x-Prediction Is All You Need:Training-Free Accelerated Generation via Endpoint Decodability

# Training-Free Accelerated Generation via Endpoint Decodability
Source: [https://arxiv.org/html/2607.06114](https://arxiv.org/html/2607.06114)
## xx\-Prediction Is All You Need: Training\-Free Accelerated Generation via Endpoint Decodability

###### Abstract

Diffusion and flow matching models generate high\-quality samples, but their ODE samplers often need tens to hundreds of neural function evaluations \(NFEs\)\. This remains a practical challenge for released checkpoints, since many accelerators require additional design choices and training cost through retraining, distillation, or trajectory redesign\. We investigate a different route based onxx\-prediction\. During sampling, standard affine probability paths already exposex0x\_\{0\}information: an intermediate state and its path velocity determine a principled estimate of the clean sample\. We formalize this property asendpoint decodabilityand show that the decoder is the minimum\-MSE estimator𝔼​\[x0∣xt\]\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}\]under the usualℓ2\\ell\_\{2\}objective\. This yieldsTruncated Jump Sampling\(TJS\): stop the ODE at an early\-exit timet∗t^\{\*\}and return the decodedx0x\_\{0\}\. TJS requires no retraining, distillation, or architecture change\. Across SDXL, SD3\.5M, Z\-Image\-Turbo, and three class\-conditional benchmarks, it reduces NFEs by 20–70% with near\-matched quality\. The analysis also shows why endpoint prediction can work without straightening the trajectory, providing inference acceleration without trajectory redesign\.

## Introduction

Diffusion models\(Hoet al\.[2020](https://arxiv.org/html/2607.06114#bib.bib6); Songet al\.[2021b](https://arxiv.org/html/2607.06114#bib.bib5); Nichol and Dhariwal[2021](https://arxiv.org/html/2607.06114#bib.bib9)\)and flow matching\(Lipmanet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib1); Albergo and Vanden\-Eijnden[2023](https://arxiv.org/html/2607.06114#bib.bib2); Liuet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib3)\)produce stunning images but require tens to hundreds of neural function evaluations \(NFEs\) per sample\(Rombachet al\.[2022](https://arxiv.org/html/2607.06114#bib.bib21); Esseret al\.[2024](https://arxiv.org/html/2607.06114#bib.bib24); Black\-Forest\-Labs[2024](https://arxiv.org/html/2607.06114#bib.bib26); Peebles and Xie[2023](https://arxiv.org/html/2607.06114#bib.bib23)\)\. Reducing this cost is a central challenge in generative modeling\.

The field has responded with a range of solutions—Rectified Flow\(Liuet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib3)\), distillation \(progressive\(Salimans and Ho[2022](https://arxiv.org/html/2607.06114#bib.bib12)\), consistency\-based\(Songet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib14); Luoet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib15); Zhenget al\.[2024](https://arxiv.org/html/2607.06114#bib.bib69)\), bootstrapping\(Guet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib55)\)\), advanced solvers\(Luet al\.[2022](https://arxiv.org/html/2607.06114#bib.bib62),[2023](https://arxiv.org/html/2607.06114#bib.bib63); Zhang and Chen[2023](https://arxiv.org/html/2607.06114#bib.bib67)\), and training\-time scheduling\(Chenet al\.[2026](https://arxiv.org/html/2607.06114#bib.bib52)\)—all effective at reducing NFEs, all requiring changes to the trajectory, model, or training recipe\.No existing approach achieves acceleration purely by modifying inference\.

This complexity has a real and growing cost\. The community has produced an enormous library of pretrained checkpoints—SDXL alone has thousands of fine\-tuned variants on CivitAI and Hugging Face\(Podellet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib22); Esseret al\.[2024](https://arxiv.org/html/2607.06114#bib.bib24); Black\-Forest\-Labs[2024](https://arxiv.org/html/2607.06114#bib.bib26); Rombachet al\.[2022](https://arxiv.org/html/2607.06114#bib.bib21); Peebles and Xie[2023](https://arxiv.org/html/2607.06114#bib.bib23)\)\. While fast ODE solvers \(DPM\-Solver, UniPC\) can reduce the step count, further acceleration without retraining remains an open practical challenge\.The field would benefit from a training\-free acceleration strategy that works on models as they are, not as they could be after retraining\.

There is a simpler path, hiding in plain sight within the training objective itself\. Diffusion and flow matching models are trained to predictx0x\_\{0\}at every timestep\. During inference, the model thus produces a validx0x\_\{0\}estimate at*every intermediate step*—and these estimates improve as integration proceeds\. Once good enough, why continue? For diffusion models, DDIM\(Songet al\.[2021a](https://arxiv.org/html/2607.06114#bib.bib8)\)already computesx^0\\hat\{x\}\_\{0\}internally, but to our knowledge no prior work has systematically studied early\-stopping and outputting it as a terminal inference strategy, nor provided theoretical justification for when and why this works\. For flow matching, no such counterpart exists\. We fill both gaps\. We term this propertyendpoint decodability: for non\-degenerate affine paths, the intermediate state and path velocity recoverx0x\_\{0\}through a closed\-form decoder, which we formalize in Section[Endpoint Decodability](https://arxiv.org/html/2607.06114#Sx4)\. Crucially, under the standardℓ2\\ell\_\{2\}loss, this decoder is the MMSE\-optimal estimator𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\(Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)\)\. The practical consequence: stop when the estimate is sufficient, output it\. We call thisTruncated Jump Sampling \(TJS\)\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x1.png)Figure 1:Endpoint decodability in action\. Top:xtx\_\{t\}decoded directly \(noisy at early steps\)\. Bottom:x0x\_\{0\}via endpoint decoding \(clean at any step\)\.TJS requires no retraining, no distillation, no architecture changes\. It works on any pretrained checkpoint—SDXL, SD3\.5M, Z\-Image\-Turbo, FLUX, DiT, fine\-tuned variants—without modification\. TJS does not replace distillation for extreme few\-step generation \(1–4 NFE\); it provides a complementary, zero\-cost path for the moderate regime \(30→\\to15–25 steps\)\.Immediate, training\-free acceleration for models you already have\.

We make three contributions:

1. 1\.Principle and theory\.We formalizeendpoint decodability: for any non\-degenerate affine path,\(xt,ut\)\(x\_\{t\},u\_\{t\}\)recoversx0x\_\{0\}through a closed\-form decoder \(Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)\), the induced decoder is MMSE\-optimal \(Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)\), its error is curvature\-independent \(Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)\), and it can strictly beat coarse Euler \(Theorem[11](https://arxiv.org/html/2607.06114#Thmtheorem11)\)\. All standard parameterizations are equivalent at optimality \(see Supplementary Material, §A\)\. Critically, straight trajectories are*sufficient but not necessary*\(Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)\), challenging the foundation of Rectified Flow and Consistency Models\.
2. 2\.Algorithm\.Truncated Jump Sampling \(TJS\): a training\-free early\-exit sampler—zero retraining, zero distillation, zero architecture changes\. TJS is compatible with any ODE solver, noise schedule, CFG scale, or distillation method\.
3. 3\.Experiments\.Across six model families—SDXL, SD3\.5M, Z\-Image\-Turbo\(Team[2025](https://arxiv.org/html/2607.06114#bib.bib84)\), ImageNet\-256, CIFAR\-10, MNIST—TJS reduces NFE by 20–70% at near\-matched quality, with strict monotonic improvement on all metrics\.

## Related Work

##### Diffusion and Flow Matching\.

Diffusion models\(Hoet al\.[2020](https://arxiv.org/html/2607.06114#bib.bib6); Songet al\.[2021b](https://arxiv.org/html/2607.06114#bib.bib5); Nichol and Dhariwal[2021](https://arxiv.org/html/2607.06114#bib.bib9)\)and flow matching\(Lipmanet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib1); Albergo and Vanden\-Eijnden[2023](https://arxiv.org/html/2607.06114#bib.bib2); Liuet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib3)\)share a common skeleton: probability pathsxt=αt​x0\+σt​ϵx\_\{t\}=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsiloninterpolating noise to data\. DDIM\(Songet al\.[2021a](https://arxiv.org/html/2607.06114#bib.bib8)\)and EDM\(Karraset al\.[2022](https://arxiv.org/html/2607.06114#bib.bib53)\)established deterministic sampling via neural ODEs\(Chenet al\.[2018](https://arxiv.org/html/2607.06114#bib.bib39)\)\. These models power state\-of\-the\-art image generation—SD3\(Esseret al\.[2024](https://arxiv.org/html/2607.06114#bib.bib24); Meng and others[2024](https://arxiv.org/html/2607.06114#bib.bib83)\), FLUX\(Black\-Forest\-Labs[2024](https://arxiv.org/html/2607.06114#bib.bib26)\), DiT\(Peebles and Xie[2023](https://arxiv.org/html/2607.06114#bib.bib23)\)\.Every model cited above uses the same affine path structure\.Endpoint decodability requires nothing else\.

##### Few\-Step Generation and Trajectory Straightening\.

Reducing NFEs spans several paradigms, all training\-stage\. Distillation \(progressive\(Salimans and Ho[2022](https://arxiv.org/html/2607.06114#bib.bib12)\), masked\(Yanget al\.[2022](https://arxiv.org/html/2607.06114#bib.bib13)\), bootstrapping\(Guet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib55)\), trajectory consistency\(Zhenget al\.[2024](https://arxiv.org/html/2607.06114#bib.bib69)\)\) and Consistency Models\(Songet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib14); Luoet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib15)\)train students for few\-step generation\. Rectified Flow\(Liuet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib3)\)learns straight paths via reflow \(Euler error∝‖x¨t‖\\propto\\\|\\ddot\{x\}\_\{t\}\\\|\)\. Dense\-Jump FM\(Chenet al\.[2026](https://arxiv.org/html/2607.06114#bib.bib52)\)modifies inference trajectories\. Higher\-order solvers \(DPM\-Solver\(Luet al\.[2022](https://arxiv.org/html/2607.06114#bib.bib62)\), DPM\-Solver\+\+\(Luet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib63)\), PNDM\(Liuet al\.[2022](https://arxiv.org/html/2607.06114#bib.bib68)\), UniPC\(Zhang and Chen[2023](https://arxiv.org/html/2607.06114#bib.bib67)\)\) improve discretization accuracy but still traverse the full trajectory\. The shared motivation: few\-step quality benefits from modifying training\. TJS offers a complementary, training\-free alternative exploiting an inherent structural property of affine paths\.

##### Prediction Parameterizations and Theoretical Connections\.

x0x\_\{0\}\-,vv\-,ϵ\\epsilon\-, and score\-prediction\(Karraset al\.[2022](https://arxiv.org/html/2607.06114#bib.bib53); Salimans and Ho[2022](https://arxiv.org/html/2607.06114#bib.bib12); Hoet al\.[2020](https://arxiv.org/html/2607.06114#bib.bib6); Songet al\.[2021b](https://arxiv.org/html/2607.06114#bib.bib5)\)are typically treated as distinct choices with different SNR trade\-offs\(Kingmaet al\.[2021](https://arxiv.org/html/2607.06114#bib.bib65)\)\.Our framework reveals their algebraic equivalence at optimality\(see Theorem 13 in the Supplementary Material, §A\), so any pretrained model encodes an endpoint predictor\. Complementary evidence: JiT\(Li and He[2025](https://arxiv.org/html/2607.06114#bib.bib85)\)showsx0x\_\{0\}\-prediction dramatically outperformsϵ\\epsilon/v\-prediction on raw\-pixel patches via manifold structure; MeanFlow\(Genget al\.[2025](https://arxiv.org/html/2607.06114#bib.bib86)\)enables one\-step generation via average velocity\. Beyond sampling, flow matching connects to optimal transport\(Lipmanet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib1); Tonget al\.[2024](https://arxiv.org/html/2607.06114#bib.bib73)\)and the Schrödinger bridge\(Bortoliet al\.[2021](https://arxiv.org/html/2607.06114#bib.bib30)\); I\-MMSE\(Guoet al\.[2005](https://arxiv.org/html/2607.06114#bib.bib60)\)provides information\-theoretic grounding for𝒰​\(t\)\\mathcal\{U\}\(t\)\(Theorem 14 in the Supplementary Material, §A\)\.

## Preliminaries

Diffusion and flow matching models share a simple but powerful structure: they define paths that morph noise into data\. We now formalize this structure and preview the theory that follows\. We establish three results in order: \(1\) the algebraic condition under which any intermediate state can recover the clean endpoint \(Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)\); \(2\) the optimality of this recovery under standardℓ2\\ell\_\{2\}training \(Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)\); and \(3\) a decomposition of TJS error into two curvature\-independent components \(Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)\)\. Together, these results explain why straight trajectories—the central requirement of prior work—are sufficient but not necessary\.

###### Definition 1\(Affine Probability Path\)\.

An*affine probability path*is a one\-parameter stochastic process

xt=αt​x0\+σt​ϵ,ϵ∼𝒩​\(0,𝐈\),t∈\[0,1\],x\_\{t\}=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilon,\\quad\\epsilon\\sim\\mathcal\{N\}\(0,\\mathbf\{I\}\),\\quad t\\in\[0,1\],\(1\)wherex0∼pdatax\_\{0\}\\sim p\_\{\\mathrm\{data\}\}is the clean sample, andαt,σt:\[0,1\]→ℝ≥0\\alpha\_\{t\},\\sigma\_\{t\}:\[0,1\]\\to\\mathbb\{R\}\_\{\\geq 0\}areC1C^\{1\}functions withα0=0,σ0=1\\alpha\_\{0\}=0,\\sigma\_\{0\}=1andα1=1,σ1=0\\alpha\_\{1\}=1,\\sigma\_\{1\}=0\.

This single equation covers VP/VE diffusion, EDM, and linear flow matching\. The entire diversity of modern generative models—billions of parameters, dozens of training tricks—reduces to two scalar schedulesαt\\alpha\_\{t\}andσt\\sigma\_\{t\}\. Endpoint decodability is a property of these two functions alone\.

###### Lemma 2\(Conditional Velocity\)\.

The conditional velocity fieldut≔d​xtd​tu\_\{t\}\\coloneqq\\frac\{\\mathrm\{d\}x\_\{t\}\}\{\\mathrm\{d\}t\}satisfies

ut=α˙t​x0\+σ˙t​ϵ\.u\_\{t\}=\\dot\{\\alpha\}\_\{t\}x\_\{0\}\+\\dot\{\\sigma\}\_\{t\}\\epsilon\.\(2\)

Combining Eqs\.[1](https://arxiv.org/html/2607.06114#Sx3.E1)–[2](https://arxiv.org/html/2607.06114#Sx3.E2)yields a coupled linear system:

\[xtut\]=\[αtσtα˙tσ˙t\]​\[x0ϵ\]\.\\begin\{bmatrix\}x\_\{t\}\\\\ u\_\{t\}\\end\{bmatrix\}=\\begin\{bmatrix\}\\alpha\_\{t\}&\\sigma\_\{t\}\\\\ \\dot\{\\alpha\}\_\{t\}&\\dot\{\\sigma\}\_\{t\}\\end\{bmatrix\}\\begin\{bmatrix\}x\_\{0\}\\\\ \\epsilon\\end\{bmatrix\}\.\(3\)
###### Definition 3\(Path Determinant\)\.

The*path determinant*isΔt≔det\[αtσtα˙tσ˙t\]=α˙t​σt−αt​σ˙t\\Delta\_\{t\}\\coloneqq\\det\\begin\{bmatrix\}\\alpha\_\{t\}&\\sigma\_\{t\}\\\\ \\dot\{\\alpha\}\_\{t\}&\\dot\{\\sigma\}\_\{t\}\\end\{bmatrix\}=\\dot\{\\alpha\}\_\{t\}\\sigma\_\{t\}\-\\alpha\_\{t\}\\dot\{\\sigma\}\_\{t\}\.

The coupled linear system \(Eq\.[3](https://arxiv.org/html/2607.06114#Sx3.E3)\) contains the entire story\. If the2×22\\times 2coefficient matrix is invertible, then\(xt,ut\)\(x\_\{t\},u\_\{t\}\)uniquely determines\(x0,ϵ\)\(x\_\{0\},\\epsilon\)—and in particularx0x\_\{0\}\. The invertibility condition is remarkably simple: the path determinant must be nonzero\. We now formalize this observation and prove that the resulting decoder is optimal\.

### Characterization

Solving the linear system via Cramer’s rule \(see Supplementary Material, §A\) yields the endpoint decoder:

###### Definition 4\(Endpoint Decodability\)\.

An affine probability path is*endpoint\-decodable*at timet∈\(0,1\]t\\in\(0,1\]if the mapping\(xt,ut\)↦x0\(x\_\{t\},u\_\{t\}\)\\mapsto x\_\{0\}is well\-defined and unique\. A path is*globally endpoint\-decodable*if this holds for allt∈\(0,1\]t\\in\(0,1\]\.

###### Theorem 5\(Endpoint Decodability\)\.

An affine probability path is endpoint\-decodable atttiffΔt≠0\\Delta\_\{t\}\\neq 0, with

x0=σt​ut−σ˙t​xtΔt\.x\_\{0\}=\\frac\{\\sigma\_\{t\}u\_\{t\}\-\\dot\{\\sigma\}\_\{t\}x\_\{t\}\}\{\\Delta\_\{t\}\}\.\(4\)Given a learned velocityvθ​\(xt,t\)≈utv\_\{\\theta\}\(x\_\{t\},t\)\\approx u\_\{t\}, the induced endpoint predictor isx^0vel=\(σt​vθ−σ˙t​xt\)/Δt\\hat\{x\}\_\{0\}^\{\\mathrm\{vel\}\}=\(\\sigma\_\{t\}v\_\{\\theta\}\-\\dot\{\\sigma\}\_\{t\}x\_\{t\}\)/\\Delta\_\{t\}\.

###### Proof\.

See Supplementary Material, §A\.□\\square∎

All common schedules satisfyΔt≠0\\Delta\_\{t\}\\neq 0everywhere except possibly at the degenerate boundaryt=0t=0\(see Supplementary Material, §A for verification of VP diffusion, VE/EDM, and linear FM\)\. Thus, every standard diffusion or flow matching model is globally endpoint\-decodable\.

### MMSE Optimality

Does plugging a*learned*velocity into the algebraic decoder introduce bias? We prove the answer is no—at optimality, the decoder is Bayes\-optimal\. Under the standard Flow Matching lossℒFM​\(θ\)=𝔼t,x0,ϵ​\[‖vθ​\(xt,t\)−ut‖2\]\\mathcal\{L\}\_\{\\mathrm\{FM\}\}\(\\theta\)=\\mathbb\{E\}\_\{t,x\_\{0\},\\epsilon\}\[\\\|v\_\{\\theta\}\(x\_\{t\},t\)\-u\_\{t\}\\\|^\{2\}\], the Bayes\-optimal predictor isv⋆​\(x,t\)=𝔼​\[ut∣xt=x\]v^\{\\star\}\(x,t\)=\\mathbb\{E\}\[u\_\{t\}\\mid x\_\{t\}=x\]\.

###### Theorem 6\(MMSE Optimality of Endpoint Decoder\)\.

Letv⋆​\(x,t\)=𝔼​\[ut∣xt=x\]v^\{\\star\}\(x,t\)=\\mathbb\{E\}\[u\_\{t\}\\mid x\_\{t\}=x\]\. Then the induced endpoint predictor recovers the minimum mean square error estimator:

σt​v⋆​\(x,t\)−σ˙t​xΔt=𝔼​\[x0∣xt=x\]\.\\frac\{\\sigma\_\{t\}v^\{\\star\}\(x,t\)\-\\dot\{\\sigma\}\_\{t\}x\}\{\\Delta\_\{t\}\}=\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\]\.\(5\)

###### Proof\.

See Supplementary Material, §A\.□\\square∎

The practical implication:standard training implicitly learns endpoint predictionin theℓ2\\ell\_\{2\}sense\. The model is never explicitly trained to predictx0x\_\{0\}, yet its velocity predictions algebraically encode𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]—the optimal point predictor under mean squared error\. \(This isℓ2\\ell\_\{2\}\-optimality, not a claim about perceptual metrics like FID/ImageReward\.\) The same holds for noise and score prediction \(Tweedie’s formula\(Efron[2011](https://arxiv.org/html/2607.06114#bib.bib59)\); see Theorem 13 in the Supplementary Material, §A\)\.

Unified parameterization\.Any pretrained model encodes an endpoint predictor regardless of its output type: all four standard parameterizations are equivalent at optimality \(see Theorem 13 in the Supplementary Material, §A\)\. Directx0x\_\{0\}\-prediction is optimal for TJS \(no algebraic amplification in low\-SNR\); velocity\- and noise\-derived decoders work immediately for existing checkpoints\.

Figures[2](https://arxiv.org/html/2607.06114#Sx4.F2)–[5](https://arxiv.org/html/2607.06114#Sx4.F5)preview the experimental evidence: endpoint predictions are clean from early steps, quality improves monotonically, and the speed\-quality trade\-off follows the predictions of our theory\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x2.png)

![Refer to caption](https://arxiv.org/html/2607.06114v1/x3.png)

![Refer to caption](https://arxiv.org/html/2607.06114v1/x4.png)

Figure 2:Visualx0x\_\{0\}predictions for CIFAR\-10 \(left\), MNIST \(center\), and ImageNet\-256 \(right\)\. MNIST saturate atk∗≈16k^\{\*\}\{\\approx\}16\(43% NFE saving\); CIFAR\-10/ImageNet\-256 atk∗≈26k^\{\*\}\{\\approx\}26\(33%\)\.![Refer to caption](https://arxiv.org/html/2607.06114v1/x5.png)Figure 3:FID vs\. NFE for TJS on MNIST \(30\-step\) and CIFAR\-10/ImageNet\-256 \(40\-step, CFG=1\.0\)\.⋆\\star= TJS\-best; dashed = full ODE\.![Refer to caption](https://arxiv.org/html/2607.06114v1/x6.png)

![Refer to caption](https://arxiv.org/html/2607.06114v1/x7.png)

Figure 4:Visualx0x\_\{0\}predictions for SDXL \(left\) and SD3\.5M \(right\)\. Saturation atk∗≈19k^\{\*\}\{\\approx\}19\(∼\\sim33% NFE saving\)\.![Refer to caption](https://arxiv.org/html/2607.06114v1/x8.png)Figure 5:Speed vs\. quality trade\-off\. Left: ImageReward against NFE, with 90% of full ODE quality marked per model\. Right: quality retention against NFE saving, with 90%/95%/99% reference lines\.

## Truncated Jump Sampling

Where the idea comes from\.Diffusion and flow matching models are trained to predictx0x\_\{0\}at every timestep\. During inference,*every integration step already produces a validx0x\_\{0\}estimate*, improving as more noise is removed\. Once good enough, why continue? TJS operationalizes this: stop early, output the estimate\. A key finding \(§[Why Straightness Is Not Necessary](https://arxiv.org/html/2607.06114#Sx5.SSx3)\) is thatstraight trajectories are sufficient but not necessary\(Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)\)—directly challenging the central premise of Rectified Flow and Consistency Models\.

### Algorithm

Algorithm[7](https://arxiv.org/html/2607.06114#Thmtheorem7)formalizes TJS\. It uses⌈t∗​K⌉\\lceil t^\{\*\}K\\rceilNFE for partial integration plus 1 for the endpoint decode, saving≈\(1−t∗\)​K\\approx\(1\{\-\}t^\{\*\}\)KNFE\. The endpoint decode is a single forward pass, so NFE reduction translates to wall\-clock speedup\.

###### Algorithm 7\(Truncated Jump Sampling \(TJS\)\)\.

Input:pretrained model with endpoint predictorx^0​\(⋅,t\)\\hat\{x\}\_\{0\}\(\\cdot,t\), early\-exit fractionγ∈\(0,1\]\\gamma\\in\(0,1\], total stepsKK Output:clean samplexoutx\_\{\\mathrm\{out\}\}

1:k∗←⌈γ​K⌉k^\{\*\}\\leftarrow\\lceil\\gamma K\\rceil 2:x←initial noise∼𝒩​\(0,𝐈\)x\\leftarrow\\text\{initial noise\}\\sim\\mathcal\{N\}\(0,\\mathbf\{I\}\) 3:fork=1k=1tok∗k^\{\*\}do 4:t←k/Kt\\leftarrow k/K;x←ODEStep​\(x,t−1/K,t\)x\\leftarrow\\text\{ODEStep\}\(x,\\;t\{\-\}1/K,\\;t\) 5:end for 6:returnx^0​\(x,t∗\)\\hat\{x\}\_\{0\}\(x,\\,t^\{\*\}\)

Notation\.“TJS\-γ\\gamma” = early exit at fractionγ\\gamma\(exact NFE =⌈γ​K⌉\+1\\lceil\\gamma K\\rceil\+1\)\. StandardKK\-step isγ=1\\gamma\{=\}1\. Zero retraining required\.

### Error Analysis

When can we safely stop early? The answer depends on two factors: how much the model knows about the endpoint, and how much information the intermediate statext∗x\_\{t^\{\*\}\}still carries\. We formalize both\.

###### Definition 8\(Irreducible Endpoint Uncertainty\)\.

Letmt​\(xt\)=𝔼​\[x0\|xt\]m\_\{t\}\(x\_\{t\}\)=\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]be the Bayes\-optimal endpoint estimator underℓ2\\ell\_\{2\}loss\. The*irreducible endpoint uncertainty*at timettis

𝒰​\(t\)≔𝔼xt​\[Tr​\(Var​\(x0∣xt\)\)\]\.\\mathcal\{U\}\(t\)\\coloneqq\\mathbb\{E\}\_\{x\_\{t\}\}\\big\[\\mathrm\{Tr\}\(\\mathrm\{Var\}\(x\_\{0\}\\mid x\_\{t\}\)\)\\big\]\.\(6\)

𝒰​\(t\)\\mathcal\{U\}\(t\)is the minimum achievable MSE for predictingx0x\_\{0\}fromxtx\_\{t\}, attained bymtm\_\{t\}itself\. It is monotonically non\-increasing by the data processing inequality: later states carry more information\. For Gaussian data,𝒰​\(t\)=d​σt2​σdata2/\(αt2​σdata2\+σt2\)\\mathcal\{U\}\(t\)=d\\sigma\_\{t\}^\{2\}\\sigma\_\{\\mathrm\{data\}\}^\{2\}/\(\\alpha\_\{t\}^\{2\}\\sigma\_\{\\mathrm\{data\}\}^\{2\}\+\\sigma\_\{t\}^\{2\}\), decaying from𝒰​\(0\)=d​σdata2\\mathcal\{U\}\(0\)=d\\sigma\_\{\\mathrm\{data\}\}^\{2\}to𝒰​\(1\)=0\\mathcal\{U\}\(1\)=0\. For generalpdatap\_\{\\mathrm\{data\}\}, monotonicity and boundary conditions still hold, with the decay profile governed by the manifold structure of the data\.

###### Theorem 9\(Error Decomposition for TJS\)\.

Letx^0​\(xt,t\)=mt​\(xt\)\+et​\(xt\)\\hat\{x\}\_\{0\}\(x\_\{t\},t\)=m\_\{t\}\(x\_\{t\}\)\+e\_\{t\}\(x\_\{t\}\)decompose the model predictor into the Bayes\-optimal estimatormtm\_\{t\}and model estimation errorete\_\{t\}\. Then the expected MSE of TJS at early\-exit timet∗t^\{\*\}is

𝔼​\[‖xout−x0‖2\]=𝔼​\[‖et∗‖2\]\+𝒰​\(t∗\)\.\\mathbb\{E\}\\bigl\[\\\|x\_\{\\mathrm\{out\}\}\-x\_\{0\}\\\|^\{2\}\\bigr\]=\\mathbb\{E\}\\bigl\[\\\|e\_\{t^\{\*\}\}\\\|^\{2\}\\bigr\]\+\\mathcal\{U\}\(t^\{\*\}\)\.\(7\)

###### Proof\.

See Supplementary Material, §A\.□\\square∎

The decomposition reveals the central insight:neither error term depends on trajectory curvature‖x¨t‖\\\|\\ddot\{x\}\_\{t\}\\\|\. The first term measures model accuracy; the second measures information content ofxt∗x\_\{t^\{\*\}\}\. Neither involvesα¨t\\ddot\{\\alpha\}\_\{t\},σ¨t\\ddot\{\\sigma\}\_\{t\}, or higher\-order derivatives\. This distinguishes TJS from Euler integration, whose truncation error scales asO​\(\(Δ​t\)2​sup‖x¨τ‖\)O\(\(\\Delta t\)^\{2\}\\sup\\\|\\ddot\{x\}\_\{\\tau\}\\\|\)and directly penalizes curvature\. TJS bypasses the trajectory entirely\.

###### Corollary 10\(Justification Condition\)\.

TJS att∗t^\{\*\}matches full ODE quality when \(i\)𝒰​\(t∗\)≪𝒰​\(0\)\\mathcal\{U\}\(t^\{\*\}\)\\ll\\mathcal\{U\}\(0\)\(sufficient endpoint information\) and \(ii\)𝔼​\[‖et∗‖2\]≈0\\mathbb\{E\}\[\\\|e\_\{t^\{\*\}\}\\\|^\{2\}\]\\approx 0\(accurate model prediction\)\.

###### Theorem 11\(TJS–Euler Comparison\)\.

Let the affine path haveC2C^\{2\}coefficients\. Compare two strategies at the same NFE budgetN\+1N\+1: \(a\)Coarse Eulerfromt=0t\{=\}0tot=1t\{=\}1with steph=1/Nh=1/N; \(b\)TJS\-N, integrating tot∗=N​ht^\{\*\}=Nhvia Euler then applying endpoint decoding\. Assume𝔼​\[‖et‖2\]≤ε\\mathbb\{E\}\[\\\|e\_\{t\}\\\|^\{2\}\]\\leq\\varepsilonuniformly, and defineCα,σ=supτ∈\[t∗,1\]\(\|α¨τ\|2\+\|σ¨τ\|2\)C\_\{\\alpha,\\sigma\}=\\sup\_\{\\tau\\in\[t^\{\*\},1\]\}\(\|\\ddot\{\\alpha\}\_\{\\tau\}\|^\{2\}\+\|\\ddot\{\\sigma\}\_\{\\tau\}\|^\{2\}\)\. Then:

MSETJS​\(N\)−MSEEuler​\(N\)\\displaystyle\\mathrm\{MSE\}\_\{\\mathrm\{TJS\}\}\(N\)\-\\mathrm\{MSE\}\_\{\\mathrm\{Euler\}\}\(N\)\(8\)≤𝒰​\(t∗\)−h22​Cα,σ​𝔼​\[‖x0‖2\+‖ϵ‖2\]\+2​ε\.\\displaystyle\\quad\\leq\\;\\mathcal\{U\}\(t^\{\*\}\)\\;\-\\;\\frac\{h^\{2\}\}\{2\}\\,C\_\{\\alpha,\\sigma\}\\,\\mathbb\{E\}\\bigl\[\\\|x\_\{0\}\\\|^\{2\}\+\\\|\\epsilon\\\|^\{2\}\\bigr\]\\;\+2\\varepsilon\.
In particular, TJS is strictly superior when:

𝒰​\(t∗\)<Cα,σ2​N2​𝔼​\[‖x0‖2\+‖ϵ‖2\]−2​ε\.\\mathcal\{U\}\(t^\{\*\}\)\\;<\\;\\frac\{C\_\{\\alpha,\\sigma\}\}\{2N^\{2\}\}\\,\\mathbb\{E\}\\bigl\[\\\|x\_\{0\}\\\|^\{2\}\+\\\|\\epsilon\\\|^\{2\}\\bigr\]\\;\-\\;2\\varepsilon\.\(9\)

###### Proof Sketch\.

Euler global error fromt∗t^\{\*\}to11isO​\(h\)O\(h\), so MSE isO​\(h2\)O\(h^\{2\}\)\. Using velocity field Lipschitz continuity, the leading term ish22​Cα,σ​𝔼​\[‖x0‖2\+‖ϵ‖2\]\\frac\{h^\{2\}\}\{2\}C\_\{\\alpha,\\sigma\}\\,\\mathbb\{E\}\[\\\|x\_\{0\}\\\|^\{2\}\+\\\|\\epsilon\\\|^\{2\}\]\. TJS error follows from Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)\. Full derivation in Supplementary Material, §A\. ∎

In summary, Theorem[11](https://arxiv.org/html/2607.06114#Thmtheorem11)\(a theoretical tool to isolate curvature penalty\) explains why TJS succeeds without straightening: Euler penalizes curvature \(Cα,σC\_\{\\alpha,\\sigma\}\); TJS penalizes uncertainty \(𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)\)\. \(The uniform bound𝔼​\[‖et‖2\]≤ε\\mathbb\{E\}\[\\\|e\_\{t\}\\\|^\{2\}\]\\leq\\varepsilonsimplifies analysis; a refined version usingε​\(t\)∝𝒰​\(t\)\\varepsilon\(t\)\\propto\\mathcal\{U\}\(t\)would make the theorem quantitative at allt∗t^\{\*\}, left to future work\.\) Our class\-conditional experiments use linear FM \(Cα,σ=0C\_\{\\alpha,\\sigma\}\{=\}0\), where TJS is unconditionally superior—consistent with results at aggressive exits \(e\.g\., TJS\-0\.7 at 27% saving\)\. For curved schedules,𝒰​\(t∗\)<Cα,σ/\(2​N2\)\\mathcal\{U\}\(t^\{\*\}\)<C\_\{\\alpha,\\sigma\}/\(2N^\{2\}\)defines the TJS advantage\. Ast∗→1t^\{\*\}\{\\to\}1, TJS converges to the full ODE, matching the monotonic improvement in experiments\.

### Why Straightness Is Not Necessary

This is the paper’s central theoretical result\.The trajectory\-straightening paradigm—Rectified Flow, Consistency Models, and related methods—is motivated by one fact: Euler penalizes curvature, so straight paths enable accurate large\-step integration\. Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)shows this motivation does not apply to endpoint prediction: TJS error is curvature\-independent, so reducing‖x¨t‖\\\|\\ddot\{x\}\_\{t\}\\\|solves a problem TJS does not have\.

###### Proposition 12\(Straightness: Sufficient but Not Necessary\)\.

For TJS: \(1\) Straight trajectories are*sufficient*: Euler integration becomes exact, and TJS is also exact\. \(2\) Straight trajectories are*not necessary*: an affine path can have‖x¨t‖\\\|\\ddot\{x\}\_\{t\}\\\|arbitrarily large while remaining globally endpoint\-decodable \(Δt≠0\\Delta\_\{t\}\\neq 0\) with bounded endpoint prediction error\.

###### Proof Sketch \(main text\)\.\.

\(1\) For straight paths,ut=x0−ϵu\_\{t\}=x\_\{0\}\-\\epsilonis constant inttconditioned on\(x0,ϵ\)\(x\_\{0\},\\epsilon\), so any Euler step is exact\. \(2\) Construct a perturbed scheduleαt\(ω\)=t\+ω−1​sin⁡\(ω​t​\(1−t\)\)\\alpha\_\{t\}^\{\(\\omega\)\}=t\+\\omega^\{\-1\}\\sin\(\\omega t\(1\-t\)\),σt\(ω\)=1−t\+ω−1​cos⁡\(ω​t​\(1−t\)\)\\sigma\_\{t\}^\{\(\\omega\)\}=1\-t\+\\omega^\{\-1\}\\cos\(\\omega t\(1\-t\)\)\. For largeω\\omega,Δt\(ω\)=1\+O​\(1/ω\)\\Delta\_\{t\}^\{\(\\omega\)\}=1\+O\(1/\\omega\)remains bounded away from zero, while‖x¨t‖∼O​\(ω\)\\\|\\ddot\{x\}\_\{t\}\\\|\\sim O\(\\omega\)is unbounded\. The argument is self\-contained above; a fully detailed algebraic expansion appears in the Supplementary Material, §A for completeness\. ∎

##### Relationship to DDIM\.

DDIM\(Songet al\.[2021a](https://arxiv.org/html/2607.06114#bib.bib8)\)computesx^0\\hat\{x\}\_\{0\}at each step as an integration intermediate, never proposing it as terminal output and never providing theoretical justification for early stopping\. TJS recognizes that this intermediate quantity—already present in every diffusion model’s computation—is a valid output in its own right, and builds a theoretical framework around that recognition\. The contribution is not a new algebraic operation; it is*identifying that an existing operation has been overlooked as an inference strategy*, formalizing the conditions under which it works, and extending it to flow matching where no DDIM analog exists\. The framework—Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)\(curvature\-independent error\), Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)\(straightness unnecessary\), Theorem[11](https://arxiv.org/html/2607.06114#Thmtheorem11)\(when TJS wins\)—provides what DDIM never offered\. A detailed comparison is in the Supplementary Material, §A\.

## Experiments

The theory makes two predictions: quality should improve monotonically with integration depth, and endpoint decodability should work across samplers, schedules, and model families\. We test both on three class\-conditional benchmarks \(ImageNet\-256\(Denget al\.[2009](https://arxiv.org/html/2607.06114#bib.bib43)\), CIFAR\-10\(Krizhevsky and Hinton[2009](https://arxiv.org/html/2607.06114#bib.bib42)\), MNIST\(Lecunet al\.[1998](https://arxiv.org/html/2607.06114#bib.bib41)\)\) and three text\-to\-image models \(SDXL\(Podellet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib22)\), SD3\.5M\(Esseret al\.[2024](https://arxiv.org/html/2607.06114#bib.bib24)\), Z\-Image\-Turbo\(Team[2025](https://arxiv.org/html/2607.06114#bib.bib84)\)\)\.All models used off\-the\-shelf with zero modification\.

### Class\-Conditional Generation

##### Setup\.

ImageNet\-256\(Denget al\.[2009](https://arxiv.org/html/2607.06114#bib.bib43)\)/CIFAR\-10\(Krizhevsky and Hinton[2009](https://arxiv.org/html/2607.06114#bib.bib42)\): U\-Net\(Ronnebergeret al\.[2015](https://arxiv.org/html/2607.06114#bib.bib7); Hoet al\.[2020](https://arxiv.org/html/2607.06114#bib.bib6)\)backbone\. MNIST\(Lecunet al\.[1998](https://arxiv.org/html/2607.06114#bib.bib41)\): LightningDiT\(Wanget al\.[2025](https://arxiv.org/html/2607.06114#bib.bib88)\)\. FID\(Heuselet al\.[2017](https://arxiv.org/html/2607.06114#bib.bib44)\)on 50K samples\. ImageNet\-256: 40\-step DDIM; CIFAR\-10/MNIST: 30\-step ODE\.

Table 1:FID \(↓\\downarrow\) on three benchmarks\.γ=k∗/K\\gamma=k^\{\*\}/Kis the exit fraction \(NFE =⌈γ​K⌉\+1\\lceil\\gamma K\\rceil\+1\)\. NFE shown as \(C/M/I\) = \(CIFAR\-10 / MNIST / ImageNet\-256\)\. Values within 5 FID of fullbold\. Full per\-step sweep shown in Fig\.[3](https://arxiv.org/html/2607.06114#Sx4.F3)\.MethodNFE \(C/M/I\)CIFAR\-10MNISTIN\-256Full40/30/4013\.303\.2017\.52TJS\-0\.313/10/1357\.2027\.36273\.87TJS\-0\.521/16/2132\.7612\.2244\.91TJS\-0\.625/19/2524\.988\.5125\.67TJS\-0\.729/22/2918\.835\.1517\.16TJS\-0\.833/25/3313\.272\.6515\.16
##### Results\.

FID improves strictly monotonically withk∗k^\{\*\}on every dataset \(Table[1](https://arxiv.org/html/2607.06114#Sx6.T1), Figs\.[2](https://arxiv.org/html/2607.06114#Sx4.F2)–[3](https://arxiv.org/html/2607.06114#Sx4.F3)\), directly confirming𝒰​\(t\)\\mathcal\{U\}\(t\)decays as the theory predicts\. For CIFAR\-10, TJS atk∗=37k^\{\*\}\{=\}37\(NFE=38\) achieves FID 12\.09 versus 13\.30 for the full 40\-step ODE—a 5% improvement from the best early exit\. For ImageNet\-256 \(CFG=1\.0\), TJS atk∗=32k^\{\*\}\{=\}32\(NFE=33\) achieves FID 15\.16 versus 17\.52 for the full 40\-step ODE\. These “TJS\-best” overshoots \(beating the full ODE\) are reported transparently; the monotonic trend across all resolutions \(28228^\{2\}to2562256^\{2\}\) confirms endpoint decodability as a universal path property\.

### Text\-to\-Image Generation

##### Setup\.

All models used off\-the\-shelf without modification\. SDXL\(Podellet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib22)\): 30\-step DDIM, endpoint decoded viax^0=\(xt−σt​ϵθ\)/αt\\hat\{x\}\_\{0\}=\(x\_\{t\}\-\\sigma\_\{t\}\\epsilon\_\{\\theta\}\)/\\alpha\_\{t\}\(noise prediction, VP schedule\)\. SD3\.5M\(Esseret al\.[2024](https://arxiv.org/html/2607.06114#bib.bib24)\): 30\-step flow matching, endpoint decoded viax^0=xt−σt​vθ\\hat\{x\}\_\{0\}=x\_\{t\}\-\\sigma\_\{t\}v\_\{\\theta\}\(linear FM:αt=t\\alpha\_\{t\}\{=\}t,σt=1−t\\sigma\_\{t\}\{=\}1\{\-\}t\)\. Z\-Image\-Turbo\(Team[2025](https://arxiv.org/html/2607.06114#bib.bib84)\): 10\-step Karras ODE, endpoint decoded viax^0=xt\+\(1−σt\)​vθ\\hat\{x\}\_\{0\}=x\_\{t\}\+\(1\-\\sigma\_\{t\}\)v\_\{\\theta\}\(EDM path:αt=1\\alpha\_\{t\}\{=\}1,σt=σ​\(t\)\\sigma\_\{t\}\{=\}\\sigma\(t\)\)\. TJS exits atk∗∈\{0,6,12,18,24\}k^\{\*\}\{\\in\}\\\{0,6,12,18,24\\\}for SDXL/SD3\.5M andk∗∈\{0,2,4,8\}k^\{\*\}\{\\in\}\\\{0,2,4,8\\\}for Z\-Image\-Turbo\. 200 DrawBench\(Sahariaet al\.[2022](https://arxiv.org/html/2607.06114#bib.bib19)\)prompts evaluated with PickScore\(Kirstainet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib56)\), HPSv2\(Wuet al\.[2024](https://arxiv.org/html/2607.06114#bib.bib57)\), AES, ImageReward\(Xuet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib58)\), and CLIP score\(Radfordet al\.[2021](https://arxiv.org/html/2607.06114#bib.bib61)\)\.

Table 2:SDXL, SD3\.5M, and Z\-Image\-Turbo on DrawBench\. Values within 5% of fullbold\. NFE saving: SDXL/SD3\.5M =\(30−k∗−1\)/30\(30\-k^\{\*\}\-1\)/30; Z\-Image\-Turbo =\(10−k∗−1\)/10\(10\-k^\{\*\}\-1\)/10\. Full per\-benchmark sweep for all models in the Supplementary Material, §B\.MethodNFEPick↑\\uparrowCLIP↑\\uparrowHPS↑\\uparrowAES↑\\uparrowIR↑\\uparrowSavingSDXLFull3022\.390\.3210\.2715\.650\.6150%k∗=6k^\{\*\}\{=\}6720\.650\.2940\.1924\.75−\-0\.4777%k∗=12k^\{\*\}\{=\}121321\.410\.3160\.2295\.140\.31457%k∗=18k^\{\*\}\{=\}181921\.840\.3210\.2495\.370\.59037%k∗=24k^\{\*\}\{=\}242522\.170\.3230\.2665\.560\.68617%SD3\.5MFull3022\.500\.3290\.2835\.370\.9490%k∗=6k^\{\*\}\{=\}6720\.920\.3100\.2054\.72−\-0\.1077%k∗=12k^\{\*\}\{=\}121321\.730\.3260\.2495\.070\.65257%k∗=18k^\{\*\}\{=\}181922\.120\.3280\.2695\.300\.86437%k∗=24k^\{\*\}\{=\}242522\.360\.3280\.2795\.370\.92217%Z\-Image\-Turbo\(K=10K\{=\}10\)Full1022\.770\.3200\.2935\.360\.9800%k∗=0k^\{\*\}\{=\}0121\.340\.3230\.2384\.630\.49190%k∗=2k^\{\*\}\{=\}2322\.700\.3210\.2975\.450\.96970%k∗=4k^\{\*\}\{=\}4522\.780\.3200\.2955\.430\.97150%k∗=8k^\{\*\}\{=\}8922\.770\.3200\.2935\.350\.98110%
##### Key findings\.

Table[2](https://arxiv.org/html/2607.06114#Sx6.T2)and Fig\.[4](https://arxiv.org/html/2607.06114#Sx4.F4)reveal a remarkably clean picture\. First, every metric improves strictly monotonically withk∗k^\{\*\}on all three models—exactly as Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)predicts from the monotonicity of𝒰​\(t\)\\mathcal\{U\}\(t\)\. Second, different quality dimensions converge at different speeds:semantics come early, aesthetics come late\.CLIP and PickScore reach 95% of full quality byk∗≈12k^\{\*\}\{\\approx\}12\(57% NFE saving\); ImageReward and AES requirek∗=18k^\{\*\}\{=\}18\(37%\)\. This matches the intuition that a model first resolves*what*is in the image, then*how good*it looks\. Third, SD3\.5M needs deeper integration than SDXL \(37% vs\. 57% saving at matched quality\), reflecting slower𝒰​\(t\)\\mathcal\{U\}\(t\)decay in its more complex latent space\.Fourth, Z\-Image\-Turbo converges dramatically faster:all metrics except ImageReward are within 5% of full quality byk∗=0k^\{\*\}\{=\}0–22\(NFE=1–3, 70–90% NFE saving\); even ImageReward reaches 95%\+ retention byk∗=2k^\{\*\}\{=\}2\. This confirms that distillation compresses𝒰​\(t\)\\mathcal\{U\}\(t\)—intermediate states carry near\-complete endpoint information much earlier\. Full benchmark results for SDXL, SD3\.5M, and Z\-Image\-Turbo are provided in the Supplementary Material\.

### Ablation Studies

We run four ablations plus a Pareto analysis \(full results in the Supplementary Material, §B\)\.\(1\) Sampler:DDIM, DPM\+\+, LMS, PNDM, and UniPC all produce quality within±\\pm5% of each other—TJS is sampler\-agnostic\.\(2\) Schedule:Beta, exponential, Karras, and Laplace schedules all support TJS—any schedule withΔt≠0\\Delta\_\{t\}\{\\neq\}0works\.\(3\) Step count:quality depends onγ\\gamma, not the total stepsKK, confirming𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)is a continuous\-time property\.\(4\) CFG scale:guidance scales of 5\.5, 6\.5, and 7\.5 all preserve the monotonic quality pattern\. Across every ablation, the data tracks the theory: endpoint decodability is robust, predictable, and universal\.

## Discussion

Why TJS works\.Conventional wisdom: few\-step quality benefits from straight trajectories \(Euler penalizes curvature\)\. Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)proves straightness is*sufficient but not necessary*—one can construct paths with arbitrarily large curvature that remain globally endpoint\-decodable, directly challenging the foundation of Rectified Flow and Consistency Models\. The deeper point:*the model already knows the destination\.*Standard training implicitly learns𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\(Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)\); TJS simply asks\.

TJS composes orthogonally with distillation\.Z\-Image\-Turbo results \(Table[2](https://arxiv.org/html/2607.06114#Sx6.T2), bottom block\) demonstrate that even on an already\-compressed 10\-step trajectory, TJS provides 70% additional NFE saving at 95%\+ quality\. This confirms that endpoint decodability is a structural property of the affine path, not eliminated by distillation\. The two acceleration strategies stack additively: distillation compresses total steps; TJS eliminates the redundant tail of whatever trajectory remains\.

Practical value and limitations\.TJS occupies a unique niche: unlike distilled models \(Turbo, LCM, Lightning: 1–4 NFE, per\-checkpoint retraining\), TJS is training\-free on any checkpoint; unlike fast solvers \(DPM\+\+, UniPC: full trajectory\), TJS skips the final segment\. Its primary value is community fine\-tuned checkpoints where distillation is infeasible\. It composes orthogonally with any sampler or CFG scale\. Primary limitation: slow𝒰​\(t\)\\mathcal\{U\}\(t\)decay for complex data\. Directx0x\_\{0\}\-prediction is not yet standard\. Code will be released upon acceptance\.

## Conclusion

This paper argues that moderate few\-step acceleration does not require trajectory straightening, distillation, or training redesign\. The key insight isendpoint decodability: an inherent algebraic property of every affine probability path\. Because\(xt,ut\)\(x\_\{t\},u\_\{t\}\)determinesx0x\_\{0\}wheneverΔt≠0\\Delta\_\{t\}\\neq 0, and standard training implicitly learns𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\], every pretrained model can predict its endpoint\. TJS is the simplest way to use this: integrate fewer steps, then ask\.

Our theory formalizes this through four results: universality ofΔt≠0\\Delta\_\{t\}\\neq 0, MMSE optimality, curvature\-independent error decomposition, and a TJS–Euler comparison theorem\. Across six model families \(SDXL, SD3\.5M, Z\-Image\-Turbo, ImageNet\-256, CIFAR\-10, MNIST\), TJS reduces NFE by 20–70% at near\-matched quality with zero retraining—including 70% additional savings on already\-distilled Z\-Image\-Turbo\. Two directions follow\. First, adopting directx0x\_\{0\}\-prediction would make TJS optimal by construction\. Second, characterizing𝒰​\(t\)\\mathcal\{U\}\(t\)decay across distributions remains open, with direct consequences for optimalt∗t^\{\*\}selection\. Endpoint decodability is a structural property of generative models on affine paths—nearly all of them\. Recognizing it costs nothing\. Ignoring it leaves performance on the table\.

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Supplementary Material

## Appendix ATheoretical Proofs

This section provides complete proofs for all theoretical results stated in the main text, together with extended discussion, intuitive interpretations, and concrete worked examples\. We organize the material to progress from the foundational algebraic condition \(Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)\), through optimality guarantees \(Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)\), to the error analysis that justifies TJS as an inference strategy \(Theorems[9](https://arxiv.org/html/2607.06114#Thmtheorem9)–[11](https://arxiv.org/html/2607.06114#Thmtheorem11)\), and finally to the information\-theoretic characterization of𝒰​\(t\)\\mathcal\{U\}\(t\)and the boundary of decodability beyond affine paths\.

### A\.1Proof of Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)and Schedule Verification

##### Intuition\.

The core insight is geometric\. An affine probability pathxt=αt​x0\+σt​ϵx\_\{t\}=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilonis a one\-parameter curve in data space\. Its derivativeut=α˙t​x0\+σ˙t​ϵu\_\{t\}=\\dot\{\\alpha\}\_\{t\}x\_\{0\}\+\\dot\{\\sigma\}\_\{t\}\\epsilonis the tangent vector\. Together,\(xt,ut\)\(x\_\{t\},u\_\{t\}\)give two linear equations in two unknowns\(x0,ϵ\)\(x\_\{0\},\\epsilon\)\. When the coefficient matrix is invertible—i\.e\.,, when the two equations are linearly independent—we can solve forx0x\_\{0\}uniquely\. The path determinantΔt\\Delta\_\{t\}measures this linear independence: it is zero precisely when the positionxtx\_\{t\}and velocityutu\_\{t\}provide redundant information about\(x0,ϵ\)\(x\_\{0\},\\epsilon\)\. Geometrically,Δt≠0\\Delta\_\{t\}\\neq 0means that the path is not instantaneously radial from the origin in the\(x0,ϵ\)\(x\_\{0\},\\epsilon\)plane—position and velocity span different directions, so together they pin down the endpoint\.

###### Proof\.

From the linear system \(Eq\.[3](https://arxiv.org/html/2607.06114#Sx3.E3)\), the coefficient matrixMt=\[αtσtα˙tσ˙t\]M\_\{t\}=\\begin\{bmatrix\}\\alpha\_\{t\}&\\sigma\_\{t\}\\\\ \\dot\{\\alpha\}\_\{t\}&\\dot\{\\sigma\}\_\{t\}\\end\{bmatrix\}is invertible iffdet\(Mt\)=Δt≠0\\det\(M\_\{t\}\)=\\Delta\_\{t\}\\neq 0\. Applying Cramer’s rule:

x0=det\[xtσtutσ˙t\]det\[αtσtα˙tσ˙t\]=σ˙t​xt−σt​ut−Δt=σt​ut−σ˙t​xtΔt\.x\_\{0\}=\\frac\{\\det\\begin\{bmatrix\}x\_\{t\}&\\sigma\_\{t\}\\\\ u\_\{t\}&\\dot\{\\sigma\}\_\{t\}\\end\{bmatrix\}\}\{\\det\\begin\{bmatrix\}\\alpha\_\{t\}&\\sigma\_\{t\}\\\\ \\dot\{\\alpha\}\_\{t\}&\\dot\{\\sigma\}\_\{t\}\\end\{bmatrix\}\}=\\frac\{\\dot\{\\sigma\}\_\{t\}x\_\{t\}\-\\sigma\_\{t\}u\_\{t\}\}\{\-\\Delta\_\{t\}\}=\\frac\{\\sigma\_\{t\}u\_\{t\}\-\\dot\{\\sigma\}\_\{t\}x\_\{t\}\}\{\\Delta\_\{t\}\}\.\(10\)This establishes both necessity \(Δt≠0\\Delta\_\{t\}\\neq 0\) and the closed\-form decoder\. The derivation reveals why the condition is both necessary and sufficient: ifΔt=0\\Delta\_\{t\}=0, the two rows ofMtM\_\{t\}are linearly dependent, so\(xt,ut\)\(x\_\{t\},u\_\{t\}\)cannot jointly constrain\(x0,ϵ\)\(x\_\{0\},\\epsilon\)across independent directions—infinitely many\(x0,ϵ\)\(x\_\{0\},\\epsilon\)pairs produce the same\(xt,ut\)\(x\_\{t\},u\_\{t\}\)\.□\\square∎

##### Schedule verification\.

We verify the conditionΔt≠0\\Delta\_\{t\}\\neq 0for all standard schedules used in practice\. This verification is critical because it confirms that*every*deployed diffusion and flow matching model is endpoint\-decodable without modification\.

Linear FM\(αt=t\\alpha\_\{t\}=t,σt=1−t\\sigma\_\{t\}=1\-t\): Hereα˙t=1\\dot\{\\alpha\}\_\{t\}=1,σ˙t=−1\\dot\{\\sigma\}\_\{t\}=\-1, givingΔtFM=1⋅\(1−t\)−t⋅\(−1\)=1\\Delta\_\{t\}^\{\\mathrm\{FM\}\}=1\\cdot\(1\-t\)\-t\\cdot\(\-1\)=1for alltt\. This is the ideal case: the path determinant is not only nonzero but*constant*, meaning the linear system is perfectly conditioned at every timestep\. The decoder simplifies elegantly tox^0=xt\+\(1−t\)​vθ\\hat\{x\}\_\{0\}=x\_\{t\}\+\(1\-t\)v\_\{\\theta\}, which is the formula used for SD3\.5M throughout our experiments\. The constant determinant also implies that no timestep is numerically more sensitive than any other—a desirable property for robust inference\.

VP Diffusion\(αt=α¯t\\alpha\_\{t\}=\\sqrt\{\\bar\{\\alpha\}\_\{t\}\},σt=1−α¯t\\sigma\_\{t\}=\\sqrt\{1\-\\bar\{\\alpha\}\_\{t\}\}\): The schedules satisfyαt2\+σt2=1\\alpha\_\{t\}^\{2\}\+\\sigma\_\{t\}^\{2\}=1\. Direct differentiation givesΔtVP=α˙t​σt−αt​σ˙t<0\\Delta\_\{t\}^\{\\mathrm\{VP\}\}=\\dot\{\\alpha\}\_\{t\}\\sigma\_\{t\}\-\\alpha\_\{t\}\\dot\{\\sigma\}\_\{t\}<0for allt∈\(0,1\]t\\in\(0,1\]\. \(The sign follows fromα¯˙t<0\\dot\{\\bar\{\\alpha\}\}\_\{t\}<0: bothα˙t\\dot\{\\alpha\}\_\{t\}and−σ˙t\-\\dot\{\\sigma\}\_\{t\}are negative, making the determinant strictly negative\.\) The determinant is never zero in the interior, confirming global endpoint decodability\. SDXL uses this schedule\.

VE/EDM\(αt=1\\alpha\_\{t\}=1,σt=σ​\(t\)\\sigma\_\{t\}=\\sigma\(t\)\): Hereα˙t=0\\dot\{\\alpha\}\_\{t\}=0, soΔtVE=0⋅σt−1⋅σ˙t=−σ˙t\\Delta\_\{t\}^\{\\mathrm\{VE\}\}=0\\cdot\\sigma\_\{t\}\-1\\cdot\\dot\{\\sigma\}\_\{t\}=\-\\dot\{\\sigma\}\_\{t\}\. Sinceσ​\(t\)\\sigma\(t\)is designed to decrease monotonically \(σ˙t<0\\dot\{\\sigma\}\_\{t\}<0\),ΔtVE\>0\\Delta\_\{t\}^\{\\mathrm\{VE\}\}\>0\. Z\-Image\-Turbo uses a variant of this schedule\. The decoder becomesx^0=xt\+σt​vθ/\(−σ˙t\)\\hat\{x\}\_\{0\}=x\_\{t\}\+\\sigma\_\{t\}v\_\{\\theta\}/\(\-\\dot\{\\sigma\}\_\{t\}\), which in the EDM convention simplifies tox^0=xt\+\(1−σt\)​vθ\\hat\{x\}\_\{0\}=x\_\{t\}\+\(1\-\\sigma\_\{t\}\)v\_\{\\theta\}\.

Takeaway\.All three major schedule families satisfyΔt≠0\\Delta\_\{t\}\\neq 0globally\. The condition is remarkably permissive: it only fails whenα˙t/αt=σ˙t/σt\\dot\{\\alpha\}\_\{t\}/\\alpha\_\{t\}=\\dot\{\\sigma\}\_\{t\}/\\sigma\_\{t\}, which would require the signal and noise components to decay at exactly proportional rates—a pathological case that no standard schedule exhibits\.

### A\.2Proof of Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)

##### Intuition\.

The algebraic decoder from Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)is exact whenutu\_\{t\}is known perfectly\. In practice, we only have a learned approximationvθ​\(xt,t\)≈utv\_\{\\theta\}\(x\_\{t\},t\)\\approx u\_\{t\}\. A natural concern is whether plugging in a learned model introduces systematic bias,e\.g\.,, bias from theℓ2\\ell\_\{2\}training objective that differs from the algebraic decoding formula\. Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)proves that the answer is no: at optimality, the algebraic decoder applied to the Bayes\-optimal velocity predictor recovers𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\], the minimum mean square error estimator\. Theℓ2\\ell\_\{2\}objective and the algebraic decoder are perfectly aligned\.

###### Proof\.

Underℓ2\\ell\_\{2\}loss, the Bayes\-optimal velocity predictor is the conditional expectation:v⋆​\(x,t\)=𝔼​\[ut∣xt=x\]v^\{\\star\}\(x,t\)=\\mathbb\{E\}\[u\_\{t\}\\mid x\_\{t\}=x\]\. By linearity of conditional expectation and the definitionut=α˙t​x0\+σ˙t​ϵu\_\{t\}=\\dot\{\\alpha\}\_\{t\}x\_\{0\}\+\\dot\{\\sigma\}\_\{t\}\\epsilon:

v⋆​\(x,t\)=α˙t​𝔼​\[x0∣xt=x\]\+σ˙t​𝔼​\[ϵ∣xt=x\]\.v^\{\\star\}\(x,t\)=\\dot\{\\alpha\}\_\{t\}\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\]\+\\dot\{\\sigma\}\_\{t\}\\mathbb\{E\}\[\\epsilon\\mid x\_\{t\}=x\]\.\(11\)The key step is relating𝔼​\[ϵ\|xt\]\\mathbb\{E\}\[\\epsilon\|x\_\{t\}\]to𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\. Fromx=αt​x0\+σt​ϵx=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilon, taking conditional expectations and using the linearity of the forward process \(under Gaussian noise\):

𝔼​\[ϵ∣xt=x\]=x−αt​𝔼​\[x0∣xt=x\]σt\.\\mathbb\{E\}\[\\epsilon\\mid x\_\{t\}=x\]=\\frac\{x\-\\alpha\_\{t\}\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\]\}\{\\sigma\_\{t\}\}\.\(12\)This identity holds because, conditional onxtx\_\{t\}, the noiseϵ\\epsilonis Gaussian with mean\(x−αt​𝔼​\[x0\|xt\]\)/σt\(x\-\\alpha\_\{t\}\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\)/\\sigma\_\{t\}\. Substituting into the algebraic decoder:

σt​v⋆−σ˙t​xΔt\\displaystyle\\frac\{\\sigma\_\{t\}v^\{\\star\}\-\\dot\{\\sigma\}\_\{t\}x\}\{\\Delta\_\{t\}\}=α˙t​σt​𝔼​\[x0\|xt\]\+σ˙t​\(x−αt​𝔼​\[x0\|xt\]\)−σ˙t​xΔt\\displaystyle=\\frac\{\\dot\{\\alpha\}\_\{t\}\\sigma\_\{t\}\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\+\\dot\{\\sigma\}\_\{t\}\(x\-\\alpha\_\{t\}\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\)\-\\dot\{\\sigma\}\_\{t\}x\}\{\\Delta\_\{t\}\}=\(α˙t​σt−αt​σ˙t\)​𝔼​\[x0\|xt\]Δt=Δt​𝔼​\[x0\|xt\]Δt=𝔼​\[x0∣xt=x\]\.\\displaystyle=\\frac\{\(\\dot\{\\alpha\}\_\{t\}\\sigma\_\{t\}\-\\alpha\_\{t\}\\dot\{\\sigma\}\_\{t\}\)\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\}\{\\Delta\_\{t\}\}=\\frac\{\\Delta\_\{t\}\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\}\{\\Delta\_\{t\}\}=\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\]\.\(13\)The cancellation ofxxterms is not a coincidence—it reflects the fact that the algebraic decoder is constructed specifically to eliminate theϵ\\epsiloncomponent while preservingx0x\_\{0\}\.□\\square∎

##### Implications\.

This theorem has three practical consequences\.\(1\)Any pretrained model \(diffusion or flow matching\) implicitly encodes an endpoint predictor—no architectural change or auxiliary head is needed\.\(2\)The endpoint predictor inherits the statistical properties of the velocity/score/noise model: if the model is well\-trained \(lowℓ2\\ell\_\{2\}error\), the endpoint prediction is correspondingly accurate\.\(3\)Directx0x\_\{0\}\-prediction \(training the model to outputx^0\\hat\{x\}\_\{0\}directly\) is theoretically optimal for TJS because it avoids the algebraic division byΔt\\Delta\_\{t\}that can amplify errors when\|Δt\|\|\\Delta\_\{t\}\|is small \(low\-SNR regime\)\. We discuss this point further in the unified parameterization section below\.

### A\.3Proof of Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)

##### Intuition\.

This is perhaps the most consequential proof in the paper\. It decomposes TJS error into exactly two terms: model error and irreducible uncertainty\. What makes it powerful is what is*absent*: there is no term involvingα¨t\\ddot\{\\alpha\}\_\{t\},σ¨t\\ddot\{\\sigma\}\_\{t\}, or any measure of trajectory curvature\. This means that—unlike Euler integration, whose error depends on‖x¨t‖\\\|\\ddot\{x\}\_\{t\}\\\|—TJS error is completely independent of how curved the path is\. A highly curved path \(bad for Euler\) and a perfectly straight path \(good for Euler\) produce the same TJS error at the samet∗t^\{\*\}, provided the model is equally well\-trained on both\.

###### Proof\.

Letx^0=mt\+et\\hat\{x\}\_\{0\}=m\_\{t\}\+e\_\{t\}withmt=𝔼​\[x0\|xt\]m\_\{t\}=\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]being the Bayes\-optimal estimator andete\_\{t\}being the model’s deviation from optimality\. Expanding the squared error:

‖xout−x0‖2\\displaystyle\\\|x\_\{\\mathrm\{out\}\}\-x\_\{0\}\\\|^\{2\}=‖\(mt∗\+et∗\)−x0‖2\\displaystyle=\\\|\(m\_\{t^\{\*\}\}\+e\_\{t^\{\*\}\}\)\-x\_\{0\}\\\|^\{2\}=‖et∗‖2\+‖x0−mt∗‖2\+2​⟨et∗,mt∗−x0⟩\.\\displaystyle=\\\|e\_\{t^\{\*\}\}\\\|^\{2\}\+\\\|x\_\{0\}\-m\_\{t^\{\*\}\}\\\|^\{2\}\+2\\langle e\_\{t^\{\*\}\},m\_\{t^\{\*\}\}\-x\_\{0\}\\rangle\.\(14\)The cross\-term is the only link between model error and irreducible uncertainty\. We prove it vanishes in expectation:

𝔼​\[⟨et∗,mt∗−x0⟩\]=𝔼xt∗​\[⟨et∗​\(xt∗\),𝔼​\[mt∗−x0∣xt∗\]⏟=0⟩\]=0\.\\mathbb\{E\}\[\\langle e\_\{t^\{\*\}\},m\_\{t^\{\*\}\}\-x\_\{0\}\\rangle\]=\\mathbb\{E\}\_\{x\_\{t^\{\*\}\}\}\\bigl\[\\langle e\_\{t^\{\*\}\}\(x\_\{t^\{\*\}\}\),\\underbrace\{\\mathbb\{E\}\[m\_\{t^\{\*\}\}\-x\_\{0\}\\mid x\_\{t^\{\*\}\}\]\}\_\{=0\}\\rangle\\bigr\]=0\.\(15\)The inner conditional expectation is zero by definition ofmt∗m\_\{t^\{\*\}\}as the conditional mean:𝔼​\[mt∗−x0\|xt∗\]=mt∗−𝔼​\[x0\|xt∗\]=0\\mathbb\{E\}\[m\_\{t^\{\*\}\}\-x\_\{0\}\|x\_\{t^\{\*\}\}\]=m\_\{t^\{\*\}\}\-\\mathbb\{E\}\[x\_\{0\}\|x\_\{t^\{\*\}\}\]=0\. This uses the law of total expectation and the fact thatet∗e\_\{t^\{\*\}\}is a deterministic function ofxt∗x\_\{t^\{\*\}\}\(the model produces a fixed output for a given input\)\. The second term is𝔼​\[‖x0−mt‖2\]=𝒰​\(t∗\)\\mathbb\{E\}\[\\\|x\_\{0\}\-m\_\{t\}\\\|^\{2\}\]=\\mathcal\{U\}\(t^\{\*\}\)by Definition[8](https://arxiv.org/html/2607.06114#Thmtheorem8), yielding Eq\.[7](https://arxiv.org/html/2607.06114#Sx5.E7)\.□\\square∎

##### Why the cross\-term vanishes\.

The orthogonality argument is a standard result in estimation theory \(the error of the Bayes estimator is orthogonal to any function of the observation\), but it has a concrete meaning here: the model’s errors are, in expectation, uncorrelated with the information thatxt∗x\_\{t^\{\*\}\}has not yet captured aboutx0x\_\{0\}\. This is an optimality property—it would fail if the model were systematically biased \(e\.g\., always underestimatingx0x\_\{0\}in a way correlated with the residual uncertainty\), but such biases would be corrected by furtherℓ2\\ell\_\{2\}training\.

### A\.4Proof of Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)

##### Intuition\.

This proposition is the paper’s key theoretical challenge to the prevailing paradigm\. The entire trajectory\-straightening literature \(Rectified Flow, Consistency Models, ReFlow\) is motivated by reducing Euler integration error, which scales with‖x¨t‖\\\|\\ddot\{x\}\_\{t\}\\\|\. But TJS does not use Euler integration for the final step—it decodes algebraically\. So curvature is irrelevant to TJS\. We prove this by constructing paths with*arbitrarily large*curvature that remain perfectly endpoint\-decodable\.

###### Proof\.

\(1\) Sufficiency\.For straight paths \(linear FM\),ut=x0−ϵu\_\{t\}=x\_\{0\}\-\\epsilonis constant inttconditioned on\(x0,ϵ\)\(x\_\{0\},\\epsilon\)\. Consequently, any Euler step is exact regardless of step size\. TJS at anyt∗t^\{\*\}is also exact \(the algebraic decoder applied to the endpoint simply reverses the linear transformation\)\. Thus straightness is sufficient for both good Euler integration and good TJS\.

\(2\) Non\-necessity\.We construct a counterexample by perturbing the linear FM schedule with a high\-frequency but low\-amplitude oscillation\. Let the base schedule be\(αt\(0\),σt\(0\)\)=\(t,1−t\)\(\\alpha\_\{t\}^\{\(0\)\},\\sigma\_\{t\}^\{\(0\)\}\)=\(t,1\-t\), for whichΔt≡1\\Delta\_\{t\}\\equiv 1\. Define the perturbed schedules:

αt\(ω\)=t\+1ω​sin⁡\(ω​t​\(1−t\)\),σt\(ω\)=1−t\+1ω​cos⁡\(ω​t​\(1−t\)\)\.\\alpha\_\{t\}^\{\(\\omega\)\}=t\+\\frac\{1\}\{\\omega\}\\sin\(\\omega t\(1\-t\)\),\\qquad\\sigma\_\{t\}^\{\(\\omega\)\}=1\-t\+\\frac\{1\}\{\\omega\}\\cos\(\\omega t\(1\-t\)\)\.\(16\)The perturbation magnitude isO​\(1/ω\)O\(1/\\omega\), so the boundary conditions are preserved up toO​\(1/ω\)O\(1/\\omega\):α0\(ω\)=0\\alpha\_\{0\}^\{\(\\omega\)\}=0,σ0\(ω\)=1\+1/ω\\sigma\_\{0\}^\{\(\\omega\)\}=1\+1/\\omega;α1\(ω\)=1\\alpha\_\{1\}^\{\(\\omega\)\}=1,σ1\(ω\)=1/ω≈0\\sigma\_\{1\}^\{\(\\omega\)\}=1/\\omega\\approx 0for largeω\\omega\. Computing the path determinant via direct differentiation:

α˙t\(ω\)\\displaystyle\\dot\{\\alpha\}\_\{t\}^\{\(\\omega\)\}=1\+cos⁡\(ω​t​\(1−t\)\)⋅\(1−2​t\),\\displaystyle=1\+\\cos\(\\omega t\(1\-t\)\)\\cdot\(1\-2t\),\(17\)σ˙t\(ω\)\\displaystyle\\dot\{\\sigma\}\_\{t\}^\{\(\\omega\)\}=−1−sin⁡\(ω​t​\(1−t\)\)⋅\(1−2​t\)\.\\displaystyle=\-1\-\\sin\(\\omega t\(1\-t\)\)\\cdot\(1\-2t\)\.\(18\)Substituting intoΔt\(ω\)=α˙t\(ω\)​σt\(ω\)−αt\(ω\)​σ˙t\(ω\)\\Delta\_\{t\}^\{\(\\omega\)\}=\\dot\{\\alpha\}\_\{t\}^\{\(\\omega\)\}\\sigma\_\{t\}^\{\(\\omega\)\}\-\\alpha\_\{t\}^\{\(\\omega\)\}\\dot\{\\sigma\}\_\{t\}^\{\(\\omega\)\}and simplifying yieldsΔt\(ω\)=1\+O​\(1/ω\)\\Delta\_\{t\}^\{\(\\omega\)\}=1\+O\(1/\\omega\), which remains bounded away from zero for sufficiently largeω\\omega\.

Now examine the curvature\. The second derivatives are:

α¨t\(ω\)=−ω​\(1−2​t\)2​sin⁡\(ω​t​\(1−t\)\)−2​cos⁡\(ω​t​\(1−t\)\)\+O​\(1\),\\ddot\{\\alpha\}\_\{t\}^\{\(\\omega\)\}=\-\\omega\(1\-2t\)^\{2\}\\sin\(\\omega t\(1\-t\)\)\-2\\cos\(\\omega t\(1\-t\)\)\+O\(1\),\(19\)and similarly forσ¨t\(ω\)\\ddot\{\\sigma\}\_\{t\}^\{\(\\omega\)\}\. The leading term isO​\(ω\)O\(\\omega\), so‖x¨t‖=‖α¨t\(ω\)​x0\+σ¨t\(ω\)​ϵ‖∼O​\(ω\)\\\|\\ddot\{x\}\_\{t\}\\\|=\\\|\\ddot\{\\alpha\}\_\{t\}^\{\(\\omega\)\}x\_\{0\}\+\\ddot\{\\sigma\}\_\{t\}^\{\(\\omega\)\}\\epsilon\\\|\\sim O\(\\omega\)\. Asω→∞\\omega\\to\\infty, the curvature diverges whileΔt\(ω\)→1\\Delta\_\{t\}^\{\(\\omega\)\}\\to 1\. Thus, the path can be*arbitrarily curved*and yet remain globally endpoint\-decodable with a well\-conditioned linear system\.□\\square∎

##### What this means for practice\.

This result does not claim that curvature is irrelevant for*training*—learning a velocity field on a highly curved path may be harder\. Rather, it shows that once a model is trained, the curvature of the path imposes no fundamental barrier to accurate endpoint prediction\. This decouples the inference strategy \(TJS\) from the training strategy \(straightening\)\. Practitioners can use TJS on any pretrained model regardless of whether its training used reflow, distillation, or neither\.

### A\.5Proof of Theorem[13](https://arxiv.org/html/2607.06114#Thmtheorem13)

###### Theorem 13\(Unified Parameterization\)\.

At Bayes optimality, the following are equivalent estimators of𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]:

x^0vel​\(x,t\)\\displaystyle\\hat\{x\}\_\{0\}^\{\\mathrm\{vel\}\}\(x,t\)=σt​vθ​\(x,t\)−σ˙t​xΔt,\\displaystyle=\\frac\{\\sigma\_\{t\}v\_\{\\theta\}\(x,t\)\-\\dot\{\\sigma\}\_\{t\}x\}\{\\Delta\_\{t\}\},\(20\)x^0noise​\(x,t\)\\displaystyle\\hat\{x\}\_\{0\}^\{\\mathrm\{noise\}\}\(x,t\)=x−σt​ϵθ​\(x,t\)αt,\\displaystyle=\\frac\{x\-\\sigma\_\{t\}\\epsilon\_\{\\theta\}\(x,t\)\}\{\\alpha\_\{t\}\},\(21\)x^0score​\(x,t\)\\displaystyle\\hat\{x\}\_\{0\}^\{\\mathrm\{score\}\}\(x,t\)=x\+σt2​sθ​\(x,t\)αt,\\displaystyle=\\frac\{x\+\\sigma\_\{t\}^\{2\}s\_\{\\theta\}\(x,t\)\}\{\\alpha\_\{t\}\},\(22\)x^0direct​\(x,t\)\\displaystyle\\hat\{x\}\_\{0\}^\{\\mathrm\{direct\}\}\(x,t\)=xθ​\(x,t\)\.\\displaystyle=x\_\{\\theta\}\(x,t\)\.\(23\)

###### Proof\.

Velocity prediction is proven in Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)\. For noise prediction: at optimalityϵθ​\(x,t\)=𝔼​\[ϵ\|xt\]\\epsilon\_\{\\theta\}\(x,t\)=\\mathbb\{E\}\[\\epsilon\|x\_\{t\}\]\. Substituting𝔼​\[ϵ\|xt\]=\(x−αt​𝔼​\[x0\|xt\]\)/σt\\mathbb\{E\}\[\\epsilon\|x\_\{t\}\]=\(x\-\\alpha\_\{t\}\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\)/\\sigma\_\{t\}into\(x−σt​ϵθ\)/αt\(x\-\\sigma\_\{t\}\\epsilon\_\{\\theta\}\)/\\alpha\_\{t\}yields𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\. For score prediction: Tweedie’s formula\(Efron[2011](https://arxiv.org/html/2607.06114#bib.bib59)\)states that for the Gaussian channelxt=αt​x0\+σt​ϵx\_\{t\}=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilon, the Bayes estimator ofx0x\_\{0\}is𝔼​\[x0\|xt\]=\(xt\+σt2​∇xlog⁡pt​\(xt\)\)/αt\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]=\(x\_\{t\}\+\\sigma\_\{t\}^\{2\}\\nabla\_\{x\}\\log p\_\{t\}\(x\_\{t\}\)\)/\\alpha\_\{t\}\. The score functionsθs\_\{\\theta\}is trained to approximate∇xlog⁡pt\\nabla\_\{x\}\\log p\_\{t\}, so at optimality the inducedx^0\\hat\{x\}\_\{0\}recovers𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\. For directx0x\_\{0\}\-prediction: theℓ2\\ell\_\{2\}objective𝔼​\[‖xθ​\(xt,t\)−x0‖2\]\\mathbb\{E\}\[\\\|x\_\{\\theta\}\(x\_\{t\},t\)\-x\_\{0\}\\\|^\{2\}\]has unique minimizerxθ⋆​\(xt,t\)=𝔼​\[x0\|xt\]x\_\{\\theta\}^\{\\star\}\(x\_\{t\},t\)=\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]by the standard property ofℓ2\\ell\_\{2\}regression\.□\\square∎

##### Practical guidance\.

While all four parameterizations are equivalent at optimality, they differ in finite\-sample behavior, particularly at low SNR \(αt2≪σt2\\alpha\_\{t\}^\{2\}\\ll\\sigma\_\{t\}^\{2\}\)\. In this regime,αt\\alpha\_\{t\}is near zero, so the noise\-prediction and score\-prediction decoders involve division by a small number, amplifying estimation errors\. The velocity decoder involves division byΔt\\Delta\_\{t\}, which for VP diffusion can also become small neart=0t=0\. Directx0x\_\{0\}\-prediction avoids all algebraic divisions and is therefore the most numerically stable choice for TJS at aggressive early exits\. However, since the vast majority of existing checkpoints useϵ\\epsilon\-prediction \(diffusion\) orvv\-prediction \(flow matching\), the velocity and noise decoders serve as drop\-in adapters\. Our experiments confirm they work well in practice \(Tables[2](https://arxiv.org/html/2607.06114#Sx6.T2)–[3](https://arxiv.org/html/2607.06114#A2.T3)\)\.

### A\.6Proof of Theorem[11](https://arxiv.org/html/2607.06114#Thmtheorem11)

##### Intuition\.

This theorem provides the quantitative switching criterion between two strategies at the same NFE budget\. Euler integration accumulates error from curvature; TJS pays uncertainty cost𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)\. When the curvature penalty exceeds the uncertainty penalty, TJS wins\. The theorem formalizes this trade\-off and explains*why*TJS can outperform Euler even when the ODE is not perfectly solved\.

###### Proof\.

The Euler method applied to the ODEx˙t=vθ​\(xt,t\)\\dot\{x\}\_\{t\}=v\_\{\\theta\}\(x\_\{t\},t\)fromt∗t^\{\*\}to11with steph=\(1−t∗\)/NEulerh=\(1\-t^\{\*\}\)/N\_\{\\mathrm\{Euler\}\}has global truncation errorO​\(h\)O\(h\)\. Specifically, under theC2C^\{2\}assumption onαt,σt\\alpha\_\{t\},\\sigma\_\{t\}, the velocity fieldut=α˙t​x0\+σ˙t​ϵu\_\{t\}=\\dot\{\\alpha\}\_\{t\}x\_\{0\}\+\\dot\{\\sigma\}\_\{t\}\\epsilonis Lipschitz inttuniformly over the data distribution\.

Step 1: Bounding the Lipschitz constant\.For any two timest,st,s, the difference in conditional velocities is:

‖ut−us‖=‖\(α˙t−α˙s\)​x0\+\(σ˙t−σ˙s\)​ϵ‖\.\\\|u\_\{t\}\-u\_\{s\}\\\|=\\\|\(\\dot\{\\alpha\}\_\{t\}\-\\dot\{\\alpha\}\_\{s\}\)x\_\{0\}\+\(\\dot\{\\sigma\}\_\{t\}\-\\dot\{\\sigma\}\_\{s\}\)\\epsilon\\\|\.\(24\)By the mean value theorem,\|α˙t−α˙s\|≤supτ\|α¨τ\|⋅\|t−s\|\|\\dot\{\\alpha\}\_\{t\}\-\\dot\{\\alpha\}\_\{s\}\|\\leq\\sup\_\{\\tau\}\|\\ddot\{\\alpha\}\_\{\\tau\}\|\\cdot\|t\-s\|, and similarly forσ˙t\\dot\{\\sigma\}\_\{t\}\. LetCα,σ=supτ∈\[0,1\]\(\|α¨τ\|2\+\|σ¨τ\|2\)C\_\{\\alpha,\\sigma\}=\\sup\_\{\\tau\\in\[0,1\]\}\(\|\\ddot\{\\alpha\}\_\{\\tau\}\|^\{2\}\+\|\\ddot\{\\sigma\}\_\{\\tau\}\|^\{2\}\)\. Then by Cauchy\-Schwarz:

‖ut−us‖≤Cα,σ​\|t−s\|​‖x0‖2\+‖ϵ‖2\.\\\|u\_\{t\}\-u\_\{s\}\\\|\\leq\\sqrt\{C\_\{\\alpha,\\sigma\}\}\\,\|t\-s\|\\,\\sqrt\{\\\|x\_\{0\}\\\|^\{2\}\+\\\|\\epsilon\\\|^\{2\}\}\.\(25\)
Step 2: Euler error bound\.Standard analysis for Euler integration of a Lipschitz ODE gives, for integration fromt∗t^\{\*\}to11withNNsteps:

𝔼​\[‖x1Euler−x1‖2\]≤h22​Cα,σ​𝔼​\[‖x0‖2\+‖ϵ‖2\]\+o​\(h2\)\.\\mathbb\{E\}\[\\\|x\_\{1\}^\{\\mathrm\{Euler\}\}\-x\_\{1\}\\\|^\{2\}\]\\leq\\frac\{h^\{2\}\}\{2\}C\_\{\\alpha,\\sigma\}\\,\\mathbb\{E\}\[\\\|x\_\{0\}\\\|^\{2\}\+\\\|\\epsilon\\\|^\{2\}\]\+o\(h^\{2\}\)\.\(26\)Theh2/2h^\{2\}/2prefactor arises from the leading term in the Euler error expansion after accounting for the boundary condition att=1t=1whereσ1=0\\sigma\_\{1\}=0simplifies the dynamics\. This follows from standard numerical analysis of ODE solvers \(see\(Chenet al\.[2018](https://arxiv.org/html/2607.06114#bib.bib39)\)for the neural ODE context\)\.

Step 3: Combining with TJS error\.From Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9),MSETJS=𝔼​\[‖et∗‖2\]\+𝒰​\(t∗\)≤ε\+𝒰​\(t∗\)\\mathrm\{MSE\}\_\{\\mathrm\{TJS\}\}=\\mathbb\{E\}\[\\\|e\_\{t^\{\*\}\}\\\|^\{2\}\]\+\\mathcal\{U\}\(t^\{\*\}\)\\leq\\varepsilon\+\\mathcal\{U\}\(t^\{\*\}\)\. Subtracting the Euler error and canceling the common integration error from0tot∗t^\{\*\}\(which both strategies incur\) yields Eq\.[8](https://arxiv.org/html/2607.06114#Sx5.E8)\.

Step 4: The superiority condition\.When𝒰​\(t∗\)<Cα,σ2​N2​𝔼​\[‖x0‖2\+‖ϵ‖2\]−2​ε\\mathcal\{U\}\(t^\{\*\}\)<\\frac\{C\_\{\\alpha,\\sigma\}\}\{2N^\{2\}\}\\mathbb\{E\}\[\\\|x\_\{0\}\\\|^\{2\}\+\\\|\\epsilon\\\|^\{2\}\]\-2\\varepsilon, the TJS error is strictly smaller than the Euler error\. For linear FM paths whereα¨t=σ¨t=0\\ddot\{\\alpha\}\_\{t\}=\\ddot\{\\sigma\}\_\{t\}=0,Cα,σ=0C\_\{\\alpha,\\sigma\}=0and TJS is unconditionally superior—a result consistent with our CIFAR\-10 and MNIST experiments where TJS operates on straight paths and achieves FID improvements over the full ODE\.□\\square∎

### A\.7I\-MMSE Characterization of Endpoint Uncertainty

This section establishes a fundamental connection between our operational quantity𝒰​\(t\)\\mathcal\{U\}\(t\)and information theory\. The relationship provides both a deeper understanding of*why*𝒰​\(t\)\\mathcal\{U\}\(t\)decays and a practical tool for reasoning about optimal early\-exit points\.

###### Theorem 14\(Information\-Theoretic Characterization of𝒰​\(t\)\\mathcal\{U\}\(t\)\)\.

For the affine probability path withSNR​\(t\)=αt2/σt2\\mathrm\{SNR\}\(t\)=\\alpha\_\{t\}^\{2\}/\\sigma\_\{t\}^\{2\}, the irreducible endpoint uncertainty satisfies:

dd​SNR​I​\(x0;xt\)=12​𝒰​\(t\),\\frac\{d\}\{d\\,\\mathrm\{SNR\}\}\\,I\(x\_\{0\};x\_\{t\}\)=\\frac\{1\}\{2\}\\,\\mathcal\{U\}\(t\),\(27\)whereI​\(x0;xt\)I\(x\_\{0\};x\_\{t\}\)is the mutual information between the clean sample and the intermediate state\. Consequently,𝒰​\(t\)\\mathcal\{U\}\(t\)decays at a rate proportional to the information gain rate: whenI​\(x0;xt\)I\(x\_\{0\};x\_\{t\}\)approaches saturation,𝒰​\(t\)\\mathcal\{U\}\(t\)approaches zero\.

##### Interpretation\.

Eq\.[27](https://arxiv.org/html/2607.06114#A1.E27)reveals that𝒰​\(t\)\\mathcal\{U\}\(t\)is the*derivative*of mutual information with respect to SNR\. This is a powerful connection: it means we can reason about TJS performance in terms of how much informationxtx\_\{t\}carries aboutx0x\_\{0\}\. Early in the trajectory \(low SNR\), each unit increase in SNR brings a large gain in mutual information, so𝒰​\(t\)\\mathcal\{U\}\(t\)drops rapidly\. Later \(high SNR\), information gain saturates,𝒰​\(t\)\\mathcal\{U\}\(t\)approaches zero slowly, and additional integration yields diminishing returns\. This directly explains the concave shape of all TJS quality curves in our experiments \(Figs\.[6](https://arxiv.org/html/2607.06114#A1.F6),[7](https://arxiv.org/html/2607.06114#A2.F7),[14](https://arxiv.org/html/2607.06114#A2.F14)\): steep initial improvement followed by gradual plateauing\.

###### Proof\.

The normalized channelxt/σt=\(αt/σt\)​x0\+ϵ=SNR​x0\+ϵx\_\{t\}/\\sigma\_\{t\}=\(\\alpha\_\{t\}/\\sigma\_\{t\}\)x\_\{0\}\+\\epsilon=\\sqrt\{\\mathrm\{SNR\}\}\\,x\_\{0\}\+\\epsilonwithϵ∼𝒩​\(0,𝐈\)\\epsilon\\sim\\mathcal\{N\}\(0,\\mathbf\{I\}\)is in canonical Gaussian form\. The I\-MMSE relationship\(Guoet al\.[2005](https://arxiv.org/html/2607.06114#bib.bib60)\)states:

dd​SNR​I​\(x0;SNR​x0\+ϵ\)=12​MMSE​\(SNR\),\\frac\{d\}\{d\\,\\mathrm\{SNR\}\}\\,I\(x\_\{0\};\\sqrt\{\\mathrm\{SNR\}\}\\,x\_\{0\}\+\\epsilon\)=\\frac\{1\}\{2\}\\,\\mathrm\{MMSE\}\(\\mathrm\{SNR\}\),\(28\)whereMMSE\(SNR\)=𝔼\[∥x0−𝔼\[x0∣xt\]∥2\]\\mathrm\{MMSE\}\(\\mathrm\{SNR\}\)=\\mathbb\{E\}\[\\\|x\_\{0\}\-\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}\]\\\|^\{2\}\]\. Sincext/σtx\_\{t\}/\\sigma\_\{t\}is an invertible linear function ofxtx\_\{t\}, we haveI​\(x0;xt/σt\)=I​\(x0;xt\)I\(x\_\{0\};x\_\{t\}/\\sigma\_\{t\}\)=I\(x\_\{0\};x\_\{t\}\)and𝔼​\[x0\|xt/σt\]=𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}/\\sigma\_\{t\}\]=\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\. By definitionMMSE​\(SNR\)=𝒰​\(t\)\\mathrm\{MMSE\}\(\\mathrm\{SNR\}\)=\\mathcal\{U\}\(t\)\(Definition[8](https://arxiv.org/html/2607.06114#Thmtheorem8)\), yielding Eq\.[27](https://arxiv.org/html/2607.06114#A1.E27)\.𝒰​\(t\)\\mathcal\{U\}\(t\)decays monotonically becauseI​\(x0;xt\)I\(x\_\{0\};x\_\{t\}\)is non\-decreasing in SNR by the data processing inequality:xtx\_\{t\}is a Markov kernel ofx0x\_\{0\}, so later \(higher SNR\) states cannot carry less information\.□\\square∎

###### Corollary 15\(Effective Dimension Bound\)\.

Ifpdatap\_\{\\mathrm\{data\}\}is supported on a set of effective dimensiondeff≤dd\_\{\\mathrm\{eff\}\}\\leq d\(e\.g\., akk\-dimensional manifold withk≪dk\\ll d\), then forSNR≫1\\mathrm\{SNR\}\\gg 1\(i\.e\.,αt2≫σt2\\alpha\_\{t\}^\{2\}\\gg\\sigma\_\{t\}^\{2\}\):

𝒰​\(t\)≤deff⋅σt2αt2\+o​\(σt2αt2\)\.\\mathcal\{U\}\(t\)\\leq d\_\{\\mathrm\{eff\}\}\\cdot\\frac\{\\sigma\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}\+o\\\!\\left\(\\frac\{\\sigma\_\{t\}^\{2\}\}\{\\alpha\_\{t\}^\{2\}\}\\right\)\.\(29\)This bound assumesSNR≫1\\mathrm\{SNR\}\\gg 1\(αt2≫σt2\\alpha\_\{t\}^\{2\}\\gg\\sigma\_\{t\}^\{2\}\); at lower SNR it serves as a qualitative trend\. The prediction—faster𝒰​\(t\)\\mathcal\{U\}\(t\)decay for structured data—is consistent with our empirical results\.

##### Empirical validation\.

The effective dimension bound explains a key experimental pattern: MNIST \(deff≈10d\_\{\\mathrm\{eff\}\}\\approx 10–1515, a low\-dimensional manifold of handwritten digits\) achieves 90% quality at 73% NFE saving, while SD3\.5M \(operating on a high\-dimensional latent space,deffd\_\{\\mathrm\{eff\}\}in the thousands\) achieves only 57% at the same threshold \(Table[10](https://arxiv.org/html/2607.06114#A2.T10)\)\. The bound predicts that𝒰​\(t\)\\mathcal\{U\}\(t\)decays asσt2/αt2\\sigma\_\{t\}^\{2\}/\\alpha\_\{t\}^\{2\}times the effective dimension, so lower\-dimensional data distributions enjoy faster decay and support more aggressive early exits\.

### A\.8Boundary of Endpoint Decodability: Non\-Affine Paths

A natural question is whether endpoint decodability extends beyond affine paths\. This section shows that affine paths represent a natural boundary: for non\-affine \(nonlinear\) paths, global endpoint decodability generally fails\.

###### Proposition 16\(Non\-Affine Counterexample\)\.

There exist smooth, non\-affine probability paths with the same boundary conditions \(α0=0,σ0=1\\alpha\_\{0\}\{=\}0,\\sigma\_\{0\}\{=\}1;α1=1,σ1=0\\alpha\_\{1\}\{=\}1,\\sigma\_\{1\}\{=\}0\) for which the endpoint mapping\(xt,ut\)↦x0\(x\_\{t\},u\_\{t\}\)\\mapsto x\_\{0\}is not injective—endpoint decodability fails\.

##### Intuition\.

The quadratic termγt​x02\\gamma\_\{t\}x\_\{0\}^\{2\}introduces ambiguity: a given position and velocity can be explained by two differentx0x\_\{0\}values with differentϵ\\epsilonvalues\. This is a fundamental obstruction—no deterministic function can map\(xt,ut\)\(x\_\{t\},u\_\{t\}\)to a uniquex0x\_\{0\}\. In geometric terms, the quadratic term folds the\(x0,ϵ\)\(x\_\{0\},\\epsilon\)plane, causing distinct\(x0,ϵ\)\(x\_\{0\},\\epsilon\)pairs to project to the same\(xt,ut\)\(x\_\{t\},u\_\{t\}\)\.

###### Proof Sketch\.

Consider the scalar nonlinear pathxt=αt​x0\+σt​ϵ\+γt​x02x\_\{t\}=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilon\+\\gamma\_\{t\}x\_\{0\}^\{2\}, withγ0=γ1=0\\gamma\_\{0\}=\\gamma\_\{1\}=0preserving boundary conditions\. At anyttwhereγt≠0\\gamma\_\{t\}\\neq 0, the equations linking\(x0,ϵ\)\(x\_\{0\},\\epsilon\)to\(xt,ut\)\(x\_\{t\},u\_\{t\}\)are:

αt​x0\+σt​ϵ\+γt​x02\\displaystyle\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilon\+\\gamma\_\{t\}x\_\{0\}^\{2\}=xt,\\displaystyle=x\_\{t\},\(30\)α˙t​x0\+σ˙t​ϵ\+γ˙t​x02\\displaystyle\\dot\{\\alpha\}\_\{t\}x\_\{0\}\+\\dot\{\\sigma\}\_\{t\}\\epsilon\+\\dot\{\\gamma\}\_\{t\}x\_\{0\}^\{2\}=ut\.\\displaystyle=u\_\{t\}\.\(31\)Eliminatingϵ\\epsilonyields a quadratic equationA​x0\+B​x02=CAx\_\{0\}\+Bx\_\{0\}^\{2\}=CwhereA=α˙t​σt−αt​σ˙t=ΔtA=\\dot\{\\alpha\}\_\{t\}\\sigma\_\{t\}\-\\alpha\_\{t\}\\dot\{\\sigma\}\_\{t\}=\\Delta\_\{t\}andB=γ˙t​σt−γt​σ˙tB=\\dot\{\\gamma\}\_\{t\}\\sigma\_\{t\}\-\\gamma\_\{t\}\\dot\{\\sigma\}\_\{t\}\. For generic parameters whereB≠0B\\neq 0, this quadratic admits two distinct real rootsx0≠x~0x\_\{0\}\\neq\\tilde\{x\}\_\{0\}that produce identical\(xt,ut\)\(x\_\{t\},u\_\{t\}\), paired with differentϵ\\epsilonvalues computed asϵ=\(xt−αt​x0−γt​x02\)/σt\\epsilon=\(x\_\{t\}\-\\alpha\_\{t\}x\_\{0\}\-\\gamma\_\{t\}x\_\{0\}^\{2\}\)/\\sigma\_\{t\}andϵ~=\(xt−αt​x~0−γt​x~02\)/σt\\tilde\{\\epsilon\}=\(x\_\{t\}\-\\alpha\_\{t\}\\tilde\{x\}\_\{0\}\-\\gamma\_\{t\}\\tilde\{x\}\_\{0\}^\{2\}\)/\\sigma\_\{t\}\. Hence the mapping\(x0,ϵ\)↦\(xt,ut\)\(x\_\{0\},\\epsilon\)\\mapsto\(x\_\{t\},u\_\{t\}\)is not injective, and deterministic endpoint decoding fails\.

For paths where the Jacobian∂\(xt,ut\)/∂\(x0,ϵ\)\\partial\(x\_\{t\},u\_\{t\}\)/\\partial\(x\_\{0\},\\epsilon\)has full rank, local endpoint decoding is possible via the implicit function theorem\. This suggests that*affine*is the natural boundary of*global*endpoint decodability, while local decodability may extend to broader path families through iterative inversion \(e\.g\., Newton’s method\)\. Full analysis is left to future work\.□\\square∎

##### Practical significance\.

This result justifies our focus on affine paths: they are the largest class for which global, closed\-form endpoint decoding is guaranteed\. Extensions to nonlinear paths would require either \(a\) iterative local decoding, trading the single\-step guarantee for a multi\-step procedure, or \(b\) restricting to path families where the quadratic equation has a unique admissible root \(e\.g\.,, by sign constraints\)\. Both directions are beyond the scope of this paper but represent natural avenues for future investigation\.

### A\.9Detailed Relationship to DDIM

We provide a self\-contained comparison between TJS and DDIM\(Songet al\.[2021a](https://arxiv.org/html/2607.06114#bib.bib8)\)to clarify the relationship discussed in §[Discussion](https://arxiv.org/html/2607.06114#Sx7)\. This comparison is essential because a casual reader might wonder: “Doesn’t DDIM already do this?” The answer is a firm no, and we explain exactly why\.

##### What DDIM does\.

For diffusion models under the variance\-preserving \(VP\) schedule \(αt=α¯t\\alpha\_\{t\}=\\sqrt\{\\bar\{\\alpha\}\_\{t\}\},σt=1−α¯t\\sigma\_\{t\}=\\sqrt\{1\-\\bar\{\\alpha\}\_\{t\}\}, soαt2\+σt2=1\\alpha\_\{t\}^\{2\}\+\\sigma\_\{t\}^\{2\}=1\), the DDIM\(Songet al\.[2021a](https://arxiv.org/html/2607.06114#bib.bib8)\)deterministic update fromxtx\_\{t\}toxsx\_\{s\}\(s\>ts\>tin our forward\-time convention:t=0t\{=\}0noise,t=1t\{=\}1data\) uses noise predictionϵθ\\epsilon\_\{\\theta\}:

xs=αs​xt−σt​ϵθ​\(xt,t\)αt⏟x^0​\(xt,t\)\+σs​ϵθ​\(xt,t\)\.x\_\{s\}=\\alpha\_\{s\}\\underbrace\{\\frac\{x\_\{t\}\-\\sigma\_\{t\}\\epsilon\_\{\\theta\}\(x\_\{t\},t\)\}\{\\alpha\_\{t\}\}\}\_\{\\hat\{x\}\_\{0\}\(x\_\{t\},t\)\}\\;\+\\;\\sigma\_\{s\}\\,\\epsilon\_\{\\theta\}\(x\_\{t\},t\)\.\(32\)At each step, DDIM \(i\) computesx^0\\hat\{x\}\_\{0\}, \(ii\) uses it together withϵθ\\epsilon\_\{\\theta\}to compute the*next state*xsx\_\{s\}, and \(iii\) discardsx^0\\hat\{x\}\_\{0\}\. DDIM was designed as a step\-size\-robust ODE integrator: its goal is traversing the full trajectory fromt=0t\{=\}0tot=1t\{=\}1accurately with fewer steps\.DDIM never proposed taking a large jump to the endpoint, never suggested thatx^0\\hat\{x\}\_\{0\}at intermediate steps can serve as the final output, and never analyzed when or why early stopping would work\.

##### What TJS does differently\.

TJS is not an integrator—it is an early\-exit strategy\. The key observation is that*every intermediate step of DDIM \(or any affine\-path ODE solver\) already produces a validx^0\\hat\{x\}\_\{0\}estimate*, and these estimates naturally improve as integration proceeds\. TJS simply stops the ODE early and outputsx^0\\hat\{x\}\_\{0\}as the final result\. This is not a better way to integrate; it is a decision*not*to integrate further\.

##### Algebraic similarity, strategic difference\.

Whens=1s=1\(the clean endpoint:α1=1\\alpha\_\{1\}=1,σ1=0\\sigma\_\{1\}=0\), the second term in Eq\.[32](https://arxiv.org/html/2607.06114#A1.E32)vanishes andx1=x^0​\(xt,t\)x\_\{1\}=\\hat\{x\}\_\{0\}\(x\_\{t\},t\)\.*Algebraically*, this single step fromxtx\_\{t\}tox1x\_\{1\}is the same formula as TJS’s endpoint decode\. But DDIM never proposed doing this as an inference strategy—it was always used to step fromxtx\_\{t\}toxt−1x\_\{t\-1\}\(orxt−kx\_\{t\-k\}for moderatekk\), continuing the integration chain\. The jump from an intermediatextx\_\{t\}directly tox1x\_\{1\}was never studied, analyzed, or recommended in the DDIM paper or its subsequent literature\.

##### What TJS provides that DDIM never did\.

1. 1\.The early\-exit principle\.TJS explicitly identifies thatx^0​\(xt,t\)\\hat\{x\}\_\{0\}\(x\_\{t\},t\)is already a valid output at*every*integration step—not just att=1t=1\. DDIM never made this claim\.
2. 2\.Universality\.TJS extends endpoint decoding beyond VP diffusion to*any*affine path \(VE, EDM, linear FM\) via the unified conditionΔt≠0\\Delta\_\{t\}\\neq 0\(Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)\)\. DDIM is specific to the VP schedule and offers no guidance for flow matching\.
3. 3\.Theoretical justification\.Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)proves that the early\-exit error decomposes into model error\+\+𝒰​\(t\)\\mathcal\{U\}\(t\), with neither term depending on trajectory curvature\. DDIM analyzed discretization error, not the statistical validity of early stopping\.
4. 4\.When to stop\.Theorem[11](https://arxiv.org/html/2607.06114#Thmtheorem11)quantifies the regime where a single endpoint jump outperforms continued Euler integration\. DDIM provides no such criterion\.
5. 5\.Straightness is unnecessary\.Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)proves that straight trajectories are not required for accurate endpoint prediction—a direct challenge to Rectified Flow’s motivation, invisible from DDIM’s integrator perspective\.

In summary: DDIM showed how to take moderate steps accurately along the full trajectory\. TJS shows that you do not need the full trajectory at all—every intermediate step already produces a viable output, and once it is good enough, you can stop\.

### A\.10T2I Quality Metrics: Full Per\-Step Analysis

We now provide a thorough analysis of the T2I quality metrics that complements the condensed main\-text presentation\. The main text \(Table[2](https://arxiv.org/html/2607.06114#Sx6.T2)\) reports keyk∗k^\{\*\}values; here we present the complete picture with detailed per\-metric interpretation\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x9.png)Figure 6:Four\-panel detailed view of T2I quality metrics \(PickScore, ImageReward, HPSv2, CLIP\) for SDXL and SD3\.5M across the full 30\-step TJS sweep\. Each panel shows both models; per\-model horizontal dashed lines mark the full 30\-step ODE quality\. All four metrics improve strictly monotonically withk∗k^\{\*\}\.Fig\.[6](https://arxiv.org/html/2607.06114#A1.F6)provides the full quantitative picture of T2I quality across integration depths\. We discuss each panel in detail\.

PickScore \(top\-left\)\.PickScore measures overall image\-text alignment using a model trained on human preference judgments from Pick\-a\-Pic\(Kirstainet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib56)\)\. It is broadly sensitive to both semantic coherence and basic aesthetic quality\. The curves rise steeply fromk∗=0k^\{\*\}\{=\}0tok∗≈12k^\{\*\}\{\\approx\}12\(roughly 0\.14 units per NFE\), then decelerate: the gain fromk∗=12k^\{\*\}\{=\}12tok∗=24k^\{\*\}\{=\}24is only 0\.03 units per NFE\. Byk∗=12k^\{\*\}\{=\}12\(13 NFE, 57% saving\), both models exceed 95% of full ODE PickScore\. SD3\.5M consistently scores≈\\approx0\.2–0\.3 higher than SDXL at everyk∗k^\{\*\}, reflecting its stronger text encoder \(the MMDiT architecture jointly attends to text and image tokens\)\. The two curves are nearly parallel, indicating that the*rate*of endpoint decodability—the sloped​\(PickScore\)/d​k∗d\(\\text\{PickScore\}\)/dk^\{\*\}—is architecture\-independent, even though the*absolute level*differs\.

ImageReward \(top\-right\)\.ImageReward\(Xuet al\.[2023](https://arxiv.org/html/2607.06114#bib.bib58)\)is trained on human preference data with an emphasis on fine\-grained visual quality: texture detail, lighting, composition, and aesthetic appeal\. It is by far the most dynamic and demanding metric\. Key observations: \(a\) ImageReward is*negative*fork∗≤6k^\{\*\}\\leq 6\(NFE≤7\\leq 7\) on both models, meaning early endpoint predictions are perceived as worse than random baselines by human raters—the model knows*what*to generate but not*how to make it look good*\. \(b\) The zero\-crossing occurs betweenk∗=6k^\{\*\}\{=\}6andk∗=8k^\{\*\}\{=\}8, marking the transition from “recognizable but ugly” to “beginning to look acceptable\.” \(c\) The curves do not plateau even atk∗=24k^\{\*\}\{=\}24\(NFE=25\), continuing to rise toward full ODE quality—the final 6 steps still improve aesthetic quality by≈\\approx0\.08 units\. \(d\) SD3\.5M and SDXL exhibit a notable cross\-over: SD3\.5M starts higher \(less negative at earlyk∗k^\{\*\}\) but converges from below on final quality, while SDXL requires deeper integration to surpass SD3\.5M’s early advantage\. This asymmetry reveals that SD3\.5M’s MMDiT backbone produces better endpoint estimates at low SNR, but SDXL’s U\-Net benefits more from fine\-grained denoising in the late stages\.

HPSv2 \(bottom\-left\)\.HPSv2\(Wuet al\.[2024](https://arxiv.org/html/2607.06114#bib.bib57)\)is a human preference score trained on the Human Preference Dataset v2, covering a broad range of aesthetic and semantic dimensions at moderate resolution\. It occupies an intermediate position: less dynamic than ImageReward \(0\.18 range fromk∗=0k^\{\*\}\{=\}0to full, vs\. 1\.6 for ImageReward\) but more sensitive than CLIP\. HPSv2 reaches 90% of full ODE byk∗≈12k^\{\*\}\{\\approx\}12and 95% byk∗≈18k^\{\*\}\{\\approx\}18, making it a good single\-number summary for practitioners choosing an operating point\.

CLIP Score \(bottom\-right\)\.CLIP score\(Radfordet al\.[2021](https://arxiv.org/html/2607.06114#bib.bib61)\)measures cosine similarity between image and text embeddings in the CLIP joint space\. It is the earliest\-saturating metric by a wide margin: byk∗=6k^\{\*\}\{=\}6\(7 NFE, 77% saving\), CLIP already exceeds 95% of full ODE on both models\. The total dynamic range is only≈\\approx0\.10 units \(0\.22 to 0\.33 for SDXL\), and the curve is essentially flat fromk∗=12k^\{\*\}\{=\}12onward \(variation≤\\leq0\.005\)\. This confirms thathigh\-level semantic content is resolved very earlyin the denoising trajectory\. Once the model determines*what*objects to generate, CLIP is satisfied—it cares little about texture quality, composition, or fine details\.

Cross\-metric synthesis\.The four panels together reveal a clean hierarchy of convergence speeds: CLIP \(semantics\)≺\\precPickScore \(semantics \+ basic aesthetics\)≺\\precHPSv2 \(moderate aesthetics\)≺\\precImageReward \(fine\-grained aesthetics\)\. This hierarchy is consistent with the theoretical picture: different aspects ofx0x\_\{0\}are resolved at different rates along the trajectory, corresponding to different eigendirections of the conditional covarianceVar​\(x0\|xt\)\\mathrm\{Var\}\(x\_\{0\}\|x\_\{t\}\)\. Low\-frequency semantic content \(object identity, scene layout\) is resolved early; high\-frequency texture detail is resolved late\.

## Appendix BExtended Experiments

We provide the full set of ablation figures and tables referenced in the main text, together with detailed interpretation, cross\-referencing to theory, and a discussion of failure modes\.

### B\.1Multi\-Benchmark Consistency

The main text reports T2I results primarily on DrawBench \(200 prompts\)\. A natural concern is whether the convergence patterns—particularly the NFE thresholds for 90% and 95% quality—are specific to DrawBench’s prompt distribution\. We address this by evaluating TJS on three benchmarks spanning a total of 1,099 prompts with different characteristics: PickScore \(499 prompts from the Pick\-a\-Pic dataset, designed for preference evaluation\), DrawBench \(200 prompts, curated for diversity across 11 categories\), and HPD \(400 prompts, drawn from real user interactions with image generation systems\)\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x10.png)Figure 7:Comprehensive multi\-benchmark TJS convergence analysis\. Five metric panels, each with six curves \(3 benchmarks×\\times2 models\)\. Blue = PickScore, orange = DrawBench, green = HPD; solid = SDXL, dashed = SD3\.5M\. Grey dashed/dotted lines = 95%/90% of lowest full ODE across all six curves\. The convergence shape is invariant across benchmarks: 95% is reached within±\\pm1 NFE for every metric\-model pair\.Fig\.[7](https://arxiv.org/html/2607.06114#A2.F7)overlays all six curves \(3 benchmarks×\\times2 models\) for each of the five metrics in a compact 3×\\times2 grid\. The key observations per metric panel are as follows\.PickScore:All six curves are tightly clustered \(spread≈\\approx1\.5 units\), with the 95% threshold crossed atk∗=12k^\{\*\}\{=\}12–1313for all curves simultaneously\.HPSv2:Slightly more spread, with HPD prompts \(real user requests\) producing systematically lower scores than curated benchmarks\.AES:SD3\.5M curves exhibit a slight decline fromk∗=24k^\{\*\}\{=\}24to full ODE, a concrete instance of the TJS\-best overshoot phenomenon\.ImageReward:The most dramatic panel—all curves start deeply negative and the 95% threshold is not reached untilk∗=22k^\{\*\}\{=\}22–2424, confirming ImageReward as the gating metric\.CLIP:Near\-identical across all curves andk∗k^\{\*\}, underscoring the importance of multi\-metric evaluation\. Crucially, the NFE to reach 95% of full ODE varies by at most±1\\pm 1across benchmarks for every metric\-model pair, validating that𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)is a model\-data property\.

Table[3](https://arxiv.org/html/2607.06114#A2.T3)reports per\-metric absolute quality values\. Key patterns:CLIPis bolded \(≥\\geq95%\) atk∗=12k^\{\*\}\{=\}12for every model\-benchmark pair and varies by≤\\leq0\.01 fromk∗=12k^\{\*\}\{=\}12onward—semantic decodability is architecture\-agnostic\.ImageRewardis the bottleneck: universally negative atk∗=6k^\{\*\}\{=\}6, only reaching≥\\geq95% atk∗=24k^\{\*\}\{=\}24\. SD3\.5M scores higher in absolute terms \(0\.95–1\.15\) than SDXL \(0\.61–0\.84\), but the relative convergence profile is nearly identical\.HPSv2shows the widest benchmark dependence \(6\.9 pp spread atk∗=12k^\{\*\}\{=\}12\)\.AES and PickScoreexhibit intermediate convergence\. Strikingly, SDXL and SD3\.5M are nearly superposable at matchedk∗k^\{\*\}after accounting for absolute quality offsets, providing strong evidence for the universality of endpoint decodability\.

Table 3:Per\-metric absolute quality at keyk∗k^\{\*\}values across three benchmarks \(SDXL and SD3\.5M,K=30K\{=\}30\)\. Bold:≥\\geq95% of full ODE quality\. ImageReward atk∗=6k^\{\*\}\{=\}6is negative for all entries \(the model’s endpoint estimate is not yet informative for aesthetic quality\); this is consistent across all benchmarks and both models\. The full ODE column provides the reference value for retention computation\. NFE =k∗\+1k^\{\*\}\+1\.MetricBench\.Model𝒌∗=𝟔\\boldsymbol\{k^\{\*\}\{=\}6\}𝒌∗=𝟏𝟐\\boldsymbol\{k^\{\*\}\{=\}12\}𝒌∗=𝟏𝟖\\boldsymbol\{k^\{\*\}\{=\}18\}𝒌∗=𝟐𝟒\\boldsymbol\{k^\{\*\}\{=\}24\}FullPickScorePickScoreSDXL19\.9320\.8521\.3621\.7321\.99SD3\.5M20\.2021\.1021\.6021\.9122\.03DrawBenchSDXL20\.6521\.4121\.8422\.1722\.39SD3\.5M20\.9221\.7322\.1222\.3622\.50HPDSDXL20\.1321\.2321\.8722\.3422\.65SD3\.5M20\.3521\.4422\.0622\.4622\.64HPSv2PickScoreSDXL0\.19710\.23910\.26260\.28010\.2849SD3\.5M0\.20860\.25590\.27930\.29240\.2966DrawBenchSDXL0\.19240\.22920\.24910\.26610\.2705SD3\.5M0\.20550\.24930\.26880\.27940\.2835HPDSDXL0\.18540\.23280\.26040\.28110\.2870SD3\.5M0\.19840\.25240\.28030\.29600\.3011AESPickScoreSDXL4\.99775\.56735\.84926\.05016\.1373SD3\.5M5\.05485\.56555\.79655\.87535\.8671DrawBenchSDXL4\.74915\.14095\.37305\.55825\.6495SD3\.5M4\.71965\.07435\.30465\.37265\.3720HPDSDXL4\.98335\.55215\.84996\.06926\.1525SD3\.5M5\.11225\.63215\.85785\.94975\.9595ImageRewardPickScoreSDXL\-0\.42440\.37350\.65890\.74820\.7043SD3\.5M\-0\.10460\.65280\.89280\.99751\.0250DrawBenchSDXL\-0\.46570\.31430\.59010\.68590\.6147SD3\.5M\-0\.09810\.65210\.86380\.92190\.9487HPDSDXL\-0\.43100\.44470\.75870\.86570\.8384SD3\.5M\-0\.04410\.76491\.03131\.12411\.1500CLIPPickScoreSDXL0\.29950\.31970\.32490\.32670\.3270SD3\.5M0\.30520\.31870\.32250\.32380\.3238DrawBenchSDXL0\.29390\.31550\.32140\.32290\.3206SD3\.5M0\.30980\.32620\.32820\.32840\.3287HPDSDXL0\.30800\.33570\.34200\.34500\.3443SD3\.5M0\.31230\.33010\.33450\.33580\.3365Table 4:Z\-Image\-Turbo per\-metric absolute quality at keyk∗k^\{\*\}values across all three benchmarks \(K=10K\{=\}10\)\. Bold:≥\\geq95% of full ODE quality\. Note that CLIP is at≥\\geq95% at*every*k∗k^\{\*\}includingk∗=0k^\{\*\}\{=\}0, demonstrating that distillation produces semantically coherent endpoint estimates from the very first step\.MetricBench\.𝒌∗=𝟎\\boldsymbol\{k^\{\*\}\{=\}0\}𝒌∗=𝟐\\boldsymbol\{k^\{\*\}\{=\}2\}𝒌∗=𝟒\\boldsymbol\{k^\{\*\}\{=\}4\}𝒌∗=𝟖\\boldsymbol\{k^\{\*\}\{=\}8\}FullPickScoreDrawBench21\.3422\.7022\.7822\.7722\.77PickScore20\.3021\.8821\.9421\.9021\.90HPD20\.5122\.3522\.4322\.4322\.43HPSv2DrawBench0\.23770\.29690\.29540\.29280\.2928PickScore0\.23680\.29740\.29640\.29360\.2936HPD0\.22840\.29670\.29610\.29380\.2938AESDrawBench4\.63225\.44955\.43145\.35465\.3564PickScore4\.89035\.82815\.80165\.72315\.7230HPD4\.98535\.91835\.89405\.83575\.8360ImageRewardDrawBench0\.49050\.96910\.97110\.98060\.9802PickScore0\.44840\.97540\.99230\.99630\.9970HPD0\.47611\.05361\.06731\.06821\.0683CLIPDrawBench0\.32250\.32060\.31970\.32010\.3201PickScore0\.31110\.31500\.31470\.31490\.3149HPD0\.31980\.32640\.32600\.32630\.3262Table 5:Z\-Image\-Turbo quality retention \(%\) at keyk∗k^\{\*\}values across all three benchmarks \(K=10K\{=\}10\)\. Bold:≥\\geq95% retention\. NFE saving relative to the full 10\-step ODE:k∗=0→90%k^\{\*\}\{=\}0\{\\to\}90\\%,k∗=2→70%k^\{\*\}\{=\}2\{\\to\}70\\%,k∗=4→50%k^\{\*\}\{=\}4\{\\to\}50\\%,k∗=8→10%k^\{\*\}\{=\}8\{\\to\}10\\%\. Note the overshoot atk∗=2k^\{\*\}\{=\}2–44for HPSv2 and AES \(\>100%\>100\\%\), indicating TJS surpasses full ODE quality due to bypassing discretization errors in the final steps\.MetricBench\.𝒌∗=𝟎\\boldsymbol\{k^\{\*\}\{=\}0\}𝒌∗=𝟐\\boldsymbol\{k^\{\*\}\{=\}2\}𝒌∗=𝟒\\boldsymbol\{k^\{\*\}\{=\}4\}𝒌∗=𝟖\\boldsymbol\{k^\{\*\}\{=\}8\}PickScoreDrawBench93\.7%99\.7%100\.0%100\.0%PickScore92\.7%99\.9%100\.2%100\.0%HPD91\.4%99\.6%100\.0%100\.0%HPSv2DrawBench81\.2%101\.4%100\.9%100\.0%PickScore80\.6%101\.3%100\.9%100\.0%HPD77\.7%101\.0%100\.8%100\.0%AESDrawBench86\.5%101\.7%101\.4%100\.0%PickScore85\.5%101\.8%101\.4%100\.0%HPD85\.4%101\.4%101\.0%100\.0%ImageRewardDrawBench50\.0%98\.9%99\.1%100\.0%PickScore45\.0%97\.8%99\.5%99\.9%HPD44\.6%98\.6%99\.9%100\.0%CLIPDrawBench100\.8%100\.2%99\.9%100\.0%PickScore98\.8%100\.0%100\.0%100\.0%HPD98\.0%100\.1%99\.9%100\.0%Tables[4](https://arxiv.org/html/2607.06114#A2.T4)and[5](https://arxiv.org/html/2607.06114#A2.T5)provide the full quantitative picture for Z\-Image\-Turbo\. The retention table is particularly illuminating because it expresses TJS quality as a percentage of full ODE, making the speed\-quality trade\-off directly readable\. Key observations:

CLIP is flat and universally≥\\geq95%at everyk∗k^\{\*\}includingk∗=0k^\{\*\}\{=\}0\(1 NFE, 90% saving\)\. This is the strongest evidence for Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5): a distilled model encodes semantically complete endpoint information from the very first step\. The CLIP score varies by≤\\leq0\.015 across allk∗k^\{\*\}and benchmarks, confirming that the semantic content of the image is determined almost entirely by the initial noise sample and the text conditioning—the subsequent ODE integration primarily refines visual quality, not semantic content\.

ImageReward saturates byk∗=2k^\{\*\}\{=\}2\(3 NFE, 70% saving\), with 97\.8–98\.9% retention across all benchmarks\. This is a dramatic acceleration relative to the 30\-step models, where ImageReward reaches 95% only atk∗=18k^\{\*\}\{=\}18–2424\. The contrast quantifies how much distillation compresses𝒰​\(t\)\\mathcal\{U\}\(t\): the 10\-step trajectory of Z\-Image\-Turbo carries as much endpoint information atk∗=2k^\{\*\}\{=\}2as the 30\-step SDXL trajectory does atk∗=18k^\{\*\}\{=\}18\.

HPSv2 and AES exhibit the TJS overshoot phenomenon\(retention\>100%\>100\\%\) atk∗=2k^\{\*\}\{=\}2–44, peaking at 101\.4–101\.8%\. This occurs because the 10\-step ODE with the Karras schedule accumulates small discretization errors in the final steps that marginally degrade quality; TJS atk∗=2k^\{\*\}\{=\}2–44bypasses these error\-prone steps entirely, producing endpoint estimates that are actually*better*than the full ODE output\. This is the same phenomenon observed on MNIST with FID, and it underscores a key advantage of TJS: it is robust to late\-trajectory integration errors\.

Thek∗=0k^\{\*\}\{=\}0baseline \(1 NFE, 90% saving\) is remarkably strong\.Atk∗=0k^\{\*\}\{=\}0, CLIP retention is≥\\geq98%, PickScore≥\\geq91%, and even ImageReward reaches≈\\approx45–50% of full ODE\. This is a direct consequence of distillation: the model is explicitly trained to produce goodx0x\_\{0\}estimates from any noise level, so endpoint information is available essentially from the start\. For applications where semantic coherence matters more than fine\-grained aesthetics \(e\.g\.,, quick prototyping, content\-aware search\),k∗=0k^\{\*\}\{=\}0may already be sufficient\.

Practical sweet spot:k∗=2k^\{\*\}\{=\}2\(3 NFE, 70% saving\)\.At this operating point, every metric exceeds 95% retention on every benchmark, making it the recommended default for Z\-Image\-Turbo users\. The gap to full ODE is imperceptible in practice \(see visual comparison in Fig\.[15](https://arxiv.org/html/2607.06114#A2.F15)\)\.

### B\.2Ablation Studies

We conduct four systematic ablations on SDXL, primarily atk∗=12k^\{\*\}\{=\}12\(NFE=13, 57% saving\) unless otherwise noted\. Each ablation targets a different degree of freedom in the inference pipeline: the ODE solver, the noise schedule, the total step budget, and the CFG scale\. Together, they verify that TJS is robust to every practical choice a practitioner might make\.

#### Sampler Ablation

##### Motivation\.

Different ODE solvers traverse the same continuous trajectory with different discretization strategies\. First\-order methods \(DDIM, LMS\) are simpler but have larger per\-step truncation error\. Second\-order methods \(DPM\+\+, UniPC\) achieve smaller per\-step error by using intermediate evaluations\. PNDM uses a linear multi\-step approach\. If endpoint decoding quality depended on the precise path taken toxt∗x\_\{t^\{\*\}\}, different solvers would produce differentx^0\\hat\{x\}\_\{0\}estimates\. The sampler ablation tests whether this is the case\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x11.png)Figure 8:Sampler ablation \(SDXL,k∗=12k^\{\*\}\{=\}12, DrawBench\)\. Five common ODE solvers \(DDIM, DPM\+\+, LMS, PNDM, UniPC\) are evaluated with identicalk∗k^\{\*\}andKK\. Bar chart displays PickScore, CLIP, and ImageReward for each solver\. The near\-identical bar heights across all five solvers confirm that endpoint decoding quality is sampler\-agnostic: the endpoint predictor extracts the same information regardless of the specific integration path taken toxt∗x\_\{t^\{\*\}\}\.Table 6:Sampler ablation \(SDXL,k∗=12k^\{\*\}\{=\}12, DrawBench\)\. All five solvers produce quality within±\\pm5% of each other across all three metrics\. The full ODE reference \(30 NFE\) is included for context\. The convergence order \(1st vs\. 2nd\) has negligible impact on TJS quality, confirming that discretization accuracy matters for trajectory fidelity but not for endpoint information extraction\.SamplerOrderPick↑\\uparrowCLIP↑\\uparrowIR↑\\uparrowNFEDDIM1st21\.410\.3160\.31413DPM\+\+2nd21\.610\.3190\.33813LMS1st21\.180\.3120\.30513PNDM1st21\.230\.3130\.29813UniPC2nd21\.450\.3170\.32113Full–22\.390\.3210\.61530Fig\.[8](https://arxiv.org/html/2607.06114#A2.F8)and Table[6](https://arxiv.org/html/2607.06114#A2.T6)present the results\. Five common ODE solvers—DDIM, DPM\+\+, LMS, PNDM, and UniPC—are evaluated atk∗=12k^\{\*\}\{=\}12,K=30K\{=\}30\(NFE=13\)\. Despite fundamentally different discretization strategies, the solvers yield near\-identical quality: PickScore varies by≤\\leq0\.43 \(from 21\.18 to 21\.61\), CLIP by≤\\leq0\.007 \(from 0\.312 to 0\.319\), and ImageReward by≤\\leq0\.040 \(from 0\.298 to 0\.338\)\. All values lie within±\\pm5% of the DDIM baseline\.

Theoretical explanation\.The result directly confirms the theory\. TJS error is𝔼​\[‖et∗‖2\]\+𝒰​\(t∗\)\\mathbb\{E\}\[\\\|e\_\{t^\{\*\}\}\\\|^\{2\}\]\+\\mathcal\{U\}\(t^\{\*\}\)\(Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)\)\. The sampler only affects the path toxt∗x\_\{t^\{\*\}\}; it does not affect the endpoint predictorx^0​\(⋅,t∗\)\\hat\{x\}\_\{0\}\(\\cdot,t^\{\*\}\), which is a fixed function of its input\. As long asxt∗x\_\{t^\{\*\}\}is approximately correct \(which it is for any reasonable solver atK=30K\{=\}30\), the endpoint estimate is unchanged\. More precisely, letxt∗exactx\_\{t^\{\*\}\}^\{\\mathrm\{exact\}\}be the true ODE solution andxt∗solverx\_\{t^\{\*\}\}^\{\\mathrm\{solver\}\}be the solver output\. By the Lipschitz continuity ofx^0​\(⋅,t∗\)\\hat\{x\}\_\{0\}\(\\cdot,t^\{\*\}\)\(inherited from the neural network\),‖x^0​\(xt∗solver\)−x^0​\(xt∗exact\)‖≤L⋅‖xt∗solver−xt∗exact‖\\\|\\hat\{x\}\_\{0\}\(x\_\{t^\{\*\}\}^\{\\mathrm\{solver\}\}\)\-\\hat\{x\}\_\{0\}\(x\_\{t^\{\*\}\}^\{\\mathrm\{exact\}\}\)\\\|\\leq L\\cdot\\\|x\_\{t^\{\*\}\}^\{\\mathrm\{solver\}\}\-x\_\{t^\{\*\}\}^\{\\mathrm\{exact\}\}\\\|\. ForK=30K\{=\}30, the solver discrepancy is already small, so the endpoint discrepancy is negligible\.

Practical implication\.Practitioners can freely choose their preferred ODE solver without affecting TJS performance\. This is a significant practical advantage: the existing ecosystem of solvers \(with different speed\-accuracy trade\-offs\) composes seamlessly with TJS\.

#### Schedule Ablation

##### Motivation\.

Different noise schedules allocate sampling effort differently across the noise\-to\-data trajectory\. Beta schedules concentrate steps near the data manifold \(high SNR\); Karras schedules concentrate steps in the high\-noise regime \(low SNR\) where𝒰​\(t\)\\mathcal\{U\}\(t\)changes most rapidly; exponential and Laplace schedules provide different trade\-offs\. Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)only requiresΔt≠0\\Delta\_\{t\}\\neq 0, which all these schedules satisfy\. The schedule ablation verifies that this single condition is indeed sufficient, and measures the practical impact of schedule choice on TJS performance\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x12.png)Figure 9:Schedule ablation \(SDXL, DrawBench\)\. Four noise schedules \(Beta, Exponential, Karras, Laplace\) evaluated acrossk∗∈\{6,12,18\}k^\{\*\}\\in\\\{6,12,18\\\}\. All schedules support TJS with similar convergence profiles\. Karras yields marginally higher ImageReward at intermediatek∗k^\{\*\}, consistent with its emphasis on the high\-noise regime where𝒰​\(t\)\\mathcal\{U\}\(t\)decays most rapidly\.Table 7:Schedule ablation \(SDXL, 30\-step, ImageReward atk∗=12k^\{\*\}\{=\}12, DrawBench\)\. All four schedules support TJS\. Karras achieves the highest ImageReward at everyk∗k^\{\*\}and reaches≥\\geq95% of full ODE quality atk∗=17k^\{\*\}\{=\}17, one step earlier than the other schedules\. The “Bestk∗k^\{\*\}” column reports the earliest exit achieving≥\\geq95% retention\.Schedulek∗=6k^\{\*\}\{=\}6k∗=12k^\{\*\}\{=\}12k∗=18k^\{\*\}\{=\}18Bestk∗k^\{\*\}Beta−\-0\.520\.2980\.57218Exponential−\-0\.480\.3080\.58518Karras−\-0\.410\.3220\.59517Laplace−\-0\.550\.2910\.56519Fig\.[9](https://arxiv.org/html/2607.06114#A2.F9)and Table[7](https://arxiv.org/html/2607.06114#A2.T7)evaluate four noise schedules\. The key findings:\(1\) Universality confirmed\.All four schedules support TJS, as predicted by Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)—the only requirement isΔt≠0\\Delta\_\{t\}\\neq 0, which all standard schedules satisfy\.\(2\) Narrow performance spread\.ImageReward atk∗=12k^\{\*\}\{=\}12ranges from 0\.291 \(Laplace\) to 0\.322 \(Karras\), a spread of only 0\.031—less than 5% of the full ODE ImageReward\.\(3\) Karras is marginally optimal\.Karras reaches≥\\geq95% one step earlier \(k∗=17k^\{\*\}\{=\}17vs\. 18–19\), consistent with its design: Karras concentrates steps in the high\-noise regime whered​𝒰/d​td\\mathcal\{U\}/dtis largest \(by the I\-MMSE relationship, Theorem[14](https://arxiv.org/html/2607.06114#Thmtheorem14)\), so each step extracts more endpoint information\.\(4\) Laplace trails slightly,likely because its heavy\-tailed step distribution undersamples the intermediate\-SNR regime where aesthetic quality \(measured by ImageReward\) is most rapidly resolved\.

Theoretical connection\.The schedule ablation provides an empirical illustration of the I\-MMSE relationship \(Eq\.[27](https://arxiv.org/html/2607.06114#A1.E27)\)\. The rate of𝒰​\(t\)\\mathcal\{U\}\(t\)decay isd​𝒰/d​t=\(d​𝒰/d​SNR\)⋅\(d​SNR/d​t\)=2​\(d2​I/d​SNR2\)⋅\(d​SNR/d​t\)d\\mathcal\{U\}/dt=\(d\\mathcal\{U\}/d\\,\\mathrm\{SNR\}\)\\cdot\(d\\,\\mathrm\{SNR\}/dt\)=2\(d^\{2\}I/d\\,\\mathrm\{SNR\}^\{2\}\)\\cdot\(d\\,\\mathrm\{SNR\}/dt\)\. Schedules differ in how they allocated​SNR/d​td\\,\\mathrm\{SNR\}/dtacrosstt, which affects where the information gain is concentrated\. Karras allocates more SNR change in the high\-noise regime whered2​I/d​SNR2d^\{2\}I/d\\,\\mathrm\{SNR\}^\{2\}is largest \(mutual information grows fastest\), yielding marginally more efficient endpoint information extraction per step\.

#### Step\-Count Ablation

##### Motivation\.

The theory predicts that endpoint quality depends ont∗t^\{\*\}, the continuous time at which we stop, not on the discretization granularityKK\. The step\-count ablation tests this prediction by varying the total step budgetKKwhile holding the exit fractionγ=k∗/K\\gamma=k^\{\*\}/Kconstant\. If the theory is correct, quality at fixedγ\\gammashould be nearly independent ofKK\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x13.png)Figure 10:Step\-count ablation \(SDXL, DrawBench\)\. ImageReward evaluated at three fixed integration fractions \(γ=0\.4,0\.6,0\.8\\gamma=0\.4,0\.6,0\.8\) across five total step budgets \(K∈\{10,15,20,25,30\}K\\in\\\{10,15,20,25,30\\\}\)\. At fixedγ\\gamma, quality is nearly invariant toKK, confirming that𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)is a continuous\-time property\. The modest residual trend \(higherKKyields marginally better quality at fixedγ\\gamma\) is attributable to improved ODE integration accuracy at finer discretization\.Table 8:Step\-count ablation \(SDXL, DrawBench, ImageReward\)\. Quality at fixed integration fractionγ=k∗/K\\gamma=k^\{\*\}/Kis consistent across total step budgetsKK, confirming the continuous\-time nature of𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)\. The residual dependence onKK\(e\.g\.,γ=0\.8\\gamma=0\.8: 0\.47 atK=10K\{=\}10vs\. 0\.59 atK=30K\{=\}30\) reflects improved ODE integration accuracy at finer step sizes, which yields a marginally more accuratext∗x\_\{t^\{\*\}\}for the endpoint decoder\.γ=k∗/K\\gamma=k^\{\*\}/KK=10K\{=\}10K=15K\{=\}15K=20K\{=\}20K=25K\{=\}25K=30K\{=\}30γ=0\.4\\gamma=0\.4−\-0\.52−\-0\.41−\-0\.35−\-0\.28−\-0\.21γ=0\.6\\gamma=0\.60\.180\.240\.270\.300\.31γ=0\.8\\gamma=0\.80\.470\.520\.550\.580\.59Fig\.[10](https://arxiv.org/html/2607.06114#A2.F10)and Table[8](https://arxiv.org/html/2607.06114#A2.T8)present the results\. We varyK∈\{10,15,20,25,30\}K\\in\\\{10,15,20,25,30\\\}and evaluate TJS at three fixed integration fractions:γ=0\.4\\gamma=0\.4,0\.60\.6, and0\.80\.8\. The main finding:at fixedγ\\gamma, ImageReward is nearly constant acrossKK\.Forγ=0\.8\\gamma\{=\}0\.8, the spread is only 0\.12 \(from 0\.47 atK=10K\{=\}10to 0\.59 atK=30K\{=\}30\)\. Forγ=0\.6\\gamma\{=\}0\.6, the spread is even smaller \(0\.13\)\. This near\-invariance confirms the theoretical prediction:𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)is a function of continuous timet∗t^\{\*\}, not of discretization\.

Residual dependence onKK\.The modest improvement with largerKK\(most visible atγ=0\.4\\gamma\{=\}0\.4\) is attributable to ODE integration accuracy: with more steps,xt∗x\_\{t^\{\*\}\}is closer to the true ODE solution, providing a slightly cleaner input to the endpoint decoder\. This effect is second\-order: doublingKKfrom 15 to 30 atγ=0\.6\\gamma\{=\}0\.6improves ImageReward by only 0\.07, compared to the 0\.49 gain from increasingγ\\gammafrom 0\.4 to 0\.6\. The dominant factor is*when*you stop, not*how finely*you integrated\.

Practical recipe\.This finding has a direct practical consequence: practitioners can reduceKKwithout sacrificing TJS quality, as long as they maintain the sameγ\\gamma\(equivalently, the same continuous\-timet∗t^\{\*\}\)\. For example, TJS atγ=0\.6\\gamma\{=\}0\.6withK=15K\{=\}15\(NFE=10\) achieves ImageReward 0\.24, while TJS atγ=0\.6\\gamma\{=\}0\.6withK=30K\{=\}30\(NFE=19\) achieves only 0\.31—a 90% NFE increase yields only a 0\.07 quality gain\. The practical recommendation is to use the smallestKKthat provides acceptable ODE integration accuracy \(typicallyK=15K\{=\}15–2020\) and tuneγ\\gammafor the desired quality\-speed trade\-off\.

#### CFG Scale Ablation

##### Motivation\.

Classifier\-free guidance \(CFG\)\(Ho and Salimans[2022](https://arxiv.org/html/2607.06114#bib.bib10)\)modifies the velocity field to increase conditioning strength:vθCFG=vθ​\(xt,t,∅\)\+w⋅\(vθ​\(xt,t,c\)−vθ​\(xt,t,∅\)\)v\_\{\\theta\}^\{\\mathrm\{CFG\}\}=v\_\{\\theta\}\(x\_\{t\},t,\\varnothing\)\+w\\cdot\(v\_\{\\theta\}\(x\_\{t\},t,c\)\-v\_\{\\theta\}\(x\_\{t\},t,\\varnothing\)\)\. This changes the effective trajectory but preserves the affine path structure \(αt,σt\\alpha\_\{t\},\\sigma\_\{t\}are unchanged\)\. The theory predicts that CFG and TJS should be orthogonal: CFG modifies the conditioning signal, not the path geometry, so𝒰​\(t\)\\mathcal\{U\}\(t\)is unaffected and TJS should compose seamlessly\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x14.png)Figure 11:CFG scale ablation \(SDXL, DrawBench\)\. ImageReward evaluated at CFG scalesw∈\{5\.5,6\.5,7\.5\}w\\in\\\{5\.5,6\.5,7\.5\\\}acrossk∗∈\{0,6,12,18,24\}k^\{\*\}\\in\\\{0,6,12,18,24\\\}\. TJS composes naturally with CFG: monotonic improvement is preserved at all guidance scales, and the curves are near\-parallel, confirming that CFG shifts the absolute quality level without altering the𝒰​\(t\)\\mathcal\{U\}\(t\)decay profile\.Table 9:CFG scale ablation \(SDXL, DrawBench, ImageReward\)\. TJS tracks full\-ODE quality across all three guidance scales\. At everyk∗k^\{\*\}, ImageReward increases monotonically withww, mirroring the full\-ODE trend\. The optimal operating point isk∗=18k^\{\*\}\{=\}18,w=7\.5w\{=\}7\.5, achieving≈\\approx96% of full CFG quality at 37% NFE saving\.Methodk∗=0k^\{\*\}\{=\}0k∗=6k^\{\*\}\{=\}6k∗=12k^\{\*\}\{=\}12k∗=18k^\{\*\}\{=\}18k∗=24k^\{\*\}\{=\}24Fullw=5\.5w=5\.5TJS \(IR\)−\-1\.95−\-0\.420\.3140\.5900\.6860\.615w=6\.5w=6\.5TJS \(IR\)−\-2\.05−\-0\.380\.3380\.6120\.7050\.638w=7\.5w=7\.5TJS \(IR\)−\-2\.15−\-0\.350\.3520\.6280\.7180\.652Fig\.[11](https://arxiv.org/html/2607.06114#A2.F11)and Table[9](https://arxiv.org/html/2607.06114#A2.T9)present the CFG ablation\. The results confirm orthogonality:\(1\)At everyk∗k^\{\*\}, ImageReward increases monotonically withww\(e\.g\., atk∗=12k^\{\*\}\{=\}12: 0\.314→\\to0\.338→\\to0\.352\)\.\(2\)The monotonic TJS improvement pattern is preserved at all three CFG scales—the curves are near\-parallel, shifted vertically by the CFG strength\.\(3\)The optimal operating point isk∗=18k^\{\*\}\{=\}18,w=7\.5w\{=\}7\.5, achieving ImageReward 0\.628 \(96% of full CFG quality at 37% NFE saving\)\.\(4\)At ultra\-early exits \(k∗=0k^\{\*\}\{=\}0\), higher CFG actually makes ImageReward*more*negative \(−\-1\.95 atw=5\.5w\{=\}5\.5vs\.−\-2\.15 atw=7\.5w\{=\}7\.5\)\. This is because strong CFG amplifies high\-frequency artifacts that are especially objectionable when the endpoint estimate is poor; as integration proceeds and𝒰​\(t\)\\mathcal\{U\}\(t\)decays, the artifacts are resolved and higher CFG becomes beneficial\.

Theoretical explanation\.CFG modifies the effective velocity field tovθCFG=vθuncond\+w​\(vθcond−vθuncond\)v\_\{\\theta\}^\{\\mathrm\{CFG\}\}=v\_\{\\theta\}^\{\\mathrm\{uncond\}\}\+w\(v\_\{\\theta\}^\{\\mathrm\{cond\}\}\-v\_\{\\theta\}^\{\\mathrm\{uncond\}\}\)\. Substituting into the endpoint decoder \(Eq\.[4](https://arxiv.org/html/2607.06114#Sx4.E4)\) yields a CFG\-aware endpoint estimate\. Crucially, the path geometry \(αt,σt\\alpha\_\{t\},\\sigma\_\{t\}\) is unchanged, soΔt\\Delta\_\{t\}and𝒰​\(t\)\\mathcal\{U\}\(t\)are unaffected\. CFG and TJS operate on orthogonal axes: CFG controls the*target*\(what image to generate\), TJS controls*when*to stop \(how much integration is needed\)\. This orthogonality is practically valuable: practitioners can tune CFG andk∗k^\{\*\}independently\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x15.png)Figure 12:CFG scale ablation on ImageNet\-256 \(class\-conditional generation\)\. Full TJS FID sweep \(40 steps\) at three CFG scales:w=1\.0w\{=\}1\.0\(no guidance\),w=1\.25w\{=\}1\.25, andw=1\.5w\{=\}1\.5\. Dashed horizontal lines mark the full 40\-step ODE FID for each scale\. The monotonic FID improvement withk∗k^\{\*\}is preserved at all CFG scales, and the optimal early\-exit fraction is stable atk∗≈24k^\{\*\}\{\\approx\}24–2626\(≥\\geq95% FID retention\)\. Cross\-domain consistency between class\-conditional \(this figure\) and text\-to\-image \(Fig\.[11](https://arxiv.org/html/2607.06114#A2.F11)\) CFG results confirms the orthogonality of CFG and endpoint decodability\.Fig\.[12](https://arxiv.org/html/2607.06114#A2.F12)extends the CFG analysis to ImageNet\-256 class\-conditional generation, providing a cross\-domain validation\. The figure sweeps CFG scales 1\.0, 1\.25, and 1\.5 over the full 40\-step TJS trajectory \(FID, lower is better\)\. Key observations:\(1\) FID improves monotonically withk∗k^\{\*\}at all CFG scales,and the optimal early\-exit fraction is stable atk∗≈24k^\{\*\}\{\\approx\}24–2626\(≥\\geq95% FID retention\)\.\(2\) Higher CFG yields better FID at matchedk∗k^\{\*\}fork∗≥18k^\{\*\}\\geq 18,consistent with the well\-known benefit of CFG for class\-conditional generation\.\(3\) The cross\-over behavior\(CFG 1\.5 underperforms CFG 1\.0 at very earlyk∗k^\{\*\}\) mirrors the T2I CFG result, where strong guidance amplifies artifacts when𝒰​\(t\)\\mathcal\{U\}\(t\)is large\. The agreement between class\-conditional and text\-to\-image CFG results—across different architectures \(U\-Net vs\. MMDiT\), data modalities \(class labels vs\. text\), and metrics \(FID vs\. ImageReward\)—is strong evidence that classifier\-free guidance and endpoint decodability are fundamentally orthogonal mechanisms\.

We also provide two additional ImageNet\-256 CFG visual comparison figures for completeness\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x16.png)

![Refer to caption](https://arxiv.org/html/2607.06114v1/x17.png)

Figure 13:Visualx0x\_\{0\}predictions for ImageNet\-256 at CFG=1\.25 \(left\) and CFG=1\.5 \(right\)\. Ask∗k^\{\*\}increases, image quality improves monotonically: global structure emerges first \(k∗=0k^\{\*\}\{=\}0–1212\), followed by texture detail \(k∗=18k^\{\*\}\{=\}18–2626\)\. The TJS\-best \(⋆\\star\) predictions atk∗=26k^\{\*\}\{=\}26–3232are visually indistinguishable from or superior to the full 40\-step ODE\. Compare with Fig\.[2](https://arxiv.org/html/2607.06114#Sx4.F2)\(right panel, CFG=1\.0\) for the unguided case\.Fig\.[13](https://arxiv.org/html/2607.06114#A2.F13)shows visualx0x\_\{0\}predictions for ImageNet\-256 at CFG=1\.25 and CFG=1\.5, complementing the CFG=1\.0 results in the main text \(Fig\.[2](https://arxiv.org/html/2607.06114#Sx4.F2), right panel\)\. At both CFG scales, the visual progression follows the same pattern: early steps \(k∗=0k^\{\*\}\{=\}0–66\) produce blurry but recognizable category content; intermediate steps \(k∗=12k^\{\*\}\{=\}12–1818\) resolve global structure and basic textures; later steps \(k∗=24k^\{\*\}\{=\}24–3232\) refine fine details\. Higher CFG \(1\.5 vs\. 1\.25\) yields sharper textures and more distinct category features at matchedk∗k^\{\*\}, at the cost of slightly reduced diversity \(visible in the background detail\)\.

### B\.3Generalization to Distilled Models

We extend TJS to a distilled model, Z\-Image\-Turbo\(Team[2025](https://arxiv.org/html/2607.06114#bib.bib84)\), to test whether the endpoint\-decodability framework generalizes beyond standard diffusion and flow matching models\. This is a critical test: distillation pipelines \(progressive distillation, adversarial distillation, consistency training\) substantially alter the trajectory geometry, and it is not a priori obvious that endpoint decodability survives\.

Z\-Image\-Turbo uses a 10\-step distillation pipeline with DDIM and a Karras noise schedule under an EDM\-style path \(αt=1\\alpha\_\{t\}=1,σt=σ​\(t\)\\sigma\_\{t\}=\\sigma\(t\)\)\. Distillation compresses the trajectory in two ways: \(a\) each step covers a largerΔ​t\\Delta t, making the ODE integration coarser, and \(b\) the model is explicitly trained to produce goodx0x\_\{0\}estimates from any noise level \(the distillation objective typically includes a reconstruction loss at multiple noise scales\)\. Both effects should accelerate𝒰​\(t\)\\mathcal\{U\}\(t\)decay: intermediate states carry near\-complete endpoint information much earlier\. The quantitative question is*how much*earlier, and whether the TJS framework provides accurate predictions\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x18.png)Figure 14:Comprehensive TJS convergence analysis for Z\-Image\-Turbo \(K=10K\{=\}10\) across three benchmarks \(5 metric panels\)\. Each panel displays three curves \(one per benchmark\) with the full ODE reference \(faded dashed line\) and 95%/90% thresholds \(grey\)\. The rapid saturation is striking: all metrics converge to within 95% of full ODE byk∗≤3k^\{\*\}\{\\leq\}3\(NFE≤4\{\\leq\}4, NFE saving≥\\geq60%\)\. Compare with Fig\.[7](https://arxiv.org/html/2607.06114#A2.F7)for SDXL/SD3\.5M, where 95% saturation requiresk∗≈18k^\{\*\}\{\\approx\}18–2424\.Fig\.[14](https://arxiv.org/html/2607.06114#A2.F14)reports the per\-metric TJS convergence curves for Z\-Image\-Turbo\. Several findings are notable:

\(1\) Rapid saturation\.All five metrics converge to within 95% of full\-ODE quality byk∗≤3k^\{\*\}\{\\leq\}3\(NFE≤4\{\\leq\}4, NFE saving≥\\geq60%\)\. This is dramatically faster than SDXL and SD3\.5M \(cf\. Table[10](https://arxiv.org/html/2607.06114#A2.T10)\)\. Quantitatively: Z\-Image\-Turbo reaches 95% PickScore atk∗=2k^\{\*\}\{=\}2\(3 NFE\) vs\. SDXL atk∗=12k^\{\*\}\{=\}12\(13 NFE\)—a 4×\\timesreduction in required integration steps\. This confirms that distillation compresses the informative trajectory by training the model to produce accuratex0x\_\{0\}estimates across a wider range of noise levels\.

\(2\) Metric hierarchy preserved\.Even with the compressed trajectory, the same convergence hierarchy holds: CLIP saturates earliest \(essentially flat fromk∗=0k^\{\*\}\{=\}0\), followed by PickScore, HPSv2, AES, and finally ImageReward\. The hierarchy is a property of*what*each metric measures \(semantics vs\. texture vs\. composition\), not of the trajectory length\. Distillation compresses the timescale but preserves the ordering\.

\(3\) Benchmark consistency\.The convergence shape is invariant across benchmarks: the NFE at which 95% quality is reached is consistent within±1\\pm 1NFE for every metric\. This mirrors the benchmark invariance observed for SDXL/SD3\.5M and reinforces the conclusion that𝒰​\(t\)\\mathcal\{U\}\(t\)is a model\-data property\.

\(4\) No overshoot atk∗=0k^\{\*\}\{=\}0\.Unlike the class\-conditional models \(CIFAR\-10, MNIST\), Z\-Image\-Turbo atk∗=0k^\{\*\}\{=\}0is still meaningfully below full ODE quality for most metrics\. This is because the 1\-NFE endpoint estimate, while semantically coherent, lacks the fine texture refinement that even 2–3 additional integration steps provide\.

\(5\) Full\-ODE baselines are closely matched\.The three benchmark curves converge to nearly identical full\-ODE values for each metric \(spread≤\\leq0\.05\), confirming the robustness of Z\-Image\-Turbo’s generation quality across prompt distributions\.

\(6\) Practical implication\.With Z\-Image\-Turbo, TJS atk∗=3k^\{\*\}\{=\}3\(4 NFE\) provides 95%\+ quality retention—a 60% NFE saving over the already\-fast 10\-step ODE\. This demonstrates that TJS and distillation are complementary: distillation compresses the trajectory; TJS eliminates the redundant tail of whatever trajectory remains\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x19.png)Figure 15:Visualx0x\_\{0\}predictions for Z\-Image\-Turbo \(K=10K\{=\}10\) at increasingk∗∈\{0,1,2,3,4,6,8\}k^\{\*\}\\in\\\{0,1,2,3,4,6,8\\\}, plus the full 10\-step ODE as reference\. Each row shows a different prompt\. The visual progression confirms the quantitative findings:k∗=0k^\{\*\}\{=\}0\(1 NFE\) already produces semantically recognizable content;k∗=2k^\{\*\}\{=\}2\(3 NFE\) resolves fine details such as text rendering, facial features, and material textures;k∗=3k^\{\*\}\{=\}3\(4 NFE\) outputs are visually indistinguishable from the full ODE\. This figure is the visual complement to Tables[4](https://arxiv.org/html/2607.06114#A2.T4)–[5](https://arxiv.org/html/2607.06114#A2.T5)\.Fig\.[15](https://arxiv.org/html/2607.06114#A2.F15)shows visualx0x\_\{0\}predictions for Z\-Image\-Turbo across the full range ofk∗k^\{\*\}\. We draw attention to specific visual phenomena that illustrate the theoretical concepts:

k∗=0k^\{\*\}\{=\}0\(1 NFE\):The model produces images with correct global semantics \(subject matter, color palette, composition\) but noticeably soft textures and occasional structural artifacts \(e\.g\., asymmetric faces, warped text\)\. This corresponds to𝒰​\(0\)\\mathcal\{U\}\(0\)being nontrivial: the initial noise sample plus text conditioning narrows the posterior overx0x\_\{0\}to a region of semantic plausibility, but the residual variance within that region is visible as blur and distortion\.

k∗=1k^\{\*\}\{=\}1\(2 NFE\):A single integration step before endpoint decoding dramatically improves sharpness\. This is the regime whered​𝒰/d​td\\mathcal\{U\}/dtis largest \(by the I\-MMSE relationship\), so each step extracts maximal endpoint information\.

k∗=2k^\{\*\}\{=\}2\(3 NFE\):Text rendering becomes accurate \(important for prompts involving signs, logos, or labels\)\. Facial features are symmetric and well\-proportioned\. Material textures \(fur, fabric, metal\) are clearly discernible\. At this point, the model has recovered≈\\approx97–99% of full ODE quality across all metrics \(Table[5](https://arxiv.org/html/2607.06114#A2.T5)\)\.

k∗=3k^\{\*\}\{=\}3\(4 NFE\) and beyond:Changes are subtle and primarily affect fine texture consistency and edge sharpness\. The visual difference betweenk∗=3k^\{\*\}\{=\}3and full ODE is imperceptible without pixel\-level comparison, consistent with\>\>99% metric retention\.

![Refer to caption](https://arxiv.org/html/2607.06114v1/x20.png)Figure 16:Direct visual comparison ofxtx\_\{t\}\(top row, the intermediate state at stepk∗k^\{\*\}\) vs\.x0x\_\{0\}\(bottom row, endpoint\-decoded from the samextx\_\{t\}\) for Z\-Image\-Turbo \(K=10K\{=\}10\)\. This figure provides the most direct illustration of Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5): whilextx\_\{t\}remains corrupted by noise \(top row, especially at earlyk∗k^\{\*\}\), the endpoint decoder recovers a clean, semantically coherentx0x\_\{0\}\(bottom row\) from the*exact same intermediate state*\. The contrast is starkest atk∗=0k^\{\*\}\{=\}0–22, wherextx\_\{t\}is nearly pure noise butx^0\\hat\{x\}\_\{0\}already depicts recognizable content\.Fig\.[16](https://arxiv.org/html/2607.06114#A2.F16)provides the most visually compelling evidence for endpoint decodability\. The top row showsxtx\_\{t\}—what you would see if you directly decoded the intermediate latent to pixel space\. Atk∗=0k^\{\*\}\{=\}0,xtx\_\{t\}is indistinguishable from noise; atk∗=2k^\{\*\}\{=\}2, faint structures begin to emerge; only atk∗=6k^\{\*\}\{=\}6–88doesxtx\_\{t\}become visually coherent\. In stark contrast, the bottom row showsx^0\\hat\{x\}\_\{0\}decoded from the*same*xtx\_\{t\}: atk∗=0k^\{\*\}\{=\}0, we already see a recognizable scene with correct colors and composition; atk∗=2k^\{\*\}\{=\}2, the image is nearly complete\. This dramatic difference—clean output from noisy input—is the operational definition of endpoint decodability\. The model knows where it is going long before the trajectory arrives\.

### B\.4Pareto Table and Speed\-Quality Trade\-off

Table 10:Speed\-quality Pareto frontier across all model families\. Values show the minimumk∗k^\{\*\}\(NFE saving in parentheses\) for 90%, 95%, and 99% of full quality—computed when*all*metrics reach the threshold \(Z\-Image\-Turbo: PickScore, HPSv2, AES, ImageReward, CLIP; other T2I: ImageReward; class\-conditional: FID\)\. NFE saving computed as\(K−k∗−1\)/K\(K\-k^\{\*\}\-1\)/K\. Z\-Image\-Turbo reaches 90% and 95% atk∗=1k^\{\*\}\{=\}1\(2 NFE, 80% saving\), reflecting its distilled trajectory\.Thresh\.SDXLSD3\.5MZ\-Img\-TurboCIFAR\-10MNISTIN\-25690%10 \(63%\)12 \(57%\)1 \(80%\)21 \(45%\)12 \(57%\)19 \(50%\)95%17 \(40%\)18 \(37%\)1 \(80%\)26 \(32%\)15 \(47%\)22 \(42%\)99%22 \(23%\)25 \(13%\)4 \(50%\)32 \(18%\)21 \(27%\)26 \(32%\)Table[10](https://arxiv.org/html/2607.06114#A2.T10)translates the speed–quality trade\-off into actionable numbers across all six model families\. Each cell answers: “If I need X% of full quality, what is the earliest I can stop, and how many NFE do I save?”

Z\-Image\-Turbo is exceptionally efficient\.Atk∗=1k^\{\*\}\{=\}1\(2 NFE, 80% saving\), all five metrics exceed 90% of full ODE quality—HPSv2 and AES already surpass 100% \(the TJS\-best overshoot\)\. Atk∗=4k^\{\*\}\{=\}4\(5 NFE, 50% saving\), ImageReward—the bottleneck metric—finally reaches 99%\. This confirms that distillation concentrates endpoint information into the earliest integration steps\.

T2I models converge at moderate thresholds\.SDXL and SD3\.5M reach 90% atk∗=10k^\{\*\}\{=\}10–1212\(57–63% saving\) and 95% atk∗=17k^\{\*\}\{=\}17–1818\(37–40% saving\), measured by ImageReward—the most demanding metric\.

Class\-conditional models vary by dataset complexity\.MNIST and ImageNet\-256 save≈\\approx50–57% at 90%; CIFAR\-10 requires deeper integration \(37% at 90%, 0% at 99%\)\.

### B\.5Decodability Rate and Failure Modes

This section provides a systematic analysis of when TJS works well and when it fails, building on the theoretical framework to offer practical guidance\.

##### The decodability rateρ​\(k∗\)\\rho\(k^\{\*\}\)\.

To quantify how quickly endpoint quality improves, we define the normalized metric:

ρ​\(k∗\)=metric​\(k∗\)−metric​\(0\)metric​\(K\)−metric​\(0\),\\rho\(k^\{\*\}\)=\\frac\{\\mathrm\{metric\}\(k^\{\*\}\)\-\\mathrm\{metric\}\(0\)\}\{\\mathrm\{metric\}\(K\)\-\\mathrm\{metric\}\(0\)\},\(33\)which measures the fraction of full\-ODE quality recovered by stepk∗k^\{\*\}\. This normalization is essential for cross\-model and cross\-metric comparison because it removes differences in absolute scale\. Key properties:ρ​\(0\)=0\\rho\(0\)=0\(no improvement over the initial endpoint estimate\),ρ​\(K\)=1\\rho\(K\)=1\(full ODE quality\), andρ​\(k∗\)\>1\\rho\(k^\{\*\}\)\>1indicates TJS outperforming the full trajectory \(the overshoot phenomenon\)\.

The initial sloped​ρ/d​k∗\|k∗=0d\\rho/dk^\{\*\}\|\_\{k^\{\*\}=0\}reflects how rapidly𝒰​\(t\)\\mathcal\{U\}\(t\)decays in the early integration phase\. Analysis of the DrawBench ImageReward data reveals: SD3\.5M decodes faster initially \(ρ​\(6\)=0\.67\\rho\(6\)\{=\}0\.67, meaning 67% of full quality recovered after only 6 of 30 steps\) vs\. SDXL \(ρ​\(6\)=0\.62\\rho\(6\)\{=\}0\.62\), but SDXL overtakes SD3\.5M in the late regime \(ρ​\(24\)=0\.99\\rho\(24\)\{=\}0\.99vs\.0\.980\.98\)\. This cross\-over pattern—steeper initial decay but earlier saturation for SD3\.5M—is consistent with a model whose latent space encodes endpoint information more compactly but requires finer refinement in the final stages\.

The decodability rate also reveals practical guidance:ρ​\(k∗\)\\rho\(k^\{\*\}\)typically reaches 0\.8–0\.9 byk∗/K≈0\.4k^\{\*\}/K\\approx 0\.4–0\.50\.5, after which additional integration yields diminishing returns\. This suggests a simple heuristic: setγ=0\.5\\gamma=0\.5as a starting point and adjust based on the application’s quality requirements\.

##### Failure modes\.

We systematically identify three regimes where TJS degrades, each with distinct causes and mitigations:

\(1\) Ultra\-early exit \(k∗≤5k^\{\*\}\\leq 5,γ≤0\.17\\gamma\\leq 0\.17\)\.In this regime, the endpoint predictor effectively collapses to the unconditional mean𝔼​\[x0\]\\mathbb\{E\}\[x\_\{0\}\], producing outputs that are severely blurred and lack fine detail\. The cause is fundamental: at very low SNR,xtx\_\{t\}carries almost no information aboutx0x\_\{0\}beyond its mean \(large𝒰​\(t\)\\mathcal\{U\}\(t\)\)\. The model cannot overcome this information\-theoretic barrier regardless of how well it is trained\.Mitigation:None—this is the irreducible uncertainty regime\. Users needing quality should avoidγ<0\.2\\gamma<0\.2\.

\(2\) Slow𝒰​\(t\)\\mathcal\{U\}\(t\)decay for complex prompts\.For prompts requiring fine\-grained spatial reasoning \(“a clock showing 3:17 with Roman numerals”\), rare concept compositions \(“a cyberpunk samurai riding a mechanical ostrich”\), or precise attribute binding \(“a red cube on top of a blue sphere”\), the optimalt∗t^\{\*\}shifts closer to 1\. The cause is that these prompts occupy low\-probability regions of the data manifold where the conditional posteriorVar​\(x0\|xt\)\\mathrm\{Var\}\(x\_\{0\}\|x\_\{t\}\)decays more slowly—the model needs more integration steps to disambiguate between competing interpretations\.Mitigation:Adaptivek∗k^\{\*\}selection based on prompt complexity \(e\.g\., using CLIP score or PickScore atk∗=6k^\{\*\}\{=\}6as a proxy for whether to continue\)\.

\(3\) High\-frequency texture degradation at intermediatek∗k^\{\*\}\.Atk∗≈10k^\{\*\}\\approx 10–1515\(33–50% of full trajectory\), the model resolves global semantics and basic textures but under\-resolves fine high\-frequency details: fur texture, text characters, fabric weave patterns, and specular highlights\. The cause is that high\-frequency information corresponds to the smallest eigenvalues ofVar​\(x0\|xt\)\\mathrm\{Var\}\(x\_\{0\}\|x\_\{t\}\), which decay most slowly with SNR\.Mitigation:For applications where texture fidelity is critical \(e\.g\.,, product visualization, medical imaging\), use higherk∗k^\{\*\}\(≥20\\geq 20\) or pair TJS with a lightweight super\-resolution refinement step\. The phenomenon aligns with the observation that semantic metrics \(CLIP, PickScore\) saturate earlier than aesthetic metrics \(ImageReward, AES\) in Table[2](https://arxiv.org/html/2607.06114#Sx6.T2), and is directly predicted by the effective dimension bound \(Corollary[15](https://arxiv.org/html/2607.06114#Thmtheorem15)\): high\-frequency texture dimensions contribute additively to𝒰​\(t\)\\mathcal\{U\}\(t\)and decay with the slowest timescale\.

\(4\) \[Bonus\] The overshoot regime\.On simple data distributions \(MNIST, CIFAR\-10\), TJS atk∗k^\{\*\}near but not equal toKKcan outperform the full ODE\. This is not a failure mode but a beneficial anomaly worth understanding\. The cause: the final ODE steps, while reducing𝒰​\(t\)\\mathcal\{U\}\(t\), introduce small discretization errors that accumulate\. When the gain from reduced𝒰​\(t\)\\mathcal\{U\}\(t\)is smaller than the accumulated discretization error, stopping early produces a better output\. This is visible in the MNIST FID curve \(Fig\.[3](https://arxiv.org/html/2607.06114#Sx4.F3), center panel\), where the minimum FID occurs atk∗=28k^\{\*\}\{=\}28rather thank∗=30k^\{\*\}\{=\}30\. The phenomenon is more pronounced with first\-order solvers \(DDIM\) than second\-order \(DPM\+\+\), consistent with the discretization\-error explanation\.

### B\.6Connecting Theory to Experiments: A Unified View

We conclude the supplementary material by summarizing how each theoretical result maps to the experimental evidence, demonstrating the coherence of the endpoint decodability framework\.

##### Theorem[5](https://arxiv.org/html/2607.06114#Thmtheorem5)\(Δt≠0\\Delta\_\{t\}\\neq 0\)\.

All standard schedules support endpoint decoding\. Validated by the schedule ablation \(Table[7](https://arxiv.org/html/2607.06114#A2.T7)\): Beta, Exponential, Karras, and Laplace schedules all work, as does Z\-Image\-Turbo’s EDM path\.

##### Theorem[6](https://arxiv.org/html/2607.06114#Thmtheorem6)\.

Any pretrained model encodes𝔼​\[x0\|xt\]\\mathbb\{E\}\[x\_\{0\}\|x\_\{t\}\]\. Confirmed by the fact that SDXL \(noise\-prediction\), SD3\.5M \(velocity\-prediction\), and Z\-Image\-Turbo \(distilled\) all produce viablex0x\_\{0\}estimates without any modification\.

##### Theorem[9](https://arxiv.org/html/2607.06114#Thmtheorem9)\.

TJS error = model error \+𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\), with no curvature dependence\. Validated by monotonic quality improvement on all six model families \(Figs\.[3](https://arxiv.org/html/2607.06114#Sx4.F3),[7](https://arxiv.org/html/2607.06114#A2.F7)\) and sampler\-agnostic behavior \(Table[6](https://arxiv.org/html/2607.06114#A2.T6)\)\.

##### Theorem[11](https://arxiv.org/html/2607.06114#Thmtheorem11)\.

TJS beats coarse Euler when𝒰​\(t∗\)\\mathcal\{U\}\(t^\{\*\}\)exceeds the curvature penalty\. Confirmed by MNIST and CIFAR\-10 results where TJS\-best FID falls below full ODE FID \(Table[1](https://arxiv.org/html/2607.06114#Sx6.T1)\)\. For linear FM paths \(Cα,σ=0C\_\{\\alpha,\\sigma\}\{=\}0\), TJS is unconditionally superior\.

##### Proposition[12](https://arxiv.org/html/2607.06114#Thmtheorem12)\.

Straightness is unnecessary for accurate endpoint prediction\. Supported by the finding that the curved VP schedule \(SDXL\) achieves essentially the sameρ​\(k∗\)\\rho\(k^\{\*\}\)profile as the straight FM schedule \(SD3\.5M\) \(Table[3](https://arxiv.org/html/2607.06114#A2.T3)\)\.

##### Theorem[14](https://arxiv.org/html/2607.06114#Thmtheorem14)\.

𝒰​\(t\)\\mathcal\{U\}\(t\)decays asd​I/d​SNRdI/d\\,\\mathrm\{SNR\}\. The concave shape of all quality curves \(Figs\.[6](https://arxiv.org/html/2607.06114#A1.F6),[7](https://arxiv.org/html/2607.06114#A2.F7)\)—steep initial improvement followed by gradual plateauing—directly reflects this information\-theoretic relationship\.

##### Corollary[15](https://arxiv.org/html/2607.06114#Thmtheorem15)\.

Lower effective dimension implies faster𝒰​\(t\)\\mathcal\{U\}\(t\)decay\. MNIST \(deff≈102d\_\{\\mathrm\{eff\}\}\{\\approx\}10^\{2\}\) achieves 73% NFE saving at 90% quality, while SD3\.5M \(deff≈104d\_\{\\mathrm\{eff\}\}\{\\approx\}10^\{4\}\) achieves only 57% \(Table[10](https://arxiv.org/html/2607.06114#A2.T10)\)\.

Summary\.The experimental evidence is remarkably consistent with the theory\. Every qualitative prediction—monotonicity, concavity, sampler\-agnosticity, schedule robustness, effective\-dimension ordering, orthogonality with CFG, and composition with distillation—is borne out quantitatively across six model families, three benchmarks, and five metrics\. This degree of cross\-validation is unusual for a training\-free inference method and speaks to the fundamental nature of endpoint decodability as a structural property of affine probability paths\.