CacheScope: Exact-Compute Allocation for Training-Free Diffusion Acceleration

# CacheScope: Exact-Compute Allocation for Training-Free Diffusion Acceleration
Source: [https://arxiv.org/html/2610.12530](https://arxiv.org/html/2610.12530)
Jusheng ZhangZhixing ZhangHao LiuZijie LiangJin JiangKeze Wang\\corresponding

###### Abstract

Training\-free feature reuse accelerates diffusion transformer inference by replacing full denoising computation at selected steps with cached approximations\. Existing methods mainly improve the reconstruction of skipped features, but we find that once a stable first\-order reconstructor is used, increasing reconstruction complexity brings limited gains, whereasthe placement of exact\-computation steps has a much larger effect on fidelity\. More importantly,absolute shallow\-probe variation is jointly affected by input\-trajectory displacement and network response strength, and therefore does not directly reflect cache risk\. We address this confounding by normalizing probe\-output variation with the corresponding input displacement, yielding a directional gain per unit trajectory movement\. Based on this signal, we proposeCacheScope, which identifies high\-risk reuse steps online and selectively falls back to full denoising computation while leaving the underlying cache reconstructor unchanged\. CacheScope requires no additional training, gradients, auxiliary networks, or prompt\-specific offline schedule optimization\. Across FLUX\.1\-dev, Wan2\.1\-T2V\-1\.3B, and Open\-Sora 1\.2, CacheScope achieves up to3\.57×3\.57\\timesacceleration and reduces median LPIPS by up to70\.1%70\.1\\%over the strongest probe\-based baseline\.

Sun Yat\-sen University

## Introduction

Diffusion and flow\-based generative models\([Ho et al\. 2020](https://arxiv.org/html/2610.12530#bib.bib20);[Song et al\. 2021b](https://arxiv.org/html/2610.12530#bib.bib21);[Rombach et al\. 2022](https://arxiv.org/html/2610.12530#bib.bib22);[Lipman et al\. 2023](https://arxiv.org/html/2610.12530#bib.bib24);[Liu et al\. 2023](https://arxiv.org/html/2610.12530#bib.bib25)\)have become a dominant paradigm for high\-fidelity image and video synthesis\([Ho et al\. 2022](https://arxiv.org/html/2610.12530#bib.bib26);[Yang et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib32);[Kong et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib31);[Wan Team 2025](https://arxiv.org/html/2610.12530#bib.bib29);[Wang et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib62)\)\. Related efforts also target controllable video generation and embodied action models\([Zeng et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib2);[Zhan et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib4);[Zhang et al\. 2026c](https://arxiv.org/html/2610.12530#bib.bib58);[Zhang et al\. 2025h](https://arxiv.org/html/2610.12530#bib.bib54);[Zhang et al\. 2025k](https://arxiv.org/html/2610.12530#bib.bib48)\)\. Their generation process, however, requires repeatedly evaluating a large denoising network along a discretized trajectory\. For modern diffusion transformers \(DiTs\)\([Peebles and Xie 2023](https://arxiv.org/html/2610.12530#bib.bib23)\), each evaluation performs expensive attention and feed\-forward computation over long spatial or spatiotemporal token sequences, making inference latency and computational cost remain substantial\. Training\-free feature reuse provides a practical acceleration route by replacing selected denoising evaluations with inexpensive approximations constructed from previously cached states\. Since it requires neither retraining nor modification of the pretrained model weights, feature reuse can be directly applied to rapidly evolving diffusion backbones\.

A training\-free reuse method must answer two fundamental questions:*how to reuse*previously computed information, and*when to reuse*it\. The former determines how the output of a skipped denoising evaluation is reconstructed, while the latter determines which steps should retain exact computation\. Both questions are important\. A weak reconstruction rule can immediately introduce visible errors and cause deviations to accumulate along the generation trajectory\. Consequently, a large body of recent work has explored increasingly sophisticated reconstruction strategies, ranging from direct feature reuse and scalar correction to trajectory extrapolation and multi\-history regression\([Ma et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib11);[Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15);[Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16);[Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)\.

Figure 1:Overview of the CacheScope insight\.Under a fixed cache reconstructor and a fixed exact\-compute budget, more flexible reconstruction rules do not consistently outperform a bounded first\-order reconstructor, while the placement of exact evaluations dominates fidelity\. Raw probe variationΔ​yt\\Delta y\_\{t\}conflates trajectory displacement with network response; CacheScope instead schedules exact computation using the trajectory\-conditioned sensitivityst=Δ​yt/\(Δ​xt\+ϵ\)s\_\{t\}=\\Delta y\_\{t\}/\(\\Delta x\_\{t\}\+\\epsilon\)\.Nevertheless, our controlled experiments reveal a clear asymmetry between these two design dimensions\. Once a stable bounded first\-order reconstructor is available, the more flexible reconstruction formulations we examine do not consistently outperform it\. Extending the residual history, introducing higher\-order fitting rules, or using more flexible regression formulations improves fidelity in some settings but degrades it in others\. On video diffusion transformers, unconstrained multi\-history reconstruction can even become unstable and perform substantially worse than a simple first\-order alternative\. We therefore do not argue that reconstruction is unimportant, nor that no better reconstructor exists; rather, within the examined design space, we did not find a formulation that consistently outperforms the lightweight first\-order rule\.

In contrast, we find thatthe placement of exact computation remains highly consequential: even when the cache reconstructor, probe depth, warm\-up policy, and total number of exact denoiser evaluations are all fixed, different reuse schedules can produce markedly different generation quality\. Thus, the central bottleneck is not merely whether a sufficiently strong reconstruction rule is used, but whether a limited exact\-compute budget is allocated to the right locations along the denoising trajectory\. This observation shifts the focus of diffusion caching from designing increasingly elaborate approximators to a more fundamental allocation problem:

> *Under a fixed exact\-computation budget, which denoising steps are most valuable to compute exactly?*

Existing adaptive schedulers commonly address this problem by measuring the raw variation of a shallow probe output\. Intuitively, a large probe change is interpreted as evidence that the current step is difficult to approximate and should therefore receive an exact forward pass\. However, this interpretation implicitly assumes that transitions at different denoising steps are directly comparable\. We find that this assumption does not hold along diffusion trajectories:the observed probe\-output variation is jointly determined by two factors—how far the input trajectory moves between adjacent steps, and how strongly the network responds to that movement\(Figure[1](https://arxiv.org/html/2610.12530#Sx1.F1)\)\. A large output change may therefore arise simply because the latent input has moved a long distance, rather than because the network is locally sensitive\. Conversely, a small trajectory movement can trigger a disproportionately strong network response while still producing only a moderate absolute variation\.

This distinction can be seen through a local first\-order view\. Letxtx\_\{t\}denote the shallow\-probe input,yt=g⁡\(xt\)y\_\{t\}=g\(x\_\{t\}\)its output, andJtJ\_\{t\}the local Jacobian of the probe\. Then

yt−yt−1≈Jt​\(xt−xt−1\)\.y\_\{t\}\-y\_\{t\-1\}\\approx J\_\{t\}\(x\_\{t\}\-x\_\{t\-1\}\)\.\(1\)The magnitude of the observed output change therefore entangles the trajectory displacementxt−xt−1x\_\{t\}\-x\_\{t\-1\}with the directional network response induced by that displacement\. Raw probe variation alone cannot reliably distinguish a large but proportionate response from a small movement that is strongly amplified by the model\. As a result, it is not a directly comparable measure of the value of exact computation across denoising steps—that is, it does not directly reflect the risk of reusing cache at a given step\.

Motivated by this insight, we introduceCacheScope, a lightweight drop\-in scheduling rule for online exact\-compute allocation\. Rather than ranking steps by absolute probe variation, CacheScope measures the network response relative to the corresponding local trajectory movement\. Specifically, it constructs a trajectory\-conditioned sensitivity score from the ratio between relative probe\-output variation and relative probe\-input movement:

st=Δ​ytΔ​xt\+ϵ\.s\_\{t\}=\\frac\{\\Delta y\_\{t\}\}\{\\Delta x\_\{t\}\+\\epsilon\}\.\(2\)This response\-per\-movement quantity serves as a lightweight proxy for the local directional sensitivity of the denoising network\. CacheScope accumulates the sensitivity score online and performs an exact denoiser evaluation when the accumulated utility reaches a predefined threshold; otherwise, the step is approximated using the existing cache reconstructor\. In this way, CacheScope converts trajectory\-aware sensitivity estimation into a budgeted allocation policy for exact computation\.

Importantly, CacheScope changes the semantics of the scheduling decision without increasing the complexity of the reuse mechanism\. It requires no training, gradients, auxiliary prediction network, online optimization, or offline schedule search\. It leaves the underlying cache reconstructor and cached history unchanged, and introduces only negligible scalar computation beyond an existing shallow probe\. CacheScope can therefore be incorporated into existing probe\-based caching pipelines without altering their reconstruction components\.

We evaluate CacheScope on the image diffusion transformer FLUX\.1\-dev and the video diffusion transformers Wan2\.1\-T2V\-1\.3B and Open\-Sora 1\.2\. Under strictly matched exact\-computation budgets, CacheScope consistently improves perceptual fidelity across all backbones and multiple acceleration regimes\. It establishes a new end\-to\-end speed–fidelity frontier among the evaluated training\-free feature\-reuse methods \(Figure\), achieving up to3\.57×3\.57\\timesacceleration and reducing median LPIPS by up to70\.1%70\.1\\%over the strongest probe\-based baseline\. At about1\.40×1\.40\\timesacceleration on Open\-Sora 1\.2, CacheScope reduces LPIPS from0\.05660\.0566to0\.01690\.0169; at about2\.95×2\.95\\timeson FLUX, from0\.13530\.1353to0\.08160\.0816; and at about3\.46×3\.46\\timeson Wan2\.1, from0\.20270\.2027to0\.16070\.1607\. Controlled comparisons with identical reconstruction rules and matched skip counts further confirm that these gains arise from allocating exact computation to more consequential denoising steps, rather than from using more computation or a stronger reconstructor\. Beyond the main comparisons, the supplementary material provides an extensive measurement record, including full paired analyses with confidence intervals, a multiple\-comparison audit, independent\-seed replication, additional operating points, schedule\-placement analysis, cost\-model validation, and a codec\-independence check\.

Our contributions are summarized as follows:

- •We conduct controlled studies of the two fundamental dimensions of training\-free feature reuse\. The results show that, within the examined design space, more flexible reconstruction rules do not consistently outperform a bounded first\-order reconstructor and can even be unstable, while exact\-step placement remains a major source of quality variation\.
- •We show that raw shallow\-probe variation is not directly comparable across denoising steps because it conflates trajectory displacement with network response\. We further demonstrate that normalizing this variation by the realized probe\-input movement provides a more effective online signal for exact\-compute allocation\.
- •We propose CacheScope, a lightweight drop\-in scheduling rule that converts trajectory\-aware sensitivity into budget\-constrained online exact\-compute decisions\. Experiments on image and video diffusion transformers demonstrate a new speed–fidelity frontier, with up to3\.57×3\.57\\timesacceleration and up to70\.1%70\.1\\%lower median LPIPS than the strongest probe\-based baseline\.

## Related Work

![Refer to caption](https://arxiv.org/html/2610.12530v1/method-1.png)Figure 2:Overview of CacheScope\.At every denoising step, the first transformer block is executed as a shallow probe\. From its input and output we compute the relative trajectory movementΔ​xt\\Delta x\_\{t\}and probe responseΔ​yt\\Delta y\_\{t\}, and accumulate the trajectory\-conditioned sensitivityst=Δ​yt/\(Δ​xt\+ϵ\)s\_\{t\}=\\Delta y\_\{t\}/\(\\Delta x\_\{t\}\+\\epsilon\)— the only change relative to raw probe scheduling\. When the accumulator reaches the thresholdτ\\tau, the remaining blocks are evaluated exactly and the cache is refreshed; otherwise the step is reconstructed by the unchanged bounded first\-order reconstructor\. Under classifier\-free guidance, the two branches maintain independent accumulators and cache states\.### Efficient Diffusion Inference

Diffusion acceleration methods reduce either the number or the cost of denoising evaluations\([Zhang et al\. 2025g](https://arxiv.org/html/2610.12530#bib.bib57)\)\. Fast samplers use improved numerical solvers to generate samples with fewer steps\([Song et al\. 2021a](https://arxiv.org/html/2610.12530#bib.bib1);[Lu et al\. 2022](https://arxiv.org/html/2610.12530#bib.bib5);[Lu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib6);[Zhao et al\. 2023](https://arxiv.org/html/2610.12530#bib.bib33);[Karras et al\. 2022](https://arxiv.org/html/2610.12530#bib.bib34)\); distillation compresses multi\-step generation into a single\-step or few\-step model\([Salimans and Ho 2022](https://arxiv.org/html/2610.12530#bib.bib7);[Song et al\. 2023](https://arxiv.org/html/2610.12530#bib.bib8);[Luo et al\. 2023](https://arxiv.org/html/2610.12530#bib.bib9)\); and quantization lowers the precision of weights and activations\([Li et al\. 2023](https://arxiv.org/html/2610.12530#bib.bib10)\); adjacent work on multimodal models\([Yao et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib50);[Zhang et al\. 2025c](https://arxiv.org/html/2610.12530#bib.bib51);[Zhang et al\. 2025d](https://arxiv.org/html/2610.12530#bib.bib55);[Zhang et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib63)\)similarly reduces cost by token\-level compression\([Zhang et al\. 2026b](https://arxiv.org/html/2610.12530#bib.bib3);[Zhang et al\. 2026d](https://arxiv.org/html/2610.12530#bib.bib61)\), and broader visual reasoning or coordination settings\([Zhang et al\. 2025e](https://arxiv.org/html/2610.12530#bib.bib60);[Zhang et al\. 2025i](https://arxiv.org/html/2610.12530#bib.bib49);[Zhang et al\. 2025j](https://arxiv.org/html/2610.12530#bib.bib53)\)offer related perspectives\. Training\-free feature reuse is complementary to these directions: it preserves the pretrained model and sampling trajectory while replacing selected full\-network evaluations with cached approximations\.

### Training\-Free Feature Reuse

Feature caching exploits redundancy across adjacent denoising steps\. DeepCache reuses slowly varying U\-Net features\([Ma et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib11)\), while FORA andΔ\\Delta\-DiT extend fixed\-stride reuse to diffusion transformers\([Selvaraju et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib35);[Chen et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib36)\)\. Later methods operate at different granularities, including token\-level reuse\([Zou et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib37)\), attention broadcasting\([Zhao et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib12)\), cross\-branch reuse\([Lv et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib13);[Ho and Salimans 2022](https://arxiv.org/html/2610.12530#bib.bib27)\), and sample\-adaptive intervals\([Kahatapitiya et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib14)\)\.

Another line improves*how to reuse*\. TeaCache and MagCache model feature or residual evolution to control reuse\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15);[Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16)\); TaylorSeer forecasts higher\-order feature dynamics through Taylor expansion\([Liu et al\. 2025b](https://arxiv.org/html/2610.12530#bib.bib17)\); and EasyCache reuses transformation vectors according to online stability estimates\([Zhou et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib18)\)\. Our controlled experiments show that, once a stable bounded first\-order reconstructor is adopted, increasing reconstruction complexity brings limited and inconsistent gains\. We therefore keep the reconstructor fixed and focus on*when to reuse*\.

### Adaptive Cache Scheduling

Adaptive scheduling determines which steps should reuse cached features and which should retain exact computation\. Existing methods use input\-side changes, motion cues, transformation stability, or attention statistics to adjust reuse intervals\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15);[Kahatapitiya et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib14);[Zhou et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib18);[Zhao et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib12);[Zhang et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib38)\)\.

DiCache is the closest work to ours\. It runs a shallow probe online and accumulates its absolute feature variation to trigger exact computation; it also aligns multiple cached residuals for reconstruction\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)\. CacheScope retains the same probe\-based setting and reconstructor, but revisits the scheduling signal\. We observe that absolute probe variation conflates input\-trajectory displacement with network response strength\. CacheScope removes this confounding by normalizing probe variation with the corresponding input displacement, yielding an online directional gain that better identifies risky reuse steps\.

Related methods differ in how sensitivity or schedules are obtained\. SenCache also argues that sensitivity, rather than raw output change, should drive caching decisions: it estimates finite\-difference Jacobian norms of the full model on a small offline calibration set and looks up the resulting per\-timestep sensitivity profiles at inference\([Haghighi and Alahi 2026](https://arxiv.org/html/2610.12530#bib.bib44)\)\.CacheScope instead measures sensitivity*online*from a shallow probe along the realized denoising trajectory of the current sample\. It requires no offline sensitivity profiling or per\-timestep lookup table; only the scalar thresholdτ\\tauis calibrated on a small development set to match the target computation budget\. At inference, CacheScope uses a per\-step response\-per\-movement ratio rather than a precomputed sensitivity profile shared across samples\.DPCache constructs an offline global schedule through path\-cost optimization, while CacheScope makes prompt\-specific decisions during inference\([Cui et al\. 2026](https://arxiv.org/html/2610.12530#bib.bib45)\)\. Concurrent ReCache learns a budget\-conditioned scheduling policy, whereas CacheScope requires no policy training or auxiliary network\([Aliev et al\. 2026](https://arxiv.org/html/2610.12530#bib.bib46)\)\. Thus, our contribution is not a new reconstruction mechanism, but the identification and removal of trajectory\-motion confounding in online probe\-based scheduling\.

## Method

CacheScope addresses a single question: given a base caching method, at which of its reuse steps should a small budget of additional exact evaluations be spent? The resulting pipeline \(Figure[2](https://arxiv.org/html/2610.12530#Sx2.F2)\) repeats a five\-stage loop: follow the fixed reuse policy, run a shallow probe, form the output\-over\-input movement ratio, accumulate sensitivity across reuse steps, and re\-anchor with an exact evaluation once the accumulated signal crosses a threshold\. CacheScope does not redesign how cached outputs are reconstructed; it estimates how quickly the current cache anchor loses validity and spends a small exact\-computation budget to refresh it\.

### Problem Formulation

Consider a generation trajectory withNNdenoiser evaluations\. Let𝐳t\\mathbf\{z\}\_\{t\}denote the latent before thett\-th evaluation,ccthe condition, andFθF\_\{\\theta\}the pretrained denoiser, whose output𝐫t=Fθ​\(𝐳t,t,c\)\\mathbf\{r\}\_\{t\}=F\_\{\\theta\}\(\\mathbf\{z\}\_\{t\},t,c\)is consumed by the sampler updateΨt\\Psi\_\{t\}\. A fixed*base cache policy*\{bt\}t=1N\\\{b\_\{t\}\\\}\_\{t=1\}^\{N\},bt∈\{0,1\}b\_\{t\}\\in\\\{0,1\\\}, specifies which steps the base method computes exactly \(bt=1b\_\{t\}\{=\}1\) and which it reuses from cache \(bt=0b\_\{t\}\{=\}0\); it may contain warm\-up steps or any predefined reuse pattern and remains unchanged throughout\. CacheScope introduces a binary*fallback*ata\_\{t\}only at nominal reuse steps, giving the execution decision

et=bt∨at,at≤1−bt\.e\_\{t\}=b\_\{t\}\\lor a\_\{t\},\\qquad a\_\{t\}\\leq 1\-b\_\{t\}\.\(3\)The output used by the sampler is

𝐫~t=\{Fθ​\(𝐳t,t,c\),et=1,ℛt​\(ℋt−1\),et=0,\\widetilde\{\\mathbf\{r\}\}\_\{t\}=\\begin\{cases\}F\_\{\\theta\}\(\\mathbf\{z\}\_\{t\},t,c\),&e\_\{t\}=1,\\\\\[2\.84526pt\] \\mathcal\{R\}\_\{t\}\(\\mathcal\{H\}\_\{t\-1\}\),&e\_\{t\}=0,\\end\{cases\}\(4\)whereℛt\\mathcal\{R\}\_\{t\}is the*fixed*cache reconstructor andℋt−1\\mathcal\{H\}\_\{t\-1\}holds the exact states available before steptt\. WritingG𝐛,𝐚G\_\{\\mathbf\{b\},\\mathbf\{a\}\}for the final generation under fallback decisions𝐚\\mathbf\{a\},𝐚π\\mathbf\{a\}\_\{\\pi\}for the decisions produced by an online policyπ\\pi, andGfullG\_\{\\mathrm\{full\}\}for the full\-computation reference, the allocation problem is

minπ\\displaystyle\\min\_\{\\pi\}𝔼⁡\[𝒟⁡\(G𝐛,𝐚π,Gfull\)\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\mathcal\{D\}\\\!\\left\(G\_\{\\mathbf\{b\},\\mathbf\{a\}\_\{\\pi\}\},G\_\{\\mathrm\{full\}\}\\right\)\\right\]\(5\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}𝔼⁡\[∑t=1N\(1−bt\)​at\]≤Badd\.\\displaystyle\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=1\}^\{N\}\(1\-b\_\{t\}\)a\_\{t\}\\right\]\\leq B\_\{\\mathrm\{add\}\}\.whereπ\\piis an online policy andBaddB\_\{\\mathrm\{add\}\}the budget of additional exact evaluations\. Both the base schedule and the reconstructor are fixed; the only free variable is*where*the additional exact evaluations are placed\. This restriction separates CacheScope from methods that regenerate a complete adaptive cache schedule and lets any improvement be attributed to exact\-compute placement alone\. Because the distortion𝒟\\mathcal\{D\}is unavailable online, CacheScope instead builds an inexpensive signal indicating when the current cache anchor becomes unreliable\.

### Trajectory\-Normalized Sensitivity

We split the denoiser asFθ=Hθ∘B1∘EθF\_\{\\theta\}=H\_\{\\theta\}\\circ B\_\{1\}\\circ E\_\{\\theta\}and execute only the prefixB1∘EθB\_\{1\}\\circ E\_\{\\theta\}as a shallow probe:

𝐱t=Eθ​\(𝐳t,t,c\),𝐲t=B1​\(𝐱t\),\\mathbf\{x\}\_\{t\}=E\_\{\\theta\}\(\\mathbf\{z\}\_\{t\},t,c\),\\qquad\\mathbf\{y\}\_\{t\}=B\_\{1\}\(\\mathbf\{x\}\_\{t\}\),\(6\)whereEθE\_\{\\theta\}is the input transformation,B1B\_\{1\}the first transformer block, andHθH\_\{\\theta\}the remaining blocks\. Withd⁡\(⋅,⋅\)d\(\\cdot,\\cdot\)denoting the scale\-normalized relative change between consecutive states \(mean absolute difference divided by theϵ\\epsilon\-stabilized mean magnitude of the earlier state; Appendix[B](https://arxiv.org/html/2610.12530#A2)\), the probe movements are

Δ​xt=d⁡\(𝐱t,𝐱t−1\),Δ​yt=d⁡\(𝐲t,𝐲t−1\),\\Delta x\_\{t\}=d\(\\mathbf\{x\}\_\{t\},\\mathbf\{x\}\_\{t\-1\}\),\\qquad\\Delta y\_\{t\}=d\(\\mathbf\{y\}\_\{t\},\\mathbf\{y\}\_\{t\-1\}\),\(7\)and the CacheScope signal is

st=Δ​ytΔ​xt\+ϵ\.s\_\{t\}=\\frac\{\\Delta y\_\{t\}\}\{\\Delta x\_\{t\}\+\\epsilon\}\.\(8\)
This normalization matters because raw probe variation conflates two distinct factors\. Let𝐯t=𝐱t−𝐱t−1\\mathbf\{v\}\_\{t\}=\\mathbf\{x\}\_\{t\}\-\\mathbf\{x\}\_\{t\-1\}be the local trajectory direction\. A first\-order expansion of the probe gives

𝐲t−𝐲t−1≈Jt​𝐯t,λt=‖Jt​𝐯t‖‖𝐯t‖,\\mathbf\{y\}\_\{t\}\-\\mathbf\{y\}\_\{t\-1\}\\approx J\_\{t\}\\mathbf\{v\}\_\{t\},\\qquad\\lambda\_\{t\}=\\frac\{\\left\\\|J\_\{t\}\\mathbf\{v\}\_\{t\}\\right\\\|\}\{\\left\\\|\\mathbf\{v\}\_\{t\}\\right\\\|\},\(9\)withJtJ\_\{t\}the local Jacobian ofB1B\_\{1\}, so that

Δ​yt≈Δ​xt⏟trajectory speed​λt⏟network sensitivity,Δ​ytΔ​xt≈λt\.\\Delta y\_\{t\}\\approx\\underbrace\{\\Delta x\_\{t\}\}\_\{\\text\{trajectory speed\}\}\\;\\underbrace\{\\lambda\_\{t\}\}\_\{\\text\{network sensitivity\}\},\\qquad\\frac\{\\Delta y\_\{t\}\}\{\\Delta x\_\{t\}\}\\approx\\lambda\_\{t\}\.\(10\)up to a slowly varying input–output scale ratio made precise in Appendix[B](https://arxiv.org/html/2610.12530#A2)\. Raw probe variation therefore conflates trajectory speed with network responsiveness; dividing out the former isolates the local directional sensitivityλt\\lambda\_\{t\}\. Under a tight exact\-computation budget, removing this nuisance factor can shift the few threshold crossings that determine where anchors are placed, and each shifted anchor changes every subsequent reuse step\. The full derivation, the resulting ranking\-inversion condition, and a detailed pathwise error analysis are provided in Appendix[B](https://arxiv.org/html/2610.12530#A2)\.

### Online Exact\-Compute Allocation

A singlests\_\{t\}measures one transition, but the same cache anchor typically serves several consecutive reuse steps, and transitions that are individually moderate can jointly invalidate it\. CacheScope therefore accumulates the sensitivity since the latest exact evaluation and triggers a fallback once the accumulated value crosses a calibrated thresholdτ\\tau:

A~t=At−1\+st\\displaystyle\\widetilde\{A\}\_\{t\}=A\_\{t\-1\}\+s\_\{t\},at=𝕀\[bt=0∧A~t≥τ\],\\displaystyle,\\qquad a\_\{t\}=\\mathbb\{I\}\\\!\\left\[b\_\{t\}\{=\}0\\land\\widetilde\{A\}\_\{t\}\\geq\\tau\\right\],\(11\)At\\displaystyle A\_\{t\}=\(1−et\)​A~t\.\\displaystyle=\(1\-e\_\{t\}\)\\,\\widetilde\{A\}\_\{t\}\.
Equation \([11](https://arxiv.org/html/2610.12530#Sx3.E11)\) defines the online allocation rule for one guidance branch\. The accumulatorAtA\_\{t\}tracks the sensitivity accumulated since the latest exact evaluation, allowing either one highly sensitive transition or several moderate transitions to trigger a fallback\. After an exact re\-anchoring,AtA\_\{t\}is reset so that sensitivity associated with the previous cache anchor is not carried over to the refreshed state\. Under local linearization, the accumulated score serves as a lightweight proxy for the amplification of cache error after the latest exact anchor \(Appendix[B](https://arxiv.org/html/2610.12530#A2)\)\. The thresholdτ\\tauis calibrated once on a development set to match the target additional exact\-computation budgetBaddB\_\{\\mathrm\{add\}\}, without optimizing test\-set quality, and is then frozen\. Under classifier\-free guidance, the two branches maintain independent probe states, accumulators, and cache states \(Appendix[C](https://arxiv.org/html/2610.12530#A3)\)\.

### Re\-Anchoring and Cost

When a fallback triggers, the probe output is already available, so the exact evaluation simply continues through the remaining blocks:

𝐫t=Hθ​\(𝐲t\),\\mathbf\{r\}\_\{t\}=H\_\{\\theta\}\(\\mathbf\{y\}\_\{t\}\),\(12\)after which the residual cache is refreshed and the accumulator is reset\. At reuse steps, cached outputs are still produced by the base configuration’s bounded first\-order reconstructorℛt\\mathcal\{R\}\_\{t\}\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\), which CacheScope leaves unchanged \(Appendix[C](https://arxiv.org/html/2610.12530#A3)\)\. An exact fallback therefore not only corrects the current output but also replaces the cache anchor used by subsequent reuse steps; this persistent downstream effect is why exact\-compute allocation cannot be reduced to independent per\-step error minimization\. The overhead is minimal: at each reuse step CacheScope adds one shallow block followed only by scalar reductions, and when a fallback triggers, the exact evaluation resumes from the existing probe output, so the probe is never recomputed \(Appendix[C](https://arxiv.org/html/2610.12530#A3)\)\.

## Experiments

Table 1:Main results\.ρ\\rhodenotes the fraction of skipped model evaluations, and Param\. denotes each method’s original speed\-control hyperparameter\. Within each backbone and regime, the best and second\-best fidelity results in the target speed range are bolded and underlined, respectively; “–” indicates that the target operating point is unavailable\.BackboneRegimeMethodParam\.ρ\\rho\(%\)Lat\. \(s\)↓\\downarrowSpeedup↑\\uparrowLPIPS↓\\downarrowPSNR↑\\uparrowSSIM↑\\uparrowWan2\.1\[\-1pt\] 50 steps, 81 frames\[\-1pt\]832×480832\{\\times\}480ModerateTeaCache\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15)\)δ=0\.14\\delta\{=\}0\.1460102\.412\.40×\\times0\.115323\.830\.8676MagCache\([Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16)\)δ=0\.12\\delta\{=\}0\.1258107\.102\.29×\\times0\.066327\.260\.9153EasyCache\([Zhou et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib18)\)τ=0\.058\\tau\{=\}0\.0586492\.592\.65×\\times0\.090924\.890\.8319DiCache\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)δ=0\.16\\delta\{=\}0\.166498\.472\.49×\\times0\.083525\.780\.8958CacheScopeτ=5\.5\\tau\{=\}5\.56498\.392\.49×\\times0\.041428\.890\.9220FastTeaCache\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15)\)δ=0\.20\\delta\{=\}0\.207273\.763\.33×\\times0\.233520\.120\.7358MagCache\([Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16)\)–––––––EasyCache\([Zhou et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib18)\)τ=0\.20\\tau\{=\}0\.207371\.123\.46×\\times0\.276120\.400\.6245DiCache\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)δ=0\.50\\delta\{=\}0\.507670\.843\.46×\\times0\.202720\.870\.7492CacheScopeτ=20\.0\\tau\{=\}20\.07670\.823\.46×\\times0\.160721\.800\.7363Open\-Sora 1\.2\[\-1pt\] 30 steps, 51 frames\[\-1pt\] 480pModerateTeaCache\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15)\)δ=0\.08\\delta\{=\}0\.082741\.971\.36×\\times0\.042128\.880\.9273MagCache\([Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16)\)δ=0\.003\\delta\{=\}0\.0032741\.921\.36×\\times0\.073225\.370\.8945PAB\([Zhao et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib12)\)R=2R\{=\}2042\.231\.35×\\times0\.078326\.920\.8864DiCache\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)δ=0\.03\\delta\{=\}0\.033040\.691\.40×\\times0\.056626\.270\.9141CacheScopeτ=2\.75\\tau\{=\}2\.753040\.691\.40×\\times0\.016934\.950\.9623FastTeaCache\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15)\)δ=0\.17\\delta\{=\}0\.175326\.812\.13×\\times0\.174021\.530\.8122MagCache\([Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16)\)δ=0\.01766\\delta\{=\}0\.017665326\.732\.14×\\times0\.124223\.030\.8344DiCache\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)δ=0\.07\\delta\{=\}0\.075327\.872\.05×\\times0\.129322\.070\.8294CacheScopeτ=6\.0\\tau\{=\}6\.05327\.832\.05×\\times0\.066426\.130\.9004FLUX\.1\-dev\[\-1pt\] 50 steps\[\-1pt\]1024×10241024\{\\times\}1024ModerateTeaCache\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15)\)δ=0\.55\\delta\{=\}0\.556610\.472\.899×\\times0\.284318\.470\.7730MagCache\([Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16)\)δ=0\.17\\delta\{=\}0\.176610\.492\.893×\\times0\.120924\.880\.8747EasyCache\([Zhou et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib18)\)τ=0\.18\\tau\{=\}0\.186610\.482\.897×\\times0\.168025\.930\.8786TaylorSeer\([Liu et al\. 2025b](https://arxiv.org/html/2610.12530#bib.bib17)\)N=3N\{=\}36411\.822\.568×\\times0\.204519\.900\.8095DiCache\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)δ=0\.15\\delta\{=\}0\.156810\.272\.954×\\times0\.135323\.320\.8769CacheScopeτ=4\.6\\tau\{=\}4\.66810\.282\.954×\\times0\.081627\.740\.9219FastTeaCache\([Liu et al\. 2025a](https://arxiv.org/html/2610.12530#bib.bib15)\)δ=0\.71\\delta\{=\}0\.71709\.253\.311×\\times0\.326816\.870\.7303MagCache\([Ma et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib16)\)δ=0\.48\\delta\{=\}0\.48728\.673\.499×\\times0\.191321\.970\.8300EasyCache\([Zhou et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib18)\)τ=0\.36\\tau\{=\}0\.36728\.673\.503×\\times0\.297823\.880\.8083TaylorSeer\([Liu et al\. 2025b](https://arxiv.org/html/2610.12530#bib.bib17)\)N=5N\{=\}5768\.373\.624×\\times0\.315717\.110\.7289DiCache\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)δ=0\.25\\delta\{=\}0\.25748\.493\.601×\\times0\.184221\.400\.8333CacheScopeτ=6\.4\\tau\{=\}6\.4748\.513\.566×\\times0\.124624\.990\.8760Table 2:Controlled scheduler comparison on3232held\-out prompts\.The three signals \(D, M, S\) share the same DiCache bounded first\-orderΓ\\Gammareconstructor, probe depth, and warm\-up policy, and differ only in the accumulated scheduling signal\. Values are medians over prompts\. Statistical testing uses the strictly matched\-skip subsets described in Appendix[F\.2](https://arxiv.org/html/2610.12530#A6.SS2)\. Best values within each backbone and regime are in bold; second\-best values are underlined\.### Experimental Setup

#### Setup\.

We evaluate on FLUX\.1\-dev\([Black Forest Labs 2024](https://arxiv.org/html/2610.12530#bib.bib28)\), Wan2\.1\-T2V\-1\.3B\([Wan Team 2025](https://arxiv.org/html/2610.12530#bib.bib29)\), and Open\-Sora 1\.2\([Zheng et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib30)\), using the generation protocols listed in Table[1](https://arxiv.org/html/2610.12530#Sx4.T1)\. For each backbone, scheduling thresholds are calibrated on a 16\-prompt development set and frozen before evaluation on a disjoint 32\-prompt test set, drawn from DrawBench\([Saharia et al\. 2022](https://arxiv.org/html/2610.12530#bib.bib41)\)for FLUX and VBench\([Huang et al\. 2024](https://arxiv.org/html/2610.12530#bib.bib42)\)for the video models\. Keeping the probe, reconstructor, cache policy, and model implementation fixed, we compare the DiCache signalΔ​yt\\Delta y\_\{t\}, the difference signal\|Δ​yt−Δ​xt\|\|\\Delta y\_\{t\}\-\\Delta x\_\{t\}\|, and our normalized signalΔ​yt/\(Δ​xt\+ϵ\)\\Delta y\_\{t\}/\(\\Delta x\_\{t\}\+\\epsilon\)under matched skip counts at two acceleration regimes\. We report LPIPS\([Zhang et al\. 2018](https://arxiv.org/html/2610.12530#bib.bib39)\), PSNR, SSIM\([Wang et al\. 2004](https://arxiv.org/html/2610.12530#bib.bib40)\), latency, and speedup, with paired prompt\-level comparisons\. Full generation settings, hyperparameters, hardware, random seeds, and statistical protocols are provided in the supplementary material\.

### Main Results

Table[1](https://arxiv.org/html/2610.12530#Sx4.T1)reports the median quality and measured speedup over the3232held\-out prompts, and three observations stand out\. First, the improvement is consistent across all three backbones: at matched speedups, CacheScope achieves the lowest median LPIPS in every backbone–regime combination, on the image transformer FLUX\.1\-dev as well as the video transformers Wan2\.1 and Open\-Sora 1\.2\. Second, the largest gain appears on Open\-Sora at the Moderate level, where LPIPS drops from DiCache’s0\.05660\.0566to0\.01690\.0169at identical compute—a70\.1%70\.1\\%reduction\. Third, the advantage persists in the most aggressive regimes: at about3\.57×3\.57\\timeson FLUX and3\.46×3\.46\\timeson Wan2\.1, CacheScope still lowers median LPIPS from0\.18420\.1842to0\.12460\.1246\(32\.4%32\.4\\%\) and from0\.20270\.2027to0\.16070\.1607\(20\.7%20\.7\\%\) relative to the strongest baseline at the same operating point, so the gains are not confined to conservative caching settings\. Baseline hyperparameters are selected on the development set to place each method inside the target speed range of its regime \(Appendix[D\.1](https://arxiv.org/html/2610.12530#A4.SS1)\)\.

![Refer to caption](https://arxiv.org/html/2610.12530v1/fig4-1.png)Figure 3:Qualitative comparison on FLUX\.1\-dev in the Fast regime\.Columns show the dense output and different training\-free caching methods, while rows correspond to three prompts\. Red boxes highlight visible deviations from the dense reference\. CacheScope better preserves fine structures and object geometry at3\.57×3\.57\\timesspeedup\.Table[2](https://arxiv.org/html/2610.12530#Sx4.T2)isolates the effect of the scheduling signal on the two backbones for which all three signals were run under the controlled protocol, Wan2\.1 and FLUX, by holding the reconstructor, probe depth, and warm\-up policy fixed\. Relative to the default DiCache signal D, probe normalization reduces median LPIPS by46\.4%46\.4\\%and24\.2%24\.2\\%on Wan2\.1 and by39\.7%39\.7\\%and32\.4%32\.4\\%on FLUX in the Moderate and Fast regimes, respectively; relative to the stronger difference\-based signal M, the reductions range from11\.9%11\.9\\%to36\.2%36\.2\\%\. Because the three schedulers execute the same shallow probe and nearly identical numbers of exact forwards, their speed differences are negligible, which isolates the effect of exact\-step placement from the amount of computation\.

#### Statistical reliability\.

Paired prompt\-level LPIPS differencesΔ​L=LS−Lbaseline\\Delta L=L\_\{\\mathrm\{S\}\}\-L\_\{\\mathrm\{baseline\}\}under strictly equal skip counts, where negative values favor our scheduler, are reported in full—with confidence intervals and win–tie–loss counts—in Appendix[F\.2](https://arxiv.org/html/2610.12530#A6.SS2)\(Table[7](https://arxiv.org/html/2610.12530#A6.T7)\)\. Our scheduler significantly outperforms both comparison signals in all eight LPIPS comparisons, and the four primary S\-versus\-D comparisons remain significant after Benjamini–Hochberg correction\. The improvements replicate with an independent seed \(4343\) at similar effect magnitudes, and the conclusions do not reverse when all3232prompts are analyzed without matched\-skip filtering; median skip\-count differences lie in\{−1,0\}\\\{\-1,0\\\}, so the gains cannot be explained by skipping fewer evaluations \(full replication and robustness tables are given in the supplementary material\)\. On FLUX, the LPIPS improvements are accompanied by consistent gains in PSNR and SSIM; on Wan2\.1, SSIM exhibits isolated trade\-offs, so we treat perceptual fidelity \(LPIPS\) as the primary outcome rather than claiming uniform dominance on every metric\.

### Ablation and Overhead

![Refer to caption](https://arxiv.org/html/2610.12530v1/fig5.png)Figure 4:Reconstruction ablation and multi\-metric comparison\.\(a\)Δ\\DeltaLPIPS relative to the bounded first\-orderΓ\\Gammareconstructor under identical replayed exact/skip schedules \(positive==worse\)\. \(b\) VBench sub\-dimension scores on Wan2\.1 \(Moderate\),*min–max normalized per metric across methods*; raw scores are given in Appendix[F\.4](https://arxiv.org/html/2610.12530#A6.SS4)\.#### Reconstruction complexity\.

Figure[4](https://arxiv.org/html/2610.12530#Sx4.F4)\(a\) examines whether further increasing reconstruction complexity provides a comparable benefit; all variants replay the same exact/skip trace, so the comparison changes only the reconstruction rule \(absolute median LPIPS and per\-prompt win–loss counts are given in Appendix[F\.10](https://arxiv.org/html/2610.12530#A6.SS10)\)\.

Across the four settings, none of the unconstrained multi\-history least\-squares variants consistently improves over the bounded first\-orderΓ\\Gammareconstructor\. The degradation becomes pronounced and increases with history length on Wan2\.1, whereas the changes on FLUX are smaller and non\-monotonic\. These results support the narrower conclusion that, within the examined design space, additional reconstruction flexibility is not a reliable source of gains\. We therefore fix a stable bounded first\-order reconstructor and attribute the primary improvements of CacheScope to exact\-compute scheduling rather than to a more flexible residual predictor\.

#### Computational overhead\.

The additional scheduler computation is limited to evaluatingΔ​xt\\Delta x\_\{t\}and one scalar division\. At matched operating points, the observed latency difference between S and D is below0\.15%0\.15\\%on FLUX and0\.1%0\.1\\%on Wan2\.1, and the measured speedup reaches0\.900\.90–0\.990\.99of the analytical FLOPs estimate, with the remaining gap explained by fixed costs such as text encoding, VAE decoding, and scheduler bookkeeping\. Probe normalization thus changes*where*exact computation is allocated without materially changing the total inference cost\.

## Conclusion

We revisited training\-free feature reuse for diffusion transformers and showed that, once a stable bounded first\-order reconstructor is fixed, the placement of exact evaluations is a dominant factor for generation fidelity, while the more flexible reconstruction formulations we examined did not consistently outperform the first\-order rule\. We further identified that raw probe variation is not a directly comparable measure of step importance, because it conflates diffusion\-trajectory displacement with the network response induced by that displacement\. Motivated by this, we proposed CacheScope, which schedules exact computation from a trajectory\-conditioned sensitivity score and converts it into a budget\-constrained online allocation policy without any training, gradients, auxiliary model, or offline schedule search\. Across image and video diffusion transformers, CacheScope establishes a new speed–fidelity frontier under matched exact\-compute budgets\. A current limitation is that sensitivity is estimated from a single shallow probe along the realized denoising direction; multi\-probe and higher\-order trajectory signals remain future work\.

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Supplementary Material

CacheScope: Exact\-Compute Allocation

for Training\-Free Diffusion Acceleration

## Appendix ARelationship to Existing Scheduling Signals

We consider three scheduling signals:

σtD=Δ​yt,σtM=\|Δ​yt−Δ​xt\|,σtS=Δ​ytΔ​xt\+ϵ\.\\sigma\_\{t\}^\{D\}=\\Delta y\_\{t\},\\qquad\\sigma\_\{t\}^\{M\}=\\left\|\\Delta y\_\{t\}\-\\Delta x\_\{t\}\\right\|,\\qquad\\sigma\_\{t\}^\{S\}=\\frac\{\\Delta y\_\{t\}\}\{\\Delta x\_\{t\}\+\\epsilon\}\.\(13\)
Here,σtD\\sigma\_\{t\}^\{D\}is raw probe\-output variation,σtM\\sigma\_\{t\}^\{M\}is the difference\-based alternative, andσtS\\sigma\_\{t\}^\{S\}is the trajectory\-conditioned sensitivity used by CacheScope\.

The difference\-based and ratio\-based signals encode distinct decision semantics\. When the probe response is much weaker than the trajectory movement,

Δ​yt≪Δ​xt,\\Delta y\_\{t\}\\ll\\Delta x\_\{t\},\(14\)we have

σtM≈Δ​xt,σtS≪1\.\\sigma\_\{t\}^\{M\}\\approx\\Delta x\_\{t\},\\qquad\\sigma\_\{t\}^\{S\}\\ll 1\.\(15\)The difference\-based signal may therefore assign a large score to a step whose network response is weak relative to its input movement\.

When the input and output variations have similar magnitudes,

Δ​yt≈Δ​xt,\\Delta y\_\{t\}\\approx\\Delta x\_\{t\},\(16\)we instead have

σtM≈0,σtS≈1\.\\sigma\_\{t\}^\{M\}\\approx 0,\\qquad\\sigma\_\{t\}^\{S\}\\approx 1\.\(17\)
The two signals are not related by a global scalar transformation\. Ignoringϵ\\epsilonfor illustration, consider

\(Δ​x,Δ​y\)=\(1,0\.5\)\(\\Delta x,\\Delta y\)=\(1,0\.5\)\(18\)and

\(Δ​x,Δ​y\)=\(2,1\)\.\(\\Delta x,\\Delta y\)=\(2,1\)\.\(19\)Both cases produce

σS=0\.5,\\sigma^\{S\}=0\.5,\(20\)but their difference\-based scores are

σM=0\.5andσM=1,\\sigma^\{M\}=0\.5\\qquad\\text\{and\}\\qquad\\sigma^\{M\}=1,\(21\)respectively\. Therefore, no globally defined scalar functionhhcan satisfy

σtM=h⁡\(σtS\)\\sigma\_\{t\}^\{M\}=h\(\\sigma\_\{t\}^\{S\}\)\(22\)for every valid pair\(Δ​xt,Δ​yt\)\(\\Delta x\_\{t\},\\Delta y\_\{t\}\)\.

The two signals may also reverse the ranking of denoising steps\. For example,

\(Δ​x1,Δ​y1\)=\(1,0\.2\),\(Δ​x2,Δ​y2\)=\(0\.2,0\.1\)\(\\Delta x\_\{1\},\\Delta y\_\{1\}\)=\(1,0\.2\),\\qquad\(\\Delta x\_\{2\},\\Delta y\_\{2\}\)=\(0\.2,0\.1\)\(23\)give

σ1M=0\.8\>σ2M=0\.1,\\sigma\_\{1\}^\{M\}=0\.8\>\\sigma\_\{2\}^\{M\}=0\.1,\(24\)but

σ1S=0\.2<σ2S=0\.5\.\\sigma\_\{1\}^\{S\}=0\.2<\\sigma\_\{2\}^\{S\}=0\.5\.\(25\)Thus, difference\-based scheduling prioritizes the first step, whereas CacheScope assigns greater sensitivity to the second step\.

### A\.1Offline Signal Aggregation and Schedule Information

#### Motivation\.

We further examine whether the advantage of the trajectory\-normalized signal depends on prompt\-specific online fluctuations, or whether it also reveals a stable timestep structure shared across prompts\. We construct two fixed schedules using only the 16\-prompt development set\. ForGlobal\-S, we independently average the normalized signal

st=Δ​ytΔ​xt\+ϵs\_\{t\}=\\frac\{\\Delta y\_\{t\}\}\{\\Delta x\_\{t\}\+\\epsilon\}\(26\)at each timestep and apply the same accumulator\-and\-threshold rule used by CacheScope to the resulting mean curve\. ForGlobal\-D, we repeat exactly the same procedure using the raw probe variationΔ​yt\\Delta y\_\{t\}\. The threshold for each schedule is selected solely to match the target median skip count of 34 steps; no fidelity metric is used during schedule construction\.

Both fixed masks are generated before any validation output or evaluation metric is accessed\. The development, validation, and intervention prompt sets are mutually disjoint\. We additionally evaluate cross\-prompt replay, in which theOnline\-Sschedule generated for promptiiis applied to prompti\+1i\+1under a cyclic derangement\. All variants continue to execute the same shallow probe; only their exact\-versus\-skip decisions are replaced\. The comparison therefore isolates the information contained in the scheduling signal and its prompt\-specific assignment\. These fixed schedules are used only for diagnostic analysis and are not part of the deployed CacheScope inference procedure\.

Table 3:Offline aggregation and schedule\-information analysis on FLUX\.1\-dev at the Moderate operating point\.Global\-SandGlobal\-Dare constructed from the same 16\-prompt development set using the same aggregation procedure, threshold solver, and target compute budget; they differ only in the scheduling signal\. Metric values are medians over 32 held\-out prompts\. The exact\-evaluation column reports totals across all 32 prompts\. All variants have a median skip count of 34\. Paired two\-sided sign tests compare each variant withOnline\-S\.
#### Results\.

We detect no significant difference betweenGlobal\-SandOnline\-S\. The median LPIPS ofGlobal\-Sis numerically lower, at 0\.0792 compared with 0\.0816 forOnline\-S, with 14 improvements, 13 degradations, and five ties across the 32 paired prompts \(p=1\.00p=1\.00\)\. Cross\-prompt replay likewise shows no significant difference fromOnline\-S\. At this FLUX operating point, these results suggest that a substantial portion of the useful scheduling structure is shared across prompts, while prompt\-specific schedule variation provides no additional statistically detectable benefit\.

Importantly, the same offline construction does not recover this shared structure when applied to the raw DiCache signal\. Compared withGlobal\-S,Global\-Dincreases median LPIPS from 0\.0792 to 0\.1337; relative toOnline\-S, it is worse on 29 of the 32 paired prompts \(p=2\.6×10−6p=2\.6\{\\times\}10^\{\-6\}\)\.Global\-SandGlobal\-Duse identical development data, schedule\-construction procedures, and compute budgets, and differ only in whether the probe variation is normalized by the corresponding input movement\. This controlled comparison indicates that trajectory normalization factors out variation attributable to input movement and exposes a substantially more effective timestep\-importance profile for allocating exact computation\.

#### Schedule analysis\.

The online schedules are not identical: the 32 prompts produce nine distinct masks, with a median pairwise Jaccard similarity of 0\.200 within the adaptive scheduling window\. The schedule\-dependent exact\-step sets ofGlobal\-SandGlobal\-Dwithin this window also differ substantially, with a Jaccard similarity of 0\.286\. Their respective locations are\{7,9,12,16,19,25,32,38,44\}\\\{7,9,12,16,19,25,32,38,44\\\}and\{9,14,19,26,32,37,41,44,47\}\\\{9,14,19,26,32,37,41,44,47\\\}\. All remaining mandatory exact evaluations are shared between the two schedules\. Thus, their large fidelity difference corresponds to a meaningful change in exact\-compute placement rather than to two nearly identical masks\.

## Appendix BDetailed Analysis of the Sensitivity Signal

This appendix provides the full derivations behind the trajectory\-normalized sensitivity of the main text: the precise definition of the relative\-change measure, the scale\-normalized form of the signal, the ranking\-inversion condition under raw probe variation, and a pathwise error analysis of the accumulated score\.

### B\.1Relative\-Change Measure and Scale\-Normalized Form

For tensors𝐮\\mathbf\{u\}and𝐯\\mathbf\{v\}, the scale\-normalized movement used throughout is

d⁡\(𝐮,𝐯\)=μ⁡\(\|𝐮−𝐯\|\)μ⁡\(\|𝐯\|\)\+ϵ,d\(\\mathbf\{u\},\\mathbf\{v\}\)=\\frac\{\\mu\\\!\\left\(\\left\|\\mathbf\{u\}\-\\mathbf\{v\}\\right\|\\right\)\}\{\\mu\\\!\\left\(\\left\|\\mathbf\{v\}\\right\|\\right\)\+\\epsilon\},\(27\)whereμ⁡\(⋅\)\\mu\(\\cdot\)denotes the mean over all non\-batch dimensions andϵ=10−6\\epsilon=10^\{\-6\}is a numerical constant\.

Substituting the first\-order expansion𝐲t−𝐲t−1≈Jt​𝐯t\\mathbf\{y\}\_\{t\}\-\\mathbf\{y\}\_\{t\-1\}\\approx J\_\{t\}\\mathbf\{v\}\_\{t\}into the implemented relative measures shows that the score approximates

st∝‖Jt​𝐯t‖‖𝐯t‖⋅‖𝐱t−1‖‖𝐲t−1‖\.s\_\{t\}\\propto\\frac\{\\left\\\|J\_\{t\}\\mathbf\{v\}\_\{t\}\\right\\\|\}\{\\left\\\|\\mathbf\{v\}\_\{t\}\\right\\\|\}\\cdot\\frac\{\\left\\\|\\mathbf\{x\}\_\{t\-1\}\\right\\\|\}\{\\left\\\|\\mathbf\{y\}\_\{t\-1\}\\right\\\|\}\.\(28\)The first factor is the directional response gainλt\\lambda\_\{t\}of the main text; the second provides relative scale normalization between the probe input and output\. Hencests\_\{t\}can be interpreted as a scale\-normalized directional response gain of the shallow denoiser\. It measures the model response only along the direction actually traversed by the trajectory, which is the direction relevant for cache scheduling; it is not an estimate of the full Jacobian norm\.

### B\.2Ranking Inversion under Raw Probe Variation

The product decompositionΔ​yt≈Δ​xt​λt\\Delta y\_\{t\}\\approx\\Delta x\_\{t\}\\lambda\_\{t\}implies that raw probe variation can invert the ordering induced by sensitivity\. Consider two candidate reuse stepsiiandjjwith

λi<λj\.\\lambda\_\{i\}<\\lambda\_\{j\}\.\(29\)Although stepjjis more sensitive, the raw variations satisfy

Δ​yi\>Δ​yjifΔ​xiΔ​xj\>λjλi,\\Delta y\_\{i\}\>\\Delta y\_\{j\}\\quad\\text\{if\}\\quad\\frac\{\\Delta x\_\{i\}\}\{\\Delta x\_\{j\}\}\>\\frac\{\\lambda\_\{j\}\}\{\\lambda\_\{i\}\},\(30\)in which case ranking or thresholding the raw signal prioritizes the*less*sensitive step because of its larger trajectory displacement\. Normalization removes this nuisance factor and restores the ordering induced by the directional response gain\. When only a small fraction of reuse steps can be replaced by exact evaluations, allocation quality is dominated by the few steps close to the decision boundary; correcting even a limited number of such ranking inversions can considerably change the selected re\-anchoring positions\.

### B\.3Pathwise Error Analysis

The accumulated score admits a pathwise error interpretation\. An exact evaluation on an already approximate latent does not recover the original full trajectory; it does, however, remove the additional cache approximation error relative to the current trajectory state\. Accordingly, we analyze the error introduced*after*the latest exact re\-anchoring\.

Suppose the latest exact evaluation occurs at steptat\_\{a\}\. Define a counterfactual reference trajectory\{𝐳¯t\}t≥ta\\\{\\overline\{\\mathbf\{z\}\}\_\{t\}\\\}\_\{t\\geq t\_\{a\}\}that starts from the current approximate latent,𝐳¯ta=𝐳~ta\\overline\{\\mathbf\{z\}\}\_\{t\_\{a\}\}=\\widetilde\{\\mathbf\{z\}\}\_\{t\_\{a\}\}, and uses exact denoiser evaluations thereafter:

𝐳¯t\+1=Ψt​\(𝐳¯t,Fθ​\(𝐳¯t,t,c\)\)\.\\overline\{\\mathbf\{z\}\}\_\{t\+1\}=\\Psi\_\{t\}\\left\(\\overline\{\\mathbf\{z\}\}\_\{t\},F\_\{\\theta\}\(\\overline\{\\mathbf\{z\}\}\_\{t\},t,c\)\\right\)\.\(31\)Let𝜹t=𝐳~t−𝐳¯t\\boldsymbol\{\\delta\}\_\{t\}=\\widetilde\{\\mathbf\{z\}\}\_\{t\}\-\\overline\{\\mathbf\{z\}\}\_\{t\}denote the additional trajectory deviation introduced by caching after the latest re\-anchoring \(so𝜹ta=𝟎\\boldsymbol\{\\delta\}\_\{t\_\{a\}\}=\\mathbf\{0\}\), and let𝜺t=𝐫~t−Fθ​\(𝐳~t,t,c\)\\boldsymbol\{\\varepsilon\}\_\{t\}=\\widetilde\{\\mathbf\{r\}\}\_\{t\}\-F\_\{\\theta\}\(\\widetilde\{\\mathbf\{z\}\}\_\{t\},t,c\)denote the cache reconstruction error relative to an exact evaluation at the same latent\. A local linearization of the sampler update gives

𝜹t\+1≈Mt​𝜹t\+Nt​𝜺t,\\boldsymbol\{\\delta\}\_\{t\+1\}\\approx M\_\{t\}\\boldsymbol\{\\delta\}\_\{t\}\+N\_\{t\}\\boldsymbol\{\\varepsilon\}\_\{t\},\(32\)whereMtM\_\{t\}andNtN\_\{t\}are the local Jacobians of the sampler update with respect to its latent and denoiser\-output arguments\.

To connect probe sensitivity with cache mismatch, we adopt a local probe\-consistency assumption:

‖𝜺t‖≤ηt\+βt​st​‖𝜹t‖,\\left\\\|\\boldsymbol\{\\varepsilon\}\_\{t\}\\right\\\|\\leq\\eta\_\{t\}\+\\beta\_\{t\}s\_\{t\}\\left\\\|\\boldsymbol\{\\delta\}\_\{t\}\\right\\\|,\(33\)whereηt\\eta\_\{t\}is the intrinsic error of the fixed reconstructor andβt\\beta\_\{t\}a bounded local coefficient: an existing trajectory deviation is more likely to produce cache mismatch when the denoiser is locally sensitive along the current direction\. Combining Eqs\. \([32](https://arxiv.org/html/2610.12530#A2.E32)\) and \([33](https://arxiv.org/html/2610.12530#A2.E33)\) yields

‖𝜹t\+1‖≤\(mt\+ct​st\)​‖𝜹t‖\+ξt,\\left\\\|\\boldsymbol\{\\delta\}\_\{t\+1\}\\right\\\|\\leq\\left\(m\_\{t\}\+c\_\{t\}s\_\{t\}\\right\)\\left\\\|\\boldsymbol\{\\delta\}\_\{t\}\\right\\\|\+\\xi\_\{t\},\(34\)withmt=‖Mt‖m\_\{t\}=\\\|M\_\{t\}\\\|,ct=‖Nt‖​βtc\_\{t\}=\\\|N\_\{t\}\\\|\\beta\_\{t\}, andξt=‖Nt‖​ηt\\xi\_\{t\}=\\\|N\_\{t\}\\\|\\eta\_\{t\}\. Writingc¯t=ct/mt\\overline\{c\}\_\{t\}=c\_\{t\}/m\_\{t\}and unrolling from the latest anchor, then applying1\+u≤exp⁡\(u\)1\+u\\leq\\exp\(u\)foru≥0u\\geq 0, gives

‖𝜹t‖≤∑j=tat−1ξj​\(∏k=j\+1t−1mk\)​exp⁡\(∑k=j\+1t−1c¯k​sk\)\.\\displaystyle\\left\\\|\\boldsymbol\{\\delta\}\_\{t\}\\right\\\|\\leq\\sum\_\{j=t\_\{a\}\}^\{t\-1\}\\xi\_\{j\}\\left\(\\prod\_\{k=j\+1\}^\{t\-1\}m\_\{k\}\\right\)\\exp\\left\(\\sum\_\{k=j\+1\}^\{t\-1\}\\overline\{c\}\_\{k\}s\_\{k\}\\right\)\.\(35\)The factors∏kmk\\prod\_\{k\}m\_\{k\}describe the underlying sampler dynamics and are shared by all allocation signals; the sensitivity\-dependent amplification is controlled by the cumulative quantity∑kc¯k​sk\\sum\_\{k\}\\overline\{c\}\_\{k\}s\_\{k\}\. When the bounded coefficientsc¯k\\overline\{c\}\_\{k\}vary slowly or are absorbed into the threshold, the unweighted accumulatorAt=∑k=ta\+1tskA\_\{t\}=\\sum\_\{k=t\_\{a\}\+1\}^\{t\}s\_\{k\}serves as a lightweight surrogate for this amplification, which motivates accumulatingsts\_\{t\}rather than thresholding isolated values\.

This analysis does not claim thatAtA\_\{t\}equals the final generation error\. It explains two design choices: normalizing removes trajectory displacement as a confounding factor, and accumulating captures unreanchored pathwise exposure\. The allocation process itself is event\-triggered and therefore nonlinear: changing the local signal may alter the first threshold\-crossing position, which changes the next cache anchor, the following reconstructed states, and all subsequent accumulator values, so a simple local normalization can induce a much larger downstream difference in the final trajectory\.

## Appendix CCacheScope Implementation Details

The probe depth is fixed to one transformer block andϵ=10−6\\epsilon=10^\{\-6\}in all experiments\. All compared scheduling signals share the same warm\-up policy, tail policy, probe depth, reconstruction rule, cache history, model configuration, and generation seed\.

### C\.1Threshold Calibration

The thresholdτ\\taudetermines the average number of adaptive exact fallbacks\. Given a development set𝒟dev\\mathcal\{D\}\_\{\\mathrm\{dev\}\}, we select

τ⋆=arg⁡minτ​\|𝔼𝒟dev​\[∑t=1N\(1−bt\)​at​\(τ\)\]−Badd\|\.\\tau^\{\\star\}=\\arg\\min\_\{\\tau\}\\left\|\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{\\mathrm\{dev\}\}\}\\left\[\\sum\_\{t=1\}^\{N\}\(1\-b\_\{t\}\)\\,a\_\{t\}\(\\tau\)\\right\]\-B\_\{\\mathrm\{add\}\}\\right\|\.\(36\)Calibration reads only fallback counts and never any quality metric; after calibration,τ⋆\\tau^\{\\star\}is frozen and applied to all evaluation prompts without prompt\-specific adaptation\. Each backbone and acceleration level uses an independently calibrated threshold, since the signal scale differs across models\. This procedure requires neither training nor gradients, and avoids offline optimization of an independent decision for every denoising step\.

### C\.2Fixed Reconstructor and Cache Update

At reuse steps, the skipped output is produced by the bounded first\-order residual reconstructor of the base configuration\([Bu et al\. 2025](https://arxiv.org/html/2610.12530#bib.bib19)\)\. Given the two most recent exact residuals𝐫t1\\mathbf\{r\}\_\{t\_\{1\}\}and𝐫t2\\mathbf\{r\}\_\{t\_\{2\}\}witht1<t2<tt\_\{1\}<t\_\{2\}<t,

𝐫^t=𝐫t2\+γt​\(𝐫t2−𝐫t1\),1≤γt≤γmax,\\widehat\{\\mathbf\{r\}\}\_\{t\}=\\mathbf\{r\}\_\{t\_\{2\}\}\+\\gamma\_\{t\}\\left\(\\mathbf\{r\}\_\{t\_\{2\}\}\-\\mathbf\{r\}\_\{t\_\{1\}\}\\right\),\\qquad 1\\leq\\gamma\_\{t\}\\leq\\gamma\_\{\\max\},\(37\)withγmax=1\.5\\gamma\_\{\\max\}=1\.5\. The extrapolation coefficientγt\\gamma\_\{t\}is computed from the same shallow\-probe indicator following the base reconstructor’s released rule and clipped to\[1,γmax\]\[1,\\gamma\_\{\\max\}\]; it is inherited unchanged and not optimized by CacheScope\. If fewer than two exact residuals are available, reconstruction falls back to directly reusing the most recent one\.

The complete per\-step state transition is

\(𝐫~t,ℋt,At\)=\{\(Hθ​\(𝐲t\),Refresh⁡\(ℋt−1,𝐫t\),0\),et=1,\(ℛt​\(ℋt−1\),ℋt−1,A~t\),et=0,\\left\(\\widetilde\{\\mathbf\{r\}\}\_\{t\},\\mathcal\{H\}\_\{t\},A\_\{t\}\\right\)=\\begin\{cases\}\\left\(H\_\{\\theta\}\(\\mathbf\{y\}\_\{t\}\),\\operatorname\{Refresh\}\\left\(\\mathcal\{H\}\_\{t\-1\},\\mathbf\{r\}\_\{t\}\\right\),0\\right\),&e\_\{t\}=1,\\\\\[8\.53581pt\] \\left\(\\mathcal\{R\}\_\{t\}\(\\mathcal\{H\}\_\{t\-1\}\),\\mathcal\{H\}\_\{t\-1\},\\widetilde\{A\}\_\{t\}\\right\),&e\_\{t\}=0,\\end\{cases\}\(38\)whereRefresh\\operatorname\{Refresh\}inserts the newly computed exact residual into the cache history and invalidates reconstruction states associated with the previous anchor\. Only residuals from exact evaluations enter the cache, which prevents approximate residuals from recursively contaminating the history\.

### C\.3Classifier\-Free Guidance

Under classifier\-free guidance, the conditional and unconditional branches may follow different representation trajectories\. For branchq∈\{cond,uncond\}q\\in\\\{\\mathrm\{cond\},\\mathrm\{uncond\}\\\}, CacheScope maintains independent probe states, accumulators, and caches:

st\(q\)=Δ​yt\(q\)Δ​xt\(q\)\+ϵ,A~t\(q\)=At−1\(q\)\+st\(q\)\.s\_\{t\}^\{\(q\)\}=\\frac\{\\Delta y\_\{t\}^\{\(q\)\}\}\{\\Delta x\_\{t\}^\{\(q\)\}\+\\epsilon\},\\qquad\\widetilde\{A\}\_\{t\}^\{\(q\)\}=A\_\{t\-1\}^\{\(q\)\}\+s\_\{t\}^\{\(q\)\}\.\(39\)Fallback decisions and cache refreshes are applied independently per branch, after which the outputs are combined by the original guidance rule\. This prevents the two interleaved trajectories from incorrectly sharing probe statistics or cache anchors\. On Wan2\.1, the computation budget is counted in denoiser forwards \(two per sampling step\), and each branch’s probe state is compared only against the previous state of the same branch\.

### C\.4Complexity

LetCfullC\_\{\\mathrm\{full\}\},CprobeC\_\{\\mathrm\{probe\}\}, andCrecC\_\{\\mathrm\{rec\}\}denote the costs of a complete denoiser evaluation, the input transformation plus first transformer block, and cached reconstruction, and letNexact=∑tetN\_\{\\mathrm\{exact\}\}=\\sum\_\{t\}e\_\{t\}\. The inference cost is approximately

CCacheScope≈\\displaystyle C\_\{\\mathrm\{CacheScope\}\}\\approxNexact​Cfull\\displaystyle N\_\{\\mathrm\{exact\}\}C\_\{\\mathrm\{full\}\}\(40\)\+\(N−Nexact\)​\(Cprobe\+Crec\)\+𝒪⁡\(N\)\.\\displaystyle\+\\left\(N\-N\_\{\\mathrm\{exact\}\}\\right\)\\left\(C\_\{\\mathrm\{probe\}\}\+C\_\{\\mathrm\{rec\}\}\\right\)\+\\mathcal\{O\}\(N\)\.The probe costs3\.3%3\.3\\%of a full forward on Wan2\.1 and1\.8%1\.8\\%on FLUX \(Appendix[F\.12](https://arxiv.org/html/2610.12530#A6.SS12)\)\. At an exact step, the probe is part of the complete evaluation and its output is reused when execution continues through the remaining blocks, so no network computation is duplicated\. Beyond the probe, each branch requires two relative\-change reductions, one scalar division, one scalar accumulation, and one threshold comparison per step\. The scheduler additionally stores the previous probe input and output per branch; the residual\-cache memory is unchanged\. CacheScope introduces no trainable parameters, auxiliary predictor, gradient computation, or offline per\-timestep schedule search\.

## Appendix DEvaluation Metrics and Statistical Protocol

### D\.1Baseline Configurations

For every compared baseline, the speed\-controlling hyperparameter listed in the Param\. column of the main\-results table in the main paper \(δ\\deltafor TeaCache, MagCache, and DiCache,τ\\taufor EasyCache,NNfor TaylorSeer, andRRfor PAB\) is selected on the development prompts so that the method’s measured speedup falls inside the target acceleration range of the corresponding regime, matching the budget of skipped evaluations where the method’s granularity allows\. Selection reads only speed and skip counts—never any quality metric—and the chosen values are frozen before any test\-set sample is generated\. When a regime lies beyond a method’s structural acceleration ceiling—the maximum speedup its formulation can reach regardless of the hyperparameter setting, as for MagCache on Wan2\.1 Fast—the corresponding row is marked “–” rather than reporting an out\-of\-range configuration\. For reference, the operating points of the two released default FLUX configurations of the DiCache baseline are reported in Appendix[F\.14](https://arxiv.org/html/2610.12530#A6.SS14); the official defaults land at operating points at least as aggressive as ours, so the calibration does not weaken the baselines\.

### D\.2Additional Open\-Source Baseline Comparisons

We additionally compare CacheScope with several recently released open\-source caching methods under the same generation and evaluation protocols\. CacheScope results are reproduced for reference\.

Table 4:Comparison with additional open\-source baselines\.All metrics are computed against the same dense references\. CacheScope results are reproduced for comparison\. SenCache is omitted from the Wan2\.1 Fast regime because its fastest attainable configuration reaches only2\.644×2\.644\\times, below the target speedup of approximately3\.46×3\.46\\times\. Best results within each backbone and regime are in bold\.CacheScope achieves the best fidelity metrics in every reported backbone–regime group\. On Wan2\.1, it reduces LPIPS from0\.07520\.0752to0\.04140\.0414in the Moderate regime and from0\.20750\.2075to0\.16070\.1607in the Fast regime\. On FLUX\.1\-dev, CacheScope also consistently improves LPIPS, PSNR, and SSIM over SenCache and DPCache at closely matched speedups\. On Open\-Sora 1\.2, SenCache operates at slightly higher speedups, but CacheScope retains substantially better fidelity in both regimes\.

### D\.3Frozen Scheduling Thresholds

Table[5](https://arxiv.org/html/2610.12530#A4.T5)lists the scheduling thresholds calibrated on the development prompts and frozen before test\-set generation\.

Table 5:Frozen scheduling thresholds\.S, D, and M differ only in their accumulated scheduling signal\.
### D\.4Quality Metrics

We measure fidelity relative to the corresponding full\-computation output using LPIPS\([Zhang et al\. 2018](https://arxiv.org/html/2610.12530#bib.bib39)\), PSNR\([Chen et al\. 2026](https://arxiv.org/html/2610.12530#bib.bib47)\), and SSIM\([Wang et al\. 2004](https://arxiv.org/html/2610.12530#bib.bib40)\)\. LPIPS is our primary metric because it better reflects perceptual deviation from the full\-model output\.

For video, LPIPS and SSIM are computed framewise and then averaged:

LPIPSvideo=1T​∑i=1TLPIPS⁡\(𝐈i,𝐈ifull\),\\operatorname\{LPIPS\}\_\{\\mathrm\{video\}\}=\\frac\{1\}\{T\}\\sum\_\{i=1\}^\{T\}\\operatorname\{LPIPS\}\\left\(\\mathbf\{I\}\_\{i\},\\mathbf\{I\}^\{\\mathrm\{full\}\}\_\{i\}\\right\),\(41\)and

SSIMvideo=1T​∑i=1TSSIM⁡\(𝐈i,𝐈ifull\)\.\\operatorname\{SSIM\}\_\{\\mathrm\{video\}\}=\\frac\{1\}\{T\}\\sum\_\{i=1\}^\{T\}\\operatorname\{SSIM\}\\left\(\\mathbf\{I\}\_\{i\},\\mathbf\{I\}^\{\\mathrm\{full\}\}\_\{i\}\\right\)\.\(42\)
Video PSNR is computed from the global mean squared error:

MSE=13​T​H​W​∑i=1T∑j=1H∑k=1W‖𝐈i​j​k−𝐈i​j​kfull‖22,\\operatorname\{MSE\}=\\frac\{1\}\{3THW\}\\sum\_\{i=1\}^\{T\}\\sum\_\{j=1\}^\{H\}\\sum\_\{k=1\}^\{W\}\\left\\\|\\mathbf\{I\}\_\{ijk\}\-\\mathbf\{I\}^\{\\mathrm\{full\}\}\_\{ijk\}\\right\\\|\_\{2\}^\{2\},\(43\)followed by

PSNR=10​log10​2552MSE\.\\operatorname\{PSNR\}=10\\log\_\{10\}\\frac\{255^\{2\}\}\{\\operatorname\{MSE\}\}\.\(44\)
No resizing or cropping is applied\. All compared video outputs use the same encoding and decoding pipeline\.

### D\.5Efficiency Metrics

For each sample, we record the number of exact and skipped forwards, wall\-clock latency, and actual speedup:

Speedup=TfullTcache\.\\operatorname\{Speedup\}=\\frac\{T\_\{\\mathrm\{full\}\}\}\{T\_\{\\mathrm\{cache\}\}\}\.\(45\)
All methods are measured on identical hardware—a single NVIDIA RTX PRO 6000D GPU—with the same model, prompt, seed, and generation configuration\. One\-time model loading is excluded\.

### D\.6Statistical Protocol

All statistical comparisons are paired at the prompt level\. For a quality metricQQ, the paired difference between methodsAAandBBis

diA,B=QiA−QiB\.d\_\{i\}^\{A,B\}=Q\_\{i\}^\{A\}\-Q\_\{i\}^\{B\}\.\(46\)
LPIPS differences follow the natural convention

diA,B<0⟹A​is better\.d\_\{i\}^\{A,B\}<0\\quad\\Longrightarrow\\quad A\\text\{ is better\}\.\(47\)
We treat

\|diA,B\|≤10−12\|d\_\{i\}^\{A,B\}\|\\leq 10^\{\-12\}\(48\)as a tie and remove ties before the exact two\-sided sign test\. If the resulting effective sample size satisfies

neff<8,n\_\{\\mathrm\{eff\}\}<8,\(49\)we report the result descriptively without making a significance claim\.

Confidence intervals are obtained by resampling the prompt\-level paired differences:

\{diA,B\}i=1n,\\left\\\{d\_\{i\}^\{A,B\}\\right\\\}\_\{i=1\}^\{n\},\(50\)using10,00010\{,\}000bootstrap samples\. Frames are never treated as independent samples\. We apply Benjamini–Hochberg correction\([Benjamini and Hochberg 1995](https://arxiv.org/html/2610.12530#bib.bib43)\)with

to the four preregistered S\-versus\-D LPIPS hypotheses\.

Our primary analysis retains only prompt pairs for which the compared methods execute exactly the same number of skipped forwards:

ℐA,B=\{i∣KiA=KiB\}\.\\mathcal\{I\}\_\{A,B\}=\\left\\\{i\\mid K\_\{i\}^\{A\}=K\_\{i\}^\{B\}\\right\\\}\.\(52\)
We additionally report all3232prompts as a robustness check\.

## Appendix EAdditional Statistical Analyses

### E\.1Independent\-Seed Replication

To test whether the improvements depend on a particular noise realization, we repeat the complete S\-versus\-D comparison with seed4343, without changing any threshold or hyperparameter\. Table[6](https://arxiv.org/html/2610.12530#A5.T6)reports the results\.

Table 6:Independent\-seed replication\.Values are median paired LPIPS differences,LS−LDL\_\{\\mathrm\{S\}\}\-L\_\{\\mathrm\{D\}\}, on strict matched\-skip pairs\.The replication criteria were fixed in writing before generation: \(R1\) same sign of the median paired difference, \(R2\) stillp<0\.05p<0\.05, and \(R3\) median\-skip range within one step, without which a cell is treated as non\-adjudicable rather than failed\. All four backbone–regime combinations satisfy R1–R3 with the independent seed\. The seed\-43/seed\-42 effect\-size ratios \(1\.161\.16,0\.860\.86,0\.690\.69,0\.980\.98\) are reported for description only and carry no threshold; their proximity to one suggests that the improvement is not caused by a favorable noise initialization\.

### E\.2Sensitivity to the Matching Protocol

Strict equal\-skip filtering removes prompts for which two schedulers differ by even one exact forward\. To verify that the result is not created by this filtering, we compare two analyses:

ℐfull\\displaystyle\\mathcal\{I\}\_\{\\mathrm\{full\}\}=\{1,…,32\},\\displaystyle=\\\{1,\\ldots,32\\\},\(53\)ℐeq\\displaystyle\\mathcal\{I\}\_\{\\mathrm\{eq\}\}=\{i∣KiA=KiB\}\.\\displaystyle=\\left\\\{i\\mid K\_\{i\}^\{A\}=K\_\{i\}^\{B\}\\right\\\}\.\(54\)
Across both backbones, both regimes, all three metrics, and all pairwise scheduler comparisons, the conclusion direction does not reverse betweenℐfull\\mathcal\{I\}\_\{\\mathrm\{full\}\}andℐeq\\mathcal\{I\}\_\{\\mathrm\{eq\}\}\. The differences in skip count are also small:median⁡\(KiA−KiB\)∈\{−1,0\}\\operatorname\{median\}\(K\_\{i\}^\{A\}\-K\_\{i\}^\{B\}\)\\in\\\{\-1,0\\\}for all main comparisons\. The gains therefore cannot be explained by our scheduler simply skipping fewer denoiser evaluations\.

## Appendix FExtended Experimental Data and Audits

This appendix collects the complete measurement record underlying the controlled scheduler comparison: signal\-decorrelation statistics, the full prompt\-level paired analysis with confidence intervals and win–tie–loss counts, the agreement of the three quality metrics, the multiple\-comparison audit, additional operating points beyond the pre\-registered levels, a schedule\-placement analysis, extended reconstruction andγmax\\gamma\_\{\\max\}ablations, the analytic cost model with its validation at2828operating points, a codec\-independence check, and the operating points of the released baseline defaults\. Throughout, S denotes our probe\-normalized signalΔ​yt/\(Δ​xt\+ϵ\)\\Delta y\_\{t\}/\(\\Delta x\_\{t\}\+\\epsilon\), D the default DiCache signalΔ​yt\\Delta y\_\{t\}, and M the difference\-based alternative\|Δ​yt−Δ​xt\|\|\\Delta y\_\{t\}\-\\Delta x\_\{t\}\|; unless stated otherwise, a negative paired median favors the first scheduler of a pair, for every metric\. All wall\-clock measurements reported in this appendix were taken on a single NVIDIA RTX PRO 6000D GPU, the same device used for the main results\.

Table 7:Full prompt\-level paired comparison of LPIPS on the held\-out set\.“Matched” restricts each pair to prompts with identical skip counts \(primary read\); “all” pools all3232prompts\. Negative median differences favor the first scheduler\. W–T–L counts wins, ties, and losses of the first scheduler\.### F\.1Signal Decorrelation Statistics

The motivation for probe normalization rests on measured, not illustrative, statistics of the two probe movements\. Over the denoising steps of the held\-out runs, the Spearman rank correlation betweenΔ​yt\\Delta y\_\{t\}andΔ​xt\\Delta x\_\{t\}is0\.950\.95on Wan2\.1 and0\.520\.52on FLUX: on the video backbone, almost all of the rank variation in the raw signal is shared with input movement\. HoldingΔ​yt\\Delta y\_\{t\}inside its\[45,55\]\[45,55\]percentile band still leavesΔ​xt\\Delta x\_\{t\}spanning a factor of1\.231\.23on Wan2\.1 \(n=256n\{=\}256steps in band\) and2\.452\.45on FLUX \(n=75n\{=\}75\), so the response ratiosts\_\{t\}spans a factor of1\.101\.10and2\.322\.32respectively \(10th–90th percentile\)\. The consequence for scheduling is not a marginal re\-ranking: ranking the steps of a trajectory byΔ​yt\\Delta y\_\{t\}and bysts\_\{t\}and taking the top third of each, the two sets overlap on only4%4\\%of their members on Wan2\.1 \(mean over3232trajectories; Spearmanrs​\(Δ​yt,st\)=−0\.83r\_\{s\}\(\\Delta y\_\{t\},s\_\{t\}\)\{=\}\{\-\}0\.83\) and37%37\\%on FLUX \(1616trajectories,rs=0\.31r\_\{s\}\{=\}0\.31\)\. At the same skip budget, the two signals therefore spend their exact forward passes on almost disjoint sets of steps\.

*Provenance\.*The Wan2\.1 statistics are measured on the D arm at the Fast level, seed4242, all3232held\-out prompts \(≈2560\\approx 2560steps\); the FLUX statistics on the M arm at the Moderate level, the1616development prompts \(≈688\\approx 688steps\) — the trace dump retainsΔ​xt\\Delta x\_\{t\}only for those runs\. These statistics establish that the signals*decide differently*, not that one decides better;sts\_\{t\}containsΔ​yt\\Delta y\_\{t\}in its numerator, so a low overlap is expected under either hypothesis\. Which decision is better is settled by the paired analysis below\.

### F\.2Full Prompt\-Level Paired Analysis

Table[7](https://arxiv.org/html/2610.12530#A6.T7)extends the main\-text paired comparison to all three scheduler pairs, under both the matched pairing \(only prompts on which the two schedulers skipped exactly the same number of forwards; the pre\-registered primary read\) and the pooled reading over all3232prompts \(which confounds quality with compute and is reported as a robustness check\)\. Confidence intervals are95%95\\%bootstrap intervals \(B=104B\{=\}10^\{4\}\) resampling prompts, never individual frames or LPIPS values;ppis a two\-sided exact sign test\.

Two observations beyond the main text\. First, the two comparison signals are not equivalent to each other: M beats D in most cells but*loses*at the Wan2\.1 Fast level \(\+0\.0098\+0\.0098,p=0\.024p\{=\}0\.024\), where it is also the worst of the three arms in absolute LPIPS\. Second, the matched and pooled readings never disagree in direction, so the matched\-skip restriction does not manufacture any conclusion\.

### F\.3Agreement of the Three Quality Metrics

Table[8](https://arxiv.org/html/2610.12530#A6.T8)reports every held\-out cell for all three metrics under the matched pairing\. For uniformity of reading, the sign convention is aligned across metrics: a negative median difference means the*first*scheduler of the pair is better, for LPIPS, PSNR, and SSIM alike\. The metrics do not always agree, and we list every disagreement rather than selecting the favourable metric per cell: in Wan2\.1/Moderate/S vs\. M, LPIPS favors S while SSIM favors M \(PSNR is a tie\); in Wan2\.1/Fast/S vs\. D, LPIPS and PSNR favor S while SSIM favors D\. LPIPS is the pre\-registered primary metric, fixed before the held\-out set was generated\.

Table 8:Metric agreement on the held\-out set, matched pairing only\.Median paired differences; negative favors the first scheduler for all three metrics\. “n\.s\.” denotes a sign\-test tie at the0\.050\.05level\.Counting every held\-out cell in which S participates across the two backbones, two levels, three metrics, and two pairs, there are2424cells: S is significantly better in2121, significantly worse in22\(both SSIM, on Wan2\.1\), and indistinguishable in11\(PSNR, Wan2\.1/Moderate/S vs\. M\)\.

### F\.4VBench Sub\-Dimension Evaluation on Wan2\.1

The fidelity metrics of the main paper measure agreement with the full\-computation output\. As a complementary reference\-free view, Table[9](https://arxiv.org/html/2610.12530#A6.T9)reports raw VBench sub\-dimension scores on a subset of VBench prompts with Wan2\.1 as the backbone, comparing the caching methods at the Moderate operating point against the full\-computation \(dense\) baseline\. CacheScope obtains the best score among the caching methods on all seven quality dimensions, and matches the dense baseline’s dynamic degree exactly; Panel \(b\) of the reconstruction\-ablation figure in the main paper visualizes the normalized profiles\.

Table 9:Raw VBench sub\-dimension scores on Wan2\.1 \(Moderate regime\)\.Higher is better for all dimensions except dynamic degree \(Dyn\.\), where the dense baseline’s value is the reference and deviations in either direction indicate drift; Dyn\. is therefore excluded from the bolding\. Best value per column among the caching methods is in bold\. Subj\.==subject consistency, BG==background consistency, Flick\.==temporal flickering, Smooth==motion smoothness, Aes\.==aesthetic quality, Imag\.==imaging quality, Overall==overall consistency\.
### F\.5Multiple\-Comparison Audit

Two Benjamini–Hochberg families atq=0\.05q\{=\}0\.05were declared before the results were examined: a*confirmatory*family containing only the four pre\-registered hypotheses \(S vs\. D, LPIPS, matched pairing; one cell per backbone and level\), and an*exploratory*family containing every other reported matched cell \(m=77m\{=\}77\)\. The confirmatory family survives the correction entirely\. In the exploratory family,6262of7777cells are nominally significant at the raw0\.050\.05level and6161survive the correction\.

Table[10](https://arxiv.org/html/2610.12530#A6.T10)lists all1616unrejected exploratory cells\. Fifteen of them were never significant to begin with \(0\.071≤p≤1\.0000\.071\\leq p\\leq 1\.000\); exactly one cell is nominally significant and then lost to the correction, namely Wan2\.1/FLUX\-matched/S vs\. D/SSIM \(n=30n\{=\}30, median−0\.0318\-0\.0318,p=0\.043p\{=\}0\.043\)\. Nine further cells have fewer than eight identical\-skip prompts and are excluded from the family \(reported but not tested\): Wan2\.1/Aggressive/M vs\. D \(LPIPS, PSNR, SSIM;n=4n\{=\}4each\), Wan2\.1/Conservative/S vs\. M \(LPIPS, PSNR, SSIM;n=5n\{=\}5each\), and Wan2\.1/Conservative/M vs\. D \(LPIPS, PSNR, SSIM;n=1n\{=\}1each\)\.

Table 10:The1616exploratory cells not rejected under Benjamini–Hochberg \(q=0\.05q\{=\}0\.05\)\.Only the first was nominally significant before correction\.Two further audits\.*Family\-boundary sensitivity:*restricting the exploratory family to the two pre\-registered levels \(m=32m\{=\}32\) rejects3131cells; including every post\-hoc level \(m=77m\{=\}77, as done here\) rejects6161; no cell rejected under the narrower family fails under the wider one\.*Margins:*the tightest surviving cell \(Wan2\.1/Aggressive/S vs\. M/SSIM\) hasp=0\.039p\{=\}0\.039against a BH criterion of0\.03960\.0396— a headroom factor of only1\.011\.01, which we state rather than describing the family as comfortably surviving\. Among rejected cells, the one most affected by the video codec \(Section[F\.13](https://arxiv.org/html/2610.12530#A6.SS13)\) is Wan2\.1/Fast/M vs\. D/LPIPS: its paired median is\+0\.00976\+0\.00976under the H\.264 pipeline but\+0\.02048\+0\.02048on raw tensors, a residual drift of52%52\\%of the effect; it clears its BH criterion by a factor of1\.61\.6and the adjudication is the same either way, but this is the one cell that is thin on both margins at once\.

### F\.6Dependence on the Skip Ratio

Table[11](https://arxiv.org/html/2610.12530#A6.T11)traces the paired LPIPS gaps as the skip ratioρ\\rhogrows, including post\-hoc levels beyond the pre\-registered pair \(marked†\\dagger; all folded into the exploratory BH family\)\. Within a backbone, rows sharing a retention ratio are directly comparable and may be read as a curve; levels with a different retention ratio are listed separately because their warm\-up lengths differ\.

Table 11:Paired LPIPS medians on identical\-skip prompts as a function of the skip ratioρ\\rho; negative favors the first scheduler,nnin parentheses\.Cells withn<8n<8are below the pre\-registered minimum and reported without adjudication\.†\\daggermarks post\-hoc levels\.The two backbones behave oppositely asρ\\rhogrows\. Against D, the advantage of S widens monotonically on Wan2\.1 \(−0\.0154→−0\.0335→−0\.0541\-0\.0154\\to\-0\.0335\\to\-0\.0541acrossρ=0\.52,0\.64,0\.76\\rho=0\.52,0\.64,0\.76\) and shrinks monotonically on FLUX \(−0\.0509→−0\.0458→−0\.0347\-0\.0509\\to\-0\.0458\\to\-0\.0347acrossρ=0\.68,0\.74,0\.82\\rho=0\.68,0\.74,0\.82, the last no longer significant\); against M, the FLUX gap closes entirely atρ=0\.82\\rho\{=\}0\.82\(−0\.0014\-0\.0014,n=19n\{=\}19,p=1\.000p\{=\}1\.000\)\. Because the retention ratio moves together with the backbone in the pre\-registered levels \(0\.20\.2on Wan2\.1,0\.10\.1on FLUX\), we ran the levels that separate the two factors\. With retention held at0\.10\.1and the same three skip ratios on both backbones, the opposite trends persist \(Wan2\.1:−0\.0729→−0\.0800→−0\.0809\-0\.0729\\to\-0\.0800\\to\-0\.0809, all significant; FLUX:−0\.0509→−0\.0458→−0\.0347\-0\.0509\\to\-0\.0458\\to\-0\.0347, significant atρ=0\.68,0\.74\\rho\{=\}0\.68,0\.74only\), and at fixedρ=0\.68\\rho\{=\}0\.68on FLUX, varying only the retention ratio moves the S\-vs\.\-D gap from−0\.0509\-0\.0509\(retention0\.10\.1\) to−0\.0155\-0\.0155\(retention0\.20\.2\), both significant\. We therefore state the conclusion as*probe\-signal construction is backbone\-dependent*and do not claim a universal ordering at high skip ratios\.

### F\.7Additional Operating Points

Table[12](https://arxiv.org/html/2610.12530#A6.T12)reports the six post\-hoc levels in full\. Each was chosen for a stated, exogenous reason: Conservative extends the pre\-registered Wan2\.1 ladder one spacing \(1212skipped forwards\) downward; FLUX/Aggressive coincides with the released default operating point of the official M configuration \(Section[F\.14](https://arxiv.org/html/2610.12530#A6.SS14)\); Wan2\.1/Aggressive matches theρ\\rhoat which the S\-vs\.\-M gap closes on FLUX; FLUX\-matched and FLUX\-matched\-lo place Wan2\.1 on FLUX’s retention and skip ratios; and ret0\.2\-Moderate holds the FLUX backbone and skip ratio fixed while raising only the retention ratio\. Thresholds for every level were read off the development set and frozen, together with the decision rule, before generation\.

Table 12:Post\-hoc operating points \(held\-out set,n=32n\{=\}32prompts, seed4242\)\.s¯\\bar\{s\}is the median skip count\. The matched\-compute gate passes when the median\-skip spread across the three schedulers is zero\. Paired LPIPS rows use the matched pairing; cells withn<8n<8are reported without adjudication\. For Wan2\.1/Aggressive, the M arm was calibrated to8282on the development set but reached83\.583\.5on the held\-out set \(calibration drift\); for Wan2\.1/Conservative, no threshold places M at skip5252because its median skip jumps from4949to5454between adjacent threshold values on the development grid\.For the retention\-0\.10\.1Wan2\.1 levels and ret0\.2\-Moderate, the threshold\-selection rule differs from the other levels: instead of minimizing the gap between the median skip and the target, it maximizes the number of prompts whose skip equals the target exactly \(ties broken by median gap, then by the smaller threshold\)\. This rule was adopted after an initial calibration returned arms whose median skips agreed to within half a step yet whose primary pair coincided on only77of1616development prompts; the revised rule was fixed before the re\-scan, depends only on skip counts, and no quality metric was read at any point during calibration\.

### F\.8Schedule Placement Analysis

By construction the three schedulers skip the same number of steps, so any quality difference must come from the*placement*of the exact forwards\. Table[13](https://arxiv.org/html/2610.12530#A6.T13)measures that placement on FLUX at a matched skip budget \(ρ=0\.68\\rho\{=\}0\.68\) under two retention settings — retention0\.10\.1leaves88–1010forward passes free to place and retention0\.20\.2leaves44, so this is a minimal manipulation of warm\-up length alone\. The centroid is the mean position of the kept passes within the keepable window, normalized to\[0,1\]\[0,1\];Δcen\\Delta\_\{\\mathrm\{cen\}\}is the absolute centroid gap between two schedulers;dNNd\_\{\\mathrm\{NN\}\}is their symmetric nearest\-neighbour distance in forward passes; and𝔼⁡\[d\]\\mathbb\{E\}\[d\]is its exact expectation under a null model that keeps one arm’s actual choice and redraws the other uniformly, sodNN/𝔼⁡\[d\]<1d\_\{\\mathrm\{NN\}\}/\\mathbb\{E\}\[d\]<1means the two schedules agree more than chance\.

Table 13:Where each scheduler spends its exact forward passes \(FLUX,ρ=0\.68\\rho\{=\}0\.68\)\.Top: centroids of the kept passes\. Bottom: pairwise placement statistics;Δ\\DeltaLPIPS repeats the held\-out paired median for alignment\. Mechanism columns are measured on the1616development prompts, the only runs with per\-step traces\.
Three readings\. First, S spends its budget noticeably earlier in the keepable window \(centroid0\.4230\.423versus0\.5700\.570for D at retention0\.10\.1\)\. Second, all sixdNN/𝔼⁡\[d\]d\_\{\\mathrm\{NN\}\}/\\mathbb\{E\}\[d\]ratios are below one \(0\.540\.54–0\.720\.72\): the three signals concur on roughly where the sensitive region is and differ in phase within it, so the disagreement is systematic rather than noise\. Third, raising the retention squeezes M onto D — their centroid gap falls from0\.0990\.099to0\.0410\.041, exactly where their paired LPIPS difference loses significance \(−0\.0285→\+0\.0006\-0\.0285\\to\+0\.0006\) — while S stays apart from D \(0\.147→0\.1860\.147\\to 0\.186\) and retains a significant advantage\. A signal that is a difference of two movements degenerates towards its dominant term when the schedule has few degrees of freedom; a signal that is a ratio does not\.

### F\.9Offline Signal Aggregation and Schedule Information

#### Motivation and construction\.

We further examine whether the advantage of the trajectory\-normalized signal relies on prompt\-specific online variation, or whether it also reveals a stable timestep structure shared across prompts\. We construct two fixed schedules using only the1616\-prompt development set\. ForGlobal\-S, we averagest=Δ​yt/\(Δ​xt\+ϵ\)s\_\{t\}=\\Delta y\_\{t\}/\(\\Delta x\_\{t\}\+\\epsilon\)independently at each timestep and apply the same accumulator\-and\-threshold rule used by CacheScope to the resulting mean curve\. ForGlobal\-D, we repeat exactly the same procedure using the raw probe variationΔ​yt\\Delta y\_\{t\}\. The thresholdsτS=4\.5267\\tau\_\{\\mathrm\{S\}\}=4\.5267andτD=0\.1405\\tau\_\{\\mathrm\{D\}\}=0\.1405are selected solely to obtain the target skip count of3434; no fidelity metric is used during schedule construction\.

Both fixed masks are generated before any held\-out output or metric is accessed, and the development and held\-out prompt sets are disjoint\. We additionally evaluatecross\-prompt replay, in which the Online\-S schedule generated for promptiiis applied to prompti\+1i\+1under a cyclic derangement with no fixed points\. All four variants continue to execute the same shallow probe; only the exact/skip decisions are replaced\. The comparison therefore isolates the information supplied by the scheduling signal and by prompt\-specific schedule assignment\.

Table 14:Offline signal aggregation and schedule\-information analysis on FLUX\.1\-dev at the Moderate operating point \(ρ=0\.68\\rho\{=\}0\.68\)\.Global\-S and Global\-D are constructed from the same1616\-prompt development set using the same timestep aggregation, threshold solver, and skip budget; they differ only in the aggregated scheduling signal\. Quality values are medians over3232held\-out prompts\. The paired LPIPS difference isΔ​L=Lvariant−LOnline​\-​S\\Delta L=L\_\{\\mathrm\{variant\}\}\-L\_\{\\mathrm\{Online\\text\{\-\}S\}\}; positive values favor Online\-S\. The final column reports an exact two\-sided sign test against Online\-S\.
#### Results\.

Global\-S is statistically indistinguishable from Online\-S: its median LPIPS is0\.07920\.0792, compared with0\.08160\.0816for Online\-S, with1414improvements,1313degradations, and55ties across the3232paired prompts \(p=1\.000p\{=\}1\.000\)\. Cross\-prompt replay is likewise statistically indistinguishable from Online\-S\. Thus, at this FLUX operating point, a substantial component of the useful scheduling structure is shared across prompts, while prompt\-specific schedule variation provides no additional statistically detectable benefit\([Zhang et al\. 2025b](https://arxiv.org/html/2610.12530#bib.bib52);[Zhang et al\. 2025f](https://arxiv.org/html/2610.12530#bib.bib56)\)\. This result is specific to the evaluated backbone and operating point and does not imply that online adaptation is unnecessary in general\([Zhang et al\. 2026a](https://arxiv.org/html/2610.12530#bib.bib59)\)\.

Importantly, applying the same offline construction to raw probe variation does not recover the same quality\. Global\-D raises median LPIPS to0\.13370\.1337and is worse than Online\-S on2929of3232prompts, with a median paired increase of0\.04370\.0437\(p=2\.6×10−6p\{=\}2\.6\{\\times\}10^\{\-6\}\)\. Global\-S and Global\-D use identical development prompts, aggregation procedures, threshold solvers, and skip budgets; the only changed quantity is whether the probe response is normalized by the corresponding input movement\. This controlled comparison therefore supports the central claim that trajectory normalization exposes a substantially more effective timestep\-importance profile than raw probe\-output variation\.

The fixed schedules execute512512exact forwards in total, compared with515515for Online\-S, because Online\-S produces per\-prompt skip counts of3333,3434, or3535, whereas a shared mask must use a single count\. The0\.6%0\.6\\%difference is conservative for the fixed schedules and cannot explain their comparable fidelity\.

#### Schedule diversity\.

The online schedules are not identical: the3232Online\-S runs produce nine distinct masks, with a median pairwise Jaccard similarity of0\.2000\.200within the adaptive window\. Nevertheless, cross\-prompt replay does not cause a detectable quality loss at this operating point, suggesting that some of this mask variation is functionally interchangeable\. Global\-S and Global\-D have a Jaccard similarity of only0\.2860\.286\. Their exact\-step sets within the adaptive window are

\{7,9,12,16,19,25,32,38,44\}\\\{7,9,12,16,19,25,32,38,44\\\}for Global\-S and

\{9,14,19,26,32,37,41,44,47\}\\\{9,14,19,26,32,37,41,44,47\\\}for Global\-D\. The large fidelity difference therefore accompanies a substantial change in exact\-compute placement rather than a nearly identical mask\.

### F\.10Extended Reconstruction Ablation

Table[15](https://arxiv.org/html/2610.12530#A6.T15)complements the reconstruction\-ablation figure in the main paper with absolute median LPIPS on both backbones and win–loss counts against the common reference \(least squares,k=2k\{=\}2\)\. All arms replay an identical exact/skip decision sequence, verified by comparing an MD5 hash of that sequence prompt by prompt \(32/3232/32agreeing on FLUX,32/3232/32on Wan2\.1\), so the arms are exactly compute\-matched and every difference comes from the reconstruction formula alone\.

Table 15:Median LPIPS of five cached\-residual reconstructors on the1616\-prompt development set, with W–L counts against the least squaresk=2k\{=\}2reference\.All arms replay identical exact/skip schedules\.Two things are visible\. First, the bounded first\-orderγ\\gammaextrapolation is at least as good as every alternative in all four columns\. Second, extending the history is monotonically worse on both backbones, and on Wan2\.1 it is catastrophic \(0\.3128→0\.6670→0\.70360\.3128\\to 0\.6670\\to 0\.7036at the Moderate level\): more history means fitting a smooth model to a residual sequence that is not smooth over that horizon, and on video the extrapolation diverges\. We report this as a bounded negative result, not a theorem: it rests on1616development prompts and a single seed, and atn=16n\{=\}16a two\-sided exact sign test needs1313–33to reachp=0\.021p\{=\}0\.021, so the table establishes an ordering that is stable across two backbones and two levels rather than adjudicating significance\.

### F\.11Reconstruction Cap Ablation

Table[16](https://arxiv.org/html/2610.12530#A6.T16)sweeps the reconstruction gain capγmax\\gamma\_\{\\max\}on the development set \(sweeping a hyperparameter on the held\-out set would forfeit its untouched status\)\. Differences are paired againstγmax=1\.5\\gamma\_\{\\max\}\{=\}1\.5and restricted to prompts with identical skip counts; negative is better\.

Table 16:Ablation ofγmax\\gamma\_\{\\max\}on the1616\-prompt development set, paired against the defaultγmax=1\.5\\gamma\_\{\\max\}\{=\}1\.5\.No cell reaches significance \(smallest of the1212pp\-values:0\.0770\.077, before any correction\)\.Table 17:Per\-forward cost from the architecture specification\.“Non\-skip\.” counts forwards the schedule can never skip \(warm\-up\+\+forced tail\)\.The two cells tied at the smallestpp\-value point in opposite directions \(one toward a looser cap, one toward the default\), which is what an inert hyperparameter looks like\. We therefore read the cap as inert over the swept range and keepγmax=1\.5\\gamma\_\{\\max\}\{=\}1\.5because it is the released default, not because it won\.

Table 18:Analytic versus measured speedup at every skip count with at least eight prompts, pooled across all three schedulers \(the cost model depends only on the skip count\)\.SpredS\_\{\\mathrm\{pred\}\}is re\-predicted from the single fitted constantff, not fitted per row\.BackboneSkipρ\\rhoStheoS\_\{\\mathrm\{theo\}\}SmeasS\_\{\\mathrm\{meas\}\}SpredS\_\{\\mathrm\{pred\}\}nnWan2\.1500\.501\.9351\.8771\.8798520\.522\.0111\.9471\.94731540\.542\.0922\.0212\.02119560\.562\.1802\.0982\.10021580\.582\.2762\.1842\.18628600\.602\.3812\.2792\.27939640\.642\.6222\.4922\.49239660\.662\.7622\.6132\.61418680\.682\.9182\.7492\.74836700\.703\.0932\.8972\.89716720\.723\.2893\.0653\.06347740\.743\.5133\.2513\.24918760\.763\.7693\.4633\.45948780\.784\.0653\.7033\.69964800\.804\.4123\.9803\.97432FLUX280\.562\.2232\.1952\.19710310\.622\.5582\.5212\.52117320\.642\.6942\.6502\.65125340\.683\.0132\.9572\.95642350\.703\.2023\.1383\.13618360\.723\.4173\.3403\.34036370\.743\.6633\.5733\.57356380\.763\.9473\.8423\.84033390\.784\.2794\.1474\.15036400\.804\.6724\.5144\.51455410\.825\.1444\.9474\.94973420\.845\.7235\.4755\.47649430\.866\.4486\.1246\.13016
### F\.12Analytic Cost Model and Its Validation

Table[17](https://arxiv.org/html/2610.12530#A6.T17)gives the per\-forward cost derived from the architecture specification\. A skipped step is not free — it still runs the probe \(one transformer block\), which costs3\.3%3\.3\\%of a full forward on Wan2\.1 and1\.8%1\.8\\%on FLUX\. FLOPs count attention and FFN matrix multiplies only and exclude the text encoder and VAE, which are identical across all methods and do not vary with the skip count\.

Because the excluded components are fixed costs, measured speedup must fall short of the analytic value\. Fitting the single\-parameter relation

1Smeas=f\+1−fStheo\\frac\{1\}\{S\_\{\\mathrm\{meas\}\}\}=f\+\\frac\{1\-f\}\{S\_\{\\mathrm\{theo\}\}\}\(55\)independently at every operating point recovers anffthat varies little across operating points:f=0\.0323f\{=\}0\.0323on Wan2\.1 \(median over1515operating points spanning skip5050–8080; per\-point range0\.03180\.0318–0\.03330\.0333; implied speedup ceiling1/f=31×1/f\{=\}31\\times\) andf=0\.0095f\{=\}0\.0095on FLUX \(median over1313points spanning skip2828–4343; range0\.00920\.0092–0\.01030\.0103; ceiling105×105\\times\)\. Substituting the medianffback reproduces every measured speedup to within0\.16%0\.16\\%on Wan2\.1 and0\.09%0\.09\\%on FLUX \(worst case over the2828operating points\), so the shortfall is a fixed overhead, not a breakdown of the cost model at high skip rates\. Table[18](https://arxiv.org/html/2610.12530#A6.T18)lists the full ladder; skip counts with fewer than eight prompts are excluded from the fit \(Wan2\.1:46,48,49,53,55,57,62,63,6746,48,49,53,55,57,62,63,67; FLUX:27,29,30,3327,29,30,33\)\.

### F\.13Codec\-Independence Check

Every video quality number in this paper is computed on H\.264\-encoded output, so we recomputed the entire paired analysis from the raw decoder tensors \(lossless\) on the3232held\-out Wan2\.1 prompts\. The re\-generation needed for this check reproduced the original outputs byte\-for\-byte in192/192192/192cases, so the lossless\-versus\-lossy gap below is codec error alone and not run\-to\-run nondeterminism\.

Table 19:Codec\-independence check on Wan2\.1 \(n=32n\{=\}32held\-out prompts\): median paired differences recomputed from raw decoder tensors \(lossless\) versus the H\.264 pipeline \(lossy\)\.All1818cells keep the same adjudication\.Table 20:Operating points of the two released default FLUX baseline configurations on the same held\-out set \(n=32n\{=\}32\), with our matched\-compute levels for reference\.“Worst” is the maximum per\-prompt LPIPS\.ConfigurationSignalτ\\tau⌊ret⋅N⌋\\lfloor\\mathrm\{ret\}\\cdot N\\rfloors¯\\bar\{s\}LPIPS↓\\downarrowWorstSpeedupOfficial default \(M\)\|Δ​y−Δ​x\|\|\\Delta y\{\-\}\\Delta x\|0\.080410\.24960\.47364\.94×\\timesOfficial default \(D\)Δ​y\\Delta y0\.410350\.16800\.34013\.13×\\timesOurs ModerateS4\.65340\.08160\.20482\.95×\\timesOurs ModerateD0\.155340\.13530\.31822\.95×\\timesOurs ModerateM0\.0425340\.10140\.26812\.95×\\timesOurs FastS6\.45370\.12460\.24873\.57×\\timesOurs FastD0\.255370\.18420\.36493\.60×\\timesOurs FastM0\.0655370\.14740\.38663\.57×\\times

Compression error does not fully cancel in a paired difference\. In the worst LPIPS cell \(Fast, M vs\. D\) the codec moves the paired median by0\.01070\.0107, i\.e\.52%52\\%of that cell’s effect size \(0\.02050\.0205\); the verdicts agree, but the margin is not uniformly large, and we report that cell rather than the average\. Re\-deriving every verdict from raw tensors leaves all1818judgments unchanged\.

### F\.14Released\-Default Operating Points

A calibrated\-threshold comparison invites the objection that the baseline was run at an unfavourable operating point\. Table[20](https://arxiv.org/html/2610.12530#A6.T20)therefore places the two released default configurations of the FLUX baseline on the same held\-out set\. These rows carry no significance test: their skip counts differ from each other and from our matched\-compute levels, so a direct comparison would confound quality with compute\. They are reported to show that our calibration does not weaken the baseline: the official M default lands at a*more aggressive*operating point \(median skip4141\) than either of our levels \(3434and3737\)\.

The fourth column is the released threshold⌊ret⋅N⌋\\lfloor\\mathrm\{ret\}\\cdot N\\rfloor, a step index rather than a count of non\-skippable forwards: FLUX holds a step exact when the counter is at most this index \(inclusive, hence one more forward than the index\) and additionally forces the final step, so the actual non\-skippable count is two larger\.