(m\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\(G\.64\)≤πk​𝒩​\(𝒚t;𝝁\(k\),σ~t2​𝑰\)πi​𝒩​\(𝒚t;𝝁\(i\),σ~t2​𝑰\)\+πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)\.\\displaystyle\\leq\\frac\{\\pi\_\{k\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(k\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(i\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\+\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\.\(G\.65\)
Letd∗:=arg⁡mind∈\{i,j\}​‖𝒚t−𝝁\(d\)‖2d^\{\*\}:=\\arg\\min\_\{d\\in\\\{i,j\\\}\}\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(d\)\}\|\|\_\{2\}\. Note that:

πi​𝒩​\(𝒚t;𝝁\(i\),σ~t2​𝑰\)\+πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)\\displaystyle\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(i\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\+\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)≥max⁡\{πi​𝒩​\(𝒚t;𝝁\(i\),σ~t2​𝑰\),πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)\}\\displaystyle\\geq\\max\\\{\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(i\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\),\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\\\}\(G\.66\)≥πd∗​𝒩​\(𝒚t;𝝁\(d∗\),σ~t2​𝑰\)\\displaystyle\\geq\\pi\_\{d^\{\*\}\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(d^\{\*\}\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\(G\.67\)
Incorporating[Eq\.G\.67](https://arxiv.org/html/2605.06831#A7.E67)into[Eq\.G\.65](https://arxiv.org/html/2605.06831#A7.E65)yields:

γ~k​\(𝒚t\)\\displaystyle\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)≤πk​𝒩​\(𝒚t;𝝁\(k\),σ~t2​𝑰\)πd∗​𝒩​\(𝒚t;𝝁\(d∗\),σ~t2​𝑰\)\\displaystyle\\leq\\frac\{\\pi\_\{k\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(k\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\pi\_\{d^\{\*\}\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(d^\{\*\}\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\(G\.68\)=πkπd∗​exp⁡\(−‖𝒚t−𝝁\(k\)‖222​σ~t2\)exp⁡\(−‖𝒚t−𝝁\(d∗\)‖222​σ~t2\)\\displaystyle=\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\frac\{\\exp\(\-\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(k\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\)\}\{\\exp\(\-\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(d^\{\*\}\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\)\}\(G\.69\)=πkπd∗​exp⁡\(−12​σ~t2​\(‖𝒚t−𝝁\(k\)‖22−‖𝒚t−𝝁\(d∗\)‖22\)\)\.\\displaystyle=\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\\left\(\-\\frac\{1\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\(\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(k\)\}\|\|\_\{2\}^\{2\}\-\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(d^\{\*\}\)\}\|\|\_\{2\}^\{2\}\)\\right\)\.\(G\.70\)
By[Sec\.4](https://arxiv.org/html/2605.06831#S4), we have that‖𝒚t−𝝁\(k\)‖22≥‖𝒚t−𝝁\(d∗\)‖22\+Δtα¯t\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(k\)\}\|\|\_\{2\}^\{2\}\\geq\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(d^\{\*\}\)\}\|\|\_\{2\}^\{2\}\+\\frac\{\\Delta\_\{t\}\}\{\\bar\{\\alpha\}\_\{t\}\}, which holds if and only if‖𝒚t−𝝁\(k\)‖22−‖𝒚t−𝝁\(d∗\)‖22≥Δtα¯t\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(k\)\}\|\|\_\{2\}^\{2\}\-\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(d^\{\*\}\)\}\|\|\_\{2\}^\{2\}\\geq\\frac\{\\Delta\_\{t\}\}\{\\bar\{\\alpha\}\_\{t\}\}\. Incorporating this into[Eq\.G\.70](https://arxiv.org/html/2605.06831#A7.E70)yields:

γ~k​\(𝒚t\)\\displaystyle\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)≤πkπd∗​exp⁡\(−Δt2​α¯t​σ~t2\)\\displaystyle\\leq\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\frac\{\\Delta\_\{t\}\}\{2\\bar\{\\alpha\}\_\{t\}\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\)\(G\.71\)≤πkπd∗​exp⁡\(−κ\),[Sec\.4](https://arxiv.org/html/2605.06831#S4)\\displaystyle\\leq\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\),\\text\{ \\lx@cref\{creftype~refnum\}\{assump:two\_mode\_dominance\}\}\(G\.72\)
Next, we have that:

\|δξt\|=\|1ℓ​⟨𝑬t,𝒖⟩\|\\displaystyle\|\\delta\_\{\\xi\_\{t\}\}\|=\|\\frac\{1\}\{\\ell\}\\langle\\bm\{E\}\_\{t\},\\bm\{u\}\\rangle\|\(G\.73\)≤1ℓ​‖𝑬t‖2​‖𝒖‖2,Cauchy\-Schwartz\\displaystyle\\leq\\frac\{1\}\{\\ell\}\|\|\\bm\{E\}\_\{t\}\|\|\_\{2\}\|\|\\bm\{u\}\|\|\_\{2\},\\text\{ Cauchy\-Schwartz\}\(G\.74\)=1ℓ​‖𝑬t‖2\\displaystyle=\\frac\{1\}\{\\ell\}\|\|\\bm\{E\}\_\{t\}\|\|\_\{2\}\(G\.75\)=1ℓ​‖∑k∉\{i,j\}γ~k​\(𝒚t\)​\(𝝁\(k\)−𝝁\(i\)\)‖2\\displaystyle=\\frac\{1\}\{\\ell\}\|\|\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\)\|\|\_\{2\}\(G\.76\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2ℓ​∑k∉\{i,j\}γ~k​\(𝒚t\)\\displaystyle\\leq\\frac\{\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\}\{\\ell\}\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(G\.77\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2ℓ​\(N−2\)​πkπd∗​exp⁡\(−κ\),[Eq\.G\.72](https://arxiv.org/html/2605.06831#A7.E72)\\displaystyle\\leq\\frac\{\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\}\{\\ell\}\(N\-2\)\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\),\\text\{ \\lx@cref\{creftype~refnum\}\{eq:bound\_on\_resp\}\}\(G\.78\)≤𝒪​\(N​exp⁡\(−κ\)\)\\displaystyle\\leq\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)\(G\.79\)
Similarly, we have that:

‖𝜹𝒘t‖2\\displaystyle\|\|\\bm\{\\delta\}\_\{\\bm\{w\}\_\{t\}\}\|\|\_\{2\}≤‖Proj⟂​\(𝑬\)‖2\\displaystyle\\leq\|\|\\text\{Proj\}\_\{\\perp\}\(\\bm\{E\}\)\|\|\_\{2\}\(G\.80\)≤‖𝑬‖2,property of orth\. projection\\displaystyle\\leq\|\|\\bm\{E\}\|\|\_\{2\},\\text\{ property of orth\. projection\}\(G\.81\)≤∑k∉\{i,j\}γ~k​\(𝒚t\)​‖𝝁\(k\)−𝝁\(i\)‖2\\displaystyle\\leq\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(G\.82\)≤\(N−2\)​maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2​πiπd​exp⁡\(−κ\),[Eq\.G\.72](https://arxiv.org/html/2605.06831#A7.E72)\\displaystyle\\leq\(N\-2\)\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\frac\{\\pi\_\{i\}\}\{\\pi\_\{d\}\}\\exp\(\-\\kappa\),\\text\{ \\lx@cref\{creftype~refnum\}\{eq:bound\_on\_resp\}\}\(G\.83\)≤𝒪​\(N​exp⁡\(−κ\)\)\\displaystyle\\leq\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)\(G\.84\)
as desired\.

∎

###### Lemma G\.4\.

Assume[Sec\.4](https://arxiv.org/html/2605.06831#S4)holds\. Then, there exists a timeτ≤τ1\\tau\\leq\\tau\_\{1\}such thatξt\\xi\_\{t\}, as defined by[Eq\.G\.51](https://arxiv.org/html/2605.06831#A7.E51), lies in\[−ϵ,1\+ϵ\]\[\-\\epsilon,1\+\\epsilon\], whereϵ∈O​\(N​exp⁡\(−κ\)\)\\epsilon\\in O\(N\\exp\(\-\\kappa\)\), for all timet≤τt\\leq\\tau\.

###### Proof\.

Recall the definition of anε\\varepsilon\-extended line segment in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. By enlarging the constant inε∈O​\(N​exp⁡\(−κ\)\)\\varepsilon\\in O\(N\\exp\(\-\\kappa\)\), we may assume that\|δξt\|≤ε/2\|\\delta\_\{\\xi\_\{t\}\}\|\\leq\\varepsilon/2by[AppendixG](https://arxiv.org/html/2605.06831#A7)\. Sinceγ~j​\(𝒚t\)∈\(0,1\)\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\in\(0,1\), ifξt\>1\+ε\\xi\_\{t\}\>1\+\\varepsilonandrt:=ξt−\(1\+ε\)r\_\{t\}:=\\xi\_\{t\}\-\(1\+\\varepsilon\), thenr˙t=ξ˙t≤−rt−ε/2\\dot\{r\}\_\{t\}=\\dot\{\\xi\}\_\{t\}\\leq\-r\_\{t\}\-\\varepsilon/2; similarly, ifξt<−ε\\xi\_\{t\}<\-\\varepsilonandrt:=−ε−ξtr\_\{t\}:=\-\\varepsilon\-\\xi\_\{t\}, thenr˙t=−ξ˙t≤−rt−ε/2\\dot\{r\}\_\{t\}=\-\\dot\{\\xi\}\_\{t\}\\leq\-r\_\{t\}\-\\varepsilon/2\. Thus, forttsufficiently large,rtr\_\{t\}vanishes andξt\\xi\_\{t\}is trapped in\[−ε,1\+ε\]\[\-\\varepsilon,1\+\\varepsilon\]\. ∎

###### Lemma G\.5\.

For anyϕ∈\(0,1\)\\phi\\in\(0,1\), with probability at least1−2​exp⁡\(−\(ϖ\)​ϕ2/8\)1\-2\\exp\(\-\(\\varpi\)\\phi^\{2\}/8\), we have that‖𝐰T‖2\|\|\\bm\{w\}\_\{T\}\|\|\_\{2\}, where𝐰t\\bm\{w\}\_\{t\}is defined as in[Eq\.G\.51](https://arxiv.org/html/2605.06831#A7.E51), satisfies:

‖𝒘T‖2≤\(ϖ\)​\(1\+ϕ\)\|\|\\bm\{w\}\_\{T\}\|\|\_\{2\}\\leq\\sqrt\{\(\\varpi\)\(1\+\\phi\)\}\(G\.85\)

###### Proof\.

Apply standard Chernoff tail bound for𝒙T∼𝒩​\(0,𝑰\)\\bm\{x\}\_\{T\}\\sim\\mathcal\{N\}\(0,\\bm\{I\}\), Theorem 2\.7 inFernandez\-Granda \([2017](https://arxiv.org/html/2605.06831#bib.bib14)\)\. ∎

###### Lemma G\.6\(Brownian confinement in an interval\)\.

Let\(Bt\)t≥0\(B\_\{t\}\)\_\{t\\geq 0\}be standard one\-dimensional Brownian motion\. Then for anyV\>0V\>0anda\>0a\>0,

ℙ​\(sup0≤t≤V\|Bt\|≤a\)≤4π​exp⁡\(−π2​V8​a2\)\.\\mathbb\{P\}\\Big\(\\sup\_\{0\\leq t\\leq V\}\|B\_\{t\}\|\\leq a\\Big\)\\ \\leq\\ \\frac\{4\}\{\\pi\}\\exp\\\!\\Big\(\-\\frac\{\\pi^\{2\}V\}\{8a^\{2\}\}\\Big\)\.\(G\.86\)

###### Proof\.

See Eq 7\.15 in\(Mörters & Peres,[2010](https://arxiv.org/html/2605.06831#bib.bib22)\)and evaluate at the midpoint keeping the first eigenvalue term\. ∎

## Appendix HProofs

### H\.1Proof of Theorem[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)

###### Proof\.

Considerτ≤τ1\\tau\\leq\\tau\_\{1\}such that[AppendixG](https://arxiv.org/html/2605.06831#A7)holds, and then fixt≤τt\\leq\\tau\. Consider the dynamics in[AppendixG](https://arxiv.org/html/2605.06831#A7), which are equivalent to those in[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)and are defined with respect to the rescaled time:

u​\(t\):=∫tTβs2​σs2​𝑑s,u\(t\):=\\int\_\{t\}^\{T\}\\frac\{\\beta\_\{s\}\}\{2\\sigma\_\{s\}^\{2\}\}\\,ds,\(H\.87\)
by the fundamental theorem of calculus\. Furthermore, consider the parallel and orthogonal dynamics as defined in[AppendixG](https://arxiv.org/html/2605.06831#A7), except parameterized by theu​\(t\)u\(t\)as they were defined originally per[AppendixG](https://arxiv.org/html/2605.06831#A7)\. Note thatu​\(t\)u\(t\)is monotonicallydecreasingfor forwardtt, since:

dd​t​u​\(t\)=−βt2​σt2<0,\\frac\{d\}\{dt\}u\(t\)=\-\\frac\{\\beta\_\{t\}\}\{2\\sigma\_\{t\}^\{2\}\}<0,\(H\.88\)
by definition ofβt\\beta\_\{t\}andσt2\\sigma\_\{t\}^\{2\}; in particular, it is monotonicallyincreasingat our studied reversett\.

Accordingly, the dynamics in[AppendixG](https://arxiv.org/html/2605.06831#A7)are:

dd​u​ξu​\(t\)\\displaystyle\\frac\{d\}\{du\}\\xi\_\{u\(t\)\}=γj​\(𝒚u​\(t\)\)−ξu​\(t\)\+δξu​\(t\),\\displaystyle=\\gamma\_\{j\}\(\\bm\{y\}\_\{u\(t\)\}\)\-\\xi\_\{u\(t\)\}\+\\delta\_\{\\xi\_\{u\(t\)\}\},\(H\.89\)dd​u​𝒘u​\(t\)\\displaystyle\\frac\{d\}\{du\}\\bm\{w\}\_\{u\(t\)\}=−𝒘u​\(t\)\+𝜹𝒘u​\(t\),\\displaystyle=\-\\bm\{w\}\_\{u\(t\)\}\+\\bm\{\\delta\}\_\{\\bm\{w\}\_\{u\(t\)\}\},\(H\.90\)where\|δξu\|,‖𝜹𝒘u‖2≤ε\|\\delta\_\{\\xi\_\{u\}\}\|,\\\|\\bm\{\\delta\}\_\{\\bm\{w\}\_\{u\}\}\\\|\_\{2\}\\leq\\varepsilon\.

Then, by definition of the orthogonal distance \(and since, by[AppendixG](https://arxiv.org/html/2605.06831#A7), in𝒚\\bm\{y\}\-space the\(i,j\)\(i,j\)\-mode segment is static, i\.e\.Lu​\(t\)=L0L\_\{u\(t\)\}=L\_\{0\}for alltt:\),

d⟂​\(𝒚u​\(t\),L0,ε\(i,j\)\)=‖𝒘u​\(t\)‖2,d\_\{\\perp\}\(\\bm\{y\}\_\{u\(t\)\},L\_\{0,\\varepsilon\}^\{\(i,j\)\}\)=\\\|\\bm\{w\}\_\{u\(t\)\}\\\|\_\{2\},\(H\.91\)
since the closest point on the extended line segment is the orthogonal projection\. Note that the following inequality follows vacuously if‖𝒘u​\(t\)‖2=0\\\|\\bm\{w\}\_\{u\(t\)\}\\\|\_\{2\}=0, hence we consider‖𝒘u​\(t\)‖2≠0\\\|\\bm\{w\}\_\{u\(t\)\}\\\|\_\{2\}\\neq 0\. Defineg​\(u\):=‖𝒘u‖2g\(u\):=\\\|\\bm\{w\}\_\{u\}\\\|\_\{2\}, whereu:=u​\(t\)u:=u\(t\)\.

Defineg​\(u\):=‖𝒘u‖2g\(u\):=\\\|\\bm\{w\}\_\{u\}\\\|\_\{2\}\. In the first case thatg​\(u\)≠0g\(u\)\\neq 0, we have

g​\(u\)2=𝒘u⊤​𝒘u⟹dd​u​g​\(u\)2=dd​u​\(𝒘u⊤​𝒘u\)=2​𝒘u⊤​d​𝒘ud​u\.g\(u\)^\{2\}=\\bm\{w\}\_\{u\}^\{\\top\}\\bm\{w\}\_\{u\}\\quad\\Longrightarrow\\quad\\frac\{d\}\{du\}g\(u\)^\{2\}=\\frac\{d\}\{du\}\\big\(\\bm\{w\}\_\{u\}^\{\\top\}\\bm\{w\}\_\{u\}\\big\)=2\\,\\bm\{w\}\_\{u\}^\{\\top\}\\frac\{d\\bm\{w\}\_\{u\}\}\{du\}\.\(H\.92\)Using the orthogonal dynamicsd​𝒘ud​u=−𝒘u\+𝜹𝒘u\\frac\{d\\bm\{w\}\_\{u\}\}\{du\}=\-\\bm\{w\}\_\{u\}\+\\bm\{\\delta\}\_\{\\bm\{w\}\_\{u\}\}and‖𝜹𝒘u‖2≤ε\\\|\\bm\{\\delta\}\_\{\\bm\{w\}\_\{u\}\}\\\|\_\{2\}\\leq\\varepsilon, we obtain

dd​u​g​\(u\)2\\displaystyle\\frac\{d\}\{du\}g\(u\)^\{2\}=2​𝒘u⊤​\(−𝒘u\+𝜹𝒘u\)=−2​‖𝒘u‖22\+2​𝒘u⊤​𝜹𝒘u\\displaystyle=2\\,\\bm\{w\}\_\{u\}^\{\\top\}\(\-\\bm\{w\}\_\{u\}\+\\bm\{\\delta\}\_\{\\bm\{w\}\_\{u\}\}\)=\-2\\\|\\bm\{w\}\_\{u\}\\\|\_\{2\}^\{2\}\+2\\,\\bm\{w\}\_\{u\}^\{\\top\}\\bm\{\\delta\}\_\{\\bm\{w\}\_\{u\}\}\(H\.93\)≤−2​g​\(u\)2\+2​‖𝒘u‖2​‖𝜹𝒘u‖2≤−2​g​\(u\)2\+2​ε​g​\(u\)\.\\displaystyle\\leq\-2g\(u\)^\{2\}\+2\\\|\\bm\{w\}\_\{u\}\\\|\_\{2\}\\\|\\bm\{\\delta\}\_\{\\bm\{w\}\_\{u\}\}\\\|\_\{2\}\\leq\-2g\(u\)^\{2\}\+2\\varepsilon g\(u\)\.\(H\.94\)Sincedd​u​g​\(u\)2=2​g​\(u\)​g′​\(u\)\\frac\{d\}\{du\}g\(u\)^\{2\}=2g\(u\)g^\{\\prime\}\(u\)forg​\(u\)≠0g\(u\)\\neq 0, dividing by2​g​\(u\)2g\(u\)yields

g′​\(u\)≤−g​\(u\)\+ε\.g^\{\\prime\}\(u\)\\leq\-g\(u\)\+\\varepsilon\.\(H\.95\)
In the other case thatg​\(u\)=0g\(u\)=0, the inequality holds trivially\. Multiplying byeue^\{u\}then yields:

dd​u​\(eu​g​\(u\)\)≤ε​eu\.\\frac\{d\}\{du\}\(e^\{u\}g\(u\)\)\\leq\\varepsilon e^\{u\}\.\(H\.96\)
Integrating from0touuthen gives:

eu​g​\(u\)−g​\(0\)≤ε​\(eu−1\),e^\{u\}g\(u\)\-g\(0\)\\leq\\varepsilon\(e^\{u\}\-1\),\(H\.97\)
hence:

g​\(u\)≤e−u​g​\(0\)\+ε​\(1−e−u\)≤e−u​g​\(0\)\+ε\.g\(u\)\\leq e^\{\-u\}g\(0\)\+\\varepsilon\(1\-e^\{\-u\}\)\\leq e^\{\-u\}g\(0\)\+\\varepsilon\.\(H\.98\)
In particular, denotingz:u​\(z\)=0z:u\(z\)=0to mean the timez∈\[0,T\]z\\in\[0,T\]s\.t\.u​\(z\)=0u\(z\)=0, we have that:

‖𝒘u​\(t\)‖2≤e−u​\(t\)​‖𝒘z:u​\(z\)=0‖2\+ε​\(1−e−u​\(t\)\)≤e−u​\(t\)​‖𝒘T‖2\+ε\.\\\|\\bm\{w\}\_\{u\(t\)\}\\\|\_\{2\}\\leq e^\{\-u\(t\)\}\\\|\\bm\{w\}\_\{z:u\(z\)=0\}\\\|\_\{2\}\+\\varepsilon\(1\-e^\{\-u\(t\)\}\)\\;\\;\\leq\\;\\;e^\{\-u\(t\)\}\\\|\\bm\{w\}\_\{T\}\\\|\_\{2\}\+\\varepsilon\.\(H\.99\)
In particular, by definition ofuu,z=Tz=T, yielding:

‖𝒘u​\(t\)‖2≤e−u​\(t\)​‖𝒘T‖2\+ε​\(1−e−u​\(t\)\)≤e−u​\(t\)​‖𝒘T‖2\+ε\.\\\|\\bm\{w\}\_\{u\(t\)\}\\\|\_\{2\}\\leq e^\{\-u\(t\)\}\\\|\\bm\{w\}\_\{T\}\\\|\_\{2\}\+\\varepsilon\(1\-e^\{\-u\(t\)\}\)\\;\\;\\leq\\;\\;e^\{\-u\(t\)\}\\\|\\bm\{w\}\_\{T\}\\\|\_\{2\}\+\\varepsilon\.\(H\.100\)
By definition of𝒚u​\(t\)\\bm\{y\}\_\{u\(t\)\}in[AppendixG](https://arxiv.org/html/2605.06831#A7), we have thatd⟂​\(𝒙t,Lt,ε\(i,j\)\)=α¯t​d⟂​\(𝒚u​\(t\),L0,ε\(i,j\)\)d\_\{\\perp\}\(\\bm\{x\}\_\{t\},L\_\{t,\\varepsilon\}^\{\(i,j\)\}\)=\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\,d\_\{\\perp\}\(\\bm\{y\}\_\{u\(t\)\},L\_\{0,\\varepsilon\}^\{\(i,j\)\}\)\. Applying this to[Eq\.H\.100](https://arxiv.org/html/2605.06831#A8.E100), we obtain that:

d⟂​\(𝒙t,Lt,ε\(i,j\)\)\\displaystyle d\_\{\\perp\}\(\\bm\{x\}\_\{t\},L\_\{t,\\varepsilon\}^\{\(i,j\)\}\)=α¯t​d⟂​\(𝒚u​\(t\),L0,ε\(i,j\)\)\\displaystyle=\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\,d\_\{\\perp\}\(\\bm\{y\}\_\{u\(t\)\},L\_\{0,\\varepsilon\}^\{\(i,j\)\}\)\(H\.101\)=α¯t​‖𝒘u​\(t\)‖2\\displaystyle=\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\,\\\|\\bm\{w\}\_\{u\(t\)\}\\\|\_\{2\}\(H\.102\)≤α¯t​\(e−u​\(t\)​‖𝒘T‖2\+ε\)\\displaystyle\\leq\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\Big\(e^\{\-u\(t\)\}\\\|\\bm\{w\}\_\{T\}\\\|\_\{2\}\+\\varepsilon\\Big\)\(H\.103\)=α¯t​exp⁡\(−u​\(t\)\)​d⟂​\(𝒚T,LT,ε\(i,j\)\)\+α¯t​ε\\displaystyle=\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\exp\(\-u\(t\)\)d\_\{\\perp\}\(\\bm\{y\}\_\{T\},L\_\{T,\\varepsilon\}^\{\(i,j\)\}\)\+\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\varepsilon\(H\.104\)=α¯tα¯T​exp⁡\(−u​\(t\)\)​d⟂​\(𝒙T,LT,ε\(i,j\)\)\+α¯t​ε\\displaystyle=\\frac\{\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\}\{\\sqrt\{\\bar\{\\alpha\}\_\{T\}\}\}\\exp\(\-u\(t\)\)d\_\{\\perp\}\(\\bm\{x\}\_\{T\},L\_\{T,\\varepsilon\}^\{\(i,j\)\}\)\+\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\varepsilon\(H\.105\)=α¯tγ​exp⁡\(−u​\(t\)\)​d⟂​\(𝒙T,LT,ε\(i,j\)\)\+α¯t​ε,\\displaystyle=\\sqrt\{\\frac\{\\bar\{\\alpha\}\_\{t\}\}\{\\gamma\}\}\\exp\(\-u\(t\)\)d\_\{\\perp\}\(\\bm\{x\}\_\{T\},L\_\{T,\\varepsilon\}^\{\(i,j\)\}\)\+\\sqrt\{\\bar\{\\alpha\}\_\{t\}\}\\varepsilon,\(H\.106\)≤𝒪​\(exp⁡\(−u​\(t\)\)​d⟂​\(𝒙T,LT,ε\(i,j\)\)\+ε\)\\displaystyle\\leq\\mathcal\{O\}\(\\exp\(\-u\(t\)\)d\_\{\\perp\}\(\\bm\{x\}\_\{T\},L\_\{T,\\varepsilon\}^\{\(i,j\)\}\)\+\\varepsilon\)\(H\.107\)
by definition ofα¯T\\bar\{\\alpha\}\_\{T\}andu​\(t\)u\(t\)\. To obtain the result without the initial distance, note that‖𝒘T‖2≤‖𝒚T‖2≤ϖ​\(1\+ϕ\)α¯T=ϖ​\(1\+ϕ\)γ\|\|\\bm\{w\}\_\{T\}\|\|\_\{2\}\\leq\|\|\\bm\{y\}\_\{T\}\|\|\_\{2\}\\leq\\frac\{\\sqrt\{\\varpi\(1\+\\phi\)\}\}\{\\sqrt\{\\bar\{\\alpha\}\_\{T\}\}\}=\\frac\{\\sqrt\{\\varpi\(1\+\\phi\)\}\}\{\\sqrt\{\\gamma\}\}since𝒘T\\bm\{w\}\_\{T\}is a component of𝒚T\\bm\{y\}\_\{T\}, and by applying[Eq\.G\.85](https://arxiv.org/html/2605.06831#A7.E85); note that the extraγ\\sqrt\{\\gamma\}in the denominator is again absorbed by the𝒪\\mathcal\{O\}\. Finally, sincettwas arbitrary, this holds for allt≤τt\\leq\\tauas desired\.

∎

### H\.2Proof of Corollary[4](https://arxiv.org/html/2605.06831#S4)

###### Proof\.

From the proof of[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2), we have that the dynamics of‖𝒘u‖2\|\|\\bm\{w\}\_\{u\}\|\|\_\{2\}satisfy the differential inequality:

dd​u​‖𝒘u‖2≤−‖𝒘u‖2\+ε\\frac\{d\}\{du\}\|\|\\bm\{w\}\_\{u\}\|\|\_\{2\}\\leq\-\|\|\\bm\{w\}\_\{u\}\|\|\_\{2\}\+\\varepsilon\(H\.108\)
At the boundary of the tube, we thus have:

dd​u​‖𝒘u‖2\|‖𝒘u‖2=ε≤−ε\+ε=0\.\\frac\{d\}\{du\}\|\|\\bm\{w\}\_\{u\}\|\|\_\{2\}\\Big\|\_\{\|\|\\bm\{w\}\_\{u\}\|\|\_\{2\}=\\varepsilon\}\\leq\-\\varepsilon\+\\varepsilon=0\.\(H\.109\)
In particular, if‖𝒘u‖2≤ε\|\|\\bm\{w\}\_\{u\}\|\|\_\{2\}\\leq\\varepsilon, it cannot cross the boundary to the outside, because doing so would require a strictly positive derivative\. Note that sinceα¯u\\sqrt\{\\bar\{\\alpha\}\_\{u\}\}and its reciprocal are uniformly bounded on\[T−τ1,T\]\[T\-\\tau\_\{1\},T\], the rescaling only changes the tube radius by a multiplicative constant which we absorb intoε\\varepsilon\. Thus, together with[AppendixG](https://arxiv.org/html/2605.06831#A7), theε\\varepsilon\-tube around the line segment in[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)is an invariant set, and for reverset≤τt\\leq\\tau, if a trajectory enters this set, it stays there\. ∎

### H\.3Proof of Proposition[4](https://arxiv.org/html/2605.06831#S4)

###### Proof\.

We begin by showing existence of an instantaneously stable equilibriaξt\(i\),∗\\xi^\{\(i\),\*\}\_\{t\}in theε\\varepsilon\-ball around𝝁i\\bm\{\\mu\}\_\{i\}, without loss of generality\. The proof is symmetric for𝝁j\\bm\{\\mu\}\_\{j\}\.

By[AppendixG](https://arxiv.org/html/2605.06831#A7), we have the parallel dynamics:

ξ˙t=γ~j​\(𝒚t\)−ξt\+δξt,\\dot\{\\xi\}\_\{t\}=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\+\\delta\_\{\\xi\_\{t\}\},\(H\.110\)
where:

𝒚t=𝝁\(i\)\+ξt​ℓ​𝒖\+𝒘t,\\bm\{y\}\_\{t\}=\\bm\{\\mu\}^\{\(i\)\}\+\\xi\_\{t\}\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\},\(H\.111\)
whereℓ=‖𝝁\(j\)−𝝁\(i\)‖2,𝒖=𝝁\(j\)−𝝁\(i\)ℓ\\ell=\|\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\},\\;\\bm\{u\}=\\frac\{\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\}\{\\ell\}, and𝒘t\\bm\{w\}\_\{t\}satisfies⟨𝒘t,𝒖⟩=0\\langle\\bm\{w\}\_\{t\},\\bm\{u\}\\rangle=0\. Furthermore, since the trajectory is in theε\\varepsilon\-tube aroundLt\(i,j\)L\_\{t\}^\{\(i,j\)\}, we have that‖𝒘t‖2≤ε\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}\\leq\\varepsilon\.

Fixξ:=ξt\\xi:=\\xi\_\{t\}fort≤τ^t\\leq\\hat\{\\tau\}arbitrary\. Then, we have that:

‖𝒚t−𝝁\(i\)‖22\\displaystyle\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}=‖𝝁\(i\)−𝝁\(i\)\+ξ​ℓ​𝒖\+𝒘t‖22\\displaystyle=\|\|\\bm\{\\mu\}^\{\(i\)\}\-\\bm\{\\mu\}^\{\(i\)\}\+\\xi\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\(H\.112\)=‖ξ​ℓ​𝒖\+𝒘t‖22\\displaystyle=\|\|\\xi\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\(H\.113\)=‖ξ​ℓ​𝒖‖22\+‖𝒘t‖22\+2​⟨ξ​ℓ​𝒖,𝒘t⟩\\displaystyle=\|\|\\xi\\ell\\bm\{u\}\|\|\_\{2\}^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\+2\\langle\\xi\\ell\\bm\{u\},\\bm\{w\}\_\{t\}\\rangle\(H\.114\)=ξ2​ℓ2​‖𝒖‖22\+‖𝒘t‖22,⟨𝒖,𝒘t⟩=0\\displaystyle=\\xi^\{2\}\\ell^\{2\}\|\|\\bm\{u\}\|\|\_\{2\}^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\},\\text\{ $\\langle\\bm\{u\},\\bm\{w\}\_\{t\}\\rangle=0$\}\(H\.115\)=ξ2​ℓ2​‖𝒖‖22\+‖𝒘t‖22,‖𝒖‖2=1\.\\displaystyle=\\xi^\{2\}\\ell^\{2\}\|\|\\bm\{u\}\|\|\_\{2\}^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\},\\;\|\|\\bm\{u\}\|\|\_\{2\}=1\.\(H\.116\)
Next, we have that:

‖𝒚t−𝝁\(j\)‖22\\displaystyle\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}=‖𝝁\(i\)−𝝁\(j\)\+ξ​ℓ​𝒖\+𝒘t‖22\\displaystyle=\|\|\\bm\{\\mu\}^\{\(i\)\}\-\\bm\{\\mu\}^\{\(j\)\}\+\\xi\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\(H\.117\)=‖𝝁\(i\)−\(𝝁\(i\)\+ℓ​𝒖\)\+ξ​ℓ​𝒖\+𝒘t‖22\\displaystyle=\|\|\\bm\{\\mu\}^\{\(i\)\}\-\(\\bm\{\\mu\}^\{\(i\)\}\+\\ell\\bm\{u\}\)\+\\xi\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\(H\.118\)=‖\(ξ−1\)​ℓ​𝒖\+𝒘t‖22\\displaystyle=\|\|\(\\xi\-1\)\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\(H\.119\)=‖\(ξ−1\)​ℓ‖22\+‖𝒘t‖22\+2​⟨\(ξ−1\)​ℓ​𝒖,𝒘t⟩\\displaystyle=\|\|\(\\xi\-1\)\\ell\|\|\_\{2\}^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\+2\\langle\(\\xi\-1\)\\ell\\bm\{u\},\\bm\{w\}\_\{t\}\\rangle\(H\.120\)=\(1−ξ\)2​ℓ2\+‖𝒘t‖22,\\displaystyle=\(1\-\\xi\)^\{2\}\\ell^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|^\{2\}\_\{2\},\(H\.121\)
by the same token, since‖𝒖‖22=1\|\|\\bm\{u\}\|\|\_\{2\}^\{2\}=1and⟨𝒖,𝒘t⟩=0\\langle\\bm\{u\},\\bm\{w\}\_\{t\}\\rangle=0\.

Incorporating[Eq\.H\.116](https://arxiv.org/html/2605.06831#A8.E116)and[Eq\.H\.121](https://arxiv.org/html/2605.06831#A8.E121), we have that:

‖𝒚t−𝝁\(j\)‖22−‖𝒚t−𝝁\(i\)‖22\\displaystyle\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\-\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}=\(1−ξ\)2​ℓ2\+‖𝒘t‖22−\(ξ2​ℓ2\+‖𝒘t‖22\)\\displaystyle=\(1\-\\xi\)^\{2\}\\ell^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|^\{2\}\_\{2\}\-\(\\xi^\{2\}\\ell^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}\)\(H\.122\)=\(1−ξ\)2​ℓ2−ξ2​ℓ2\\displaystyle=\(1\-\\xi\)^\{2\}\\ell^\{2\}\-\\xi^\{2\}\\ell^\{2\}\(H\.123\)=\(\(1−ξ\)2−ξ2\)​ℓ2\\displaystyle=\(\(1\-\\xi\)^\{2\}\-\\xi^\{2\}\)\\ell^\{2\}\(H\.124\)=\(1−2​ξ\)​ℓ2\.\\displaystyle=\(1\-2\\xi\)\\ell^\{2\}\.\(H\.125\)
We then bound the responsibility of modejjas:

γ~j​\(𝒚t\)\\displaystyle\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)=πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)∑m=1Nπm​𝒩​\(𝒚t;𝝁\(m\),σ~t2​𝑰\)\\displaystyle=\\frac\{\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\sum\_\{m=1\}^\{N\}\\pi\_\{m\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(m\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\(H\.126\)≤πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)πi​𝒩​\(𝒚t;𝝁\(i\),σ~t2​𝑰\)\+πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)\\displaystyle\\leq\\frac\{\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(i\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\+\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\(H\.127\)=πj​exp⁡\(−‖𝒚t−𝝁\(j\)‖222​σ~t2\)πi​exp⁡\(−‖𝒚t−𝝁\(i\)‖222​σ~t2\)\+πj​exp⁡\(−‖𝒚t−𝝁\(j\)‖222​σ~t2\)\\displaystyle=\\frac\{\\pi\_\{j\}\\exp\\left\(\-\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\}\{\\pi\_\{i\}\\exp\\left\(\-\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\+\\pi\_\{j\}\\exp\\left\(\-\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\}\(H\.128\)=11\+πiπj\(exp\(‖𝒚t−𝝁\(j\)‖22−‖𝒚t−𝝁\(i\)‖222​σ~t2\),dividing by numerator\\displaystyle=\\frac\{1\}\{1\+\\frac\{\\pi\_\{i\}\}\{\\pi\_\{j\}\}\(\\exp\\left\(\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\-\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\},\\text\{ dividing by numerator\}\(H\.129\)≤πjπi​exp⁡\(−‖𝒚t−𝝁\(j\)‖22−‖𝒚t−𝝁\(i\)‖222​σ~t2\)\\displaystyle\\leq\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\left\(\-\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\-\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\(H\.130\)≤πjπi​exp⁡\(−\(1−2​ξ\)​ℓ22​σ~t2\)\\displaystyle\\leq\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\left\(\-\\frac\{\(1\-2\\xi\)\\ell^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\(H\.131\)
Next, suppose thatξ∈\[−a,a\]\\xi\\in\[\-a,a\]fora∈𝒪​\(N​exp⁡\(−κ\)\)a\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)satisfyinga≤14,a\\leq\\frac\{1\}\{4\},to be chosen later\. Then,1−2​a≥121\-2a\\geq\\frac\{1\}\{2\}, and thus:

εγ~\\displaystyle\\varepsilon\_\{\\tilde\{\\gamma\}\}=supξ∈\[−a,a\]γ~j​\(𝒚t\)\\displaystyle=\\sup\_\{\\xi\\in\[\-a,a\]\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\(H\.132\)≤πjπi​exp⁡\(−\(1−2​a\)​ℓ22​σ~t2\)\\displaystyle\\leq\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\left\(\-\\frac\{\(1\-2a\)\\ell^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\(H\.133\)≤πjπi​exp⁡\(−\(1−2​a\)​ℓ22​σ~t2\)\\displaystyle\\leq\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\left\(\-\\frac\{\(1\-2a\)\\ell^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\(H\.134\)≤πjπi​exp⁡\(−\(1−2​a\)​2​κ\),[Sec\.4](https://arxiv.org/html/2605.06831#S4)\\displaystyle\\leq\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\left\(\-\(1\-2a\)2\\kappa\\right\),\\;\\text\{\\lx@cref\{creftype~refnum\}\{assump:modes\_far\_apart\}\}\(H\.135\)∈𝒪​\(exp⁡\(−κ\)\),1−2​a≥12\.\\displaystyle\\in\\mathcal\{O\}\(\\exp\(\-\\kappa\)\),\\;1\-2a\\geq\\frac\{1\}\{2\}\.\(H\.136\)
Then, by the above,εγ~∈𝒪​\(N​exp⁡\(−κ\)\)\\varepsilon\_\{\\tilde\{\\gamma\}\}\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)\. Furthermore, by[AppendixG](https://arxiv.org/html/2605.06831#A7), we have thatεδξ:=supξ∈\[−a,a\]\|δξ\|≤supξ∈\[−a,1\+a\]\|δξ\|∈𝒪​\(N​exp⁡\(−κ\)\)\\varepsilon\_\{\\delta\_\{\\xi\}\}:=\\sup\_\{\\xi\\in\[\-a,a\]\}\|\\delta\_\{\\xi\}\|\\leq\\sup\_\{\\xi\\in\[\-a,1\+a\]\}\|\\delta\_\{\\xi\}\|\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)\. Thus, leta:=2​\(εδξ\+εγ~\)a:=2\(\\varepsilon\_\{\\delta\_\{\\xi\}\}\+\\varepsilon\_\{\\tilde\{\\gamma\}\}\), which satisfiesa∈𝒪​\(N​exp⁡\(−κ\)\)a\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)\. Sincea∈𝒪​\(N​exp⁡\(−κ\)\)a\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)there exists aCa\>0C\_\{a\}\>0s\.t\.a≤Ca​N​exp⁡\(−κ\)a\\leq C\_\{a\}N\\exp\(\-\\kappa\)forNNsufficiently large; choosingκ≥log⁡N\+log⁡\(4​Ca\)\\kappa\\geq\\log N\+\\log\(4C\_\{a\}\)ensures thata≤14a\\leq\\frac\{1\}\{4\}\.

We then evaluate the behavior ofξ\\xiat the endpoints\. Atξ=−a\\xi=\-a, we have that:

Ft​\(−a\)\\displaystyle F\_\{t\}\(\-a\)=γ~j​\(𝒚t​\(−a\)\)\+a\+δ−a\\displaystyle=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\(\-a\)\)\+a\+\\delta\_\{\-a\}\(H\.137\)≥a\+δ−a,γ~j≥0\\displaystyle\\geq a\+\\delta\_\{\-a\},\\text\{ $\\tilde\{\\gamma\}\_\{j\}\\geq 0$\}\(H\.138\)≥a−\|δa\|,x≥−\|x\|forx∈ℝ\\displaystyle\\geq a\-\|\\delta\_\{a\}\|,\\;\\text\{ $x\\geq\-\|x\|$ for $x\\in\\mathbb\{R\}$\}\(H\.139\)≥a−εδa,δa≤εδa\\displaystyle\\geq a\-\\varepsilon\_\{\\delta\_\{a\}\},\\text\{$\\delta\_\{a\}\\leq\\varepsilon\_\{\\delta\_\{a\}\}$\}\(H\.140\)=2​εδa\+2​εγ~−εδa\\displaystyle=2\\varepsilon\_\{\\delta\_\{a\}\}\+2\\varepsilon\_\{\\tilde\{\\gamma\}\}\-\\varepsilon\_\{\\delta\_\{a\}\}\(H\.141\)≥εδa\+εγ~\\displaystyle\\geq\\varepsilon\_\{\\delta\_\{a\}\}\+\\varepsilon\_\{\\tilde\{\\gamma\}\}\(H\.142\)\>0\.\\displaystyle\>0\.\(H\.143\)
Next, atξ=a\\xi=a, we have that:

Ft​\(a\)\\displaystyle F\_\{t\}\(a\)=γ~j​\(𝒚t​\(a\)\)−a\+δa\\displaystyle=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\(a\)\)\-a\+\\delta\_\{a\}\(H\.144\)≤εγ~−a\+εδa,γ~j≤εγ~​and​δa≤εδa\\displaystyle\\leq\\varepsilon\_\{\\tilde\{\\gamma\}\}\-a\+\\varepsilon\_\{\\delta\_\{a\}\},\\;\\tilde\{\\gamma\}\_\{j\}\\leq\\varepsilon\_\{\\tilde\{\\gamma\}\}\\text\{ and\}\\;\\delta\_\{a\}\\leq\\varepsilon\_\{\\delta\_\{a\}\}\(H\.145\)=εγ~−2​εδa−2​εγ~\+εδa\\displaystyle=\\varepsilon\_\{\\tilde\{\\gamma\}\}\-2\\varepsilon\_\{\\delta\_\{a\}\}\-2\\varepsilon\_\{\\tilde\{\\gamma\}\}\+\\varepsilon\_\{\\delta\_\{a\}\}\(H\.146\)=−\(εδa\+εγ~\)\\displaystyle=\-\(\\varepsilon\_\{\\delta\_\{a\}\}\+\\varepsilon\_\{\\tilde\{\\gamma\}\}\)\(H\.147\)<0\.\\displaystyle<0\.\(H\.148\)
SinceFtF\_\{t\}is continuous inξ\\xi, since it is a composition of continuous functions \(responsibilities are compositions of continuous Gaussian p\.d\.f\.s, and hence continuous\), by the intermediate value theorem there exists aξt∗,i∈\(−a,a\)\\xi\_\{t\}^\{\*,i\}\\in\(\-a,a\)s\.t\.Ft​\(ξt∗,i\)=0F\_\{t\}\(\\xi^\{\*,i\}\_\{t\}\)=0\. Furthermore:

‖𝒚t​\(ξt∗,i\)−𝝁\(i\)‖2\\displaystyle\|\|\\bm\{y\}\_\{t\}\(\\xi\_\{t\}^\{\*,i\}\)\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}≤\|ξt∗,i\|​ℓ\+‖𝒘‖2,triangle inequality\\displaystyle\\leq\|\\xi\_\{t\}^\{\*,i\}\|\\ell\+\|\|\\bm\{w\}\|\|\_\{2\},\\text\{ triangle inequality\}\(H\.149\)≤a​ℓ\+ε\\displaystyle\\leq a\\ell\+\\varepsilon\(H\.150\)∈𝒪​\(N​exp⁡\(−κ\)\),\\displaystyle\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\),\(H\.151\)
i\.e\. there exists an instantaneous equilibria of[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50)in aε\\varepsilon\-ball of𝝁\(i\)\\bm\{\\mu\}^\{\(i\)\}, whereε∈𝒪​\(N​exp⁡\(−κ\)\)\\varepsilon\\in\\mathcal\{O\}\(N\\exp\(\-\\kappa\)\)\. Sinceξ\\xiwas arbitrary, this holds for allt≤τ^t\\leq\\hat\{\\tau\}\.

Next, we show the \(uniform\) instantaneous stability ofξt∗,i\\xi^\{\*,i\}\_\{t\}, without loss of generality; the proof is symmetric forξt∗,j\\xi^\{\*,j\}\_\{t\}\. Fixt≤τ^t\\leq\\hat\{\\tau\}\. Then, we have that:

dd​ξt​Ft​\(ξt∗,i\)=\(dd​ξt​γ~j​\(𝒚t\)\+dd​ξt​δξt−1\)\|ξt=ξt∗,i\.\\displaystyle\\frac\{d\}\{d\\xi\_\{t\}\}F\_\{t\}\(\\xi\_\{t\}^\{\*,i\}\)=\\left\(\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\+\\frac\{d\}\{d\\xi\_\{t\}\}\\delta\_\{\\xi\_\{t\}\}\-1\\right\)\\Huge\|\_\{\\xi\_\{t\}=\\xi\_\{t\}^\{\*,i\}\}\.\(H\.152\)
We demonstrate thatFt′​\(ξt∗,i\)<0F\_\{t\}^\{\\prime\}\(\\xi\_\{t\}^\{\*,i\}\)<0\. Fix timet≤τ^t\\leq\\hat\{\\tau\}\. We have that:

∇𝒚tγ~j​\(𝒚t\)=γ~j​\(𝒚t\)σ~t2​\(𝝁\(j\)−𝝁^\)\\nabla\_\{\\bm\{y\}\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)=\\frac\{\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\(\\bm\{\\mu\}^\{\(j\)\}\-\\hat\{\\bm\{\\mu\}\}\)\(H\.153\)
where𝝁^\\hat\{\\bm\{\\mu\}\}is the same as that in[AppendixG](https://arxiv.org/html/2605.06831#A7)\. Hence, we have that:

dd​ξt​γ~j​\(𝒚t\)\\displaystyle\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)=⟨∇𝒚tγ~j​\(𝒚t\),d​𝒚td​ξt⟩,chain rule\\displaystyle=\\langle\\nabla\_\{\\bm\{y\}\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\),\\frac\{d\\bm\{y\}\_\{t\}\}\{d\\xi\_\{t\}\}\\rangle,\\text\{ chain rule\}\(H\.154\)=ℓ​γ~j​\(𝒚t\)σ~t2⟨𝝁\(j\)−𝝁^\(𝒚\)\),𝒖⟩[Eq\.H\.153](https://arxiv.org/html/2605.06831#A8.E153)\\displaystyle=\\frac\{\\ell\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\langle\\bm\{\\mu\}^\{\(j\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\)\),\\bm\{u\}\\rangle\\;\\text\{\\lx@cref\{creftype~refnum\}\{eq:nabla\_y\}\}\(H\.155\)
To bound the inner product, note that, by linearity of the inner product:

⟨𝝁\(j\)−𝝁^​\(𝒚t\),𝒖⟩=⟨𝝁\(j\)−𝝁\(i\),𝒖⟩\+⟨𝝁\(i\)−𝝁^​\(𝒚t\),𝒖⟩\\displaystyle\\langle\\bm\{\\mu\}^\{\(j\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\),\\bm\{u\}\\rangle=\\langle\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\},\\bm\{u\}\\rangle\+\\langle\\bm\{\\mu\}^\{\(i\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\),\\bm\{u\}\\rangle\(H\.156\)
This implies:

\|⟨𝝁\(j\)−𝝁^​\(𝒚t\),𝒖⟩\|\\displaystyle\|\\langle\\bm\{\\mu\}^\{\(j\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\),\\bm\{u\}\\rangle\|≤\|⟨𝝁\(j\)−𝝁\(i\),𝒖⟩\|\+\|⟨𝝁\(i\)−𝝁^​\(𝒚t\),𝒖⟩\|,triangle inequality\\displaystyle\\leq\|\\langle\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\},\\bm\{u\}\\rangle\|\+\|\\langle\\bm\{\\mu\}^\{\(i\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\),\\bm\{u\}\\rangle\|,\\;\\text\{triangle inequality\}\(H\.157\)≤‖𝝁\(j\)−𝝁\(i\)‖2​‖𝒖‖2\+‖𝝁\(i\)−𝝁^​\(𝒚t\)‖2​‖𝒖‖2,Cauchy\-Schwartz\\displaystyle\\leq\|\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\|\|\\bm\{u\}\|\|\_\{2\}\+\|\|\\bm\{\\mu\}^\{\(i\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\|\|\_\{2\}\|\|\\bm\{u\}\|\|\_\{2\},\\text\{ Cauchy\-Schwartz\}\(H\.158\)≤ℓ\+\|\|∑k∈\[N\]γ~k\(𝒚t\)\(𝝁\(k\)−𝝁\(i\)\|\|2,\|\|𝒖\|\|2=1\\displaystyle\\leq\\ell\+\|\|\\sum\_\{k\\in\[N\]\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\},\\;\|\|\\bm\{u\}\|\|\_\{2\}=1\(H\.159\)≤ℓ\+γ~j\(𝒚t\)\|\|𝝁\(j\)−𝝁\(i\)\|\|2\+\|\|∑k∈\[N\]γ~k\(𝒚t\)\(𝝁\(k\)−𝝁\(i\)\|\|2,triangle inequality\\displaystyle\\leq\\ell\+\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\|\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\|\|\\sum\_\{k\\in\[N\]\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\},\\text\{triangle inequality\}\(H\.160\)≤ℓ\+γ~j​ℓ\+maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2​∑k∉\{i,j\}γ~k​\(𝒚t\)\\displaystyle\\leq\\ell\+\\tilde\{\\gamma\}\_\{j\}\\ell\+\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(H\.161\)≤2​ℓ\+maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2​∑k∉\{i,j\}γ~k​\(𝒚t\)\\displaystyle\\leq 2\\ell\+\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(H\.162\)
Incorporating[Eq\.H\.162](https://arxiv.org/html/2605.06831#A8.E162)into[Eq\.H\.155](https://arxiv.org/html/2605.06831#A8.E155)yields:

\|dd​ξt​γ~j​\(𝒚t\)\|\\displaystyle\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\|≤2​ℓ2σ~t2​γ~j​\(𝒚t\)\+m​a​xk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2​ℓσ~t2​γ~j​\(𝒚t\)​∑k∉\{i,j\}γ~k​\(𝒚t\)\\displaystyle\\leq\\frac\{2\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\+\\frac\{max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\ell\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(H\.163\)≤2​ℓ2σ~t2​γ~j​\(𝒚t\)\+m​a​xk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2​ℓ2σ~t2​γ~j​\(𝒚t\)​∑k∉\{i,j\}γ~k​\(𝒚t\)\.\\displaystyle\\leq\\frac\{2\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\+\\frac\{max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\.\(H\.164\)
Recall thatξt∗,i∈\(−a,a\)\\xi\_\{t\}^\{\*,i\}\\in\(\-a,a\)wherea≤14a\\leq\\frac\{1\}\{4\}, and thus we have\(1−2​a\)≥12\(1\-2a\)\\geq\\frac\{1\}\{2\}\. Letx:=ℓ22​σ~t2x:=\\frac\{\\ell^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\. We then have, by[Eq\.H\.135](https://arxiv.org/html/2605.06831#A8.E135), that:

γ~j\(𝒚t\)≤πjπi\(exp\(−\(1−2a\)x\)≤πjπiexp\(−x/2\)\.\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\leq\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\(\\exp\(\-\(1\-2a\)x\)\\leq\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\(\-x/2\)\.\(H\.165\)
Thus:

2​ℓ2σ~t2​γj~​\(𝒚t\)=4​x​γ~j​\(𝒚t\)≤4​πjπi​x​exp⁡\(−x/2\)\.\\frac\{2\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\_\{j\}\}\(\\bm\{y\}\_\{t\}\)=4x\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\leq 4\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}x\\exp\(\-x/2\)\.\(H\.166\)
Furthermore, by[Sec\.4](https://arxiv.org/html/2605.06831#S4), we have thatℓ22​σ~t2≥2​κ\\frac\{\\ell^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\geq 2\\kappa, i\.e\.x≥2​κx\\geq 2\\kappa\. Therefore, we have thatx​exp⁡\(−x/2\)≤supx≥2​κx​exp⁡\(−x/2\)=\(2​κ\)​exp⁡\(−2​κ/2\)=2​κ​exp⁡\(−κ\)x\\exp\(\-x/2\)\\leq\\sup\_\{x\\geq 2\\kappa\}x\\exp\(\-x/2\)=\(2\\kappa\)\\exp\(\-2\\kappa/2\)=2\\kappa\\exp\(\-\\kappa\)\. Then, incorporating this and[Eq\.H\.166](https://arxiv.org/html/2605.06831#A8.E166)into[Eq\.H\.164](https://arxiv.org/html/2605.06831#A8.E164), we have that:

supξ∈\[−a,a\]\|dd​ξt​γ~j​\(𝒚t\)\|\\displaystyle\\sup\_\{\\xi\\in\[\-a,a\]\}\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\|≤supξ∈\[−a,a\]\(2​ℓ2σ~t2​γ~j​\(𝒚t\)\+m​a​xk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2​ℓ2σ~t2​γ~j​\(𝒚t\)​∑k∉\{i,j\}γ~k​\(𝒚t\)\)\\displaystyle\\leq\\sup\_\{\\xi\\in\[\-a,a\]\}\\left\(\\frac\{2\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\+\\frac\{max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\\right\)\(H\.167\)≤2​supξ∈\[−a,a\]supℓ22​σt2~≥κℓ2σt2~​γj~​\(𝒚t\)\+maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2​supξ∈\[−a,a\]supℓ22​σt2~≥κℓ2σ~t2​γ~j​\(𝒚t\)​∑k∉\{i,j\}γ~k​\(𝒚t\)\\displaystyle\\leq 2\\sup\_\{\\xi\\in\[\-a,a\]\}\\sup\_\{\\frac\{\\ell^\{2\}\}\{2\\tilde\{\\sigma\_\{t\}^\{2\}\}\}\\geq\\kappa\}\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\_\{t\}^\{2\}\}\}\\tilde\{\\gamma\_\{j\}\}\(\\bm\{y\}\_\{t\}\)\+\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\sup\_\{\\xi\\in\[\-a,a\]\}\\sup\_\{\\frac\{\\ell^\{2\}\}\{2\\tilde\{\\sigma\_\{t\}^\{2\}\}\}\\geq\\kappa\}\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(H\.168\)≤8​πjπi​κ​exp⁡\(−κ\)\+maxk∉\{i,j\}⁡2​‖𝝁\(k\)−𝝁\(i\)‖2​κ​exp⁡\(−κ\)​∑k∉\{i,j\}γ~k​\(𝒚t\)\\displaystyle\\leq 8\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\kappa\\exp\(\-\\kappa\)\+\\max\_\{k\\notin\\\{i,j\\\}\}2\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\kappa\\exp\(\-\\kappa\)\\sum\_\{k\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\(H\.169\)≤8​πjπi​κ​exp⁡\(−κ\)\+maxk∉\{i,j\}⁡2​‖𝝁\(k\)−𝝁\(i\)‖2​κ​exp⁡\(−κ\)​\(N−2\)​maxk∉\{i,j\}⁡πkπd∗​exp⁡\(−κ\),[Eq\.G\.72](https://arxiv.org/html/2605.06831#A7.E72)\.\\displaystyle\\leq 8\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\kappa\\exp\(\-\\kappa\)\+\\max\_\{k\\notin\\\{i,j\\\}\}2\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\kappa\\exp\(\-\\kappa\)\(N\-2\)\\max\_\{k\\notin\\\{i,j\\\}\}\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\),\\text\{ \\lx@cref\{creftype~refnum\}\{eq:bound\_on\_resp\}\}\.\(H\.170\)
We next bound the derivative ofδξt\\delta\_\{\\xi\_\{t\}\}forξt∈\(−a,a\)\\xi\_\{t\}\\in\(\-a,a\)arbitrary\.

We have that:

dd​ξt​δξt=1ℓ​∑k∉\{i,j\}dd​ξt​γ~k​\(𝒚t\)​⟨𝝁\(k\)−𝝁\(i\),𝒖⟩\\frac\{d\}\{d\\xi\_\{t\}\}\\delta\_\{\\xi\_\{t\}\}=\\frac\{1\}\{\\ell\}\\sum\_\{k\\notin\\\{i,j\\\}\}\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\\langle\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\},\\bm\{u\}\\rangle\(H\.171\)
which yields that:

\|dd​ξt​δξt\|\\displaystyle\|\\frac\{d\}\{d\\xi\_\{t\}\}\\delta\_\{\\xi\_\{t\}\}\|≤1ℓ​∑k∉\{i,j\}\|dd​ξt​γ~k​\(𝒚t\)\|​\|⟨𝝁\(k\)−𝝁\(i\),𝒖⟩\|,triangle inequality\\displaystyle\\leq\\frac\{1\}\{\\ell\}\\sum\_\{k\\notin\\\{i,j\\\}\}\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\|\|\\langle\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\},\\bm\{u\}\\rangle\|,\\text\{ triangle inequality\}\(H\.172\)≤1ℓ​∑k∉\{i,j\}\|dd​ξt​γ~k​\(𝒚t\)\|​‖𝝁\(k\)−𝝁\(i\)‖2​‖𝒖‖2\\displaystyle\\leq\\frac\{1\}\{\\ell\}\\sum\_\{k\\notin\\\{i,j\\\}\}\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\|\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\|\|\\bm\{u\}\|\|\_\{2\}\(H\.173\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2ℓ​\(N−2\)​maxk∉\{i,j\}⁡\|dd​ξt​γk~​\(𝒚t\)\|\.\\displaystyle\\leq\\frac\{\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\}\{\\ell\}\(N\-2\)\\max\_\{k\\notin\\\{i,j\\\}\}\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\_\{k\}\}\(\\bm\{y\}\_\{t\}\)\|\.\(H\.174\)
We then have that:

\|dd​ξt​γ~k​\(𝒚t\)\|\\displaystyle\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\|=\|⟨∇𝒚tγ~k​\(𝒚t\),d​𝒚td​ξt⟩\|,chain rule\\displaystyle=\|\\langle\\nabla\_\{\\bm\{y\}\_\{t\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\),\\frac\{d\\bm\{y\}\_\{t\}\}\{d\\xi\_\{t\}\}\\rangle\|,\\text\{ chain rule\}\(H\.175\)=ℓ​γ~k​\(𝒚t\)σt~2​\|⟨𝝁\(k\)−𝝁^​\(𝒚t\),𝒖⟩\|,\\displaystyle=\\frac\{\\ell\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\}\{\\tilde\{\\sigma\_\{t\}\}^\{2\}\}\|\\langle\\bm\{\\mu\}^\{\(k\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\),\\bm\{u\}\\rangle\|,\(H\.176\)≤ℓ​γ~k​\(𝒚t\)σt~2​‖𝝁\(k\)−𝝁^​\(𝒚t\)‖2,Cauchy\-Schwartz,‖𝒖‖2=1\.\\displaystyle\\leq\\frac\{\\ell\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\}\{\\tilde\{\\sigma\_\{t\}\}^\{2\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\|\|\_\{2\},\\text\{ Cauchy\-Schwartz, $\|\|\\bm\{u\}\|\|\_\{2\}=1$\}\.\(H\.177\)≤ℓσ~t2​πkπd∗​exp⁡\(−κ\)​‖𝝁\(k\)−𝝁^​\(𝒚t\)‖2,[Eq\.G\.72](https://arxiv.org/html/2605.06831#A7.E72)\\displaystyle\\leq\\frac\{\\ell\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\)\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\|\|\_\{2\},\\text\{ \\lx@cref\{creftype~refnum\}\{eq:bound\_on\_resp\}\}\(H\.178\)
Then, we have that:

‖𝝁\(k\)−𝝁^​\(𝒚t\)‖2\\displaystyle\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\|\|\_\{2\}=‖𝝁\(k\)−𝝁^​\(𝒚t\)−𝝁\(i\)\+𝝁\(i\)‖2\\displaystyle=\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\-\\bm\{\\mu\}^\{\(i\)\}\+\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(H\.179\)≤‖𝝁\(k\)−𝝁\(i\)‖2\+‖𝝁^​\(𝒚t\)−𝝁\(i\)‖2\\displaystyle\\leq\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\|\|\\hat\{\\bm\{\\mu\}\}\(\\bm\{y\}\_\{t\}\)\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(H\.180\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2\+‖∑m∈\[N\]γ~m​\(𝒚t\)​𝝁\(m\)−𝝁\(i\)‖2\\displaystyle\\leq\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\|\|\\sum\_\{m\\in\[N\]\}\\tilde\{\\gamma\}\_\{m\}\(\\bm\{y\}\_\{t\}\)\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(H\.181\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2\+γ~j​\(𝒚t\)​‖𝝁\(j\)−𝝁\(i\)‖2\+‖∑m∉\{i,j\}γ~m​\(𝒚t\)​𝝁\(m\)−𝝁\(i\)‖2\\displaystyle\\leq\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\|\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\|\|\\sum\_\{m\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{m\}\(\\bm\{y\}\_\{t\}\)\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(H\.182\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2\+πjπi​exp⁡\(−\(1−2​ξ\)​ℓ22​σ~t2\)​‖𝝁\(j\)−𝝁\(i\)‖2\+‖∑m∉\{i,j\}γ~m​\(𝒚t\)​𝝁\(m\)−𝝁\(i\)‖2,[Eq\.H\.131](https://arxiv.org/html/2605.06831#A8.E131)\\displaystyle\\leq\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\(\-\\frac\{\(1\-2\\xi\)\\ell^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\)\|\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\|\|\\sum\_\{m\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{m\}\(\\bm\{y\}\_\{t\}\)\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\},\\text\{ \\lx@cref\{creftype~refnum\}\{eq:temp\_stable\_4\}\}\(H\.183\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2\+πjπi​ℓ​exp⁡\(−κ\)\+‖∑m∉\{i,j\}γ~m​\(𝒚t\)​𝝁\(m\)−𝝁\(i\)‖2,ξt∈\(−a,a\),1−2​a≥12\\displaystyle\\leq\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\ell\\exp\(\-\\kappa\)\+\|\|\\sum\_\{m\\notin\\\{i,j\\\}\}\\tilde\{\\gamma\}\_\{m\}\(\\bm\{y\}\_\{t\}\)\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\},\\text\{ $\\xi\_\{t\}\\in\(\-a,a\)$, $1\-2a\\geq\\frac\{1\}\{2\}$\}\(H\.184\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2\+πjπi​ℓ​exp⁡\(−κ\)\+maxm∉\{i,j\}​‖𝝁\(m\)−𝝁\(i\)‖2​\(N−2\)​exp⁡\(−κ\),[Eq\.G\.72](https://arxiv.org/html/2605.06831#A7.E72)\.\\displaystyle\\leq\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\ell\\exp\(\-\\kappa\)\+\\max\_\{m\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(N\-2\)\\exp\(\-\\kappa\),\\text\{ \\lx@cref\{creftype~refnum\}\{eq:bound\_on\_resp\}\}\.\(H\.185\)
Incorporating[Eq\.H\.185](https://arxiv.org/html/2605.06831#A8.E185)into[Eq\.H\.178](https://arxiv.org/html/2605.06831#A8.E178)yields:

\|dd​ξt​γ~k​\(𝒚t\)\|\\displaystyle\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\|≤\|supξt∈\(−a,a\)dd​ξt​γ~k​\(𝒚t\)\|\\displaystyle\\leq\|\\sup\_\{\\xi\_\{t\}\\in\(\-a,a\)\}\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{k\}\(\\bm\{y\}\_\{t\}\)\|\(H\.186\)≤\|supξt∈\(−a,a\)ℓσ~t2​πkπd∗​\(maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2\)\|\\displaystyle\\leq\|\\sup\_\{\\xi\_\{t\}\\in\(\-a,a\)\}\\frac\{\\ell\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\(\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\)\|\(H\.187\)≤ℓσ~t2πkπd∗exp\(−κ\)\(maxk∉\{i,j\}\|\|𝝁\(k\)−𝝁\(i\)\|\|2\+πjπiℓexp\(−κ\)\\displaystyle\\leq\\frac\{\\ell\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\)\(\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\ell\\exp\(\-\\kappa\)\(H\.188\)\+maxm∉\{i,j\}\|\|𝝁\(m\)−𝝁\(i\)\|\|2\(N−2\)exp\(−κ\)\)\\displaystyle\+\\max\_\{m\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(N\-2\)\\exp\(\-\\kappa\)\)\(H\.189\)
Then, incorporating[Eq\.H\.189](https://arxiv.org/html/2605.06831#A8.E189)into[Eq\.H\.174](https://arxiv.org/html/2605.06831#A8.E174)yields:

\|dd​ξt​δξt\|\\displaystyle\|\\frac\{d\}\{d\\xi\_\{t\}\}\\delta\_\{\\xi\_\{t\}\}\|≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2ℓ​\(N−2\)​maxk∉\{i,j\}⁡\|dd​ξt​γk~​\(𝒚t\)\|\\displaystyle\\leq\\frac\{\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\}\{\\ell\}\(N\-2\)\\max\_\{k\\notin\\\{i,j\\\}\}\|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\_\{k\}\}\(\\bm\{y\}\_\{t\}\)\|\(H\.190\)≤maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2ℓ\(N−2\)ℓσ~t2maxk∉\{i,j\}πkπd∗exp\(−κ\)\(maxk∉\{i,j\}\|\|𝝁\(k\)−𝝁\(i\)\|\|2\+πjπiℓexp\(−κ\)\\displaystyle\\leq\\frac\{\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\}\{\\ell\}\(N\-2\)\\frac\{\\ell\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\max\_\{k\\notin\\\{i,j\\\}\}\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\)\(\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\ell\\exp\(\-\\kappa\)\(H\.191\)\+maxm∉\{i,j\}\|\|𝝁\(m\)−𝝁\(i\)\|\|2\(N−2\)exp\(−κ\)\)\\displaystyle\+\\max\_\{m\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(N\-2\)\\exp\(\-\\kappa\)\)\(H\.192\)
Incorporating[Eq\.H\.170](https://arxiv.org/html/2605.06831#A8.E170)and[Eq\.H\.192](https://arxiv.org/html/2605.06831#A8.E192)into[Eq\.H\.152](https://arxiv.org/html/2605.06831#A8.E152)yields:

dd​ξt​Ft​\(ξt∗,i\)\\displaystyle\\frac\{d\}\{d\\xi\_\{t\}\}F\_\{t\}\(\\xi\_\{t\}^\{\*,i\}\)≤\|dd​ξt​Ft​\(ξt∗,i\)\|\\displaystyle\\leq\|\\frac\{d\}\{d\\xi\_\{t\}\}F\_\{t\}\(\\xi\_\{t\}^\{\*,i\}\)\|\(H\.193\)=\|\(dd​ξtγ~j\(𝒚t\)\+dd​ξtδξt−1\)\|ξt=ξt∗,i\|\\displaystyle=\\left\|\\left\(\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\+\\frac\{d\}\{d\\xi\_\{t\}\}\\delta\_\{\\xi\_\{t\}\}\-1\\right\)\\Large\|\_\{\\xi\_\{t\}=\\xi\_\{t\}^\{\*,i\}\}\\right\|\(H\.194\)≤supξt∈\(−a,a\)\|\(dd​ξt​γ~j​\(𝒚t\)\+dd​ξt​δξt−1\)\|\\displaystyle\\leq\\sup\_\{\\xi\_\{t\}\\in\(\-a,a\)\}\\ \\left\|\\left\(\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\+\\frac\{d\}\{d\\xi\_\{t\}\}\\delta\_\{\\xi\_\{t\}\}\-1\\right\)\\right\|\(H\.195\)≤supξt∈\(−a,a\)\|dd​ξt​γ~j​\(𝒚t\)\|\+supξt∈\(−a,a\)\|dd​ξt​δξt\|−1\\displaystyle\\leq\\sup\_\{\\xi\_\{t\}\\in\(\-a,a\)\}\\ \|\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\|\+\\sup\_\{\\xi\_\{t\}\\in\(\-a,a\)\}\\ \|\\frac\{d\}\{d\\xi\_\{t\}\}\\delta\_\{\\xi\_\{t\}\}\|\-1\(H\.196\)≤8​πjπi​κ​exp⁡\(−κ\)\+maxk∉\{i,j\}⁡2​‖𝝁\(k\)−𝝁\(i\)‖2​κ​exp⁡\(−κ\)​\(N−2\)​maxk∉\{i,j\}⁡πkπd∗​exp⁡\(−κ\)\\displaystyle\\leq 8\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\kappa\\exp\(\-\\kappa\)\+\\max\_\{k\\notin\\\{i,j\\\}\}2\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\\kappa\\exp\(\-\\kappa\)\(N\-2\)\\max\_\{k\\notin\\\{i,j\\\}\}\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\)\(H\.197\)\+maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2ℓ\(N−2\)ℓσ2maxk∉\{i,j\}πkπd∗exp\(−κ\)\(maxk∉\{i,j\}\|\|𝝁\(k\)−𝝁\(i\)\|\|2\+πjπiℓexp\(−κ\)\\displaystyle\+\\frac\{\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\}\{\\ell\}\(N\-2\)\\frac\{\\ell\}\{\\sigma^\{2\}\}\\max\_\{k\\notin\\\{i,j\\\}\}\\frac\{\\pi\_\{k\}\}\{\\pi\_\{d^\{\*\}\}\}\\exp\(\-\\kappa\)\(\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\+\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\ell\\exp\(\-\\kappa\)\(H\.198\)\+maxm∉\{i,j\}\|\|𝝁\(m\)−𝝁\(i\)\|\|2\(N−2\)exp\(−κ\)\)−1\\displaystyle\+\\max\_\{m\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(m\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}\(N\-2\)\\exp\(\-\\kappa\)\)\-1\(H\.199\)
since1σ~t2≤1mint∈\[t,τ^\)⁡σ~t2=1σ2\\frac\{1\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\leq\\frac\{1\}\{\\min\_\{t\\in\[t,\\hat\{\\tau\}\)\}\\tilde\{\\sigma\}\_\{t\}^\{2\}\}=\\frac\{1\}\{\\sigma^\{2\}\}\.

DefineWi,j:=πjπiW\_\{i,j\}:=\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\},M:=maxk∉\{i,j\}​‖𝝁\(k\)−𝝁\(i\)‖2M:=\\max\_\{k\\notin\\\{i,j\\\}\}\|\|\\bm\{\\mu\}^\{\(k\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}, andR:=maxk∉\{i,j\}⁡Wk,d∗R:=\\max\_\{k\\notin\\\{i,j\\\}\}W\_\{k,d^\{\*\}\}\. We then must chooseκ\\kappas\.t\.

8​Wi,j​κ​exp⁡\(−κ\)\+2​M​κ​exp⁡\(−κ\)​\(N−2\)​R​exp⁡\(−κ\)\\displaystyle 8W\_\{i,j\}\\kappa\\exp\(\-\\kappa\)\+2M\\kappa\\exp\(\-\\kappa\)\(N\-2\)R\\exp\(\-\\kappa\)\(H\.200\)\+M​\(N−2\)​Rσ2​\(M​exp⁡\(−κ\)\+Wi,j​ℓ​exp⁡\(−2​κ\)\+M​\(N−2\)​exp⁡\(−2​κ\)\)<1\\displaystyle\+\\frac\{M\(N\-2\)R\}\{\\sigma^\{2\}\}\(M\\exp\(\-\\kappa\)\+W\_\{i,j\}\\ell\\exp\(\-2\\kappa\)\+M\(N\-2\)\\exp\(\-2\\kappa\)\)<1\(H\.201\)
Denote:

T1​\(κ\)=8​Wi,j​κ​exp⁡\(−κ\),T\_\{1\}\(\\kappa\)=8W\_\{i,j\}\\kappa\\exp\(\-\\kappa\),\(H\.202\)
T2​\(κ\)=2​M​κ​exp⁡\(−κ\)​\(N−2\)​R​exp⁡\(−κ\),T\_\{2\}\(\\kappa\)=2M\\kappa\\exp\(\-\\kappa\)\(N\-2\)R\\exp\(\-\\kappa\),\(H\.203\)
and,

T3​\(κ\)=M​\(N−2\)​Rσ2​\(M​exp⁡\(−κ\)\+Wi,j​ℓ​exp⁡\(−2​κ\)\+M​\(N−2\)​exp⁡\(−2​κ\)\)T\_\{3\}\(\\kappa\)=\\frac\{M\(N\-2\)R\}\{\\sigma^\{2\}\}\(M\\exp\(\-\\kappa\)\+W\_\{i,j\}\\ell\\exp\(\-2\\kappa\)\+M\(N\-2\)\\exp\(\-2\\kappa\)\)\(H\.204\)
We first boundT1T\_\{1\}:

T1​\(κ\)\\displaystyle T\_\{1\}\(\\kappa\)=8​Wi,j​κ​exp⁡\(−κ\)\\displaystyle=8W\_\{i,j\}\\kappa\\exp\(\-\\kappa\)\(H\.205\)≤8​Wi,j​exp⁡\(−κ/2\)<13\\displaystyle\\leq 8W\_\{i,j\}\\exp\(\-\\kappa/2\)<\\frac\{1\}\{3\}\(H\.206\)⇔exp⁡\(−κ/2\)<124​Wi,j\\displaystyle\\iff\\exp\(\-\\kappa/2\)<\\frac\{1\}\{24W\_\{i,j\}\}\(H\.207\)⇔κ\>2​log⁡\(24​Wi,j\)\\displaystyle\\iff\\kappa\>2\\log\(24W\_\{i,j\}\)\(H\.208\)∈2​log⁡N\+Θ​\(1\)\\displaystyle\\in 2\\log N\+\\Theta\(1\)\(H\.209\)
Next, we boundT2T\_\{2\}\. Sinceκ\>1\\kappa\>1, we have thatκ​exp⁡\(−κ\)≤exp⁡\(−1\)\\kappa\\exp\(\-\\kappa\)\\leq\\exp\(\-1\), henceκ​exp⁡\(−2​κ\)=exp⁡\(−κ\)​\(κ​exp⁡\(−κ\)\)≤exp⁡\(−1\)​exp⁡\(−κ\)\\kappa\\exp\(\-2\\kappa\)=\\exp\(\-\\kappa\)\(\\kappa\\exp\(\-\\kappa\)\)\\leq\\exp\(\-1\)\\exp\(\-\\kappa\)\. Thus:

T2​\(κ\)\\displaystyle T\_\{2\}\(\\kappa\)=2​M​κ​exp⁡\(−2​κ\)​\(N−2\)​R\\displaystyle=2M\\kappa\\exp\(\-2\\kappa\)\(N\-2\)R\(H\.210\)≤2​M​exp⁡\(−1\)​exp⁡\(−κ\)​\(N−2\)​R<13\\displaystyle\\leq 2M\\exp\(\-1\)\\exp\(\-\\kappa\)\(N\-2\)R<\\frac\{1\}\{3\}\(H\.211\)⇔exp⁡\(−κ\)<exp⁡\(1\)6​M​R​\(N−2\)\\displaystyle\\iff\\exp\(\-\\kappa\)<\\frac\{\\exp\(1\)\}\{6MR\(N\-2\)\}\(H\.212\)⇔κ\>log⁡\(6​exp⁡\(−1\)​M​R​\(N−2\)\)\\displaystyle\\iff\\kappa\>\\log\(6\\exp\(\-1\)MR\(N\-2\)\)\(H\.213\)∈log⁡N\+Θ​\(1\)\\displaystyle\\in\\log N\+\\Theta\(1\)\(H\.214\)
Finally, we boundT3T\_\{3\}\. Since the inverse exponential is monotonically decreasing,exp⁡\(−2​κ\)≤exp⁡\(−κ\)\\exp\(\-2\\kappa\)\\leq\\exp\(\-\\kappa\)\. Hence,Mexp\(−κ\)\+Wi,jℓ\(exp\(−2κ\)\+M\(n2\)exp\(−2κ\)≤\(M\+Wi,jℓ\+M\(N−2\)\)exp\(−κ\)M\\exp\(\-\\kappa\)\+W\_\{i,j\}\\ell\(\\exp\(\-2\\kappa\)\+M\(n\_\{2\}\)\\exp\(\-2\\kappa\)\\leq\(M\+W\_\{i,j\}\\ell\+M\(N\-2\)\)\\exp\(\-\\kappa\)\. Define the constant:

A:=M​\(N−2\)​Rσ2​\(M\+Wi,j​ℓ\+M​\(N−2\)\)A:=\\frac\{M\(N\-2\)R\}\{\\sigma^\{2\}\}\(M\+W\_\{i,j\}\\ell\+M\(N\-2\)\)\(H\.215\)
Then:

T3​\(κ\)\\displaystyle T\_\{3\}\(\\kappa\)=M​\(N−2\)​Rσ2​\(M​exp⁡\(−κ\)\+Wi,j​ℓ​exp⁡\(−2​κ\)\+M​\(N−2\)​exp⁡\(−2​κ\)\)\\displaystyle=\\frac\{M\(N\-2\)R\}\{\\sigma^\{2\}\}\(M\\exp\(\-\\kappa\)\+W\_\{i,j\}\\ell\\exp\(\-2\\kappa\)\+M\(N\-2\)\\exp\(\-2\\kappa\)\)\(H\.216\)≤A​exp⁡\(−κ\)<13\\displaystyle\\leq A\\exp\(\-\\kappa\)<\\frac\{1\}\{3\}\(H\.217\)⇔exp⁡\(−κ\)<13​A\\displaystyle\\iff\\exp\(\-\\kappa\)<\\frac\{1\}\{3A\}\(H\.218\)⇔κ\>log⁡\(3​A\)\\displaystyle\\iff\\kappa\>\\log\(3A\)\(H\.219\)⇔κ\>log⁡\(3​M​\(N−2\)​Rσ2​\(M\+Wi,j​ℓ\+M​\(N−2\)\)\)\\displaystyle\\iff\\kappa\>\\log\(3\\frac\{M\(N\-2\)R\}\{\\sigma^\{2\}\}\(M\+W\_\{i,j\}\\ell\+M\(N\-2\)\)\)\(H\.220\)∈2​log⁡N\+Θ​\(1\)\\displaystyle\\in 2\\log N\+\\Theta\(1\)\(H\.221\)
Furthermore,κ≥log⁡N\+log⁡\(4​Ca\)∈2​log⁡N\+Θ​\(1\)\\kappa\\geq\\log N\+\\log\(4C\_\{a\}\)\\in 2\\log N\+\\Theta\(1\)\. Thus, the maximum over all these constraints is also in2​log⁡N\+Θ​\(1\)2\\log N\+\\Theta\(1\), as desired\. Finally, note thatt∈\[0,τ^\)t\\in\[0,\\hat\{\\tau\}\)was arbitrary, and the bound of the derivative ofFt​\(ξt\)F\_\{t\}\(\\xi\_\{t\}\)evaluated at the equilibria does not depend on time, soξt∗,i\\xi\_\{t\}^\{\*,i\}is uniformly instantaneously stable for allt∈\[0,τ^\)t\\in\[0,\\hat\{\\tau\}\), as desired\. ∎

### H\.4Proof of Corollary[4](https://arxiv.org/html/2605.06831#S4)

###### Proof\.

Fixτ^=min⁡\{τ,τ2\}\\hat\{\\tau\}=\\min\\\{\\tau,\\tau\_\{2\}\\\}as in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. By[Sec\.4](https://arxiv.org/html/2605.06831#S4), there exists an instantaneously stable equilibriumξt∗,i\\xi^\{\\ast,i\}\_\{t\}of[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50)within aε\\varepsilon\-ball of modeμ\(i\)\\mu^\{\(i\)\}, for allt<τ^t<\\hat\{\\tau\}\. Moreover, the proof of[Sec\.4](https://arxiv.org/html/2605.06831#S4)shows that the stability is*uniform*overt∈\[0,τ^\]t\\in\[0,\\hat\{\\tau\}\], in the sense that there exists a constantm0\>0m\_\{0\}\>0\(independent oftt\) such that

Ft′​\(ξt∗,i\)≤−m0∀t∈\[0,τ^\]\.F\_\{t\}^\{\\prime\}\(\\xi^\{\\ast,i\}\_\{t\}\)\\leq\-m\_\{0\}\\qquad\\forall t\\in\[0,\\hat\{\\tau\}\]\.\(H\.222\)
Sinceγ~j\\tilde\{\\gamma\}\_\{j\}is a smooth composition of Gaussian p\.d\.f\.s andΔξt\\varDelta\_\{\\xi\_\{t\}\}is differentiable inξ\\xi\(as used throughout the[Sec\.4](https://arxiv.org/html/2605.06831#S4)proof\), we have thatFtF\_\{t\}isC1C^\{1\}inξ\\xifor each fixedtt\. SinceFt​\(ξt∗,i\)=0F\_\{t\}\(\\xi\_\{t\}^\{\\ast,i\}\)=0and∂ξFt​\(ξt∗,i\)≤−m0<0\\partial\_\{\\xi\}F\_\{t\}\(\\xi\_\{t\}^\{\\ast,i\}\)\\leq\-m\_\{0\}<0, the implicit function theorem implies that the equilibria can be chosen as a continuous brancht↦ξt∗,it\\mapsto\\xi\_\{t\}^\{\\ast,i\}on\[0,τ^\]\[0,\\hat\{\\tau\}\]\.

Hence, by continuity ofFt′F\_\{t\}^\{\\prime\}inξ\\xifor each fixedt∈\[0,τ^\]t\\in\[0,\\hat\{\\tau\}\]there exists a radiusϰt\>0\\varkappa\_\{t\}\>0such that

Ft′​\(ξ\)≤−m02∀ξ∈\[ξt∗,i−ϰt,ξt∗,i\+ϰt\]\.F\_\{t\}^\{\\prime\}\(\\xi\)\\leq\-\\frac\{m\_\{0\}\}\{2\}\\qquad\\forall\\xi\\in\[\\xi^\{\\ast,i\}\_\{t\}\-\\varkappa\_\{t\},\\;\\xi^\{\\ast,i\}\_\{t\}\+\\varkappa\_\{t\}\]\.\(H\.223\)By joint continuity in\(t,ξ\)\(t,\\xi\)and compactness of\[0,τ^\]\[0,\\hat\{\\tau\}\], the radii may be chosen so thatϰ:=inft∈\[0,τ^\]ϰt\>0\\varkappa:=\\inf\_\{t\\in\[0,\\hat\{\\tau\}\]\}\\varkappa\_\{t\}\>0\. Recall from the proof of[Sec\.4](https://arxiv.org/html/2605.06831#S4)that, fora:=2​\(εδξ\+εγ~\)a:=2\(\\varepsilon\_\{\\delta\_\{\\xi\}\}\+\\varepsilon\_\{\\tilde\{\\gamma\}\}\), we haveFt​\(−a\)\>0,Ft​\(a\)<0,\|ξt∗,i\|≤a,∀t∈\[0,τ^\]F\_\{t\}\(\-a\)\>0,F\_\{t\}\(a\)<0,\|\\xi\_\{t\}^\{\\ast,i\}\|\\leq a,\\forall t\\in\[0,\\hat\{\\tau\}\]\. Sincea∈O​\(N​exp⁡\(−κ\)\)a\\in O\(N\\exp\(\-\\kappa\)\), there existsC\>0C\>0such thata≤C​N​exp⁡\(−κ\)a\\leq CN\\exp\(\-\\kappa\)\. Thus, to ensure2​a≤ϰ2a\\leq\\varkappa, it suffices to takeκ≥log⁡\(2​C​Nϰ\),\\kappa\\geq\\log\\\!\\left\(\\frac\{2CN\}\{\\varkappa\}\\right\),, which is satisfied by the conditionκ≥2​log⁡N\+Θ​\(1\)\\kappa\\geq 2\\log N\+\\Theta\(1\)\. Therefore,\[−a,a\]⊆\[ξt∗,i−ϰt,ξt∗,i\+ϰt\],∀t∈\[0,τ^\]\[\-a,a\]\\subseteq\[\\xi\_\{t\}^\{\\ast,i\}\-\\varkappa\_\{t\},\\xi\_\{t\}^\{\\ast,i\}\+\\varkappa\_\{t\}\],\\;\\forall t\\in\[0,\\hat\{\\tau\}\]\.

Letξt\(1\)\\xi\_\{t\}^\{\(1\)\}andξt\(2\)\\xi\_\{t\}^\{\(2\)\}be two solutions of[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50)withξt0\(1\),ξt0\(2\)∈\[−a,a\]\\xi\_\{t\_\{0\}\}^\{\(1\)\},\\xi\_\{t\_\{0\}\}^\{\(2\)\}\\in\[\-a,a\]\. SinceFt​\(−a\)\>0F\_\{t\}\(\-a\)\>0andFt​\(a\)<0F\_\{t\}\(a\)<0uniformly intt, the interval\[−a,a\]\[\-a,a\]is invariant\. Hence, for everyt<t0t<t\_\{0\},

ξt\(1\),ξt\(2\)∈\[−a,a\]⊆\[ξt∗,i−ϰt,ξt∗,i\+ϰt\]\.\\xi^\{\(1\)\}\_\{t\},\\ \\xi^\{\(2\)\}\_\{t\}\\in\[\-a,a\]\\subseteq\[\\xi^\{\\ast,i\}\_\{t\}\-\\varkappa\_\{t\},\\;\\xi^\{\\ast,i\}\_\{t\}\+\\varkappa\_\{t\}\]\.\(H\.224\)DefineΔt:=ξt\(1\)−ξt\(2\)\\varDelta\_\{t\}:=\\xi^\{\(1\)\}\_\{t\}\-\\xi^\{\(2\)\}\_\{t\}\. ThenΔt\\varDelta\_\{t\}isC1C^\{1\}intt, and:

Δ˙t=Ft​\(ξt\(1\)\)−Ft​\(ξt\(2\)\)\.\\dot\{\\varDelta\}\_\{t\}\\;=\\;F\_\{t\}\(\\xi^\{\(1\)\}\_\{t\}\)\-F\_\{t\}\(\\xi^\{\(2\)\}\_\{t\}\)\.\(H\.225\)
Now, fix anyt<t0≤τ^t<t\_\{0\}\\leq\\hat\{\\tau\}\. By the mean value theorem applied to theC1C^\{1\}functionFt​\(⋅\)F\_\{t\}\(\\cdot\), there existsθt\\theta\_\{t\}lying betweenξt\(1\)\\xi^\{\(1\)\}\_\{t\}andξt\(2\)\\xi^\{\(2\)\}\_\{t\}such that:

Ft​\(ξt\(1\)\)−Ft​\(ξt\(2\)\)=Ft′​\(θt\)​\(ξt\(1\)−ξt\(2\)\)=Ft′​\(θt\)​Δt\.F\_\{t\}\(\\xi^\{\(1\)\}\_\{t\}\)\-F\_\{t\}\(\\xi^\{\(2\)\}\_\{t\}\)\\;=\\;F\_\{t\}^\{\\prime\}\(\\theta\_\{t\}\)\\,\(\\xi^\{\(1\)\}\_\{t\}\-\\xi^\{\(2\)\}\_\{t\}\)\\;=\\;F\_\{t\}^\{\\prime\}\(\\theta\_\{t\}\)\\,\\varDelta\_\{t\}\.\(H\.226\)By[Eq\.H\.224](https://arxiv.org/html/2605.06831#A8.E224), we haveθt∈\[ξt∗,i−ϰt,ξt∗,i\+ϰt\]\\theta\_\{t\}\\in\[\\xi^\{\\ast,i\}\_\{t\}\-\\varkappa\_\{t\},\\xi^\{\\ast,i\}\_\{t\}\+\\varkappa\_\{t\}\], so[Eq\.H\.223](https://arxiv.org/html/2605.06831#A8.E223)yieldsFt′​\(θt\)≤−m0/2F\_\{t\}^\{\\prime\}\(\\theta\_\{t\}\)\\leq\-m\_\{0\}/2\. Incorporating this into[Eq\.H\.225](https://arxiv.org/html/2605.06831#A8.E225)and[Eq\.H\.226](https://arxiv.org/html/2605.06831#A8.E226)yields:

Δ˙t=Ft′​\(θt\)​Δt,with​Ft′​\(θt\)≤−m02\.\\dot\{\\varDelta\}\_\{t\}\\;=\\;F\_\{t\}^\{\\prime\}\(\\theta\_\{t\}\)\\,\\varDelta\_\{t\},\\qquad\\text\{with \}F\_\{t\}^\{\\prime\}\(\\theta\_\{t\}\)\\leq\-\\frac\{m\_\{0\}\}\{2\}\.\(H\.227\)
Next, consider the absolutely continuous functiont↦\|Δt\|t\\mapsto\|\\varDelta\_\{t\}\|\. Forttsuch thatΔt≠0\\varDelta\_\{t\}\\neq 0,

dd​t​\|Δt\|=sign​\(Δt\)​Δ˙t=sign​\(Δt\)​Ft′​\(θt\)​Δt=Ft′​\(θt\)​\|Δt\|≤−m02​\|Δt\|\.\\frac\{d\}\{dt\}\|\\varDelta\_\{t\}\|\\;=\\;\\mathrm\{sign\}\(\\varDelta\_\{t\}\)\\,\\dot\{\\varDelta\}\_\{t\}\\;=\\;\\mathrm\{sign\}\(\\varDelta\_\{t\}\)\\,F\_\{t\}^\{\\prime\}\(\\theta\_\{t\}\)\\,\\varDelta\_\{t\}\\;=\\;F\_\{t\}^\{\\prime\}\(\\theta\_\{t\}\)\\,\|\\varDelta\_\{t\}\|\\;\\leq\\;\-\\frac\{m\_\{0\}\}\{2\}\\,\|\\varDelta\_\{t\}\|\.\(H\.228\)Integrating[Eq\.H\.228](https://arxiv.org/html/2605.06831#A8.E228)and applying Grönwall’s inequality yields for allt<τ^t<\\hat\{\\tau\}:

\|ξt\(1\)−ξt\(2\)\|=\|Δt\|≤exp⁡\(−m02​\(u​\(t\)−u​\(t0\)\)\)​\|Δt0\|=exp⁡\(−m02​\(u​\(t\)−u​\(t0\)\)\)​\|ξt0\(1\)−ξt0\(2\)\|\.\|\\xi^\{\(1\)\}\_\{t\}\-\\xi^\{\(2\)\}\_\{t\}\|\\;=\\;\|\\varDelta\_\{t\}\|\\;\\leq\\;\\exp\\\!\\Big\(\-\\frac\{m\_\{0\}\}\{2\}\\big\(u\(t\)\-u\(t\_\{0\}\)\\big\)\\Big\)\\,\|\\varDelta\_\{t\_\{0\}\}\|\\;=\\;\\exp\\\!\\Big\(\-\\frac\{m\_\{0\}\}\{2\}\\big\(u\(t\)\-u\(t\_\{0\}\)\\big\)\\Big\)\\,\|\\xi^\{\(1\)\}\_\{t\_\{0\}\}\-\\xi^\{\(2\)\}\_\{t\_\{0\}\}\|\.\(H\.229\)
Takingt0=τ^t\_\{0\}=\\hat\{\\tau\}andξ\(2\)=ξ¯\\xi^\{\(2\)\}=\\bar\{\\xi\}, whereξ¯τ^=0\\bar\{\\xi\}\_\{\\hat\{\\tau\}\}=0, gives a solutionξ¯t∈\[−a,a\]\\bar\{\\xi\}\_\{t\}\\in\[\-a,a\]s\.t\. for allt∈\[0,τ^\]t\\in\[0,\\hat\{\\tau\}\], every trajectory in\[−a,a\]\[\-a,a\]contracts exponentially fast toξ¯t\\bar\{\\xi\}\_\{t\}by[Eq\.H\.229](https://arxiv.org/html/2605.06831#A8.E229)\. Note that this follows symmetrically in a neighborhood ofξt∗,j\\xi\_\{t\}^\{\\ast,j\}\.

∎

### H\.5Proof of Proposition[4](https://arxiv.org/html/2605.06831#S4)

Before stating the proof, we firstly recall the parallel dynamics from[AppendixG](https://arxiv.org/html/2605.06831#A7):

ξ˙t\\displaystyle\\dot\{\\xi\}\_\{t\}=γ~j​\(𝒚t\)−ξt\+δξt,\\displaystyle=\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\+\\delta\_\{\\xi\_\{t\}\},\(H\.230\)
where:

\|δξ\|,‖𝜹𝒘‖2≤ε∈𝒪​\(N​exp⁡\(−κ\)\),\|\\delta\_\{\\xi\}\|,\|\|\\bm\{\\delta\}\_\{\\bm\{w\}\}\|\|\_\{2\}\\leq\\varepsilon\\in\\mathcal\{O\}\\left\(N\\exp\(\-\\kappa\)\\right\),\(H\.231\)
for:

𝒚t=𝝁\(i\)\+ξt​ℓ​𝒖\+𝒘t,\\bm\{y\}\_\{t\}=\\bm\{\\mu\}^\{\(i\)\}\+\\xi\_\{t\}\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\},\(H\.232\)
whereℓ=‖𝝁\(j\)−𝝁\(i\)‖2,𝒖=𝝁\(j\)−𝝁\(i\)ℓ\\ell=\|\|\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\},\\bm\{u\}=\\frac\{\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\}\{\\ell\}, and𝒘t\\bm\{w\}\_\{t\}satisfies⟨𝒘t,𝒖⟩=0\\langle\\bm\{w\}\_\{t\},\\bm\{u\}\\rangle=0, and we also consider the responsibility of modejj:

γ~j​\(𝒚t\)=πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)∑m∈\[N\]​πm𝒩​\(𝒚t;𝝁\(m\),σ~t2​𝑰\)\.\\tilde\{\\gamma\}\_\{j\}\(\\bm\{y\}\_\{t\}\)=\\frac\{\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\sum\_\{m\\in\[N\]\\pi\_\{m\}\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(m\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\.\(H\.233\)
We simplify the parallel dynamics by dropping the term bounded byε\\varepsilon, and additionally consider only the dynamics with respect to the two\-mode responsibility, i\.e\. we consider:

ξ˙t=γ~j\(i,j\)​\(𝒚t\)−ξt,\\dot\{\\xi\}\_\{t\}=\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\},\(H\.234\)
where:

γ~j\(i,j\)​\(𝒚t\):=πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)πi​𝒩​\(𝒚t;𝝁\(i\),σ~t2​𝑰\)\+πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)\.\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\):=\\frac\{\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(i\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\+\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\.\(H\.235\)
Throughout the statement and the proof, we work in the rescaled forward timeuudefined in[AppendixG](https://arxiv.org/html/2605.06831#A7)and relabel this rescaled time byttfor simplicity\. Thus, the valueξτ3\\xi\_\{\\tau\_\{3\}\}denotes the initial value of the approximate parallel dynamics over the remaining time interval of lengthτ3\\tau\_\{3\}\. Then, we prove the result:

###### Proof\.

\(1\): Since the instance is within theε\\varepsilon\-tube aroundLt\(i,j\)L\_\{t\}^\{\(i,j\)\}, we have that‖𝒘t‖≤ε\|\|\\bm\{w\}\_\{t\}\|\|\\leq\\varepsilon, just as we do[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. Thus, by the same token as in the proof of[Sec\.4](https://arxiv.org/html/2605.06831#S4), we have that:

‖𝒚t−𝝁\(i\)‖22=ξt2​ℓ2\+‖𝒘t‖22,\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}=\\xi\_\{t\}^\{2\}\\ell^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\},\(H\.236\)
and:

‖𝒚t−𝝁\(j\)‖22=‖\(ξt−1\)​ℓ​𝒖\+𝒘t‖22=\(1−ξt\)2​ℓ2\+‖𝒘t‖22,\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}=\|\|\(\\xi\_\{t\}\-1\)\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\}=\(1\-\\xi\_\{t\}\)^\{2\}\\ell^\{2\}\+\|\|\\bm\{w\}\_\{t\}\|\|\_\{2\}^\{2\},\(H\.237\)
yielding:

‖𝒚t−𝝁\(j\)‖22−‖𝒚t−𝝁\(i\)‖22=\(1−2​ξt\)​ℓ2\.\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\-\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}=\(1\-2\\xi\_\{t\}\)\\ell^\{2\}\.\(H\.238\)
From[Eq\.H\.235](https://arxiv.org/html/2605.06831#A8.E235), we have that:

γ~j\(i,j\)​\(𝒚t\)1−γ~j\(i,j\)​\(𝒚t\)\\displaystyle\\frac\{\\tilde\{\\gamma\}^\{\(i,j\)\}\_\{j\}\(\\bm\{y\}\_\{t\}\)\}\{1\-\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\}=πj​𝒩​\(𝒚t;𝝁\(j\),σ~t2​𝑰\)πi​𝒩​\(𝒚t;𝝁\(i\),σ~t2​𝑰\)\\displaystyle=\\frac\{\\pi\_\{j\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(j\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\{\\pi\_\{i\}\\mathcal\{N\}\(\\bm\{y\}\_\{t\};\\bm\{\\mu\}^\{\(i\)\},\\tilde\{\\sigma\}\_\{t\}^\{2\}\\bm\{I\}\)\}\(H\.239\)=πjπi​exp⁡\(−‖𝒚t−𝝁\(j\)‖22−‖𝒚t−𝝁\(i\)‖222​σ~t2\)\\displaystyle=\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\(\-\\frac\{\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(j\)\}\|\|\_\{2\}^\{2\}\-\|\|\\bm\{y\}\_\{t\}\-\\bm\{\\mu\}^\{\(i\)\}\|\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\)\(H\.240\)=πjπi​exp⁡\(ℓ2σ~t2​\(ξ−12\)\),[Eq\.H\.238](https://arxiv.org/html/2605.06831#A8.E238)\\displaystyle=\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\(\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\(\\xi\-\\frac\{1\}\{2\}\)\),\\text\{ \\lx@cref\{creftype~refnum\}\{eq:temp\_slowdown\_1\}\}\(H\.241\)
Sinceπi=πj\\pi\_\{i\}=\\pi\_\{j\}, atξ=12\\xi=\\frac\{1\}\{2\},[Eq\.H\.241](https://arxiv.org/html/2605.06831#A8.E241)evaluates to:

γ~j\(i,j\)​\(𝒚t\)1−γ~j\(i,j\)​\(𝒚t\)\|ξt=1/2=1\\frac\{\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\}\{1\-\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\}\\big\|\_\{\\xi\_\{t\}=1/2\}=1\(H\.242\)
yielding:

γ~j\(i,j\)​\(𝒚t\)ξt=1/2=1/2\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\_\{\\xi\_\{t\}=1/2\}=1/2\(H\.243\)
DefineFt​\(ξt\)=γ~j\(i,j\)​\(𝒚t\)−ξtF\_\{t\}\(\\xi\_\{t\}\)=\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\-\\xi\_\{t\}\. Then,Ft​\(1/2\)=0F\_\{t\}\(1/2\)=0, soξ∗=12\\xi^\{\*\}=\\frac\{1\}\{2\}is an \(instantaneous\) equilibrium of[Eq\.H\.234](https://arxiv.org/html/2605.06831#A8.E234)\.

The corresponding point inℝd\\mathbb\{R\}^\{d\}is thus:

𝒚t∗\\displaystyle\\bm\{y\}\_\{t\}^\{\*\}:=𝒚t\|ξ=1/2\\displaystyle:=\\bm\{y\}\_\{t\}\\big\|\_\{\\xi=1/2\}\(H\.244\)=𝝁\(i\)\+12​ℓ​𝒖\+𝒘t\\displaystyle=\\bm\{\\mu\}^\{\(i\)\}\+\\frac\{1\}\{2\}\\ell\\bm\{u\}\+\\bm\{w\}\_\{t\}\(H\.245\)=𝝁\(i\)\+𝝁\(j\)2\+𝒘t\.\\displaystyle=\\frac\{\\bm\{\\mu\}^\{\(i\)\}\+\\bm\{\\mu\}^\{\(j\)\}\}\{2\}\+\\bm\{w\}\_\{t\}\.\(H\.246\)
Next, we demonstrate that this is a hyperbolic unstable \(instantaneous\) equilibria\. We have that:

dd​ξt​Ft​\(ξt\)\\displaystyle\\frac\{d\}\{d\\xi\_\{t\}\}F\_\{t\}\(\\xi\_\{t\}\)=dd​ξt​γ~j\(i,j\)​\(𝒚t\)−1\\displaystyle=\\frac\{d\}\{d\\xi\_\{t\}\}\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\-1\(H\.247\)=ℓ2σ~t2​γ~j\(i,j\)​\(𝒚t\)​\(1−γ~j\(i,j\)​\(𝒚t\)\)−1,\\displaystyle=\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\(1\-\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\bm\{y\}\_\{t\}\)\)\-1,\(H\.248\)
We thus have that:

dd​ξt​Ft​\(ξt\)\|ξt=1/2=ℓ2σ~t2​12​12−1=ℓ24​σ~t2−1\.\\frac\{d\}\{d\\xi\_\{t\}\}F\_\{t\}\(\\xi\_\{t\}\)\\big\|\_\{\\xi\_\{t\}=1/2\}=\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\frac\{1\}\{2\}\\frac\{1\}\{2\}\-1=\\frac\{\\ell^\{2\}\}\{4\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\-1\.\(H\.249\)
By[Sec\.4](https://arxiv.org/html/2605.06831#S4), we thus have that:

λt:=ℓ24​σ~t2−1≥κ−1\>0\\lambda\_\{t\}:=\\frac\{\\ell^\{2\}\}\{4\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\-1\\geq\\kappa\-1\>0\(H\.250\)
sinceκ\>1\\kappa\>1\. Sinceλt\\lambda\_\{t\}has no complex components,ξt=1/2\\xi\_\{t\}=1/2thus corresponds to a hyperbolic unstable \(instantaneous\) equilibria\.

\(2\): Considert∈\[0,τ3\]t\\in\[0,\\tau\_\{3\}\]and recall the approximate parallel dynamics[Eq\.H\.234](https://arxiv.org/html/2605.06831#A8.E234)\. Note thatλ¯\\underline\{\\lambda\}andλ¯\\bar\{\\lambda\}exist by the extreme value theorem, since they are defined on a compact domain with respect to continuousλt\\lambda\_\{t\}\(sinceσ~t2\\tilde\{\\sigma\}\_\{t\}^\{2\}is continuous\)\. Within Tube\(i,j\)ε,t\{\}\_\{\\varepsilon,t\}^\{\(i,j\)\}, by[Eq\.H\.241](https://arxiv.org/html/2605.06831#A8.E241), we have the identity:

γ~j\(i,j\)​\(ξt\)1−γ~j\(i,j\)​\(ξt\)=πjπi​exp⁡\(ℓ2σ~t2​\(ξt−12\)\)\.\\frac\{\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\xi\_\{t\}\)\}\{1\-\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}\(\\xi\_\{t\}\)\}=\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\\!\\left\(\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\Bigl\(\\xi\_\{t\}\-\\frac\{1\}\{2\}\\Bigr\)\\right\)\.\(H\.251\)Noting thatπi=πj\\pi\_\{i\}=\\pi\_\{j\}, denotingat:=ℓ2σ~t2a\_\{t\}:=\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}and solving the log\-odds equation obtains the logistic form:

γ~j\(i,j\)=11\+exp⁡\(−at​\(ξt−12\)\)=:ς\(at\(ξt−12\)\),\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\}=\\frac\{1\}\{1\+\\exp\(\-a\_\{t\}\(\\xi\_\{t\}\-\\frac\{1\}\{2\}\)\)\}=:\\varsigma\\\!\\left\(a\_\{t\}\\Bigl\(\\xi\_\{t\}\-\\frac\{1\}\{2\}\\Bigr\)\\right\),\(H\.252\)whereς​\(z\)=11\+e−z\\varsigma\(z\)=\\frac\{1\}\{1\+e^\{\-z\}\}is the sigmoid function\. Sinceς′​\(z\)=ς​\(z\)​\(1−ς​\(z\)\)\\varsigma^\{\\prime\}\(z\)=\\varsigma\(z\)\(1\-\\varsigma\(z\)\)andς′′​\(z\)=ς′​\(z\)​\(1−2​ς​\(z\)\)\\varsigma^\{\\prime\\prime\}\(z\)=\\varsigma^\{\\prime\}\(z\)\(1\-2\\varsigma\(z\)\), and since0≤ς≤10\\leq\\varsigma\\leq 1implies0≤ς′≤140\\leq\\varsigma^\{\\prime\}\\leq\\frac\{1\}\{4\}and\|1−2​ς\|≤1\|1\-2\\varsigma\|\\leq 1, we have\|ς′′​\(z\)\|≤14\|\\varsigma^\{\\prime\\prime\}\(z\)\|\\leq\\frac\{1\}\{4\}\. Hence,

\|Ft′′​\(ξt\)\|=\|γ~j\(i,j\)⁣′′​\(ξt\)\|≤at24\.\\displaystyle\|F\_\{t\}^\{\\prime\\prime\}\(\\xi\_\{t\}\)\|=\\bigl\|\\tilde\{\\gamma\}\_\{j\}^\{\(i,j\)\\prime\\prime\}\(\\xi\_\{t\}\)\\bigr\|\\leq\\frac\{a\_\{t\}^\{2\}\}\{4\}\.\(H\.253\)
Next, for anaato be chosen, consider0<\|a\|≤ϑ0<\|a\|\\leq\\vartheta\. SinceFt∈C2F\_\{t\}\\in C^\{2\}, Taylor’s theorem with Lagrange remainder atξ=12\\xi=\\frac\{1\}\{2\}gives:

Ft​\(12\+a\)=Ft​\(12\)\+Ft′​\(12\)​a\+12​Ft′′​\(12\+r​a\)​a2F\_\{t\}\\\!\\left\(\\frac\{1\}\{2\}\+a\\right\)=F\_\{t\}\\\!\\left\(\\frac\{1\}\{2\}\\right\)\+F\_\{t\}^\{\\prime\}\\\!\\left\(\\frac\{1\}\{2\}\\right\)a\+\\frac\{1\}\{2\}F\_\{t\}^\{\\prime\\prime\}\\\!\\left\(\\frac\{1\}\{2\}\+ra\\right\)a^\{2\}\(H\.254\)for somer∈\(0,1\)r\\in\(0,1\)\. UsingFt​\(12\)=0F\_\{t\}\(\\frac\{1\}\{2\}\)=0andFt′​\(12\)=λtF\_\{t\}^\{\\prime\}\(\\frac\{1\}\{2\}\)=\\lambda\_\{t\}, we obtain

Ft​\(12\+a\)=λt​a\+12​Ft′′​\(12\+r​a\)​a2\.F\_\{t\}\\\!\\left\(\\frac\{1\}\{2\}\+a\\right\)=\\lambda\_\{t\}a\+\\frac\{1\}\{2\}F\_\{t\}^\{\\prime\\prime\}\\\!\\left\(\\frac\{1\}\{2\}\+ra\\right\)a^\{2\}\.\(H\.255\)By[Eq\.H\.253](https://arxiv.org/html/2605.06831#A8.E253),

\|Ft​\(12\+a\)−λt​a\|\\displaystyle\\left\|F\_\{t\}\\\!\\left\(\\frac\{1\}\{2\}\+a\\right\)\-\\lambda\_\{t\}a\\right\|≤12⋅at24​\|a\|2=at28​\|a\|2\.\\displaystyle\\leq\\frac\{1\}\{2\}\\cdot\\frac\{a\_\{t\}^\{2\}\}\{4\}\|a\|^\{2\}=\\frac\{a\_\{t\}^\{2\}\}\{8\}\|a\|^\{2\}\.\(H\.256\)Dividing by\|a\|\|a\|yields

\|Ft​\(12\+a\)a−λt\|≤at28​\|a\|≤at28​ϑ,\\displaystyle\\left\|\\frac\{F\_\{t\}\(\\frac\{1\}\{2\}\+a\)\}\{a\}\-\\lambda\_\{t\}\\right\|\\leq\\frac\{a\_\{t\}^\{2\}\}\{8\}\|a\|\\leq\\frac\{a\_\{t\}^\{2\}\}\{8\}\\vartheta,\(H\.257\)and thus

λt−at28​ϑ≤Ft​\(12\+a\)a≤λt\+at28​ϑ\.\\lambda\_\{t\}\-\\frac\{a\_\{t\}^\{2\}\}\{8\}\\vartheta\\leq\\frac\{F\_\{t\}\(\\frac\{1\}\{2\}\+a\)\}\{a\}\\leq\\lambda\_\{t\}\+\\frac\{a\_\{t\}^\{2\}\}\{8\}\\vartheta\.\(H\.258\)Sinceλt=ℓ24​σ~t2−1\\lambda\_\{t\}=\\frac\{\\ell^\{2\}\}\{4\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\-1, we haveat=ℓ2σ~t2=4​\(1\+λt\)a\_\{t\}=\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}=4\(1\+\\lambda\_\{t\}\), henceat28=2​\(1\+λt\)2\\frac\{a\_\{t\}^\{2\}\}\{8\}=2\(1\+\\lambda\_\{t\}\)^\{2\}\. Plugging into[Eq\.H\.258](https://arxiv.org/html/2605.06831#A8.E258)gives

Ft​\(12\+a\)a≤λt\+2\(1\+λt\)2ϑ≤λ¯\+2\(1\+λ¯\)2ϑ=:Λ\+,\\frac\{F\_\{t\}\(\\frac\{1\}\{2\}\+a\)\}\{a\}\\leq\\lambda\_\{t\}\+2\(1\+\\lambda\_\{t\}\)^\{2\}\\vartheta\\leq\\bar\{\\lambda\}\+2\(1\+\\bar\{\\lambda\}\)^\{2\}\\vartheta=:\\Lambda\_\{\+\},\(H\.259\)and similarly

Ft​\(12\+a\)a≥λt−2​\(1\+λt\)2​ϑ≥λ¯−2​\(1\+λ¯\)2​ϑ\.\\frac\{F\_\{t\}\(\\frac\{1\}\{2\}\+a\)\}\{a\}\\geq\\lambda\_\{t\}\-2\(1\+\\lambda\_\{t\}\)^\{2\}\\vartheta\\geq\\underline\{\\lambda\}\-2\(1\+\\bar\{\\lambda\}\)^\{2\}\\vartheta\.\(H\.260\)By our choice of radiusϑ\\varthetain the statement of[Sec\.4](https://arxiv.org/html/2605.06831#S4), we haveλ¯−2​\(1\+λ¯\)2​ϑ\>0\\underline\{\\lambda\}\-2\(1\+\\bar\{\\lambda\}\)^\{2\}\\vartheta\>0, henceFt​\(12\+a\)a\>0\\frac\{F\_\{t\}\(\\frac\{1\}\{2\}\+a\)\}\{a\}\>0, so

sign​\(Ft​\(12\+a\)\)=sign​\(a\)whenever​0<\|a\|≤ϑ,\\mathrm\{sign\}\\\!\\left\(F\_\{t\}\\\!\\left\(\\tfrac\{1\}\{2\}\+a\\right\)\\right\)=\\mathrm\{sign\}\(a\)\\qquad\\text\{whenever \}0<\|a\|\\leq\\vartheta,uniformly for allt∈\[0,τ3\]t\\in\[0,\\tau\_\{3\}\]\.

Now fix a trajectory that stays in the neighborhood0<\|ξt−12\|≤ϑ0<\|\\xi\_\{t\}\-\\frac\{1\}\{2\}\|\\leq\\varthetaand define

At:=ξt−12\.A\_\{t\}:=\\xi\_\{t\}\-\\frac\{1\}\{2\}\.ThenAt≠0A\_\{t\}\\neq 0and, using the sign property above,

dd​t​\|At\|\\displaystyle\\frac\{d\}\{dt\}\|A\_\{t\}\|=sign​\(At\)​A˙t=sign​\(At\)​ξ˙t=sign​\(At\)​Ft​\(12\+At\)=\|Ft​\(12\+At\)\|\.\\displaystyle=\\mathrm\{sign\}\(A\_\{t\}\)\\dot\{A\}\_\{t\}=\\mathrm\{sign\}\(A\_\{t\}\)\\dot\{\\xi\}\_\{t\}=\\mathrm\{sign\}\(A\_\{t\}\)F\_\{t\}\\\!\\left\(\\tfrac\{1\}\{2\}\+A\_\{t\}\\right\)=\\left\|F\_\{t\}\\\!\\left\(\\tfrac\{1\}\{2\}\+A\_\{t\}\\right\)\\right\|\.\(H\.261\)Using[Eq\.H\.259](https://arxiv.org/html/2605.06831#A8.E259)witha=Ata=A\_\{t\}yields

\|Ft​\(12\+At\)\|≤Λ\+​\|At\|,\\left\|F\_\{t\}\\\!\\left\(\\tfrac\{1\}\{2\}\+A\_\{t\}\\right\)\\right\|\\leq\\Lambda\_\{\+\}\|A\_\{t\}\|,and hence

dd​t​\|At\|≤Λ\+​\|At\|whenever​0<\|At\|≤ϑ\.\\frac\{d\}\{dt\}\|A\_\{t\}\|\\leq\\Lambda\_\{\+\}\|A\_\{t\}\|\\qquad\\text\{whenever \}0<\|A\_\{t\}\|\\leq\\vartheta\.\(H\.262\)
Next, we show by contradiction that a trajectoryξt\\xi\_\{t\}satisfying0<\|ξτ3−12\|<ϑ​exp⁡\(−Λ\+​τ3\)0<\|\\xi\_\{\\tau\_\{3\}\}\-\\frac\{1\}\{2\}\|<\\vartheta\\exp\(\-\\Lambda\_\{\+\}\\tau\_\{3\}\)never exits theϑ\\vartheta\-neighborhood on\[0,τ3\]\[0,\\tau\_\{3\}\]\. DefineAt=ξt−12A\_\{t\}=\\xi\_\{t\}\-\\frac\{1\}\{2\}and relabelssastt\. Letϱ\\varrhobe the first exit time from the neighborhood\|At\|<ϑ\|A\_\{t\}\|<\\vartheta, i\.e\.

ϱ:=inf\{t∈\[0,τ3\]:\|At\|≥ϑ\},\\varrho:=\\inf\\\{t\\in\[0,\\tau\_\{3\}\]:\|A\_\{t\}\|\\geq\\vartheta\\\},\(H\.263\)with the convention thatinf∅=\+∞\\inf\\emptyset=\+\\infty\. Suppose for contradiction thatϱ≤τ3\\varrho\\leq\\tau\_\{3\}\. Then\|At\|<ϑ\|A\_\{t\}\|<\\varthetafor allt∈\[0,ϱ\)t\\in\[0,\\varrho\)and\|Aϱ\|=ϑ\|A\_\{\\varrho\}\|=\\varthetaby continuity\. On\[0,ϱ\)\[0,\\varrho\),[Eq\.H\.262](https://arxiv.org/html/2605.06831#A8.E262)holds, so Grönwall’s inequality gives

\|At\|≤\|Aτ3\|​exp⁡\(Λ\+​t\)∀t∈\[0,ϱ\)\.\|A\_\{t\}\|\\leq\|A\_\{\\tau\_\{3\}\}\|\\exp\(\\Lambda\_\{\+\}t\)\\qquad\\forall t\\in\[0,\\varrho\)\.\(H\.264\)Takingt→ϱt\\to\\varrhoin[Eq\.H\.264](https://arxiv.org/html/2605.06831#A8.E264)and using continuity of\|At\|\|A\_\{t\}\|yields:

\|Aϱ\|\\displaystyle\|A\_\{\\varrho\}\|≤\|Aτ3\|​exp⁡\(Λ\+​ϱ\)≤\|Aτ3\|​exp⁡\(Λ\+​τ3\)<ϑ,\\displaystyle\\leq\|A\_\{\\tau\_\{3\}\}\|\\exp\(\\Lambda\_\{\+\}\\varrho\)\\leq\|A\_\{\\tau\_\{3\}\}\|\\exp\(\\Lambda\_\{\+\}\\tau\_\{3\}\)<\\vartheta,\(H\.265\)since\|Aτ3\|=\|ξτ3−12\|<ϑ​exp⁡\(−Λ\+​τ3\)\|A\_\{\\tau\_\{3\}\}\|=\\bigl\|\\xi\_\{\\tau\_\{3\}\}\-\\frac\{1\}\{2\}\\bigr\|<\\vartheta\\exp\(\-\\Lambda\_\{\+\}\\tau\_\{3\}\)\. This contradicts\|Aϱ\|=ϑ\|A\_\{\\varrho\}\|=\\vartheta\. Hence,ϱ=\+∞\\varrho=\+\\infty, and therefore\|At\|<ϑ\|A\_\{t\}\|<\\varthetafor everyt∈\[0,τ3\]t\\in\[0,\\tau\_\{3\}\]\. This thus yields:

\|ξt−12\|<ϑfor every​t∈\[0,τ3\],\\left\|\\xi\_\{t\}\-\\frac\{1\}\{2\}\\right\|<\\vartheta\\quad\\text\{for every \}t\\in\[0,\\tau\_\{3\}\],\(H\.266\)which is the persistence result\.

Finally, by contrapositive, if exit does occur by time0, necessarily:

\|ξτ3−12\|≥ϑ​exp⁡\(−Λ\+​τ3\),\\left\|\\xi\_\{\\tau\_\{3\}\}\-\\frac\{1\}\{2\}\\right\|\\geq\\vartheta\\exp\(\-\\Lambda\_\{\+\}\\tau\_\{3\}\),\(H\.267\)and thus a necessary condition for exit is:

τ3≥1Λ\+​log⁡\(ϑ\|ξτ3−12\|\),\\tau\_\{3\}\\geq\\frac\{1\}\{\\Lambda\_\{\+\}\}\\log\\\!\\left\(\\frac\{\\vartheta\}\{\|\\xi\_\{\\tau\_\{3\}\}\-\\frac\{1\}\{2\}\|\}\\right\),\(H\.268\)as desired\. ∎

### H\.6Proof of Proposition[5](https://arxiv.org/html/2605.06831#S5)

###### Proof\.

SetT:=τ3T:=\\tau\_\{3\}\. Under the time relabelingθ:=T−t\\theta:=T\-t, the proposition’s event\{\|A0\|≤ϑ\}\\\{\|A\_\{0\}\|\\leq\\vartheta\\\}corresponds to\{\|AT\|≤ϑ\}\\\{\|A\_\{T\}\|\\leq\\vartheta\\\};AT=0A\_\{T\}=0in[Eq\.4](https://arxiv.org/html/2605.06831#S5.E4)corresponds toA0=0A\_\{0\}=0; and the assumption withλrep\\lambda\_\{\\mathrm\{rep\}\}is negated in the resulting forward\-time process, which we analyze below\.

Define the continuous martingale

Mt:=∫0t2​η​\(s\)​𝑑Bs,t∈\[0,T\]\.M\_\{t\}:=\\int\_\{0\}^\{t\}\\sqrt\{2\\eta\(s\)\}\\,dB\_\{s\},\\qquad t\\in\[0,T\]\.\(H\.269\)Integrating[Eq\.4](https://arxiv.org/html/2605.06831#S5.E4)yields the decomposition

At=∫0tb​\(As,s\)​𝑑s\+Mt,t∈\[0,T\]\.A\_\{t\}=\\int\_\{0\}^\{t\}b\(A\_\{s\},s\)\\,ds\+M\_\{t\},\\qquad t\\in\[0,T\]\.\(H\.270\)
We first bound the confinement event

E2​ϑ:=\{sup0≤t≤T\|At\|≤2​ϑ\}\.E\_\{2\\vartheta\}:=\\Big\\\{\\sup\_\{0\\leq t\\leq T\}\|A\_\{t\}\|\\leq 2\\vartheta\\Big\\\}\.\(H\.271\)OnE2​ϑE\_\{2\\vartheta\}we have\|As\|≤2​ϑ\|A\_\{s\}\|\\leq 2\\varthetafor alls∈\[0,T\]s\\in\[0,T\], hence by the local drift bound,

\|b​\(As,s\)\|≤K​\(s\)​\|As\|≤2​K​\(s\)​ϑ,s∈\[0,T\]\.\|b\(A\_\{s\},s\)\|\\leq K\(s\)\\,\|A\_\{s\}\|\\leq 2K\(s\)\\vartheta,\\qquad s\\in\[0,T\]\.\(H\.272\)Therefore, for eacht∈\[0,T\]t\\in\[0,T\]and onE2​ϑE\_\{2\\vartheta\},

\|∫0tb​\(As,s\)​𝑑s\|≤∫0t\|b​\(As,s\)\|​𝑑s≤2​ϑ​∫0TK​\(s\)​𝑑s\.\\Big\|\\int\_\{0\}^\{t\}b\(A\_\{s\},s\)\\,ds\\Big\|\\leq\\int\_\{0\}^\{t\}\|b\(A\_\{s\},s\)\|\\,ds\\leq 2\\vartheta\\int\_\{0\}^\{T\}K\(s\)\\,ds\.\(H\.273\)Combining[Eqs\.H\.270](https://arxiv.org/html/2605.06831#A8.E270)and[H\.273](https://arxiv.org/html/2605.06831#A8.E273), for allt∈\[0,T\]t\\in\[0,T\]and onE2​ϑE\_\{2\\vartheta\},

\|Mt\|=\|At−∫0tb​\(As,s\)​𝑑s\|≤\|At\|\+\|∫0tb​\(As,s\)​𝑑s\|≤2​ϑ​\(1\+∫0TK​\(s\)​𝑑s\)\.\|M\_\{t\}\|=\\Big\|A\_\{t\}\-\\int\_\{0\}^\{t\}b\(A\_\{s\},s\)\\,ds\\Big\|\\leq\|A\_\{t\}\|\+\\Big\|\\int\_\{0\}^\{t\}b\(A\_\{s\},s\)\\,ds\\Big\|\\leq 2\\vartheta\\Big\(1\+\\int\_\{0\}^\{T\}K\(s\)\\,ds\\Big\)\.\(H\.274\)Hence,

E2​ϑ⊆\{sup0≤t≤T\|Mt\|≤2​\(1\+∫0TK​\(s\)​𝑑s\)​ϑ\}\.E\_\{2\\vartheta\}\\subseteq\\Big\\\{\\sup\_\{0\\leq t\\leq\{T\}\}\|M\_\{t\}\|\\leq 2\\Big\(1\+\\int\_\{0\}^\{T\}K\(s\)\\,ds\\Big\)\\vartheta\\Big\\\}\.\(H\.275\)
We next identify the quadratic variation ofMM\. Fixt∈\[0,T\]t\\in\[0,T\]and letπ=\{0=t0<t1<⋯<tn=t\}\\pi=\\\{0=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{n\}=t\\\}be a partition of\[0,t\]\[0,t\]with mesh

\|π\|:=max0≤k≤n−1⁡\(tk\+1−tk\)\.\|\\pi\|:=\\max\_\{0\\leq k\\leq n\-1\}\(t\_\{k\+1\}\-t\_\{k\}\)\.\(H\.276\)Write increments

Δ​Mk:=Mtk\+1−Mtk=∫tktk\+12​η​\(s\)​𝑑Bs\.\\Delta M\_\{k\}:=M\_\{t\_\{k\+1\}\}\-M\_\{t\_\{k\}\}=\\int\_\{t\_\{k\}\}^\{t\_\{k\+1\}\}\\sqrt\{2\\eta\(s\)\}\\,dB\_\{s\}\.\(H\.277\)Define

Q​\(π\):=∑k=0n−1\(Δ​Mk\)2\.Q\(\\pi\):=\\sum\_\{k=0\}^\{n\-1\}\(\\Delta M\_\{k\}\)^\{2\}\.\(H\.278\)By Itô isometry applied to[Eq\.H\.277](https://arxiv.org/html/2605.06831#A8.E277),

𝔼\[\(ΔMk\)2\]=∫tktk\+12η\(s\)ds=:vk\.\\mathbb\{E\}\\big\[\(\\Delta M\_\{k\}\)^\{2\}\\big\]=\\int\_\{t\_\{k\}\}^\{t\_\{k\+1\}\}2\\eta\(s\)\\,ds=:v\_\{k\}\.\(H\.279\)SinceΔ​Mk\\Delta M\_\{k\}is Gaussian with mean0and variancevkv\_\{k\},

𝔼​\[\(Δ​Mk\)4\]=3​vk2,Var​\(\(Δ​Mk\)2\)=2​vk2\.\\mathbb\{E\}\\big\[\(\\Delta M\_\{k\}\)^\{4\}\\big\]=3v\_\{k\}^\{2\},\\qquad\\mathrm\{Var\}\\big\(\(\\Delta M\_\{k\}\)^\{2\}\\big\)=2v\_\{k\}^\{2\}\.\(H\.280\)Using independence of Brownian increments,

𝔼​\[Q​\(π\)\]=∑k=0n−1vk=∫0t2​η​\(s\)​𝑑s,\\mathbb\{E\}\[Q\(\\pi\)\]=\\sum\_\{k=0\}^\{n\-1\}v\_\{k\}=\\int\_\{0\}^\{t\}2\\eta\(s\)\\,ds,\(H\.281\)and

Var​\(Q​\(π\)\)=2​∑k=0n−1vk2≤2​\(max0≤k≤n−1⁡vk\)​∑k=0n−1vk\.\\mathrm\{Var\}\(Q\(\\pi\)\)=2\\sum\_\{k=0\}^\{n\-1\}v\_\{k\}^\{2\}\\leq 2\\Big\(\\max\_\{0\\leq k\\leq n\-1\}v\_\{k\}\\Big\)\\sum\_\{k=0\}^\{n\-1\}v\_\{k\}\.\(H\.282\)Sinceη∈L1​\(\[0,T\]\)\\eta\\in L^\{1\}\(\[0,T\]\), absolute continuity of the integral impliesmaxk⁡vk→0\\max\_\{k\}v\_\{k\}\\to 0as\|π\|→0\|\\pi\|\\to 0\. Hence,

𝔼​\[\(Q​\(π\)−∫0t2​η​\(s\)​𝑑s\)2\]=Var​\(Q​\(π\)\)→0as​\|π\|→0\.\\mathbb\{E\}\\Big\[\\big\(Q\(\\pi\)\-\\int\_\{0\}^\{t\}2\\eta\(s\)\\,ds\\big\)^\{2\}\\Big\]=\\mathrm\{Var\}\(Q\(\\pi\)\)\\to 0\\qquad\\text\{as \}\|\\pi\|\\to 0\.\(H\.283\)ThereforeQ​\(π\)→∫0t2​η​\(s\)​𝑑sQ\(\\pi\)\\to\\int\_\{0\}^\{t\}2\\eta\(s\)\\,dsin probability\. In particular, the quadratic variation

\[M\]t:=lim\|π\|→0Q​\(π\)\[M\]\_\{t\}:=\\lim\_\{\|\\pi\|\\to 0\}Q\(\\pi\)\(H\.284\)satisfies

\[M\]t=∫0t2​η​\(s\)​𝑑s,t∈\[0,T\]\.\[M\]\_\{t\}=\\int\_\{0\}^\{t\}2\\eta\(s\)\\,ds,\\qquad t\\in\[0,T\]\.\(H\.285\)
Next, by Dambis–Dubins–Schwarz\(Dubins & Schwarz,[1965](https://arxiv.org/html/2605.06831#bib.bib13)\), there exists a standard Brownian motion\(Wr\)r≥0\(W\_\{r\}\)\_\{r\\geq 0\}such that

Mt=W\[M\]t,t∈\[0,T\]\.M\_\{t\}=W\_\{\[M\]\_\{t\}\},\\qquad t\\in\[0,T\]\.\(H\.286\)Since\[M\]t\[M\]\_\{t\}is nondecreasing,

sup0≤t≤T\|Mt\|=sup0≤r≤\[M\]T\|Wr\|,\[M\]T=2​∫0Tη​\(s\)​𝑑s\.\\sup\_\{0\\leq t\\leq T\}\|M\_\{t\}\|=\\sup\_\{0\\leq r\\leq\[M\]\_\{T\}\}\|W\_\{r\}\|,\\qquad\[M\]\_\{T\}=2\\int\_\{0\}^\{T\}\\eta\(s\)\\,ds\.\(H\.287\)Combining[Eqs\.H\.275](https://arxiv.org/html/2605.06831#A8.E275)and[H\.287](https://arxiv.org/html/2605.06831#A8.E287)gives

ℙ​\(E2​ϑ\)≤ℙ​\(sup0≤r≤\[M\]T\|Wr\|≤2​\(1\+∫0TK​\(s\)​𝑑s\)​ϑ\)\.\\mathbb\{P\}\(E\_\{2\\vartheta\}\)\\leq\\mathbb\{P\}\\Big\(\\sup\_\{0\\leq r\\leq\[M\]\_\{T\}\}\|W\_\{r\}\|\\leq 2\\Big\(1\+\\int\_\{0\}^\{T\}K\(s\)\\,ds\\Big\)\\vartheta\\Big\)\.\(H\.288\)Applying Lemma[G](https://arxiv.org/html/2605.06831#A7)with

V=\[M\]T=2​∫0Tη​\(s\)​𝑑s,a=2​\(1\+∫0TK​\(s\)​𝑑s\)​ϑ,V=\[M\]\_\{T\}=2\\int\_\{0\}^\{T\}\\eta\(s\)\\,ds,\\qquad a=2\\Big\(1\+\\int\_\{0\}^\{T\}K\(s\)\\,ds\\Big\)\\vartheta,\(H\.289\)yields the confinement bound used in[Eq\.7](https://arxiv.org/html/2605.06831#S5.E7)\.

We now bound the terminal event\. Let\(ℱt\)t∈\[0,T\]\(\\mathcal\{F\}\_\{t\}\)\_\{t\\in\[0,T\]\}be the natural filtration of\(At\)\(A\_\{t\}\)\. Define

Hϑ:=\{\|AT\|≤ϑ\}\.H\_\{\\vartheta\}:=\\\{\|A\_\{T\}\|\\leq\\vartheta\\\}\.Then

ℙ​\(Hϑ\)=ℙ​\(Hϑ∩E2​ϑ\)\+ℙ​\(Hϑ∩E2​ϑc\)≤ℙ​\(E2​ϑ\)\+ℙ​\(Hϑ∩E2​ϑc\)\.\\mathbb\{P\}\(H\_\{\\vartheta\}\)=\\mathbb\{P\}\(H\_\{\\vartheta\}\\cap E\_\{2\\vartheta\}\)\+\\mathbb\{P\}\(H\_\{\\vartheta\}\\cap E\_\{2\\vartheta\}^\{c\}\)\\leq\\mathbb\{P\}\(E\_\{2\\vartheta\}\)\+\\mathbb\{P\}\(H\_\{\\vartheta\}\\cap E\_\{2\\vartheta\}^\{c\}\)\.\(H\.290\)
Next, let

τ:=inf\{t∈\[0,T\]:\|At\|≥2​ϑ\}\.\\tau:=\\inf\\\{t\\in\[0,T\]:\|A\_\{t\}\|\\geq 2\\vartheta\\\}\.\(H\.291\)By convention,τ=\+∞\\tau=\+\\inftyif no such time exists\. By continuity,

E2​ϑ=\{τ\>T\},E2​ϑc=\{τ≤T\}\.E\_\{2\\vartheta\}=\\\{\\tau\>T\\\},\\qquad E\_\{2\\vartheta\}^\{c\}=\\\{\\tau\\leq T\\\}\.Hence

ℙ​\(Hϑ∩E2​ϑc\)=ℙ​\(\|AT\|≤ϑ,τ≤T\)\.\\mathbb\{P\}\(H\_\{\\vartheta\}\\cap E\_\{2\\vartheta\}^\{c\}\)=\\mathbb\{P\}\(\|A\_\{T\}\|\\leq\\vartheta,\\ \\tau\\leq T\)\.\(H\.292\)
Using conditional expectation atτ\\tauand the strong Markov property,

ℙ​\(\|AT\|≤ϑ,τ≤T\)\\displaystyle\\mathbb\{P\}\(\|A\_\{T\}\|\\leq\\vartheta,\\ \\tau\\leq T\)=𝔼​\[𝟏\{τ≤T\}​ℙτ,Aτ​\(\|AT\|≤ϑ\)\]\\displaystyle=\\mathbb\{E\}\\Big\[\\mathbf\{1\}\_\{\\\{\\tau\\leq T\\\}\}\\,\\mathbb\{P\}\_\{\\tau,A\_\{\\tau\}\}\(\|A\_\{T\}\|\\leq\\vartheta\)\\Big\]≤sups∈\[0,T\]supa∈\{±2​ϑ\}ℙs,a​\(\|AT\|≤ϑ\)\.\\displaystyle\\leq\\sup\_\{s\\in\[0,T\]\}\\ \\sup\_\{a\\in\\\{\\pm 2\\vartheta\\\}\}\\ \\mathbb\{P\}\_\{s,a\}\(\|A\_\{T\}\|\\leq\\vartheta\)\.\(H\.293\)
Fixs∈\[0,T\]s\\in\[0,T\]and considerAs=2​ϑA\_\{s\}=2\\vartheta\. Define the first return time

σ:=inf\{t∈\[s,T\]:At=ϑ\}\.\\sigma:=\\inf\\\{t\\in\[s,T\]:A\_\{t\}=\\vartheta\\\}\.\(H\.294\)By continuity,

ℙs,2​ϑ​\(\|AT\|≤ϑ\)≤ℙs,2​ϑ​\(σ≤T\)\.\\mathbb\{P\}\_\{s,2\\vartheta\}\(\|A\_\{T\}\|\\leq\\vartheta\)\\leq\\mathbb\{P\}\_\{s,2\\vartheta\}\(\\sigma\\leq T\)\.\(H\.295\)
By[Eq\.6](https://arxiv.org/html/2605.06831#S5.E6), for allt∈\[0,T\]t\\in\[0,T\],

b​\(a,t\)≥λrep​a∀a≥ϑ,b\(a,t\)\\geq\\lambda\_\{\\mathrm\{rep\}\}\\,a\\qquad\\forall a\\geq\\vartheta,\(H\.296\)and recallη​\(t\)≤ηmax\\eta\(t\)\\leq\\eta\_\{\\max\}on\[0,T\]\[0,T\]\. Forr\>0r\>0defineZt:=e−r​AtZ\_\{t\}:=e^\{\-rA\_\{t\}\}\. Applying Itô’s formula on\[s,σ∧T\]\[s,\\sigma\\wedge T\]gives

d​Zt=Zt​\(−r​b​\(At,t\)\+r2​η​\(t\)\)​d​t−r​Zt​2​η​\(t\)​d​Bt\.dZ\_\{t\}=Z\_\{t\}\\big\(\-r\\,b\(A\_\{t\},t\)\+r^\{2\}\\eta\(t\)\\big\)\\,dt\-rZ\_\{t\}\\sqrt\{2\\eta\(t\)\}\\,dB\_\{t\}\.\(H\.297\)On\[s,σ\)\[s,\\sigma\), using \([H\.296](https://arxiv.org/html/2605.06831#A8.E296)\) andη​\(t\)≤ηmax\\eta\(t\)\\leq\\eta\_\{\\max\},

−r​b​\(At,t\)\+r2​η​\(t\)≤−r​\(λrep​ϑ\)\+r2​ηmax\.\-r\\,b\(A\_\{t\},t\)\+r^\{2\}\\eta\(t\)\\leq\-r\(\\lambda\_\{\\mathrm\{rep\}\}\\vartheta\)\+r^\{2\}\\eta\_\{\\max\}\.\(H\.298\)Choose

r:=λrep​ϑ2​ηmax\.r:=\\frac\{\\lambda\_\{\\mathrm\{rep\}\}\\vartheta\}\{2\\eta\_\{\\max\}\}\.\(H\.299\)Then

−r​\(λrep​ϑ\)\+r2​ηmax=−λrep2​ϑ24​ηmax<0,\-r\(\\lambda\_\{\\mathrm\{rep\}\}\\vartheta\)\+r^\{2\}\\eta\_\{\\max\}=\-\\frac\{\\lambda\_\{\\mathrm\{rep\}\}^\{\\,2\}\\vartheta^\{2\}\}\{4\\eta\_\{\\max\}\}<0,\(H\.300\)so\(Zt∧σ\)t∈\[s,T\]\(Z\_\{t\\wedge\\sigma\}\)\_\{t\\in\[s,T\]\}is a supermartingale\. By optional stopping,

𝔼​\[Zσ∧T\]≤Zs=e−r⋅2​ϑ\.\\mathbb\{E\}\[Z\_\{\\sigma\\wedge T\}\]\\leq Z\_\{s\}=e^\{\-r\\cdot 2\\vartheta\}\.\(H\.301\)On\{σ≤T\}\\\{\\sigma\\leq T\\\}we haveZσ∧T=e−r​ϑZ\_\{\\sigma\\wedge T\}=e^\{\-r\\vartheta\}, hence

ℙs,2​ϑ​\(σ≤T\)≤e−r​ϑ=exp⁡\(−λrep​ϑ22​ηmax\)\.\\mathbb\{P\}\_\{s,2\\vartheta\}\(\\sigma\\leq T\)\\leq e^\{\-r\\vartheta\}=\\exp\\\!\\Big\(\-\\frac\{\\lambda\_\{\\mathrm\{rep\}\}\\vartheta^\{2\}\}\{2\\eta\_\{\\max\}\}\\Big\)\.\(H\.302\)
An identical argument withZ~t:=er​At\\widetilde\{Z\}\_\{t\}:=e^\{rA\_\{t\}\}yields the same bound starting fromAs=−2​ϑA\_\{s\}=\-2\\vartheta\. Therefore,

sups∈\[0,T\]supa∈\{±2​ϑ\}ℙs,a​\(\|AT\|≤ϑ\)≤2​exp⁡\(−λrep​ϑ22​ηmax\)\.\\sup\_\{s\\in\[0,T\]\}\\ \\sup\_\{a\\in\\\{\\pm 2\\vartheta\\\}\}\\ \\mathbb\{P\}\_\{s,a\}\(\|A\_\{T\}\|\\leq\\vartheta\)\\leq 2\\exp\\\!\\Big\(\-\\frac\{\\lambda\_\{\\mathrm\{rep\}\}\\vartheta^\{2\}\}\{2\\eta\_\{\\max\}\}\\Big\)\.\(H\.303\)Plugging this into \([H\.293](https://arxiv.org/html/2605.06831#A8.E293)\) and then into \([H\.290](https://arxiv.org/html/2605.06831#A8.E290)\) yields

ℙ​\(Hϑ\)≤ℙ​\(E2​ϑ\)\+2​exp⁡\(−λrep​ϑ22​ηmax\)\.\\mathbb\{P\}\(H\_\{\\vartheta\}\)\\leq\\mathbb\{P\}\(E\_\{2\\vartheta\}\)\+2\\exp\\\!\\Big\(\-\\frac\{\\lambda\_\{\\mathrm\{rep\}\}\\vartheta^\{2\}\}\{2\\eta\_\{\\max\}\}\\Big\)\.\(H\.304\)
Finally, substitute the bound onℙ​\(E2​ϑ\)\\mathbb\{P\}\(E\_\{2\\vartheta\}\)from[Eq\.H\.288](https://arxiv.org/html/2605.06831#A8.E288)\(via[AppendixG](https://arxiv.org/html/2605.06831#A7)\) into[Eq\.H\.304](https://arxiv.org/html/2605.06831#A8.E304)to obtain[Eq\.7](https://arxiv.org/html/2605.06831#S5.E7)\. ∎

### H\.7Proof of Proposition[B](https://arxiv.org/html/2605.06831#A2)

###### Proof\.

Unequal Weights

We consider[Eq\.H\.234](https://arxiv.org/html/2605.06831#A8.E234)andt≤τ^t\\leq\\hat\{\\tau\}, whereτ^\\hat\{\\tau\}is defined in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\. By[Eq\.H\.241](https://arxiv.org/html/2605.06831#A8.E241), we have that:

γ~j,t\(i​j\)​\(ξ\)1−γ~j,t\(i​j\)​\(ξ\)\\displaystyle\\frac\{\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi\)\}\{1\-\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi\)\}=πjπi​exp⁡\(−‖𝒚​\(ξ\)−𝝁\(j\)‖22−‖𝒚​\(ξ\)−𝝁\(i\)‖222​σ~t2\)\\displaystyle=\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\\!\\left\(\-\\frac\{\\\|\\bm\{y\}\(\\xi\)\-\\bm\{\\mu\}^\{\(j\)\}\\\|\_\{2\}^\{2\}\-\\\|\\bm\{y\}\(\\xi\)\-\\bm\{\\mu\}^\{\(i\)\}\\\|\_\{2\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)=πjπi​exp⁡\(ℓ2σ~t2​\(ξ−12\)\),\\displaystyle=\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\exp\\\!\\left\(\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\Big\(\\xi\-\\frac\{1\}\{2\}\\Big\)\\right\),\(H\.305\)
Define the pairwise parallel drift

Fi​j,t​\(ξ\):=γ~j,t\(i​j\)​\(ξ\)−ξ\.F\_\{ij,t\}\(\\xi\):=\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi\)\-\\xi\.\(H\.306\)Letξi​j⋆​\(t\)∈\(0,1\)\\xi^\{\\star\}\_\{ij\}\(t\)\\in\(0,1\)be an interior equilibrium, i\.e\.Fi​j,t​\(ξi​j⋆​\(t\)\)=0F\_\{ij,t\}\(\\xi^\{\\star\}\_\{ij\}\(t\)\)=0, soγ~j,t\(i​j\)​\(ξi​j⋆​\(t\)\)=ξi​j⋆​\(t\)\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi^\{\\star\}\_\{ij\}\(t\)\)=\\xi^\{\\star\}\_\{ij\}\(t\)\. Taking log\-odds in[Eq\.H\.305](https://arxiv.org/html/2605.06831#A8.E305)and substitutingγ~j,t\(i​j\)​\(ξi​j⋆​\(t\)\)=ξi​j⋆​\(t\)\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi^\{\\star\}\_\{ij\}\(t\)\)=\\xi^\{\\star\}\_\{ij\}\(t\)yields

log⁡\(ξi​j⋆​\(t\)1−ξi​j⋆​\(t\)\)=log⁡\(πjπi\)\+ℓ2σ~t2​\(ξi​j⋆​\(t\)−12\),\\log\\\!\\Big\(\\frac\{\\xi^\{\\star\}\_\{ij\}\(t\)\}\{1\-\\xi^\{\\star\}\_\{ij\}\(t\)\}\\Big\)=\\log\\\!\\Big\(\\frac\{\\pi\_\{j\}\}\{\\pi\_\{i\}\}\\Big\)\+\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\Big\(\\xi^\{\\star\}\_\{ij\}\(t\)\-\\frac\{1\}\{2\}\\Big\),\(H\.307\)which is[Eq\.B\.8](https://arxiv.org/html/2605.06831#A2.E8)\. In particular, ifπi=πj\\pi\_\{i\}=\\pi\_\{j\}thenξi​j⋆​\(t\)=12\\xi^\{\\star\}\_\{ij\}\(t\)=\\tfrac\{1\}\{2\}\. Ifπj<πi\\pi\_\{j\}<\\pi\_\{i\}, thenlog⁡\(πj/πi\)<0\\log\(\\pi\_\{j\}/\\pi\_\{i\}\)<0and[Eq\.H\.307](https://arxiv.org/html/2605.06831#A8.E307)impliesξi​j⋆​\(t\)\>12\\xi^\{\\star\}\_\{ij\}\(t\)\>\\tfrac\{1\}\{2\}\(and similarilyπj\>πi\\pi\_\{j\}\>\\pi\_\{i\}impliesξi​j⋆​\(t\)<12\\xi^\{\\star\}\_\{ij\}\(t\)<\\tfrac\{1\}\{2\}\)\.

Next, differentiating[Eq\.H\.305](https://arxiv.org/html/2605.06831#A8.E305)yields:

γ~j,t\(i​j\)⁣′​\(ξ\)=ℓ2σ~t2​γ~j,t\(i​j\)​\(ξ\)​\(1−γ~j,t\(i​j\)​\(ξ\)\),\\tilde\{\\gamma\}^\{\(ij\)\\,\\prime\}\_\{j,t\}\(\\xi\)=\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\,\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi\)\\big\(1\-\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi\)\\big\),\(H\.308\)so

Fi​j,t′​\(ξ\)=γ~j,t\(i​j\)⁣′​\(ξ\)−1\.F^\{\\prime\}\_\{ij,t\}\(\\xi\)=\\tilde\{\\gamma\}^\{\(ij\)\\,\\prime\}\_\{j,t\}\(\\xi\)\-1\.\(H\.309\)Evaluating atξi​j⋆​\(t\)\\xi^\{\\star\}\_\{ij\}\(t\)\(whereγ~j,t\(i​j\)​\(ξi​j⋆​\(t\)\)=ξi​j⋆​\(t\)\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j,t\}\(\\xi^\{\\star\}\_\{ij\}\(t\)\)=\\xi^\{\\star\}\_\{ij\}\(t\)\) yields

Fi​j,t′​\(ξi​j⋆​\(t\)\)=ℓ2σ~t2​ξi​j⋆​\(t\)​\(1−ξi​j⋆​\(t\)\)−1,F^\{\\prime\}\_\{ij,t\}\\\!\\big\(\\xi^\{\\star\}\_\{ij\}\(t\)\\big\)=\\frac\{\\ell^\{2\}\}\{\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\,\\xi^\{\\star\}\_\{ij\}\(t\)\\big\(1\-\\xi^\{\\star\}\_\{ij\}\(t\)\\big\)\-1,\(H\.310\)which is[Eq\.B\.9](https://arxiv.org/html/2605.06831#A2.E9)\. Thusξi​j⋆​\(t\)\\xi^\{\\star\}\_\{ij\}\(t\)is hyperbolic unstable wheneverFi​j,t′​\(ξi​j⋆​\(t\)\)\>0F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\\star\}\_\{ij\}\(t\)\)\>0\.

Exact Parallel DynamicsBy[AppendixG](https://arxiv.org/html/2605.06831#A7), the exact parallel dynamics in[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50)differs from[Eq\.H\.234](https://arxiv.org/html/2605.06831#A8.E234)by an error termδξt\\delta\_\{\\xi\_\{t\}\}\. By[AppendixG](https://arxiv.org/html/2605.06831#A7), we can thus write:

FN,t​\(ξ\)=Fi​j,t​\(ξ\)\+et​\(ξ\),supξ∈\[−ε,1\+ε\]\|et​\(ξ\)\|≤ε,ε∈𝒪​\(N​exp⁡\(−κ\)\)\.F\_\{N,t\}\(\\xi\)=F\_\{ij,t\}\(\\xi\)\+e\_\{t\}\(\\xi\),\\qquad\\sup\_\{\\xi\\in\[\-\\varepsilon,1\+\\varepsilon\]\}\|e\_\{t\}\(\\xi\)\|\\leq\\varepsilon,\\qquad\\varepsilon\\in\\mathcal\{O\}\(N\\exp\(\{\-\\kappa\}\)\)\.\(H\.311\)
We first show existence of an interior equilibrium forFN,tF\_\{N,t\}, the exact parallel dynamics in[Eq\.G\.50](https://arxiv.org/html/2605.06831#A7.E50)\. We denoteFi​j,tF\_\{ij,t\}as the drift of the approximate parallel dynamics in[Eq\.H\.234](https://arxiv.org/html/2605.06831#A8.E234)\. By[Sec\.4](https://arxiv.org/html/2605.06831#S4), the exact parallel dynamics has instantaneously stable equilibria nearξ=0\\xi=0andξ=1\\xi=1, denotedξt∗,i\\xi^\{\*,i\}\_\{t\}andξt∗,j\\xi^\{\*,j\}\_\{t\}, respectively\. Instantaneous stability implies that the drift points toward each equilibrium\. Hence we can pick pointsξL,ξR\\xi\_\{L\},\\xi\_\{R\}with

ξt∗,i<ξt\(L\)<ξt\(R\)<ξt∗,j,FN,t​\(ξL\)<0,FN,t​\(ξR\)\>0\.\\xi^\{\*,i\}\_\{t\}<\\xi\_\{t\}^\{\(L\)\}<\\xi\_\{t\}^\{\(R\)\}<\\xi^\{\*,j\}\_\{t\},\\qquad F\_\{N,t\}\(\\xi\_\{L\}\)<0,\\qquad F\_\{N,t\}\(\\xi\_\{R\}\)\>0\.\(H\.312\)\(For example, takeξL\\xi\_\{L\}slightly to the right ofξt∗,i\\xi^\{\*,i\}\_\{t\}andξR\\xi\_\{R\}slightly to the left ofξt∗,j\\xi^\{\*,j\}\_\{t\}\.\) SinceFN,tF\_\{N,t\}is continuous inξ\\xi, the intermediate value theorem applied to[Eq\.H\.312](https://arxiv.org/html/2605.06831#A8.E312)yields an interior rootξN⋆​\(t\)∈\(ξt\(L\),ξt\(R\)\)\\xi^\{\\star\}\_\{N\}\(t\)\\in\(\\xi\_\{t\}^\{\(L\)\},\\xi\_\{t\}^\{\(R\)\}\)withFN,t​\(ξN⋆​\(t\)\)=0F\_\{N,t\}\(\\xi^\{\\star\}\_\{N\}\(t\)\)=0\.

Assume there exists an intervalIIcontainingξi​j⋆​\(t\)\\xi^\{\\star\}\_\{ij\}\(t\)such that

Fi​j,t′​\(ξ\)≥m\>0∀ξ∈I,F^\{\\prime\}\_\{ij,t\}\(\\xi\)\\geq m\>0\\qquad\\forall\\xi\\in I,\(H\.313\)which is[Eq\.B\.10](https://arxiv.org/html/2605.06831#A2.E10)\. Define

ρ:=sup\{r\>0:\[ξi​j⋆​\(t\)−r,ξi​j⋆​\(t\)\+r\]⊂I\}\.\\rho:=\\sup\\Big\\\{r\>0:\\ \\big\[\\xi^\{\\star\}\_\{ij\}\(t\)\-r,\\ \\xi^\{\\star\}\_\{ij\}\(t\)\+r\\big\]\\subset I\\Big\\\}\.\(H\.314\)Sinceρ\>0\\rho\>0, recalling thatε≤C1​exp⁡\(−κ\)\\varepsilon\\leq C\_\{1\}\\exp\(\-\\kappa\), we chooseκ\\kappasufficiently large such that:

εm<ρ\.\\frac\{\\varepsilon\}\{m\}<\\rho\.\(H\.315\)Now setξ±:=ξi​j⋆​\(t\)±ε/m\\xi\_\{\\pm\}:=\\xi^\{\\star\}\_\{ij\}\(t\)\\pm\\varepsilon/m, so that[Eq\.H\.315](https://arxiv.org/html/2605.06831#A8.E315)implies\[ξ−,ξ\+\]⊂I\[\\xi\_\{\-\},\\xi\_\{\+\}\]\\subset I\. SinceFi​j,t​\(ξi​j⋆​\(t\)\)=0F\_\{ij,t\}\(\\xi^\{\\star\}\_\{ij\}\(t\)\)=0andFi​j,t′≥mF^\{\\prime\}\_\{ij,t\}\\geq mon\[ξ−,ξ\+\]⊂I\[\\xi\_\{\-\},\\xi\_\{\+\}\]\\subset I, the mean value theorem gives

Fi​j,t​\(ξ\+\)≥ε,Fi​j,t​\(ξ−\)≤−ε,F\_\{ij,t\}\(\\xi\_\{\+\}\)\\geq\\varepsilon,\\qquad F\_\{ij,t\}\(\\xi\_\{\-\}\)\\leq\-\\varepsilon,\(H\.316\)Combining[Eqs\.H\.311](https://arxiv.org/html/2605.06831#A8.E311)and[H\.316](https://arxiv.org/html/2605.06831#A8.E316)gives

FN,t​\(ξ\+\)≥ε−ε≥0,FN,t​\(ξ−\)≤−ε\+ε≤0\.F\_\{N,t\}\(\\xi\_\{\+\}\)\\geq\\varepsilon\-\\varepsilon\\geq 0,\\qquad F\_\{N,t\}\(\\xi\_\{\-\}\)\\leq\-\\varepsilon\+\\varepsilon\\leq 0\.By continuity, there exists a rootξN⋆​\(t\)∈\[ξ−,ξ\+\]\\xi^\{\\star\}\_\{N\}\(t\)\\in\[\\xi\_\{\-\},\\xi\_\{\+\}\], and therefore

\|ξN⋆​\(t\)−ξi​j⋆​\(t\)\|≤εm,\\big\|\\xi^\{\\star\}\_\{N\}\(t\)\-\\xi^\{\\star\}\_\{ij\}\(t\)\\big\|\\leq\\frac\{\\varepsilon\}\{m\},\(H\.317\)which is[Eq\.B\.11](https://arxiv.org/html/2605.06831#A2.E11)\.

This completes the proof\. ∎

### H\.8Proof of[AppendixC](https://arxiv.org/html/2605.06831#A3)

###### Proof\.

The exact parallel drift alongLt\(i,j\)L^\{\(i,j\)\}\_\{t\}isFi​j,t​\(ξ\)=γ~j\(i​j\)​\(𝒚​\(ξ\)\)−ξF\_\{ij,t\}\(\\xi\)=\\tilde\{\\gamma\}^\{\(ij\)\}\_\{j\}\(\\bm\{y\}\(\\xi\)\)\-\\xi, where𝒚​\(ξ\)=𝝁\(i\)\+ξ​ℓ​𝒖\\bm\{y\}\(\\xi\)=\\bm\{\\mu\}^\{\(i\)\}\+\\xi\\ell\\bm\{u\}and𝒖=\(𝝁\(j\)−𝝁\(i\)\)/ℓ\\bm\{u\}=\(\\bm\{\\mu\}^\{\(j\)\}\-\\bm\{\\mu\}^\{\(i\)\}\)/\\ell\. Under score errorψ\\psi, the perturbed PF ODE \([C\.13](https://arxiv.org/html/2605.06831#A3.E13)\) induces perturbed parallel dynamics\. Projecting onto𝒖\\bm\{u\}, the drift becomes

F~N,t​\(ξ\)=Fi​j,t​\(ξ\)\+et​\(ξ\),whereet​\(ξ\):=1ℓ​⟨ψ​\(𝒚​\(ξ\),t\),𝒖⟩\.\\tilde\{F\}\_\{N,t\}\(\\xi\)=F\_\{ij,t\}\(\\xi\)\+e\_\{t\}\(\\xi\),\\quad\\text\{where\}\\quad e\_\{t\}\(\\xi\):=\\frac\{1\}\{\\ell\}\\langle\\psi\(\\bm\{y\}\(\\xi\),t\),\\bm\{u\}\\rangle\.\(H\.318\)By Cauchy\-Schwarz and[AppendixC](https://arxiv.org/html/2605.06831#A3),

\|et​\(ξ\)\|≤‖ψ​\(𝒚​\(ξ\),t\)‖2⋅‖𝒖‖2ℓ≤ϱ​\(t\)ℓ\.\|e\_\{t\}\(\\xi\)\|\\leq\\frac\{\\\|\\psi\(\\bm\{y\}\(\\xi\),t\)\\\|\_\{2\}\\cdot\\\|\\bm\{u\}\\\|\_\{2\}\}\{\\ell\}\\leq\\frac\{\\varrho\(t\)\}\{\\ell\}\.\(H\.319\)
DefineG:\[0,T\]×\[0,1\]→ℝG:\[0,T\]\\times\[0,1\]\\to\\mathbb\{R\}byG​\(t,ξ\):=F~N,t​\(ξ\)G\(t,\\xi\):=\\tilde\{F\}\_\{N,t\}\(\\xi\)\. At the unperturbed equilibrium withϱ=0\\varrho=0, we haveG​\(t,ξi​j∗​\(t\)\)\|ϱ=0=Fi​j,t​\(ξi​j∗​\(t\)\)=0G\(t,\\xi^\{\*\}\_\{ij\}\(t\)\)\|\_\{\\varrho=0\}=F\_\{ij,t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)=0\. The partial derivative satisfies

∂G∂ξ\|ξ=ξi​j∗​\(t\)=Fi​j,t′​\(ξi​j∗​\(t\)\)\+et′​\(ξi​j∗​\(t\)\)=λt\+O​\(Lψ​\(t\)\),\\frac\{\\partial G\}\{\\partial\\xi\}\\bigg\|\_\{\\xi=\\xi^\{\*\}\_\{ij\}\(t\)\}=F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)\+e^\{\\prime\}\_\{t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)=\\lambda\_\{t\}\+O\(L\_\{\\psi\}\(t\)\),\(H\.320\)where\|et′​\(ξ\)\|=\|𝒖⊤​∇𝒙ψ​𝒖\|≤Lψ​\(t\)\|e^\{\\prime\}\_\{t\}\(\\xi\)\|=\|\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\\,\\bm\{u\}\|\\leq L\_\{\\psi\}\(t\)by[AppendixC](https://arxiv.org/html/2605.06831#A3)\. ForLψ​\(t\)≤λt/2L\_\{\\psi\}\(t\)\\leq\\lambda\_\{t\}/2, we have∂G/∂ξ≥λt/2\>0\\partial G/\\partial\\xi\\geq\\lambda\_\{t\}/2\>0, so by the implicit function theorem there exists a uniqueξθ∗​\(t\)\\xi^\{\*\}\_\{\\theta\}\(t\)nearξi​j∗​\(t\)\\xi^\{\*\}\_\{ij\}\(t\)withG​\(t,ξθ∗​\(t\)\)=0G\(t,\\xi^\{\*\}\_\{\\theta\}\(t\)\)=0\.

Taylor expandingG​\(t,ξ\)G\(t,\\xi\)aroundξi​j∗​\(t\)\\xi^\{\*\}\_\{ij\}\(t\):

0=G​\(t,ξθ∗​\(t\)\)=Fi​j,t​\(ξi​j∗​\(t\)\)⏟=0\+et​\(ξi​j∗​\(t\)\)\+\(λt\+O​\(Lψ​\(t\)\)\)​\(ξθ∗−ξi​j∗\)\+O​\(\|ξθ∗−ξi​j∗\|2\)\.0=G\(t,\\xi^\{\*\}\_\{\\theta\}\(t\)\)=\\underbrace\{F\_\{ij,t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)\}\_\{=0\}\+e\_\{t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)\+\(\\lambda\_\{t\}\+O\(L\_\{\\psi\}\(t\)\)\)\(\\xi^\{\*\}\_\{\\theta\}\-\\xi^\{\*\}\_\{ij\}\)\+O\(\|\\xi^\{\*\}\_\{\\theta\}\-\\xi^\{\*\}\_\{ij\}\|^\{2\}\)\.\(H\.321\)Rearranging and using \([H\.319](https://arxiv.org/html/2605.06831#A8.E319)\):

\|ξθ∗​\(t\)−ξi​j∗​\(t\)\|≤2​\|et​\(ξi​j∗​\(t\)\)\|λt\+O​\(ϱ​\(t\)2\)≤2​ϱ​\(t\)ℓ​λt\+O​\(ϱ​\(t\)2\)\.∎\|\\xi^\{\*\}\_\{\\theta\}\(t\)\-\\xi^\{\*\}\_\{ij\}\(t\)\|\\leq\\frac\{2\|e\_\{t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)\|\}\{\\lambda\_\{t\}\}\+O\(\\varrho\(t\)^\{2\}\)\\leq\\frac\{2\\varrho\(t\)\}\{\\ell\\lambda\_\{t\}\}\+O\(\\varrho\(t\)^\{2\}\)\.\\qed\(H\.322\)

### H\.9Proof of[AppendixC](https://arxiv.org/html/2605.06831#A3)

###### Proof\.

The perturbed parallel drift isF~N,t​\(ξ\)=Fi​j,t​\(ξ\)\+et​\(ξ\)\\tilde\{F\}\_\{N,t\}\(\\xi\)=F\_\{ij,t\}\(\\xi\)\+e\_\{t\}\(\\xi\), with derivative

F~N,t′​\(ξ\)=Fi​j,t′​\(ξ\)\+et′​\(ξ\)\.\\tilde\{F\}^\{\\prime\}\_\{N,t\}\(\\xi\)=F^\{\\prime\}\_\{ij,t\}\(\\xi\)\+e^\{\\prime\}\_\{t\}\(\\xi\)\.\(H\.323\)Sinceet​\(ξ\)=1ℓ​⟨ψ​\(𝒚​\(ξ\),t\),𝒖⟩e\_\{t\}\(\\xi\)=\\frac\{1\}\{\\ell\}\\langle\\psi\(\\bm\{y\}\(\\xi\),t\),\\bm\{u\}\\rangleandd​𝒚d​ξ=ℓ​𝒖\\frac\{d\\bm\{y\}\}\{d\\xi\}=\\ell\\bm\{u\},

et′​\(ξ\)=1ℓ​⟨∇𝒙ψ​\(𝒚​\(ξ\),t\)⋅\(ℓ​𝒖\),𝒖⟩=𝒖⊤​∇𝒙ψ​\(𝒚​\(ξ\),t\)​𝒖\.e^\{\\prime\}\_\{t\}\(\\xi\)=\\frac\{1\}\{\\ell\}\\langle\\nabla\_\{\\bm\{x\}\}\\psi\(\\bm\{y\}\(\\xi\),t\)\\cdot\(\\ell\\bm\{u\}\),\\bm\{u\}\\rangle=\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\(\\bm\{y\}\(\\xi\),t\)\\,\\bm\{u\}\.\(H\.324\)
At the perturbed equilibriumξθ∗​\(t\)\\xi^\{\*\}\_\{\\theta\}\(t\):

λθ​\(t\):=F~N,t′​\(ξθ∗​\(t\)\)\\displaystyle\\lambda\_\{\\theta\}\(t\):=\\tilde\{F\}^\{\\prime\}\_\{N,t\}\(\\xi^\{\*\}\_\{\\theta\}\(t\)\)=Fi​j,t′​\(ξθ∗​\(t\)\)\+𝒖⊤​∇𝒙ψ​\(𝒚​\(ξθ∗\),t\)​𝒖\\displaystyle=F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\*\}\_\{\\theta\}\(t\)\)\+\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\(\\bm\{y\}\(\\xi^\{\*\}\_\{\\theta\}\),t\)\\,\\bm\{u\}\(H\.325\)=Fi​j,t′​\(ξi​j∗​\(t\)\)\+O​\(\|ξθ∗−ξi​j∗\|\)\+𝒖⊤​∇𝒙ψ​\(𝒚​\(ξθ∗\),t\)​𝒖\\displaystyle=F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\*\}\_\{ij\}\(t\)\)\+O\(\|\\xi^\{\*\}\_\{\\theta\}\-\\xi^\{\*\}\_\{ij\}\|\)\+\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\(\\bm\{y\}\(\\xi^\{\*\}\_\{\\theta\}\),t\)\\,\\bm\{u\}\(H\.326\)=λt\+𝒖⊤​∇𝒙ψ​\(𝒚​\(ξθ∗\),t\)​𝒖\+O​\(ϱ​\(t\)\),\\displaystyle=\\lambda\_\{t\}\+\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\(\\bm\{y\}\(\\xi^\{\*\}\_\{\\theta\}\),t\)\\,\\bm\{u\}\+O\(\\varrho\(t\)\),\(H\.327\)where the second line uses Lipschitz continuity ofFi​j,t′​\(ξ\)F^\{\\prime\}\_\{ij,t\}\(\\xi\): by the mean value theorem,

\|Fi​j,t′​\(ξθ∗\)−Fi​j,t′​\(ξi​j∗\)\|≤supξ∈\[0,1\]\|Fi​j,t′′​\(ξ\)\|⋅\|ξθ∗−ξi​j∗\|=O​\(\|ξθ∗−ξi​j∗\|\),\|F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\*\}\_\{\\theta\}\)\-F^\{\\prime\}\_\{ij,t\}\(\\xi^\{\*\}\_\{ij\}\)\|\\leq\\sup\_\{\\xi\\in\[0,1\]\}\|F^\{\\prime\\prime\}\_\{ij,t\}\(\\xi\)\|\\cdot\|\\xi^\{\*\}\_\{\\theta\}\-\\xi^\{\*\}\_\{ij\}\|=O\(\|\\xi^\{\*\}\_\{\\theta\}\-\\xi^\{\*\}\_\{ij\}\|\),\(H\.328\)and the last line uses[AppendixC](https://arxiv.org/html/2605.06831#A3)\. The two items follow from the sign of𝒖⊤​∇𝒙ψ​𝒖\\bm\{u\}^\{\\top\}\\nabla\_\{\\bm\{x\}\}\\psi\\,\\bm\{u\}\. ∎

### H\.10Proof of[AppendixC](https://arxiv.org/html/2605.06831#A3)

###### Proof\.

Fora∈\[ϑ,2​ϑ\]a\\in\[\\vartheta,2\\vartheta\], the exact drift satisfiesb​\(a,t\)≥λrep​a≥λrep​ϑb\(a,t\)\\geq\\lambda\_\{\\mathrm\{rep\}\}a\\geq\\lambda\_\{\\mathrm\{rep\}\}\\vartheta\. The perturbed drift satisfies

bθ​\(a,t\)=b​\(a,t\)\+e~t​\(a\)≥λrep​a−\|e~t​\(a\)\|≥λrep​a−ϱ​\(t\)ℓ≥λrep​a−ϱmaxℓ\.b\_\{\\theta\}\(a,t\)=b\(a,t\)\+\\tilde\{e\}\_\{t\}\(a\)\\geq\\lambda\_\{\\mathrm\{rep\}\}a\-\|\\tilde\{e\}\_\{t\}\(a\)\|\\geq\\lambda\_\{\\mathrm\{rep\}\}a\-\\frac\{\\varrho\(t\)\}\{\\ell\}\\geq\\lambda\_\{\\mathrm\{rep\}\}a\-\\frac\{\\varrho\_\{\\max\}\}\{\\ell\}\.\(H\.329\)Under the conditionϱmax/ℓ<λrep​ϑ\\varrho\_\{\\max\}/\\ell<\\lambda\_\{\\mathrm\{rep\}\}\\vartheta, fora∈\[ϑ,2​ϑ\]a\\in\[\\vartheta,2\\vartheta\]:

bθ​\(a,t\)≥\(λrep−ϱmaxℓ​a\)​a≥\(λrep−ϱmaxℓ​ϑ\)​a=λ~rep​a,b\_\{\\theta\}\(a,t\)\\geq\\left\(\\lambda\_\{\\mathrm\{rep\}\}\-\\frac\{\\varrho\_\{\\max\}\}\{\\ell a\}\\right\)a\\geq\\left\(\\lambda\_\{\\mathrm\{rep\}\}\-\\frac\{\\varrho\_\{\\max\}\}\{\\ell\\vartheta\}\\right\)a=\\tilde\{\\lambda\}\_\{\\mathrm\{rep\}\}\\,a,\(H\.330\)whereλ~rep\>0\\tilde\{\\lambda\}\_\{\\mathrm\{rep\}\}\>0by assumption\. The symmetric argument holds fora∈\[−2​ϑ,−ϑ\]a\\in\[\-2\\vartheta,\-\\vartheta\], so the escape mechanism of[Sec\.5](https://arxiv.org/html/2605.06831#S5)applies withλrep\\lambda\_\{\\mathrm\{rep\}\}replaced byλ~rep\\tilde\{\\lambda\}\_\{\\mathrm\{rep\}\}\. ∎

### H\.11Proof of Proposition[D](https://arxiv.org/html/2605.06831#A4)

###### Proof\.

Fix anyt≤τ1t\\leq\\tau\_\{1\}and write𝒑:=𝒚t\\bm\{p\}:=\\bm\{y\}\_\{t\}for the state at timett\. We first prove the claim for a single rectangular cell\.

Consider a single grid cell with corners \(true modes\)𝝁\(1\),𝝁\(2\),𝝁\(3\),𝝁\(4\)\\bm\{\\mu\}^\{\(1\)\},\\bm\{\\mu\}^\{\(2\)\},\\bm\{\\mu\}^\{\(3\)\},\\bm\{\\mu\}^\{\(4\)\}, where the diagonal pairs are\(1,4\)\(1,4\)and\(2,3\)\(2,3\)\. Defineda2:=‖𝝁\(a\)−𝒑‖22d\_\{a\}^\{2\}:=\\\|\\bm\{\\mu\}^\{\(a\)\}\-\\bm\{p\}\\\|\_\{2\}^\{2\}\. For any rectangle, one has the identity

d12\+d42=d22\+d32\.d\_\{1\}^\{2\}\+d\_\{4\}^\{2\}=d\_\{2\}^\{2\}\+d\_\{3\}^\{2\}\.\(H\.331\)\(For example, place the rectangle at\(0,0\),\(L,0\),\(0,H\),\(L,H\)\(0,0\),\(L,0\),\(0,H\),\(L,H\)and expand the four squared distances to verify the equality for all𝒑\\bm\{p\}\.\)

Assume the dominant pair\(i,j\)\(i,j\)from[Sec\.4](https://arxiv.org/html/2605.06831#S4)corresponds to the diagonal\(1,4\)\(1,4\)of this cell\. Let

m:=min⁡\{d12,d42\},δ:=\|d12−d42\|\.m:=\\min\\\{d\_\{1\}^\{2\},d\_\{4\}^\{2\}\\\},\\qquad\\delta:=\|d\_\{1\}^\{2\}\-d\_\{4\}^\{2\}\|\.Then \(wlog, sayd12=md\_\{1\}^\{2\}=mandd42=m\+δd\_\{4\}^\{2\}=m\+\\delta\) we have

d12\+d422=m\+δ2\.\\frac\{d\_\{1\}^\{2\}\+d\_\{4\}^\{2\}\}\{2\}=m\+\\frac\{\\delta\}\{2\}\.\(H\.332\)Also, usingmin⁡\{a,b\}≤\(a\+b\)/2\\min\\\{a,b\\\}\\leq\(a\+b\)/2and[Eq\.H\.331](https://arxiv.org/html/2605.06831#A8.E331),

min⁡\{d22,d32\}\\displaystyle\\min\\\{d\_\{2\}^\{2\},d\_\{3\}^\{2\}\\\}≤d22\+d322=d12\+d422=m\+δ2\.\\displaystyle\\leq\\frac\{d\_\{2\}^\{2\}\+d\_\{3\}^\{2\}\}\{2\}=\\frac\{d\_\{1\}^\{2\}\+d\_\{4\}^\{2\}\}\{2\}=m\+\\frac\{\\delta\}\{2\}\.\(H\.333\)
By[Sec\.4](https://arxiv.org/html/2605.06831#S4)for the pair\(1,4\)\(1,4\), every non\-dominant modek∉\{1,4\}k\\notin\\\{1,4\\\}satisfies

dk2≥m\+Δt\.d\_\{k\}^\{2\}\\geq m\+\\Delta\_\{t\}\.In particular,

d22≥m\+Δt,d32≥m\+Δt\.d\_\{2\}^\{2\}\\geq m\+\\Delta\_\{t\},\\qquad d\_\{3\}^\{2\}\\geq m\+\\Delta\_\{t\}\.\(H\.334\)Combining[Eq\.H\.334](https://arxiv.org/html/2605.06831#A8.E334)with[Eq\.H\.333](https://arxiv.org/html/2605.06831#A8.E333)yields

m\+Δt≤m\+δ2⟹Δt≤δ2\.m\+\\Delta\_\{t\}\\leq m\+\\frac\{\\delta\}\{2\}\\quad\\Longrightarrow\\quad\\Delta\_\{t\}\\leq\\frac\{\\delta\}\{2\}\.\(H\.335\)
Letγ~a,t\\tilde\{\\gamma\}\_\{a,t\}denote the \(rescaled\) responsibilities as in[AppendixG](https://arxiv.org/html/2605.06831#A7); for brevity writeγa:=γ~a,t\\gamma\_\{a\}:=\\tilde\{\\gamma\}\_\{a,t\}\. By[Eq\.H\.241](https://arxiv.org/html/2605.06831#A8.E241), for anya,ba,b, we have that:

γaγb=πaπb​exp⁡\(db2−da22​σ~t2\)\.\\frac\{\\gamma\_\{a\}\}\{\\gamma\_\{b\}\}=\\frac\{\\pi\_\{a\}\}\{\\pi\_\{b\}\}\\exp\\\!\\left\(\\frac\{d\_\{b\}^\{2\}\-d\_\{a\}^\{2\}\}\{2\\tilde\{\\sigma\}\_\{t\}^\{2\}\}\\right\)\.Applying this with\(a,b\)=\(1,4\)\(a,b\)=\(1,4\)and taking logs gives

d42−d12=2​σ~t2​\(log⁡γ1γ4−log⁡π1π4\),d\_\{4\}^\{2\}\-d\_\{1\}^\{2\}=2\\tilde\{\\sigma\}\_\{t\}^\{2\}\\left\(\\log\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{4\}\}\-\\log\\frac\{\\pi\_\{1\}\}\{\\pi\_\{4\}\}\\right\),hence

δ=\|d42−d12\|≤2​σ~t2​\(\|log⁡γ1γ4\|\+\|log⁡π1π4\|\)\.\\delta=\|d\_\{4\}^\{2\}\-d\_\{1\}^\{2\}\|\\leq 2\\tilde\{\\sigma\}\_\{t\}^\{2\}\\left\(\\left\|\\log\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{4\}\}\\right\|\+\\left\|\\log\\frac\{\\pi\_\{1\}\}\{\\pi\_\{4\}\}\\right\|\\right\)\.\(H\.336\)
Plug[Eq\.H\.336](https://arxiv.org/html/2605.06831#A8.E336)into[Eq\.H\.335](https://arxiv.org/html/2605.06831#A8.E335):

Δt≤σ~t2​\(\|log⁡γ1γ4\|\+\|log⁡π1π4\|\)\.\\Delta\_\{t\}\\leq\\tilde\{\\sigma\}\_\{t\}^\{2\}\\left\(\\left\|\\log\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{4\}\}\\right\|\+\\left\|\\log\\frac\{\\pi\_\{1\}\}\{\\pi\_\{4\}\}\\right\|\\right\)\.\(H\.337\)By[Sec\.4](https://arxiv.org/html/2605.06831#S4),Δt≥2​σ~t2​κ\\Delta\_\{t\}\\geq 2\\tilde\{\\sigma\}\_\{t\}^\{2\}\\kappa, so

2​κ≤\|log⁡γ1γ4\|\+\|log⁡π1π4\|\.2\\kappa\\leq\\left\|\\log\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{4\}\}\\right\|\+\\left\|\\log\\frac\{\\pi\_\{1\}\}\{\\pi\_\{4\}\}\\right\|\.Exponentiating and usingexp⁡\(\|log⁡x\|\)=max⁡\{x,1/x\}\\exp\(\|\\log x\|\)=\\max\\\{x,1/x\\\}yields

e2​κ≤max⁡\{γ1γ4,γ4γ1\}⋅max⁡\{π1π4,π4π1\}\.e^\{2\\kappa\}\\leq\\max\\left\\\{\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{4\}\},\\frac\{\\gamma\_\{4\}\}\{\\gamma\_\{1\}\}\\right\\\}\\cdot\\max\\left\\\{\\frac\{\\pi\_\{1\}\}\{\\pi\_\{4\}\},\\frac\{\\pi\_\{4\}\}\{\\pi\_\{1\}\}\\right\\\}\.\(H\.338\)
Defineρt:=min⁡\{γ1,γ4\}\\rho\_\{t\}:=\\min\\\{\\gamma\_\{1\},\\gamma\_\{4\}\\\}\. Sinceγa∈\(0,1\)\\gamma\_\{a\}\\in\(0,1\)andγ1\+γ4≤1\\gamma\_\{1\}\+\\gamma\_\{4\}\\leq 1, we havemax⁡\{γ1,γ4\}≤1−ρt\\max\\\{\\gamma\_\{1\},\\gamma\_\{4\}\\\}\\leq 1\-\\rho\_\{t\}, hence

max⁡\{γ1γ4,γ4γ1\}=max⁡\{γ1,γ4\}min⁡\{γ1,γ4\}≤1−ρtρt\.\\max\\left\\\{\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{4\}\},\\frac\{\\gamma\_\{4\}\}\{\\gamma\_\{1\}\}\\right\\\}=\\frac\{\\max\\\{\\gamma\_\{1\},\\gamma\_\{4\}\\\}\}\{\\min\\\{\\gamma\_\{1\},\\gamma\_\{4\}\\\}\}\\leq\\frac\{1\-\\rho\_\{t\}\}\{\\rho\_\{t\}\}\.\(H\.339\)LetCπ:=max⁡\{π1/π4,π4/π1\}C\_\{\\pi\}:=\\max\\\{\\pi\_\{1\}/\\pi\_\{4\},\\pi\_\{4\}/\\pi\_\{1\}\\\}\. Combining[Eqs\.H\.338](https://arxiv.org/html/2605.06831#A8.E338)and[H\.339](https://arxiv.org/html/2605.06831#A8.E339)gives

e2​κ≤Cπ​1−ρtρt⟹ρt≤CπCπ\+e2​κ≤Cπ​e−2​κ\.e^\{2\\kappa\}\\leq C\_\{\\pi\}\\frac\{1\-\\rho\_\{t\}\}\{\\rho\_\{t\}\}\\quad\\Longrightarrow\\quad\\rho\_\{t\}\\leq\\frac\{C\_\{\\pi\}\}\{C\_\{\\pi\}\+e^\{2\\kappa\}\}\\leq C\_\{\\pi\}e^\{\-2\\kappa\}\.Sinceρt=min⁡\{γ1,γ4\}\\rho\_\{t\}=\\min\\\{\\gamma\_\{1\},\\gamma\_\{4\}\\\}andγa=γ~a,t\\gamma\_\{a\}=\\tilde\{\\gamma\}\_\{a,t\}, we have shown

min⁡\{γ~1,t,γ~4,t\}≤C​π​e−2​κ∀t≤τ1\.\\min\\\{\\tilde\{\\gamma\}\_\{1,t\},\\tilde\{\\gamma\}\_\{4,t\}\\\}\\leq C\\pi e^\{\-2\\kappa\}\\qquad\\forall\\,t\\leq\\tau\_\{1\}\.This proves the desired𝒪​\(e−2​κ\)\\mathcal\{O\}\(e^\{\-2\\kappa\}\)bound for a single diagonal pair\.

Next, we extend this to the rest of the cell\. If\(i,j\)\(i,j\)is a diagonal pair of some grid cell, let\(k,ℓ\)\(k,\\ell\)be the other two corners of that cell\. Then the rectangle identity[Eq\.H\.331](https://arxiv.org/html/2605.06831#A8.E331)holds for the quadruple\(i,k,ℓ,j\)\(i,k,\\ell,j\)for the same point𝒑=𝒚t\\bm\{p\}=\\bm\{y\}\_\{t\}, and[Sec\.4](https://arxiv.org/html/2605.06831#S4)for\(i,j\)\(i,j\)impliesdk2,dℓ2≥m\+Δtd\_\{k\}^\{2\},d\_\{\\ell\}^\{2\}\\geq m\+\\Delta\_\{t\}just as in[Eq\.H\.334](https://arxiv.org/html/2605.06831#A8.E334)\. Repeating the above argument with the relabeled indices givesmin⁡\{γ~i,t,γ~j,t\}≤Cπ,i​j​e−2​κ\\min\\\{\\tilde\{\\gamma\}\_\{i,t\},\\tilde\{\\gamma\}\_\{j,t\}\\\}\\leq C\_\{\\pi,ij\}e^\{\-2\\kappa\}whereCπ,i​j:=max⁡\{πi/πj,πj/πi\}C\_\{\\pi,ij\}:=\\max\\\{\\pi\_\{i\}/\\pi\_\{j\},\\pi\_\{j\}/\\pi\_\{i\}\\\}\. Lastly, since the cell chosen was arbitrary, this holds for all \(i,j\) in the grid, and we are done\. ∎

## Appendix IUse of the Exact Score

In what follows, we detail why we use the exact score in our theoretical analysis and how this connects to the realistic learned score setting which we aim to study\. Since the marginals under the exact score agree for the ODE and SDE as discussed in[Sec\.2](https://arxiv.org/html/2605.06831#S2), the hallucination rate for the ODE and SDE agree under the exact score \(in particular, it is 0\)\. However, we can still leverage the ODE/SDE under the exact score to obtain amechanismwhich becomes useful and predictive in the learned score setting\. In particular, assuming the exact score as regularity, we show that𝒙t\\bm\{x\}\_\{t\}under the ODE can get stuck \([Sec\.4](https://arxiv.org/html/2605.06831#S4)\) while𝒙t\\bm\{x\}\_\{t\}under the SDE can escape \([Sec\.5](https://arxiv.org/html/2605.06831#S5)\)\. We study the exact score scenario in our main paper because it provides the cleanest presentation of this vulnerability\. Our experiments then demonstrate that the trapping region in[Sec\.4](https://arxiv.org/html/2605.06831#S4)characterizes hallucination differences between DDIM and DDPM in the realistic learned score scenario \(e\.g\., in[Figure5](https://arxiv.org/html/2605.06831#S6.F5)\)\.

To make this mathematically precise, we haveptDDIM,exact=ptDDPM,exactp\_\{t\}^\{\\mathrm\{DDIM,exact\}\}=p\_\{t\}^\{\\mathrm\{DDPM,exact\}\}\. LetH:=\{mode interpolation occurs\}H:=\\\{\\text\{mode interpolation occurs\}\\\}\. Then, we havePrDDIM,exact⁡\(H\)=PrDDPM,exact⁡\(H\)\\Pr^\{\\mathrm\{DDIM,exact\}\}\(H\)=\\Pr^\{\\mathrm\{DDPM,exact\}\}\(H\)\. However, we are interested in explaining the observed differences in hallucination rates in the realistic, practical learned score setting wherePrDDIM,learned⁡\(H\)≫PrDDPM,learned⁡\(H\)\\Pr^\{\\mathrm\{DDIM,learned\}\}\(H\)\\gg\\Pr^\{\\mathrm\{DDPM,learned\}\}\(H\)\([Figure1](https://arxiv.org/html/2605.06831#S1.F1)\)\. To bridge this gap, we must look beyond unconditional marginalsptp\_\{t\}and examine the*conditional trajectory dynamics*p​\(𝒙0∣𝒙τ3\)p\(\\bm\{x\}\_\{0\}\\mid\\bm\{x\}\_\{\\tau\_\{3\}\}\), which differ entirely for DDIM and DDPM\. We find that we can extract a difference in DDIM \([Sec\.4](https://arxiv.org/html/2605.06831#S4)\) and DDPM \([Sec\.5](https://arxiv.org/html/2605.06831#S5)\) under the exact score which uncovers the trapping mechanism drivingPrDDIM,learned⁡\(H\)≫PrDDPM,learned⁡\(H\)\\Pr^\{\\mathrm\{DDIM,learned\}\}\(H\)\\gg\\Pr^\{\\mathrm\{DDPM,learned\}\}\(H\)\. Concretely, letMϑ,τ3:=\{trajectory enters​ϑ​\-midpoint neighborhood during final​τ3​steps\}M\_\{\\vartheta,\\tau\_\{3\}\}:=\\\{\\text\{trajectory enters \}\\vartheta\\text\{\-midpoint neighborhood during final \}\\tau\_\{3\}\\text\{ steps\}\\\}\. Then:

Pr⁡\(H\)=Pr⁡\(H∣Mϑ,τ3\)​Pr⁡\(Mϑ,τ3\)\+Pr⁡\(H∣Mϑ,τ3c\)​Pr⁡\(Mϑ,τ3c\)\\Pr\(H\)=\\Pr\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\)\\Pr\(M\_\{\\vartheta,\\tau\_\{3\}\}\)\+\\Pr\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}^\{c\}\)\\Pr\(M\_\{\\vartheta,\\tau\_\{3\}\}^\{c\}\)\(I\.340\)
The contributions of our exact score theory in[Sec\.4](https://arxiv.org/html/2605.06831#S4)and[Sec\.5](https://arxiv.org/html/2605.06831#S5)are twofold:

1. 1\.The midpoint neighborhood andMϑ,τ3M\_\{\\vartheta,\\tau\_\{3\}\}are driving the differences in hallucination rates between DDIM and DDPM, i\.e\.Pr⁡\(H∣Mϑ,τ3c\)\\Pr\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}^\{c\}\)is small and the first term dominates so thatPr⁡\(H\)≈Pr⁡\(H∣Mϑ,τ3\)​Pr⁡\(Mϑ,τ3\)\\Pr\(H\)\\approx\\Pr\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\)\\Pr\(M\_\{\\vartheta,\\tau\_\{3\}\}\)\. This is done in[Sec\.4](https://arxiv.org/html/2605.06831#S4)\.
2. 2\.Demonstrate thatPrDDIM,exact⁡\(H∣Mϑ,τ3\)≫PrDDPM,exact⁡\(H∣Mϑ,τ3\)\\Pr^\{\\mathrm\{DDIM,exact\}\}\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\)\\gg\\Pr^\{\\mathrm\{DDPM,exact\}\}\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\)\. This is done in[Sec\.5](https://arxiv.org/html/2605.06831#S5)\.

These results arise due to the differences in \(conditional\) dynamics, even though the marginals are the same under the exact score\. The exact score assumption in our theory allows us to demonstrate this cleanly\. Still,PrDDIM,exact⁡\(Mϑ,τ3\)=PrDDPM,exact⁡\(Mϑ,τ3\)=0\\Pr^\{\\mathrm\{DDIM,exact\}\}\(M\_\{\\vartheta,\\tau\_\{3\}\}\)=\\Pr^\{\\mathrm\{DDPM,exact\}\}\(M\_\{\\vartheta,\\tau\_\{3\}\}\)=0\.

But, in the practical learned score setting wherePrlearned⁡\(Mϑ,τ3\)≠0\\Pr^\{\\mathrm\{learned\}\}\(M\_\{\\vartheta,\\tau\_\{3\}\}\)\\neq 0for both samplers, the fundamental differencePrDDIM,learned⁡\(H∣Mϑ,τ3\)≫PrDDPM,learned⁡\(H∣Mϑ,τ3\)\\Pr^\{\\mathrm\{DDIM,learned\}\}\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\)\\gg\\Pr^\{\\mathrm\{DDPM,learned\}\}\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\), which persists in the learned score case but is uncovered via our exact score analysis, helps explainPrDDIM,learned⁡\(H\)≫PrDDPM,learned⁡\(H\)\\Pr^\{\\mathrm\{DDIM,learned\}\}\(H\)\\gg\\Pr^\{\\mathrm\{DDPM,learned\}\}\(H\)\. Empirically, in[Tab\.I\.1](https://arxiv.org/html/2605.06831#A9.T1), we find that the first term in[Eq\.I\.340](https://arxiv.org/html/2605.06831#A9.E340)remains dominant whilePrDDIM,learned⁡\(H∣Mϑ,τ3\)≫PrDDPM,learned⁡\(H∣Mϑ,τ3\)\\Pr^\{\\mathrm\{DDIM,learned\}\}\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\)\\gg\\Pr^\{\\mathrm\{DDPM,learned\}\}\(H\\mid M\_\{\\vartheta,\\tau\_\{3\}\}\)\.

Table I\.1:Hallucination probabilities in the learned score case, averaged across 100k trajectories, withτ3=11\\tau\_\{3\}=11andϑ=0\.35​ℓt\\vartheta=0\.35\\ell\_\{t\}\. Here,HHdenotes the event that the final generated sample is a mode interpolation, andM:=Mϑ,τ3M:=M\_\{\\vartheta,\\tau\_\{3\}\}denotes the event that the trajectory enters theϑ\\vartheta\-midpoint neighborhood of the nearesti,ji,j\-mode segment during the finalτ3\\tau\_\{3\}steps\. We observe thatPrDDIM⁡\(H∣M\)≫PrDDPM⁡\(H∣M\)\\Pr^\{\\mathrm\{DDIM\}\}\(H\\mid M\)\\gg\\Pr^\{\\mathrm\{DDPM\}\}\(H\\mid M\), whilePr⁡\(H∣Mc\)\\Pr\(H\\mid M^\{c\}\)remains small for both samplers\. Thus, the dominant contribution comes from the first decomposition termPr⁡\(H∣M\)​Pr⁡\(M\)\\Pr\(H\\mid M\)\\Pr\(M\), supporting the conclusion that midpoint\-neighborhood entry is the key event relevant for hallucinations in the learned score case\.
## Appendix JSymbol Table

Symbolsϖ\\varpiDimension of all vectors throughout\.pbase\\displaystyle p\_\{\\text\{base\}\}The distribution which the diffusion sampling process starts from\. Throughout, we take this to be𝒩​\(0,𝑰\)\\mathcal\{N\}\(0,\\bm\{I\}\), the standard Gaussian\.pdatap\_\{\\text\{data\}\}The distribution which the diffusion process ends at and aims to sample from\. Throughout, we take this to be aNN\-component Gaussian mixture\.𝒙t\\bm\{x\}\_\{t\}A trajectory satisfying the PF ODE given in[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)for DDIM or[Eq\.2](https://arxiv.org/html/2605.06831#S2.E2)for DDPM, given in reverse time fromt=Tt=Ttot=0t=0unless otherwise specified\.β​\(t\)\\beta\(t\)Diffusion noise schedule\.𝑾t\\bm\{W\}\_\{t\}Standardϖ\\varpi\-dimensional Brownian motionα¯t\\bar\{\\alpha\}\_\{t\}Amount of signal from data distribution retained in forward or reverse process\. Given byexp⁡\(−∫0tβ​\(s\)​𝑑s\)\\exp\(\-\\int\_\{0\}^\{t\}\\beta\(s\)\\;ds\)ptp\_\{t\}Marginal distribution of diffusion process instance𝒙t\\bm\{x\}\_\{t\}at timett; equivalent in forward and reverse process\.∇𝒙tlog⁡pt​\(𝒙t\)\\nabla\_\{\\bm\{x\}\_\{t\}\}\\log p\_\{t\}\(\\bm\{x\}\_\{t\}\)Exact score function of marginalptp\_\{t\}\.𝑰\\bm\{I\}Identity matrix\.𝒩​\(𝒙;𝝁,σ2​𝑰\)\\mathcal\{N\}\(\\bm\{x\};\\bm\{\\mu\},\\sigma^\{2\}\\bm\{I\}\)Probability density function of Gaussian distribution centered at𝝁\\bm\{\\mu\}with isotropic covariance matrixσ2​𝑰\\sigma^\{2\}\\bm\{I\}\.𝑾¯t\\bar\{\\bm\{W\}\}\_\{t\}Standard Brownian motion in reverse time\.s𝜽​\(𝒙t\)s\_\{\\bm\{\\theta\}\}\(\\bm\{x\}\_\{t\}\)Learned score function\.πi\\pi\_\{i\}Coefficient of Gaussian mixture\.NNNumber of components in Gaussian mixture\.𝝁\(i\)\\bm\{\\mu\}^\{\(i\)\}iith mode of Gaussian mixture,i∈\[N\]i\\in\[N\]\.σ2\\sigma^\{2\}Variance of Gaussian mixture, i\.e\. each component has covarianceσ2​𝑰\\sigma^\{2\}\\bm\{I\}\.𝝁t\(i\)\\bm\{\\mu\}^\{\(i\)\}\_\{t\}Time\-dependent modes ofptp\_\{t\}given Gaussian mixture target\.σt2\\sigma\_\{t\}^\{2\}Time\-dependent variance ofptp\_\{t\}given Gaussian target\.σ~t2\\tilde\{\\sigma\}\_\{t\}^\{2\}Rescaled time\-dependent variance ofptp\_\{t\}\.L\(i,j\)L^\{\(i,j\)\}Line segment joining modeiiand modejj\.Lt\(i,j\)L\_\{t\}^\{\(i,j\)\}Line segment joining time\-dependent modeiiand time\-dependent modejj\.Lt,ε\(i,j\)L\_\{t,\\varepsilon\}^\{\(i,j\)\}ε\\varepsilon\-extended line segment joining time\-dependent modeiiand time\-dependent modejj\.d⟂d\_\{\\perp\}Orthogonal distance between trajectory and \(nearby\) line segment\.Tube\(i,j\)t,ζ\{\}\_\{t,\\zeta\}^\{\(i,j\)\}Tube aroundLt\(i,j\)L\_\{t\}^\{\(i,j\)\}of radiusζ\\zeta\. Throughout,ζ=ε\\zeta=\\varepsilon, unless specified otherwise\.

SymbolsΔt\\Delta\_\{t\}Time\-dependent margin between modes\. Must be sufficiently large after time in reverse process w\.r\.t\.σ~t2\\tilde\{\\sigma\}\_\{t\}^\{2\}\.κ\\kappaMargin coefficient, defining gap betweenσ~t2\\tilde\{\\sigma\}\_\{t\}^\{2\}andΔt\\Delta\_\{t\}\. Also used for margin w\.r\.t\.ℓ2\\ell^\{2\}\.τ1\\tau\_\{1\}Critical time for[Sec\.4](https://arxiv.org/html/2605.06831#S4)to hold\.τ\\tauTime which is less thanτ1\\tau\_\{1\}; critical time for exponential convergence in[4\.2](https://arxiv.org/html/2605.06831#S4.Thmthm2)to hold\.τ2\\tau\_\{2\}Critical time for[Sec\.4](https://arxiv.org/html/2605.06831#S4)to hold\. Empirically strictly less thanτ1\\tau\_\{1\}\.τ^\\hat\{\\tau\}Minimum ofτ\\tauandτ2\\tau\_\{2\}\. Critical time for[Sec\.4](https://arxiv.org/html/2605.06831#S4)to hold\.τ3\\tau\_\{3\}Critical time less thanτ^\\hat\{\\tau\}for[Sec\.4](https://arxiv.org/html/2605.06831#S4)to hold\.ϕ\\phiUsed to define probabilities in concentration inequalities throughout\.ε\\varepsilonSize by which nearby line segment is extended; radius of containment tube; and radius of ball around modes of mixture where stable equilibria exist\.ξt\\xi\_\{t\}Used to denote parallel dynamics toLt\(i,j\)L\_\{t\}^\{\(i,j\)\}\.ξ∗\\xi^\{\*\}is an equilibrium of some kind, stated specifically in each proposition in which it appears\.𝒘t\\bm\{w\}\_\{t\}Used to denote orthogonal dynamics toLt\(i,j\)L\_\{t\}^\{\(i,j\)\}\.𝒚t∗\\bm\{y\}\_\{t\}^\{\*\}Unstable equilibria at midpoint alongLt\(i,j\)L\_\{t\}^\{\(i,j\)\}\(w\.r\.t\. approximate dynamics\)𝒚t\\bm\{y\}\_\{t\}Used for time\-rescaled dynamics in[AppendixG](https://arxiv.org/html/2605.06831#A7), where modes become fixed and effective varianceσt2\\sigma\_\{t\}^\{2\}is rescaled toσ~t2\\tilde\{\\sigma\}\_\{t\}^\{2\}\.λt\\lambda\_\{t\}Unstable eigenvalue on midpoint, or unstable equilibria onLt\(i,j\)L\_\{t\}^\{\(i,j\)\}more broadly\.ϑ\\varthetaRadius from midpoint for which, givenτ3\\tau\_\{3\}time remaining\.λ~t\\tilde\{\\lambda\}\_\{t\}Integral ofλt\\lambda\_\{t\}from timetν1t\_\{\\nu\_\{1\}\}to timetν2t\_\{\\nu\_\{2\}\}\.ℓ2\\ell^\{2\}Squared distance between \(fixed, time\-independent\) modes\.H\(i,j\)H^\{\(i,j\)\}Bisector hyperplane toL\(i,j\)L^\{\(i,j\)\}, i\.e\. the plane orthogonal to the line which passes through the midpoint of the line\.𝒖\\bm\{u\}The coordinate normal toH\(i,j\)H^\{\(i,j\)\}\.AtA\_\{t\}One dimensional dynamics of distance from midpoint along𝒖\\bm\{u\}, with respect to DDPM\.b​\(At,t\)b\(A\_\{t\},t\)Deterministic dynamics \(drift\) ofAtA\_\{t\}\.π\\piWithout a subscript, defines the usual scalarπ\\pi\.K​\(t\)K\(t\)Controls how fastb​\(At,t\)b\(A\_\{t\},t\)is\.η​\(t\)\\eta\(t\)1/2​β​\(t\)1/2\\beta\(t\)\.δ\\deltaIn the main paper, used to define the terminal confinement events\. Used to define residual terms due toN−2N\-2components in appendix proofs\.

SymbolsHδH\_\{\\delta\}Event that at the end of the reverse process, trajectory stays near the bisector hyperplane\.E2​δE\_\{2\\delta\}Event that at throughout the reverse process, one enters the midpoint neighborhood\.λrep\\lambda\_\{\\text\{rep\}\}Coefficient which controls speed of drift away from bisector hyperplane\.γk\\gamma\_\{k\}Gaussian responsibilities under[Eq\.3](https://arxiv.org/html/2605.06831#S2.E3)\.γ~k\\tilde\{\\gamma\}\_\{k\}Gaussian responsibilities under time\-rescaled dynamics in[AppendixG](https://arxiv.org/html/2605.06831#A7)\.𝝁^\(i\)\\hat\{\\bm\{\\mu\}\}^\{\(i\)\}Convex combination of static modes w\.r\.t\. time\-rescaped Gaussian responsibilities\.I​\(t\)I\(t\)Integral ofβ​\(s\)\\beta\(s\)without exponentialδξ\\delta\_\{\\xi\}Effect ofN−2N\-2modes on parallel dynamics to nearbyLt\(i,j\)L\_\{t\}^\{\(i,j\)\}\.𝑬t\\bm\{E\}\_\{t\}Convex combination of distances from modeii, given time\-rescaled modes, excluding modesiiandjj\.Proj\|\|​\(𝒗\)\\text\{Proj\}\_\{\|\|\}\(\\bm\{v\}\)Projection onto vector parallel to𝒗\\bm\{v\}\.Proj⟂​\(𝒗\)\\text\{Proj\}\_\{\\perp\}\(\\bm\{v\}\)Projection onto vector orthogonal to𝒗\\bm\{v\}\.𝜹𝒘\\bm\{\\delta\}\_\{\\bm\{w\}\}Effect ofN−2N\-2modes on orthogonal dynamics to nearbyLt\(i,j\)L\_\{t\}^\{\(i,j\)\}\.d∗d^\{\*\}Mode which for which the trajectory has minimum distance from one of the two selected \(nearby\) modesiiandjj\.𝒪\\mathcal\{O\},Ω\\Omega,Θ\\ThetaUsed for standard asymptotic notation\.MMDefines maximum non\-selected mode distance\.RRDefines maximum Gaussian mixture weights w\.r\.t\.d∗d^\{\*\}, excludingπi\\pi\_\{i\},πj\\pi\_\{j\}\.θ\\thetaUsed to rescale reverse time into forward time, i\.e\.θ=T−t\\theta=T\-t\.M,QM,QFor martingaleMM,QQis its quadratic variation\.