Entropy Can Flow, or It Can Guide. Be Entropy. LEDFlow: Introducing Entropy-guided Generation Order into Uniform Discrete Flow

# Entropy can flow, or it can guide. Be Entropy.LEDFlow: Introducing Entropy-guided Generation Order into Uniform Discrete Flow
Source: [https://arxiv.org/html/2609.25131](https://arxiv.org/html/2609.25131)
Yidong Ouyang\*Yingjia WanYing Nian WuAffiliation:University of California, Los AngelesEmail:[tk1018@ucla\.edu](mailto:)Zhijiang Guo\*\*Affiliation:Hong Kong University of Science and TechnologyEmail:[zhijiangguo@hkust\-gz\.edu\.cn](mailto:)Oscar Leong\*\*Affiliation:Shanghai Jiao Tong UniversityEmail:[oscar\.leong@sjtu\.edu\.cn](mailto:)

###### Abstract

Uniform discrete flow permits repeated updates at every generation position\. While continued revision supports correction of wrong tokens, it also exposes correct intermediate predictions to later errors\. An experiment on Sudoku puzzles shows that 9\.4% of generated cells are correct at an intermediate step but incorrect in the final output\. We introduce generation order into uniform discrete flow through selective absorption, which fixes chosen predictions while preserving the uniform\-flow velocity at active positions\. To prevent absorbing incorrect predictions, we propose Low\-Entropy Discrete Flow \(LEDFlow\), a training\-free sampler that adaptively orders absorption by local entropy\. By decomposing absorption error into joint dependence and conditional prediction terms, we show that selecting the lowest\-entropy positions under a fixed absorption budget minimizes an upper bound on the conditional term\. We further support the choice of local entropy by showing that the decision\-error bound of global lookahead grows with the lookahead window under an imperfect denoiser\. Across reasoning benchmarks,LEDFlowattains 0\.845 Nikoli Sudoku solve accuracy, with the largest gains on strongly constrained tasks\. On text\-to\-image generation it attains the best overall score, and on multimodal understanding it improves over the native sampler on all six benchmarks, at an inference cost comparable to standard flow sampling\.

## 1Introduction

Discrete flow matching allows multimodal generation by transporting a source distribution to a target distribution over discrete states through a continuous time probability velocity\[[Gat et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib2)\], while uniform discrete flow progressively transforms every token in each time step under a uniform source\. FUDOKI\[[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10)\]applies this framework to multimodal understanding and generation using a metric\-induced probability path and a kinetic\-optimal velocity\. Conditioned on a target token, the velocity moves probability mass toward tokens closer to that estimated target, so that generated positions can be updated repeatedly as sampling proceeds\.

![Refer to caption](https://arxiv.org/html/2609.25131v1/motivation_plot_v5.png)Figure 1:Entropy\-guided absorption in uniform discrete flow\.\(a\)Late revisions can overwrite correct predictions\.\(b\)Selective absorption fixes chosen predictions while other positions continue evolving\.\(c\)LEDFlowprioritizes low\-entropy positions for absorption; global lookahead aggregates potentially noisy estimates across the remaining positions\.Continued revision can replace correct intermediate tokens with incorrect predictions, especially near the end of sampling\. Under the kinetic\-optimal formulation, the schedule gradient diverges asttapproaches endpoint\. The outgoing transition rate then grows without bound when a token is semantically distant from the estimated target\. This late denoiser error exposes an otherwise correct token to a large transition rate\. A Sudoku trajectory diagnostic \(Figure[2](https://arxiv.org/html/2609.25131#S3.F2)\) using the kinetic\-optimal update shows on 100 puzzles with 64 sampling steps that 9\.4% of generated cells are correct at least once during sampling but incorrect in the final output\. This motivates us to introduce generation order into uniform discrete flow through selective absorption\. We equip each position with a binary absorption indicator that sets its outgoing rates to zero once absorbed, allowing reliable predictions to remain fixed while the other positions continue evolving\. However, fixing an incorrect token prevents correction, and a random order does not distinguish reliable predictions from uncertain ones\. Therefore, the absorption policy must account for this irreversible prediction risk\.

To address this ordering problem, we propose Low\-Entropy Discrete Flow \(LEDFlow\), a training\-free sampler that introduces generation order into uniform discrete flow through entropy\-guided absorption \(Figure[1](https://arxiv.org/html/2609.25131#S1.F1)\)\. At each sampling step,LEDFlowselects the active positions with the lowest predictive entropy under the prescribed absorption budget, and absorbs them with the most likely predicted values\. The remaining positions continue to evolve under the kinetic\-optimal velocity, with subsequent predictions conditioned on the values already fixed by absorption\. At each step, the sampler recomputes entropy for an adaptive order, allowing the policy to preserve the uniform flow velocity at active positions while prioritizing low\-entropy predictions for absorption\. This reduces the above correct\-to\-wrong reversions from 9\.4% to 2\.6% of generated cells and the fraction of token changes occurring aftert=0\.75t=0\.75from 46% to 15%\.

We then provide theoretical justification ofLEDFlowby analyzing the error introduced by irreversible absorption\. Absorbing several positions independently can discard their dependence, while previously absorbed values affect the context of subsequent predictions\. We characterize the stepwise absorption error through a Kullback\-Leibler \(KL\) decomposition into joint dependence and conditional prediction terms\. We show that selecting the lowest\-entropy positions, under entropy\-error regularity on oracle\-reachable states and a fixed absorption budget, minimizes an upper bound on the conditional error term\. We provide further analysis on global entropy lookahead\[[Yang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib36)\]which evaluates candidate absorptions through predicted successor states\. With an imperfect denoiser, aggregating future estimates can compound errors, and the resulting bound of decision error scales with the lookahead window, supporting the use of local predictive entropy to guide absorption\.

Experiments evaluateLEDFlowacross tasks with increasing structural dependence\. The gains are largest on constrained reasoning, whereLEDFlowattains 0\.845 Nikoli Sudoku solve accuracy and attains the best average rank across the nine reasoning tasks\. On text\-to\-image generationLEDFlowattains the best GenEval overall score among decode\-time samplers \(2\.8% absolute over the native sampler\), and on multimodal understanding it improves over the native sampler on all six benchmarks\. We also provide empirical support for the theoretical results: entropy\-guided absorption outperforms other absorption policies in puzzle solve accuracy, while mean absolute score error increases with the lookahead window, from 0\.17 with a single lookahead term to 1\.76 under full lookahead\. While the best performing lookahead windows would result in about 8 times of inference runtime without bringing significant improvements\. Our contributions are:

- •We analyze rapid terminal transitions and correct\-to\-wrong revisions in uniform discrete flow, and introduce explicit generation order through selective absorption to preserve chosen predictions \(§[2](https://arxiv.org/html/2609.25131#S2), §[3\.1](https://arxiv.org/html/2609.25131#S3.SS1)\)\.
- •We developLEDFlow, a training\-free entropy\-guided absorption policy\. Under entropy\-error regularity on oracle\-reachable states and a fixed absorption budget, it minimizes an upper bound on the local conditional absorption error\. We also show that the global lookahead decision\-error bound grows with its window \(§[3](https://arxiv.org/html/2609.25131#S3)\)\.
- •We evaluateLEDFlowacross reasoning, multimodal understanding, and image generation\. Controlled experiments compare absorption orders and measure how lookahead window size affects score error and position selection, while runtime measurements and transfer comparisons against random order assess efficiency and generality \(§[4](https://arxiv.org/html/2609.25131#S4)\)\.

## 2Problem Formulation

### 2\.1Preliminaries

We generate a target sequencea1=\(a11,…,a1M\)∈𝒮Ma\_\{1\}=\(a\_\{1\}^\{1\},\\ldots,a\_\{1\}^\{M\}\)\\in\\mathcal\{S\}^\{M\}over a finite vocabulary𝒮\\mathcal\{S\}andMMgenerated positions\. Letxxdenote the fixed input, set𝒞:=x\\mathcal\{C\}:=x, and writeq⁡\(a1∣𝒞\)q\(a\_\{1\}\\mid\\mathcal\{C\}\)for the target distribution,𝒞t\\mathcal\{C\}\_\{t\}the conditioning information available at timett,qqandpqp^\{q\}the oracle \(data\) distributions and their conditionals,ptp\_\{t\}the prescribed probability path, andℚθ\\mathbb\{Q\}^\{\\theta\}the learned denoiser posterior\. Discrete flow matching transports a source distributionp0p\_\{0\}to the target overt∈\[0,1\]t\\in\[0,1\]under a Continuous Time Markov Chain \(CTMC\)\[[Campbell et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib1),[Gat et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib2),[Shaul et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib3)\]\. Letaaandzzdenote the current state and a candidate destination state, respectively\. For any statexx, letx−jx^\{\-j\}denote all coordinates exceptjj\. We use the discrete delta probability mass functionδ\(x,y\):=𝟏\{x=y\}\\delta\(x,y\):=\\mathbf\{1\}\\\{x=y\\\}\. A factorized probability velocity isut​\(z,a\)=∑j=1Mδ⁡\(z−j,a−j\)​utj​\(zj,a\)u\_\{t\}\(z,a\)=\\sum\_\{j=1\}^\{M\}\\delta\(z^\{\-j\},a^\{\-j\}\)\\,u\_\{t\}^\{j\}\(z^\{j\},a\), whereutju\_\{t\}^\{j\}gives the instantaneous transition rate of coordinatejj\. For a conditional path factorized on coordinatespt\|1​\(at∣a1\)=∏jpt\|1j​\(atj∣a1j\)p\_\{t\\mid 1\}\(a\_\{t\}\\mid a\_\{1\}\)=\\prod\_\{j\}p^\{j\}\_\{t\\mid 1\}\(a\_\{t\}^\{j\}\\mid a\_\{1\}^\{j\}\), the marginal coordinate velocity follows from posterior averaging,

utj\(zj,a,𝒞t\)=𝔼a1∼p1\|t\(⋅∣a,𝒞t\)\[utj\(zj,aj∣a1j\)\]\.u\_\{t\}^\{j\}\(z^\{j\},a,\\mathcal\{C\}\_\{t\}\)=\\mathbb\{E\}\_\{a\_\{1\}\\sim p\_\{1\\mid t\}\(\\cdot\\mid a,\\mathcal\{C\}\_\{t\}\)\}\\\!\\left\[u\_\{t\}^\{j\}\(z^\{j\},a^\{j\}\\mid a\_\{1\}^\{j\}\)\\right\]\.
At generation time, the reverse processp1\|tp\_\{1\\mid t\}is approximated by the learned denoiserℚ1\|tθ\(⋅∣at,𝒞t\)\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(\\cdot\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)\. FUDOKI\[[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10)\]implements the uniform distribution framework in[Gat et al\. \[2024\]](https://arxiv.org/html/2609.25131#bib.bib2)with a metric\-induced token path:pt\|1j​\(aj∣a1j\)=softmaxaj∈𝒮⁡\[−βt​d​\(aj,a1j\)\]∝exp⁡\[−βt​d​\(aj,a1j\)\]p^\{j\}\_\{t\\mid 1\}\(a^\{j\}\\mid a\_\{1\}^\{j\}\)=\\operatorname\{softmax\}\_\{a^\{j\}\\in\\mathcal\{S\}\}\\\!\\left\[\-\\beta\_\{t\}d\(a^\{j\},a\_\{1\}^\{j\}\)\\right\]\\propto\\exp\\\!\\left\[\-\\beta\_\{t\}d\(a^\{j\},a\_\{1\}^\{j\}\)\\right\], andddcorresponds to the token distance between embeddings,βt=c​\(t/\(1−t\)\)α\\beta\_\{t\}=c\\left\(t/\(1\-t\)\\right\)^\{\\alpha\}so thatβ0=0\\beta\_\{0\}=0andβ1→∞\\beta\_\{1\}\\rightarrow\\infty, allowing the source to be uniform over the vocabulary att=0t=0and the path to concentrate on target token ast→1t\\rightarrow 1\. By denotingβ˙t\\dot\{\\beta\}\_\{t\}the derivative ofβt\\beta\_\{t\}, the kinetic optimal velocity is

utj​\(zj,at∣a1j\)=pt​\(zj∣a1j\)​β˙t​\[d⁡\(atj,a1j\)−d⁡\(zj,a1j\)\]\+,u\_\{t\}^\{j\}\(z^\{j\},a\_\{t\}\\mid a\_\{1\}^\{j\}\)=p\_\{t\}\(z^\{j\}\\mid a\_\{1\}^\{j\}\)\\,\\dot\{\\beta\}\_\{t\}\\left\[d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\)\-d\(z^\{j\},a\_\{1\}^\{j\}\)\\right\]\_\{\+\},\(1\)where\[⋅\]\+=max⁡\{⋅,0\}\[\\cdot\]\_\{\+\}=\\max\\\{\\cdot,0\\\}\. Probability mass moves fromatja\_\{t\}^\{j\}tozjz^\{j\}only whend⁡\(zj,a1j\)<d⁡\(atj,a1j\)d\(z^\{j\},a\_\{1\}^\{j\}\)<d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\), allowing the flow to progress monotonically toward the target valuea1ja^\{j\}\_\{1\}supplied to the velocity field\[[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10)\]\. However, at inference time where the true target is unavailable, FUDOKI’s Euler solver instead samples a predicted targeta^1j∼ℚ1\|tθ\(⋅∣at,𝒞t\)\\hat\{a\}\_\{1\}^\{j\}\\sim\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(\\cdot\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)at every step and measure the velocity \(Eq\.[1](https://arxiv.org/html/2609.25131#S2.E1)\) at that sample, so that every generated position, including those already holding a reliable prediction, is resampled at every step\. Detailed formulations from existing work are provided in Appendix[B\.1](https://arxiv.org/html/2609.25131#A2.SS1)\.

### 2\.2Motivation and Ordered Absorption

As the schedule gradientβ˙t\\dot\{\\beta\}\_\{t\}in Eq\.[1](https://arxiv.org/html/2609.25131#S2.E1)diverges ast→1t\\rightarrow 1, this increases transition exposure at positions that disagree with the predicted target\. With denoiser error, this results in loss of correctness\.

###### Proposition 1\(Terminal transition exposure\)\.

At timett, consider positionjjwhose current token differs from its targetatj≠a1ja^\{j\}\_\{t\}\\neq a^\{j\}\_\{1\}\. The total outflow rate under Eq\.[1](https://arxiv.org/html/2609.25131#S2.E1)isλtj=β˙t​Gtj\\lambda\_\{t\}^\{j\}=\\dot\{\\beta\}\_\{t\}\\,G\_\{t\}^\{j\}, whereGtj=∑z∈𝒮pt​\(z∣a1j\)​\[d⁡\(atj,a1j\)−d⁡\(z,a1j\)\]\+≥pt​\(a1j∣a1j\)​d​\(atj,a1j\)\>0G\_\{t\}^\{j\}=\\sum\_\{z\\in\\mathcal\{S\}\}p\_\{t\}\(z\\mid a\_\{1\}^\{j\}\)\\big\[d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\)\-d\(z,a\_\{1\}^\{j\}\)\\big\]\_\{\+\}\\geq p\_\{t\}\(a\_\{1\}^\{j\}\\mid a\_\{1\}^\{j\}\)\\,d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\)\>0\. Sinceβ˙t=c​α​tα−1​\(1−t\)−α−1→∞\\dot\{\\beta\}\_\{t\}=c\\alpha\\,t^\{\\alpha\-1\}\(1\-t\)^\{\-\\alpha\-1\}\\rightarrow\\inftywhilept​\(a1j∣a1j\)→1p\_\{t\}\(a\_\{1\}^\{j\}\\mid a\_\{1\}^\{j\}\)\\rightarrow 1ast→1t\\rightarrow 1, the rate diverges,λtj→∞\\lambda^\{j\}\_\{t\}\\rightarrow\\infty, and the probability of at least one jump over a fixed intervalh\>0h\>0converges:1−e−h​λtj→11\-e^\{\-h\\lambda\_\{t\}^\{j\}\}\\rightarrow 1\. When the predicted targeta^1j\\hat\{a\}\_\{1\}^\{j\}is redrawn, disagreement with it exposes the current token to this increasing rate\.

Figure[2](https://arxiv.org/html/2609.25131#S3.F2)illustrates this problem by pretraining a flow model with Sudoku puzzles and evaluating on a held\-out set of100100puzzles with 64 sampling steps: 46% of token changes occur aftert=0\.75t=0\.75, with 9\.4% correctly generated cells replaced incorrectly at the end\. This results in a final whole\-puzzle solve rate of 34%, exposing a tension between continued revision, which allows a wrong token to be corrected, and the preservation of correct intermediate predictions\.

Selective absorption resolves this tension by making chosen predictions terminal while the other positions keep evolving under the native flow, but it introduces a new problem: deciding which predictions are reliable enough to be made irreversible\. To prioritize reliable predictions at current state and reassess remaining positions as the context changes after each absorption, the denoiser’s own uncertainty is a natural signal, whereas a concentrated posterior has little to gain from revisions, vice versa\. Section[3](https://arxiv.org/html/2609.25131#S3)formalizes selective absorption as an augmentation of uniform discrete flow and turns this principle into an ordering policy driven by local entropy\. Appendix[B\.2](https://arxiv.org/html/2609.25131#A2.SS2)provides the proof and scope of the proposition, together with the training and evaluation settings of this diagnostic\.

## 3LEDFlow: Low\-Entropy Discrete Flow

### 3\.1Selective absorption as an absorbing\-state uniform flow

We add an inference time indicator function into uniform discrete flow to change the sampler law\. Specifically, each generated position is augmented with an absorption indicatorftj∈\{0,1\}f\_\{t\}^\{j\}\\in\\\{0,1\\\}\. Every position starts active withf0j=0f\_\{0\}^\{j\}=0and follows the uniform\-flow velocity\. Onceftj=1f\_\{t\}^\{j\}=1, its outgoing transition rates are set to zero, hence keeping uniform velocity at active positions unchanged while fixing absorbed positions:

utθ,abs,j​\(zj,at,ft,𝒞t\)=\(1−ftj\)​utθ,j​\(zj,at,𝒞t\),zj≠atj\.u\_\{t\}^\{\\theta,\\mathrm\{abs\},j\}\(z^\{j\},a\_\{t\},f\_\{t\},\\mathcal\{C\}\_\{t\}\)=\(1\-f\_\{t\}^\{j\}\)\\,u\_\{t\}^\{\\theta,j\}\(z^\{j\},a\_\{t\},\\mathcal\{C\}\_\{t\}\),\\qquad z^\{j\}\\neq a\_\{t\}^\{j\}\.\(2\)An absorbed position is a fixed point of the sampler regardless of what the denoiser later predicts for it, while every active position keeps the kinetic\-optimal velocity of Eq\.[1](https://arxiv.org/html/2609.25131#S2.E1), now evaluated at a state that contains the absorbed values\. The augmentation leaves two decisions to the sampler: how many positions to absorb at each step \(absorption budget\), and which positions to absorb \(absorption order\)\. While the budget is a schedule that can be fixed in advance, the order determines which predictions become irreversible and which values condition the rest of the generations \(Section[2\.2](https://arxiv.org/html/2609.25131#S2.SS2)\)\. Under this formulation, we introduceLEDFlow, an absorption\-ordering policy guided by local entropy, and analyze its local absorption risk\. Appendix[C\.1](https://arxiv.org/html/2609.25131#A3.SS1)details the augmented process and its training\-time counterpart for the puzzle denoisers\.

### 3\.2TheLEDFlowsampler absorbs by local entropy

Algorithm 1LEDFlowselective absorption over uniform discrete flow model1:

at0∼p0,ft0j←0​∀ja\_\{t\_\{0\}\}\\sim p\_\{0\},\\quad f\_\{t\_\{0\}\}^\{j\}\\leftarrow 0\\ \\forall j
2:for

k=0,…,K−1k=0,\\ldots,K\-1do

3:Initialize

atk\+←atka\_\{t\_\{k\}\}^\{\+\}\\leftarrow a\_\{t\_\{k\}\},

ftk\+←ftkf\_\{t\_\{k\}\}^\{\+\}\\leftarrow f\_\{t\_\{k\}\}
4:Refresh set of active positions

𝒰k←\{j:ftkj=0\}\\mathcal\{U\}\_\{k\}\\leftarrow\\\{j:f\_\{t\_\{k\}\}^\{j\}=0\\\}and state

sk←\(atk,ftk,𝒞tk\)s\_\{k\}\\leftarrow\(a\_\{t\_\{k\}\},f\_\{t\_\{k\}\},\\mathcal\{C\}\_\{t\_\{k\}\}\)
5:Compute posterior

ℚ1\|tkθ\(⋅∣sk\)\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(\\cdot\\mid s\_\{k\}\)and entropy

Hθj​\(sk\)H\_\{\\theta\}^\{j\}\(s\_\{k\}\)for all

j∈𝒰kj\\in\\mathcal\{U\}\_\{k\}
6:Determine number of active positions to be absorbed

bk←max⁡\{0,\|𝒰k\|−mk\}b\_\{k\}\\leftarrow\\max\\\{0,\|\\mathcal\{U\}\_\{k\}\|\-m\_\{k\}\\\}
7:Select

BkLEDB\_\{k\}^\{\\mathrm\{LED\}\}by Eq\. \([3](https://arxiv.org/html/2609.25131#S3.E3)\)

8:for

j∈BkLEDj\\in B\_\{k\}^\{\\mathrm\{LED\}\}do

9:Absorb with the most likely token under posterior

atkj,\+←arg⁡maxzj∈𝒮​ℚ1\|tkθ​\(a1j=zj∣sk\)a\_\{t\_\{k\}\}^\{j,\+\}\\leftarrow\\displaystyle\\arg\\max\_\{z^\{j\}\\in\\mathcal\{S\}\}\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=z^\{j\}\\mid s\_\{k\}\)
10:Update absorption indicator

ftkj,\+←1f\_\{t\_\{k\}\}^\{j,\+\}\\leftarrow 1
11:endfor

12:Evolve all

jjwith

ftkj,\+=0f\_\{t\_\{k\}\}^\{j,\+\}=0from

tkt\_\{k\}to

tk\+1t\_\{k\+1\}under the uniform flow velocity in Eq\. \([2](https://arxiv.org/html/2609.25131#S3.E2)\)

13:Keep

atk\+1j←atkj,\+a\_\{t\_\{k\+1\}\}^\{j\}\\leftarrow a\_\{t\_\{k\}\}^\{j,\+\}for all

jjwith

ftkj,\+=1f\_\{t\_\{k\}\}^\{j,\+\}=1
14:Update

ftk\+1←ftk\+f\_\{t\_\{k\+1\}\}\\leftarrow f\_\{t\_\{k\}\}^\{\+\}
15:endfor

16:return

atKa\_\{t\_\{K\}\}

LEDFlowselects active positions with low predictive entropy, absorbs their most likely values, and continues evolving the remaining positions under the uniform\-flow velocity \(illustrated in Algorithm[1](https://arxiv.org/html/2609.25131#alg1)\)\. The absorption indicator in Eq\.[2](https://arxiv.org/html/2609.25131#S3.E2)addresses late revisions in Section[2](https://arxiv.org/html/2609.25131#S2)by retaining selected predictions and preventing further transitions\. Letsk=\(atk,ftk,𝒞tk\)s\_\{k\}=\(a\_\{t\_\{k\}\},f\_\{t\_\{k\}\},\\mathcal\{C\}\_\{t\_\{k\}\}\)be the sampler state before absorption and𝒰k:=\{j:ftkj=0\}\\mathcal\{U\}\_\{k\}:=\\\{j:f\_\{t\_\{k\}\}^\{j\}=0\\\}the active positions\. Writeℚ1\|tkθ​\(a1j∣sk\)\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)for the denoiser marginal at positionjj, with predictive entropy

Hθj\(sk\):=−∑zj∈𝒮ℚ1\|tkθ\(a1j=zj∣sk\)logℚ1\|tkθ\(a1j=zj∣sk\)\.H\_\{\\theta\}^\{j\}\(s\_\{k\}\):=\-\\sum\_\{z^\{j\}\\in\\mathcal\{S\}\}\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=z^\{j\}\\mid s\_\{k\}\)\\log\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=z^\{j\}\\mid s\_\{k\}\)\.At grid pointtkt\_\{k\}, letmkm\_\{k\}be the target number of active positions and setbk:=max⁡\{0,\|𝒰k\|−mk\}b\_\{k\}:=\\max\\\{0,\|\\mathcal\{U\}\_\{k\}\|\-m\_\{k\}\\\}\. The entropy\-guided ordering policy selects thebkb\_\{k\}active positions with lowest predictive entropy and updates the indicator for every selected position,

BkLED∈arg⁡min⁡∑j∈BB⊆𝒰k\|B\|=bk⁡Hθj​\(sk\)⟹ftkj,\+=\{1,j∈BkLED,ftkj,j∉BkLED,B\_\{k\}^\{\\mathrm\{LED\}\}\\in\\arg\\min\_\{\\begin\{subarray\}\{c\}B\\subseteq\\mathcal\{U\}\_\{k\}\\\\ \|B\|=b\_\{k\}\\end\{subarray\}\}\\sum\_\{j\\in B\}H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\Longrightarrow f\_\{t\_\{k\}\}^\{j,\+\}=\\begin\{cases\}1,&j\\in B\_\{k\}^\{\\mathrm\{LED\}\},\\\\ f\_\{t\_\{k\}\}^\{j\},&j\\notin B\_\{k\}^\{\\mathrm\{LED\}\},\\end\{cases\}\(3\)atkj←arg⁡maxzj∈𝒮​ℚ1\|tkθ​\(a1j=zj∣sk\),ftkj←1,j∈BkLED\.a\_\{t\_\{k\}\}^\{j\}\\leftarrow\\arg\\max\_\{z^\{j\}\\in\\mathcal\{S\}\}\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=z^\{j\}\\mid s\_\{k\}\),\\qquad f\_\{t\_\{k\}\}^\{j\}\\leftarrow 1,\\quad j\\in B\_\{k\}^\{\\mathrm\{LED\}\}\.\(4\)This policy turns each selected posterior into a fixed token prediction, changing the sampler law while retaining the uniform flow update in Eq\.[2](https://arxiv.org/html/2609.25131#S3.E2)for active positions outsideBkLEDB\_\{k\}^\{\\mathrm\{LED\}\}\. At the next time grid, the posterior and entropy are recomputed from the updated state to reassess subsequent decisions based on absorbed values\. In the trajectory run \(Figure[2](https://arxiv.org/html/2609.25131#S3.F2)\(a–c\)\),LEDFlowreduces the fraction of token changes aftert=0\.75t=0\.75from 46% to 15% and raises the puzzle solve rate from 34% to 82%\. Specifically, this ordering policy is training free, as the absorption indicators can be directly plugged into the denoiser of the same checkpoint, whereas settingftj=0f\_\{t\}^\{j\}=0throughout recovers the base uniform discrete flow sampler\. We provide theoretical justification for the selection of low local entropy for absorption order in Section[3\.3](https://arxiv.org/html/2609.25131#S3.SS3)\. Appendix[C\.2](https://arxiv.org/html/2609.25131#A3.SS2)provides the schedule, update order, and matched\-policy details\.

\(c\) Trajectory\-run statisticsMeasurementUniform flow\(no absorption\)LEDFlow\(ours\)LEDFlow−\-Uniform flowToken changes per position1\.491\.491\.061\.06−0\.43\-0\.43\[2pt/1pt\]Fraction of positionschanged more than once0\.300\.300\.130\.13−0\.17\-0\.17\[2pt/1pt\]Fraction of token changesoccurring aftert=0\.75t=0\.750\.460\.460\.150\.15−0\.31\-0\.31\[2pt/1pt\]Mean sampling time ofa token’s last change0\.670\.670\.480\.48−0\.19\-0\.19Whole\-puzzle solve \(final\)0\.3400\.3400\.820\\mathbf\{0\.820\}\+0\.480\+0\.480*Trajectory correctness \(fraction of cells\)*Correct to wrong0\.0940\.0940\.026\\mathbf\{0\.026\}−0\.068\-0\.068

Figure 2:Ordered absorption reduces late revisions and improves puzzle solving\.\(a,b\)Token\-change timing and cell accuracy in the kinetic\-optimal trajectory run on 100 Nikoli puzzles withK=64K=64\.\(c\)Statistics from that same run: puzzle solve rate increases from 0\.34 to 0\.82, and correct\-to\-wrong reversions decrease from 0\.094 to 0\.026 of all generated cells \(cells correct at an intermediate step but incorrect at the endpoint\)\.
### 3\.3Theoretical justification

This section justifies the choice of local entropy policy inLEDFlow\. We first characterize the error of irreversible absorption, then establish the conditional\-error upper bound minimized by low\-entropy selection, and finally compare the sensitivity of local and lookahead scores to estimation error\.

##### Error induced by absorption\.

Noting that absorption is irreversible, its cost depends on the denoiser during each absorption process\. At an oracle\-reachable sampler statesks\_\{k\}, where the absorbed partial assignment has positive probability under the target distribution, letBk⊆𝒰kB\_\{k\}\\subseteq\\mathcal\{U\}\_\{k\}be the positions absorbed at that step\. Given the target distributionqq, we writep1\|skq​\(a1Bk\)p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{B\_\{k\}\}\)for the oracle posterior, with the denoiser marginal defined in Section[3\.2](https://arxiv.org/html/2609.25131#S3.SS2)\. We apply the standard KL decomposition, commonly used to quantify decoding mismatches\[[Xu et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib20),[Ben\-Hamu et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib39)\], to a stochastic reference kernel that samples selected positions independently from denoiser marginals\.

###### Proposition 2\(Decomposition of local absorption error\)\.

Conditioned on a reachable statesks\_\{k\}and an absorption blockBkB\_\{k\}, we write the errorℒabs​\(Bk,sk\):=DKL​\(p1\|skq​\(a1Bk\)∥∏j∈Bkℚ1\|tkθ​\(a1j∣sk\)\)\\mathcal\{L\}\_\{\\mathrm\{abs\}\}\(B\_\{k\};s\_\{k\}\):=D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{B\_\{k\}\}\)\\,\\middle\\\|\\,\\prod\_\{j\\in B\_\{k\}\}\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\), such that

ℒabs​\(Bk,sk\)=𝒯​𝒞q​\(a1Bk∣sk\)⏟ℰjoint​\(Bk,sk\)\+∑j∈BkDKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)⏟ℰcond​\(Bk,sk\),\\mathcal\{L\}\_\{\\mathrm\{abs\}\}\(B\_\{k\};s\_\{k\}\)=\\underbrace\{\\mathcal\{TC\}\_\{q\}\(a\_\{1\}^\{B\_\{k\}\}\\mid s\_\{k\}\)\}\_\{\\mathcal\{E\}\_\{\\mathrm\{joint\}\}\(B\_\{k\};s\_\{k\}\)\}\+\\underbrace\{\\sum\_\{j\\in B\_\{k\}\}D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\}\_\{\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)\},\(5\)where𝒯​𝒞q​\(a1Bk∣sk\)=∑j∈BkHq​\(a1j∣sk\)−Hq​\(a1Bk∣sk\)\\mathcal\{TC\}\_\{q\}\(a\_\{1\}^\{B\_\{k\}\}\\mid s\_\{k\}\)=\\sum\_\{j\\in B\_\{k\}\}H\_\{q\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\-H\_\{q\}\(a\_\{1\}^\{B\_\{k\}\}\\mid s\_\{k\}\)is the conditional total correlation of the positions absorbed together\.

###### Corollary 1\(Conditional error bounds the excess absorption risk\)\.

For a reachable statesks\_\{k\}and a blockBkB\_\{k\}with\|Bk\|=b\|B\_\{k\}\|=b, letz^j\\hat\{z\}^\{j\}be the most likely value underℚ1\|tkθ​\(a1j∣sk\)\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\),cj=p1\|skq​\(a1j=z^j\)c\_\{j\}=p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}=\\hat\{z\}^\{j\}\)its oracle probability, andcj⋆=maxz⁡p1\|skq​\(a1j=z\)c\_\{j\}^\{\\star\}=\\max\_\{z\}p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}=z\)the probability of best value of oracle\. Then

∑j∈Bk\(1−cj\)≤∑j∈Bk\(1−cj⋆\)\+2​b​ℰcond​\(Bk,sk\)\.\\sum\_\{j\\in B\_\{k\}\}\(1\-c\_\{j\}\)\\;\\leq\\;\\sum\_\{j\\in B\_\{k\}\}\(1\-c\_\{j\}^\{\\star\}\)\+\\sqrt\{2b\\,\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)\}\.\(6\)

The left side of Eq\.[6](https://arxiv.org/html/2609.25131#S3.E6)is the expected number of incorrect deterministic absorptions in the selected block, while the first term on the right refers to an unremovable marginal classification risk from oracle\. The square\-root term then bounds the difference due to denoiser error\. While larger blocks reduce sequential evaluations at the cost of unoptimized dependence error, the joint term vanishes forb=1b=1\. Appendix[C\.3](https://arxiv.org/html/2609.25131#A3.SS3)gives the derivation of the decomposition and the proof of the corollary\.

##### Minimizing the conditional error bound\.

For the active set𝒰k\\mathcal\{U\}\_\{k\}at a reachable sampler statesks\_\{k\}, fix an absorption budgetb≥1b\\geq 1and let𝔅k​\(b\)=\{B⊆𝒰k:\|B\|=b\}\\mathfrak\{B\}\_\{k\}\(b\)=\\\{B\\subseteq\\mathcal\{U\}\_\{k\}:\|B\|=b\\\}\. The oracle posterior serves as an analytical reference for selecting positions with small model error\. If it were available, the conditional\-error objective would be:

Bkoracle∈arg⁡minB∈𝔅k​\(b\)⁡∑j∈BDKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)⏟ℰcond​\(B,sk\)\.B\_\{k\}^\{\\mathrm\{oracle\}\}\\in\\arg\\min\_\{B\\in\\mathfrak\{B\}\_\{k\}\(b\)\}\\underbrace\{\\sum\_\{j\\in B\}D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\}\_\{\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B;s\_\{k\}\)\}\.\(7\)
The oracle posterior is unavailable at inference, to prioritize predictions that are currently reliable, the negative relationship between prediction entropy and reasoning accuracy in[Yang et al\. \[2026\]](https://arxiv.org/html/2609.25131#bib.bib36)motivates a condition regarding reliability and denoiser uncertainty, hence predictive entropy\.

###### Definition 1\(Entropy\-error regularity\)\.

LetHmax=log⁡\|𝒮\|H\_\{\\max\}=\\log\|\\mathcal\{S\}\|\. A denoiser is*ϕ\\phi\-regular*on the oracle\-reachable sampler statessks\_\{k\}if there exists a nondecreasing functionϕ:\[0,Hmax\]→ℝ≥0\\phi:\[0,H\_\{\\max\}\]\\to\\mathbb\{R\}\_\{\\geq 0\}, withϕ⁡\(0\)=0\\phi\(0\)=0, such thatDKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)≤ϕ⁡\(Hθj​\(sk\)\)D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\\leq\\phi\\\!\\left\(H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\right\)for anysks\_\{k\}andj∈𝒰kj\\in\\mathcal\{U\}\_\{k\}\.

Applying Def\.[1](https://arxiv.org/html/2609.25131#Thmdefinition1)to the conditional term in Eq\.[5](https://arxiv.org/html/2609.25131#S3.E5)gives, for any absorption blockBk⊆𝒰kB\_\{k\}\\subseteq\\mathcal\{U\}\_\{k\},ℰcond​\(Bk,sk\)≤Φ⁡\(Bk,sk\):=∑j∈Bkϕ⁡\(Hθj​\(sk\)\)\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)\\leq\\Phi\(B\_\{k\};s\_\{k\}\):=\\sum\_\{j\\in B\_\{k\}\}\\phi\\\!\\left\(H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\right\)\. Becauseϕ\\phiis nondecreasing, the block that minimizesΦ\\Phiunder a fixed absorption budget contains the active positions with lowest predictive entropy, so the square root term in Eq\.[6](https://arxiv.org/html/2609.25131#S3.E6)is at most2​b​Φ​\(Bk,sk\)\\sqrt\{2b\\,\\Phi\(B\_\{k\};s\_\{k\}\)\}, suggesting that using entropy for the ordering policy minimizes the upper excess risk bound on the conditional error for a fixed budget\.

###### Theorem 1\(Local surrogate minimization and the oracle limit\)\.

Given aϕ\\phi\-regular denoiser:*\(i\) Local adaptive absorption\.*For a fixed absorption budgetb≥1b\\geq 1, a block containingbbactive positions with lowest entropy minimizesΦ⁡\(B,sk\)\\Phi\(B;s\_\{k\}\)overB∈𝔅k​\(b\)B\\in\\mathfrak\{B\}\_\{k\}\(b\)\. In particular, for single position absorption,

j⋆∈arg⁡minj∈𝒰k​Hθj​\(sk\)j^\{\\star\}\\in\\arg\\min\_\{j\\in\\mathcal\{U\}\_\{k\}\}H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\(8\)minimizes the upper bound on the conditional error of the next irreversible prediction\.*\(ii\) Oracle limit\.*As the learned posterior converges to the oracle posterior on reachable states,ℰcond​\(Bk,sk\)→0\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)\\to 0for every admissible absorption block and hence for every absorption policy\.

Whenb\>1b\>1, the theorem minimizes the bound on the conditional term in Eq\.[5](https://arxiv.org/html/2609.25131#S3.E5)\. The result is local: absorbing a position changes the sampler state, and hence the errors at later absorption events\. We test Def\.[1](https://arxiv.org/html/2609.25131#Thmdefinition1)in Fig\.[2](https://arxiv.org/html/2609.25131#S3.F2)as the Sudoku denoiser has a computable oracle posterior, and observe that the condition holds on 95\.5% of sampled reachable states\. While Def\.[1](https://arxiv.org/html/2609.25131#Thmdefinition1)is stated for predictive entropy, we can apply it to local uncertainty score if the conditional error is bounded by a nondecreasing function of that score, e\.g\. maximum logit and probability margin\. We next compare between local greedy decoding against estimation over future states\. App\.[C\.3](https://arxiv.org/html/2609.25131#A3.SS3)provides proof and further details\.

##### Global lookahead can amplify estimation error\.

Global lookahead decoding \(represented by Info\-Gain\[[Yang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib36)\]\) evaluates information gain by estimating predicted uncertainty over all remaining active positions\. Let𝒰⁡\(s\)\\mathcal\{U\}\(s\)containmmactive positions at statess, and lethjq​\(s\)h\_\{j\}^\{q\}\(s\)andhjθ​\(s\)h\_\{j\}^\{\\theta\}\(s\)denote the oracle and model entropies\. DefineRr​\(s\):=∑j∈𝒰⁡\(s\)hjr​\(s\)R\_\{r\}\(s\):=\\sum\_\{j\\in\\mathcal\{U\}\(s\)\}h\_\{j\}^\{r\}\(s\),r∈\{q,θ\}r\\in\\\{q,\\theta\\\}\. For an active candidate positionj∈𝒰kj\\in\\mathcal\{U\}\_\{k\}and absorbed valuezjz^\{j\}, letsj,zjs^\{j,z^\{j\}\}denote the hypothetical state after absorption obtained by setting coordinatejjtozjz^\{j\}\. Writing the model and oracle posterior asPθj\(⋅∣s\)=ℚ1\|tθ\(a1j=⋅∣s\)P\_\{\\theta\}^\{j\}\(\\cdot\\mid s\)=\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(a\_\{1\}^\{j\}=\\cdot\\mid s\),Pqj\(⋅∣s\)=p1\|sq\(a1j=⋅\)P\_\{q\}^\{j\}\(\\cdot\\mid s\)=p^\{q\}\_\{1\\mid s\}\(a\_\{1\}^\{j\}=\\cdot\), the corresponding entropy reductions areGjr\(s\):=Rr\(s\)−𝔼zj∼Prj\(⋅∣s\)\[Rr\(sj,zj\)\]G\_\{j\}^\{r\}\(s\):=R\_\{r\}\(s\)\-\\mathbb\{E\}\_\{z^\{j\}\\sim P\_\{r\}^\{j\}\(\\cdot\\mid s\)\}\\left\[R\_\{r\}\(s^\{j,z^\{j\}\}\)\\right\],r∈\{q,θ\}r\\in\\\{q,\\theta\\\}\. Assume that at current statessand every hypothetical one\-step lookahead states′s^\{\\prime\}with positive model probability,\|hjθ​\(s′\)−hjq​\(s′\)\|≤δ\|h\_\{j\}^\{\\theta\}\(s^\{\\prime\}\)\-h\_\{j\}^\{q\}\(s^\{\\prime\}\)\|\\leq\\deltafor every activejj, and the total variation12​∑zj∈𝒮\|Pθj​\(zj∣s\)−Pqj​\(zj∣s\)\|≤ϵ\\frac\{1\}\{2\}\\sum\_\{z^\{j\}\\in\\mathcal\{S\}\}\\left\|P\_\{\\theta\}^\{j\}\(z^\{j\}\\mid s\)\-P\_\{q\}^\{j\}\(z^\{j\}\\mid s\)\\right\|\\leq\\epsilonfor every candidatejj\.

###### Lemma 1\(Error amplification in global entropy reduction\)\.

Letj^∈arg⁡maxj​Gjθ​\(s\)\\hat\{j\}\\in\\arg\\max\_\{j\}G\_\{j\}^\{\\theta\}\(s\),j⋆∈arg⁡maxj​Gjq​\(s\)j^\{\\star\}\\in\\arg\\max\_\{j\}G\_\{j\}^\{q\}\(s\), andj^ent∈arg⁡minj​hjθ​\(s\)\\hat\{j\}\_\{\\mathrm\{ent\}\}\\in\\arg\\min\_\{j\}h\_\{j\}^\{\\theta\}\(s\)\. BecauseRr​\(s\)R\_\{r\}\(s\)is common to every candidate atss, it cancels from the comparison between candidates, and for every pairj,j′j,j^\{\\prime\}the error of the decision\-relevant score difference obeys

\|\(Gjθ​\(s\)−Gj′θ​\(s\)\)−\(Gjq​\(s\)−Gj′q​\(s\)\)\|\\displaystyle\\left\|\\bigl\(G\_\{j\}^\{\\theta\}\(s\)\-G\_\{j^\{\\prime\}\}^\{\\theta\}\(s\)\\bigr\)\-\\bigl\(G\_\{j\}^\{q\}\(s\)\-G\_\{j^\{\\prime\}\}^\{q\}\(s\)\\bigr\)\\right\|≤2ηm,ηm:=\(m−1\)\(δ\+ϵlog\|𝒮\|\),\\displaystyle\\leq 2\\eta\_\{m\},\\qquad\\eta\_\{m\}:=\(m\-1\)\\bigl\(\\delta\+\\epsilon\\log\|\\mathcal\{S\}\|\\bigr\),Gj⋆q​\(s\)−Gj^q​\(s\)\\displaystyle G\_\{j^\{\\star\}\}^\{q\}\(s\)\-G\_\{\\hat\{j\}\}^\{q\}\(s\)≤2ηm,hj^entq\(s\)−minjhjq\(s\)≤2δ\.\\displaystyle\\leq 2\\eta\_\{m\},\\qquad h\_\{\\hat\{j\}\_\{\\mathrm\{ent\}\}\}^\{q\}\(s\)\-\\min\_\{j\}h\_\{j\}^\{q\}\(s\)\\leq 2\\delta\.

The bound on global lookahead selection grows with windowmmas the error of remainingm−1m\-1positions aggregate\. When2​ηm2\\eta\_\{m\}exceeds the oracle candidate margin, the bound supports using local entropy under an imperfect denoiser\. Conversely, ifηm\\eta\_\{m\}is below the margin, lookahead can be preferred\. Appendix[C\.3](https://arxiv.org/html/2609.25131#A3.SS3)provides derivation details and analysis outside oracle support\.

## 4Experiments

Setup\.We evaluateLEDFlowacross tasks with increasing structural dependence: six multimodal\-understanding benchmarks through VLMEvalKit\[[Duan et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib28)\], text\-to\-image generation on GenEval\[[Ghosh et al\., 2023](https://arxiv.org/html/2609.25131#bib.bib27)\], and reasoning on MathVista, MathVerse, GSM8K, three Sudoku sets, Latin\-square completion, graph coloring, and molecular infilling\[[Lu et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib25),[Zhang et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib18),[Cobbe et al\., 2021](https://arxiv.org/html/2609.25131#bib.bib26),[Seely et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib29),[Wang et al\., 2025a](https://arxiv.org/html/2609.25131#bib.bib30),[Polykovskiy et al\., 2020](https://arxiv.org/html/2609.25131#bib.bib9)\]\. We use FUDOKI\[[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10)\]with consistent configurations for multimodal understanding, text\-to\-image generation, and mathematics, and train task\-specific uniform flow denoisers for puzzle tasks, while the ordering policy requires no additional training\. All tasks have sampler stepK=64K=64\. Understanding, generation and reasoning evaluate on full standard splits, 553 prompts with four images per prompt, andn=500n=500with five seeds respectively\. We compare two baseline groups\. The first contains non\-absorbing decoders, including Eulerτ\\tau\-leaping, corrected, guidance, and self\-correction samplers\[[Shaul et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib3),[Campbell et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib1),[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13),[Wan et al\., 2026d](https://arxiv.org/html/2609.25131#bib.bib33),[Ouyang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib35)\]\. The second contains uniform flow with different absorption policies, including probability margin \(Top K\-margin\) and global lookahead \(Info\-Gain\)\[[Yang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib36)\]\. Appendix[D](https://arxiv.org/html/2609.25131#A4)provides experimental details\.

### 4\.1Empirical Performance

![Refer to caption](https://arxiv.org/html/2609.25131v1/fig3_qualitative.png)Figure 3:Qualitative comparison across the structure spectrum\.\(a\)Text\-to\-image on FUDOKI showsLEDFlowproduces images with improved alignment with the required descriptions\.\(b\)Sudoku: unordered Euler sampler repeats digits already present \(two ‘1’s and ‘8’s in row 7 and 9\), violating the row constraint, whileLEDFlowconditions on absorbed cells and avoids the conflict\.Table 1:Multimodal understanding scores\.White/grey/blue rows denote non\-absorbing baselines/absorbing baselines/LEDFlow\.Bold,italic, andunderliningmark first, second, and third place\. Avg\. rank is the mean benchmark rank \(↓\\downarrow\)\. Higher scores are better \(↑\\uparrow\)\. Entropy\-guided selective absorption gives numerically yet modestly higher scores\.samplerPOPE↑\\uparrowMME↑\\uparrowMMBench↑\\uparrowGQA↑\\uparrowMMMU↑\\uparrowMM\-Vet↑\\uparrowAvg\. rank↓\\downarrowEuler86\.11485\.473\.957\.634\.338\.04\.33Euler \(our replication\)87\.41494\.573\.757\.036\.238\.23\.50Guidance86\.81492\.774\.258\.235\.438\.63\.00Top K\-margin83\.51497\.465\.056\.930\.040\.04\.50Info\-Gain89\.31503\.065\.156\.429\.137\.54\.33\[2pt/1pt\]LEDFlow\(ours\)89\.01504\.674\.658\.037\.440\.51\.33

Table 2:GenEval category and overall scores\.Notation follows Table[3](https://arxiv.org/html/2609.25131#S4.T3)\.LEDFlow’s improving spread is consistent with more structural requirement on text\-to\-image tasks than text generation\.SamplerSingle↑\\uparrowTwo↑\\uparrowCount↑\\uparrowColors↑\\uparrowPos\.↑\\uparrowAttr\.↑\\uparrowOverall↑\\uparrowAvg\. rank↓\\downarrowEuler0\.96250\.83840\.48750\.88330\.72000\.63000\.75365\.42Euler \(our replication\)0\.93750\.86360\.50310\.87230\.68000\.66250\.75325\.92Time\-corrected0\.91250\.86870\.51250\.91490\.66000\.70000\.76144\.83Location\-corrected0\.97500\.86870\.50000\.90430\.67000\.64000\.75974\.08RK0\.92500\.83840\.56250\.87230\.65000\.74000\.76476\.17RK2\-Trapezoid0\.95000\.79800\.47500\.90430\.67000\.68000\.74625\.67Exact guidance0\.94000\.86000\.53000\.89000\.70000\.77000\.78003\.67Top K\-margin0\.94690\.78540\.41880\.88560\.62750\.58000\.70718\.17Info\-Gain0\.93120\.77530\.49060\.87770\.66000\.55500\.71508\.42\[2pt/1pt\]LEDFlow\(ours\)0\.95630\.87120\.52190\.90160\.71500\.72250\.78142\.67

Table 3:Constrained reasoning accuracy\.LEDFlowoutperforms in constrained reasoning task\.SamplerMathematics↑\\uparrowSudoku↑\\uparrowOther puzzles↑\\uparrowAvg\. rank↓\\downarrowMathVistaMathVerseGSM8KGeneratedNikoliExtremeLatinGraphMolec\.Euler0\.2540\.1100\.0260\.6100\.4100\.0780\.6520\.9360\.8285\.22FreeCorrection0\.2610\.1190\.0190\.6300\.4000\.1260\.6280\.9880\.7974\.72Time\-corrected0\.2410\.1110\.0220\.6100\.4100\.0780\.6520\.9360\.8285\.56Location\-corrected0\.2460\.1200\.0290\.5880\.3600\.0960\.6280\.9380\.8304\.78Top K\-margin0\.2550\.1200\.0240\.8200\.7960\.2710\.9480\.9510\.8872\.67Info\-Gain0\.2270\.1190\.0190\.8040\.7800\.2310\.9901\.0000\.9143\.50\[2pt/1pt\]LEDFlow\(ours\)0\.2550\.1230\.0270\.8650\.8450\.2690\.9981\.0000\.9061\.56

Multimodal understanding\.Adding entropy\-guided selective absorption gives numerically yet modestly higher scores than the replicated Euler baseline across all six benchmarks \(Table[3](https://arxiv.org/html/2609.25131#S4.T3)\), although Info\-Gain leads on POPE and exact guidance on GQA\. These modest gains place understanding at the weakly constrained end of the task spectrum, whereas absorption order matters less\. Further analyses, full results, and the uncertainty of these gains are provided in Appendix[E\.1](https://arxiv.org/html/2609.25131#A5.SS1)\.

Text\-to\-image generation\.Table[3](https://arxiv.org/html/2609.25131#S4.T3)givesLEDFlowthe numerically highest overall score \(0\.7814\) and average rank \(2\.67\), while being competitive against state\-of\-the\-art decode\-time exact guidance \(0\.7800\)\. This improving spread across samplers is consistent with text\-to\-image prompts imposing structure than above text\-based generation\. Figure[3](https://arxiv.org/html/2609.25131#S4.F3)provides qualitative examples and we provide further generation experiments and comparisons in Appendix[E\.2](https://arxiv.org/html/2609.25131#A5.SS2)\.

Reasoning\.Table[3](https://arxiv.org/html/2609.25131#S4.T3)reports outperforming constrained reasoning results\.LEDFlowreaches 0\.845 on Nikoli versus 0\.796 for margin and 0\.780 for lookahead ordering, and also leads these policies on the generated set while staying competitive in the Extreme dataset\. Across the nine tasks,LEDFlowis numerically higher than Info\-Gain on seven and staying competitive on molecular infilling \(0\.914 versus 0\.906\)\. Appendix[E\.5](https://arxiv.org/html/2609.25131#A5.SS5)reports detailed comparisons\.

### 4\.2Analysis and Validation of Theoretical Results

Entropy\-guided absorption \(Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\)\.Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(a\) tests the ordering principle in Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)on solve accuracy over 1,600 puzzles, showing the accuracy decreases monotonically fromLEDFlow\(0\.461\) to random \(0\.225\), the non\-absorbing Euler flow \(0\.211\), and a controlled setting that prioritizes high entropy absorption \(0\.115\)\. Panel \(b\) shows that, relative to a random absorption order through each model’s own sampler, theLEDFloworder improves LLaDA\-8B, Dream\-7B, MMaDA\-8B, and LLaDA\-V, with largest gains on the constrained reasoning benchmarks\. Grey cells means inapplicable\. Detailed transfer results with statistical tests are provided in Appendix[E\.6](https://arxiv.org/html/2609.25131#A5.SS6)\.

Error amplification in global entropy reduction \(Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)\)\.Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(c\) holds the absorbing state uniform flow and budget ofLEDFlowfixed and varies only the lookahead window aggregated by the Info\-Gain policy\[[Yang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib36)\]\. The mean absolute score error growth from 0\.17 in single lookahead term to 1\.76 in full aggregation validates score\-sensitivity mechanism in Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)\. Absorption accuracy peaks at eight aggregated windows and falls to 0\.31, showing that accumulated error alters selected coordinate under an imperfect denoiser\. Panel \(d\) shows that on GenEval,LEDFlowmatches inference time of uniform flow sampler, while Info\-Gain costs about 7 times more\. Appendix[E\.7](https://arxiv.org/html/2609.25131#A5.SS7)provides the efficiency and error\-analysis details on block\-size\.

![Refer to caption](https://arxiv.org/html/2609.25131v1/theory_aligned_ablation_2x2.png)Figure 4:Empirical analysis supports the entropy\-guided absorption policy and the predicted score sensitivity of global lookahead\.*Red \(Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\):*LEDFlow’s entropy\-guided absorption policy gives the highest puzzle accuracy \(a\), and its gain over a random absorption order transfers across pretrained models and tasks \(b\)\.*Blue \(Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)\):*aggregating more lookahead terms increases score error and eventually reduces absorption accuracy \(c\), with increased inference runtime \(d\)\.

## 5Related work

Discrete flow sampling, correction, and absorption\.Uniform discrete flows iteratively sample each position for repeated revisions\[[Gat et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib2),[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10)\], using Euler orτ\\tau\-leaping sampler on a time grid\[[Campbell et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib1),[Gat et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib2),[Shaul et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib3)\]\. Corrected samplers reduce errors caused by approximating the CTMC within a sampling interval, while uniformization removes this discretization error by simulating the process exactly\[[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13),[Wan et al\., 2026d](https://arxiv.org/html/2609.25131#bib.bib33)\]\. Separately, masked diffusion uses an absorbing state to make revealed predictions terminal\[[Austin et al\., 2021](https://arxiv.org/html/2609.25131#bib.bib4),[He et al\., 2023](https://arxiv.org/html/2609.25131#bib.bib5),[Shi et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib6)\], and uniform diffusion can be represented as a mixture of conditional absorbing processes\[[Gourevitch et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib16)\]\. Building on these, we augment a native uniform discrete flow with selective absorption and analyze its local KL error without changing the uniform flow velocity on positions that remain active\.

Reveal order in masked generation\.Masked generation has long used confidence to select predictions for irreversible decoding\[[Chang et al\., 2022](https://arxiv.org/html/2609.25131#bib.bib32),[Kim et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib15),[Ye et al\., 2025a](https://arxiv.org/html/2609.25131#bib.bib17)\], while planning and remasking methods learn which tokens to reveal or revisit\[[Peng et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib23),[Hong et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib19),[Xu et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib20)\]\. Prioritizing confidence alone can postpone high entropy reasoning forks and reduce solution coverage\[[Ni et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib22)\]\. Info\-Gain estimates lookahead quality over remaining positions\[[Yang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib36)\]which depends on denoiser quality\. EB\-Sampler\[[Ben\-Hamu et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib39)\]controls parallel unmasking block error with an entropy bound, while BoE steering\[[Saini et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib40)\]guides decoding with gradients of successor state entropy\.LEDFlowapplies entropy\-guided absorption to a revisable uniform flow with analysis of local prediction and dependence costs\. Appendix[F](https://arxiv.org/html/2609.25131#A6)provides broader scope of related work\.

## 6Conclusion

We introduce generation order into uniform discrete flow through selective absorption, which fixes chosen predictions while allowing other positions to continue evolving\. Our training\-free sampler,LEDFlow, absorbs the lowest\-entropy positions first to protect reliable predictions from later errors\. By decomposing local absorption error into joint dependence and conditional prediction terms and show that, under entropy\-error regularity, this policy minimizes an upper bound on the conditional term for a fixed absorption budget\. We also show how the decision\-error bound for global lookahead grows as it combines estimates across more positions\. Experiments on reasoning, text\-to\-image generation, and multimodal understanding show the largest gains on strongly constrained tasks, while trajectory diagnostics show fewer correct\-to\-wrong revisions\. Our analysis provides a local conditional\-error guarantee, with joint position selection, correction of absorbed predictions, and lookahead with more accurate denoisers offering directions for further study \(Appendix[A](https://arxiv.org/html/2609.25131#A1)\)\.

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## Appendix AClaim scope and limitations

Table[4](https://arxiv.org/html/2609.25131#A1.T4)summarizes the evidence and boundary associated with each main claim\. The core paper studies an entropy\-guided selective\-absorption policy added at inference time to a uniform discrete flow\. Appendix[G](https://arxiv.org/html/2609.25131#A7)separately presents a preliminary external signal that could replace the online entropy score; it is not required byLEDFlowor by our main results\.

Table 4:Evidence and scope of the main claims\.ClaimSupported byBoundarySelective absorption separates the priority decision from the native flow updateSection[3\.1](https://arxiv.org/html/2609.25131#S3.SS1)and Appendix[C\.1](https://arxiv.org/html/2609.25131#A3.SS1)Absorption deliberately changes the sampler law; it is not claimed to preserve the path law of the unmodified uniform flow\.The local absorption error separates into joint factorization and conditional prediction termsProposition[2](https://arxiv.org/html/2609.25131#Thmproposition2)and Appendix[C\.3\.1](https://arxiv.org/html/2609.25131#A3.SS3.SSS1)The equality is local to a fixed state and selected block; errors across the complete sampling trajectory also depend on subsequent states\.Lowest predictive entropy minimizes the conditional\-error upper boundTheorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)and Appendix[C\.3](https://arxiv.org/html/2609.25131#A3.SS3)The result assumes entropy\-error regularity on oracle\-reachable states; it optimizes a local surrogate and does not cover trajectories after support is lost\.Global entropy reduction accumulates estimation error with lookahead windowLemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)and Appendix[E\.7](https://arxiv.org/html/2609.25131#A5.SS7)This is an upper\-bound and controlled fixed\-state result; it does not imply that lookahead fails for every denoiser or state\.Entropy\-guided absorption transfers across tasks and pretrained modelsSection[4\.2](https://arxiv.org/html/2609.25131#S4.SS2)and Appendix[E\.6](https://arxiv.org/html/2609.25131#A5.SS6)Transfer compares with random order, not shipped decoders; several differences are not significant\. Sudoku runs share a checkpoint within each capacity; comparisons across cohorts retain their stated evaluation protocols\.External grounding supplies a usable offline signalAppendix[G](https://arxiv.org/html/2609.25131#A7)The controlled study establishes improved attention localization, not an answer\-accuracy gain or a replacement for the online entropy policy\.### A\.1Limitations and intended scope

##### Local theoretical result\.

Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)bounds the error of the current absorption decision under entropy\-error regularity\. Absorbing a position changes the state encountered later, so the theorem does not assert that a locally optimal block minimizes the final sequence loss for every model or task\. The oracle limit concerns conditional KL at a fixed oracle\-reachable state\. It does not guarantee recovery from earlier incorrect absorptions or eliminate joint dependence and oracle classification risk\.

##### Irreversible absorption\.

Once selected, a position cannot be revised\. This makes the contribution of the absorption policy measurable and preserves the native uniform\-flow update on active positions, but it can propagate an early incorrect prediction\. The method is consequently best suited to settings in which predictive entropy is informative; combining selective absorption with principled revision is an open direction\.

##### Lookahead comparison\.

Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)identifies how errors in counterfactual predictions accumulate when more entropy terms are aggregated\. The bound does not state that Info\-Gain must underperform: lookahead can remain useful when its estimates are accurate or its action margins exceed their error\. Our fixed\-state window experiment tests when the accumulated error becomes large enough to change the selected position\.

##### Baseline coverage\.

Our decode\-time baselines are those with a published implementation or a fully specified algorithm that runs on a uniform discrete flow without modification\. Two related samplers fall outside that set\. EB\-Sampler\[[Ben\-Hamu et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib39)\]controls the block size rather than the ordering, so it is complementary toLEDFlowrather than an alternative to it; we report a reimplementation on the puzzle tasks in Appendix[E\.7\.3](https://arxiv.org/html/2609.25131#A5.SS7.SSS3), labelled as such, rather than a row in the cross\-task tables\. BoE steering\[[Saini et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib40)\]requires a mask embedding to anchor its first\-order expansion, which a uniform flow does not provide; we therefore do not report a number for it rather than substitute a surrogate of our own construction\. Any future comparison against either method should be labelled as a reimplementation, compute matched by function evaluations, and reported at the best setting found in a disclosed hyperparameter sweep\.

##### Empirical coverage\.

The experiments span multimodal understanding, generation, and structured reasoning, but the largest improvements occur when outputs contain strong cross\-position constraints\. Broader nonuniform flow backbones and tasks permitting revision would further test the scope of selective absorption\. The external grounding study in Appendix[G](https://arxiv.org/html/2609.25131#A7)is deliberately complementary and makes no downstream accuracy claim\.

##### Training and decoding regimes\.

The ordering policy is training\-free\. FUDOKI weights remain frozen and its absorption indicator is sampler\-side; Sudoku denoisers are trained with indicator embeddings and ground\-truth absorbed values\. That training does not guarantee calibration after model\-generated absorptions\. Both non\-absorbing and absorbing Sudoku baselines use coordinatewise kinetic\-optimal updates, with cosine\-budget absorption for absorbing baselines\. The terminal\-rate proposition concerns the flow velocity, whereas the local absorption analysis concerns the selected absorption block\. Source\-held and singleton\-absorption studies are separate ablations\.

##### Remaining empirical attribution\.

The current experiments do not isolate selective absorption against terminal\-rate clipping, early stopping, or suppressed target resampling\. Nor do the reported timing differences among confidence scores isolate the cost of the score calculation from implementation overhead\. The existing reopening and block\-size ablations test particular implementations; they do not establish that all revision rules fail or that schedule shape is immaterial\.

## Appendix BDetails for Section[2](https://arxiv.org/html/2609.25131#S2): Problem Formulation

This appendix expands the two subsections of Section[2](https://arxiv.org/html/2609.25131#S2): Appendix[B\.1](https://arxiv.org/html/2609.25131#A2.SS1)gives the discrete\-flow background behind Section[2\.1](https://arxiv.org/html/2609.25131#S2.SS1), and Appendix[B\.2](https://arxiv.org/html/2609.25131#A2.SS2)proves the terminal\-exposure proposition and documents the Sudoku diagnostic of Section[2\.2](https://arxiv.org/html/2609.25131#S2.SS2)\.

### B\.1Preliminaries: discrete flow matching background

###### Definition 2\(Conditional CTMC\)\.

Given a rate matrixRt​\(y,z\)y,z∈𝒮MR\_\{t\}\(y,z\)\_\{y,z\\in\\mathcal\{S\}^\{M\}\}whereRt​\(y,z\)≥0R\_\{t\}\(y,z\)\\geq 0forz≠yz\\neq yand∑zRt​\(y,z\)=0\\sum\_\{z\}R\_\{t\}\(y,z\)=0, the process\{at\}\\\{a\_\{t\}\\\}is a CTMC with rateRtR\_\{t\}, if forh\>0h\>0,ℙ⁡\(at\+h=z\|at=y\)=δy​\(z\)\+Rt​\(y,z\)​h\+o⁡\(h\)\\mathbb\{P\}\(a\_\{t\+h\}=z\|a\_\{t\}=y\)=\\delta\_\{y\}\(z\)\+R\_\{t\}\(y,z\)h\+o\(h\)and its marginal densityptp\_\{t\}at timettobeys the Kolmogorov forward equationp˙t​\(y\)=∑z≠y\(pt​\(z\)​Rt​\(z,y\)−pt​\(y\)​Rt​\(y,z\)\)\\dot\{p\}\_\{t\}\(y\)=\\sum\_\{z\\neq y\}\(p\_\{t\}\(z\)R\_\{t\}\(z,y\)\-p\_\{t\}\(y\)R\_\{t\}\(y,z\)\)\.

We expand the probability\-path and velocity formulation summarized in[Section2\.1](https://arxiv.org/html/2609.25131#S2.SS1), following[Gat et al\. \[2024\]](https://arxiv.org/html/2609.25131#bib.bib2), and then give the uniform metric\-induced specialization used by FUDOKI\[[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10)\]\.

##### Conditional and marginal probability paths\.

Letπ⁡\(a0,a1∣𝒞\)\\pi\(a\_\{0\},a\_\{1\}\\mid\\mathcal\{C\}\)be a coupling whose marginals are the sourcep0​\(a0∣𝒞\)p\_\{0\}\(a\_\{0\}\\mid\\mathcal\{C\}\)and targetq⁡\(a1∣𝒞\)q\(a\_\{1\}\\mid\\mathcal\{C\}\)\. Discrete flow matching specifies a conditional path between each coupled pair and marginalizes the pair:

pt​\(a∣𝒞\)\\displaystyle p\_\{t\}\(a\\mid\\mathcal\{C\}\)=∑a0,a1pt​\(a∣a0,a1,𝒞\)​π​\(a0,a1∣𝒞\),\\displaystyle=\\sum\_\{a\_\{0\},a\_\{1\}\}p\_\{t\}\(a\\mid a\_\{0\},a\_\{1\},\\mathcal\{C\}\)\\,\\pi\(a\_\{0\},a\_\{1\}\\mid\\mathcal\{C\}\),\(9\)pt​\(a∣a0,a1,𝒞\)\\displaystyle p\_\{t\}\(a\\mid a\_\{0\},a\_\{1\},\\mathcal\{C\}\)=∏j=1Mptj​\(aj∣a0,a1,𝒞\)\.\\displaystyle=\\prod\_\{j=1\}^\{M\}p\_\{t\}^\{j\}\(a^\{j\}\\mid a\_\{0\},a\_\{1\},\\mathcal\{C\}\)\.\(10\)The boundary conditionsp0j\(⋅∣a0,a1\)=δa0jp\_\{0\}^\{j\}\(\\cdot\\mid a\_\{0\},a\_\{1\}\)=\\delta\_\{a\_\{0\}^\{j\}\}andp1j\(⋅∣a0,a1\)=δa1jp\_\{1\}^\{j\}\(\\cdot\\mid a\_\{0\},a\_\{1\}\)=\\delta\_\{a\_\{1\}^\{j\}\}ensure that the marginal path begins atp0p\_\{0\}and ends atqq\. When the source is fixed or integrated into the coordinate path, this reduces to the target\-conditioned notationpt\|1​\(a∣a1\)p\_\{t\\mid 1\}\(a\\mid a\_\{1\}\)used in the main text\.

##### Probability velocities and continuity\.

For a current stateaa, the coordinate velocityutj​\(v,a\)u\_\{t\}^\{j\}\(v,a\)defines the infinitesimal update

ℙ⁡\(At\+hj=v∣At=a\)=δaj​\(v\)\+h​utj​\(v,a\)\+o⁡\(h\)\.\\mathbb\{P\}\(A\_\{t\+h\}^\{j\}=v\\mid A\_\{t\}=a\)=\\delta\_\{a^\{j\}\}\(v\)\+h\\,u\_\{t\}^\{j\}\(v,a\)\+o\(h\)\.\(11\)It is a valid CTMC rate when

∑v∈𝒮utj​\(v,a\)=0,utj​\(v,a\)≥0for​v≠aj\.\\sum\_\{v\\in\\mathcal\{S\}\}u\_\{t\}^\{j\}\(v,a\)=0,\\qquad u\_\{t\}^\{j\}\(v,a\)\\geq 0\\quad\\text\{for \}v\\neq a^\{j\}\.\(12\)The full velocity permits one\-coordinate transitions,ut​\(z,a\)=∑jδ⁡\(z−j,a−j\)​utj​\(zj,a\)u\_\{t\}\(z,a\)=\\sum\_\{j\}\\delta\(z^\{\-j\},a^\{\-j\}\)u\_\{t\}^\{j\}\(z^\{j\},a\), with the first argument denoting the destination\. Hence the path evolves according to the Kolmogorov forward equation

p˙t​\(a∣𝒞\)=∑z∈𝒮Mpt​\(z∣𝒞\)​ut​\(a,z\)\.\\dot\{p\}\_\{t\}\(a\\mid\\mathcal\{C\}\)=\\sum\_\{z\\in\\mathcal\{S\}^\{M\}\}p\_\{t\}\(z\\mid\\mathcal\{C\}\)u\_\{t\}\(a,z\)\.\(13\)Equivalently, define the probability flux fromzztoaabyJt​\(a,z\)=pt​\(z\)​ut​\(a,z\)J\_\{t\}\(a,z\)=p\_\{t\}\(z\)u\_\{t\}\(a,z\)\. Thenp˙t​\(a\)\+diva⁡Jt=0\\dot\{p\}\_\{t\}\(a\)\+\\operatorname\{div\}\_\{a\}J\_\{t\}=0, where

diva⁡Jt=∑z≠a\[Jt​\(z,a\)−Jt​\(a,z\)\]\.\\operatorname\{div\}\_\{a\}J\_\{t\}=\\sum\_\{z\\neq a\}\\bigl\[J\_\{t\}\(z,a\)\-J\_\{t\}\(a,z\)\\bigr\]\.\(14\)Thus a velocity generatesptp\_\{t\}precisely when its flux satisfies this continuity equation and the rate conditions above\.

##### Conditional velocities and posterior marginalization\.

Supposeutj\(v,aj∣a0,a1,𝒞\)u\_\{t\}^\{j\}\(v,a^\{j\}\\mid a\_\{0\},a\_\{1\},\\mathcal\{C\}\)generates the coordinate path in \([10](https://arxiv.org/html/2609.25131#A2.E10)\)\. The corresponding marginal velocity is obtained by averaging over the target pair conditioned on the current state:

utj​\(v,a,𝒞\)\\displaystyle u\_\{t\}^\{j\}\(v,a,\\mathcal\{C\}\)=𝔼\(a0,a1\)∼pt\(⋅,⋅∣a,𝒞\)\[utj\(v,aj∣a0,a1,𝒞\)\],\\displaystyle=\\mathbb\{E\}\_\{\(a\_\{0\},a\_\{1\}\)\\sim p\_\{t\}\(\\cdot,\\cdot\\mid a,\\mathcal\{C\}\)\}\\\!\\left\[u\_\{t\}^\{j\}\(v,a^\{j\}\\mid a\_\{0\},a\_\{1\},\\mathcal\{C\}\)\\right\],\(15\)pt\(a0,a1∣a,𝒞\)\\displaystyle p\_\{t\}\(a\_\{0\},a\_\{1\}\\mid a,\\mathcal\{C\}\)=pt​\(a∣a0,a1,𝒞\)​π​\(a0,a1∣𝒞\)pt​\(a∣𝒞\)\.\\displaystyle=\\frac\{p\_\{t\}\(a\\mid a\_\{0\},a\_\{1\},\\mathcal\{C\}\)\\pi\(a\_\{0\},a\_\{1\}\\mid\\mathcal\{C\}\)\}\{p\_\{t\}\(a\\mid\\mathcal\{C\}\)\}\.\(16\)For a target\-conditioned path, onlyp1\|t​\(a1∣a,𝒞\)p\_\{1\\mid t\}\(a\_\{1\}\\mid a,\\mathcal\{C\}\)remains unknown\. It is approximated by the denoiserℚ1\|tθ\\mathbb\{Q\}\_\{1\\mid t\}^\{\\theta\}, trained with the coordinate cross\-entropy objective

ℒden​\(θ\)=−𝔼t,a0,a1,at​\[∑j=1Mlog⁡ℚ1\|tθ​\(a1j∣at,𝒞t\)\]\.\\mathcal\{L\}\_\{\\mathrm\{den\}\}\(\\theta\)=\-\\mathbb\{E\}\_\{t,a\_\{0\},a\_\{1\},a\_\{t\}\}\\left\[\\sum\_\{j=1\}^\{M\}\\log\\mathbb\{Q\}\_\{1\\mid t\}^\{\\theta\}\(a\_\{1\}^\{j\}\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)\\right\]\.\(17\)Substituting this posterior into \([15](https://arxiv.org/html/2609.25131#A2.E15)\) yields the learned velocity used at generation time\.

##### Uniform metric\-induced flow in FUDOKI\.

FUDOKI uses a distanced:𝒮×𝒮→ℝ≥0d:\\mathcal\{S\}\\times\\mathcal\{S\}\\to\\mathbb\{R\}\_\{\\geq 0\}satisfyingd⁡\(v,y\)=0d\(v,y\)=0if and only ifv=yv=y, and defines

pt\|1j​\(v∣a1j\)=exp⁡\[−βt​d​\(v,a1j\)\]∑w∈𝒮exp⁡\[−βt​d​\(w,a1j\)\],β0=0,βt→t→1∞\.p\_\{t\\mid 1\}^\{j\}\(v\\mid a\_\{1\}^\{j\}\)=\\frac\{\\exp\[\-\\beta\_\{t\}d\(v,a\_\{1\}^\{j\}\)\]\}\{\\sum\_\{w\\in\\mathcal\{S\}\}\\exp\[\-\\beta\_\{t\}d\(w,a\_\{1\}^\{j\}\)\]\},\\qquad\\beta\_\{0\}=0,\\quad\\beta\_\{t\}\\xrightarrow\[t\\to 1\]\{\}\\infty\.\(18\)The initial coordinate distribution is therefore uniform over𝒮\\mathcal\{S\}, while the target concentrates ona1ja\_\{1\}^\{j\}\. To select a velocity for this prescribed path, write the conditional flux fromrrtovvasJtj​\(v,r∣a1j\)J\_\{t\}^\{j\}\(v,r\\mid a\_\{1\}^\{j\}\)and minimize its kinetic energy subject to continuity, nonnegative off\-diagonal flux, and the path boundaries:

minJtj∫01∑v≠rωt\(v,r\)Jtj​\(v,r∣a1j\)2pt\|1j​\(r∣a1j\)dt,ωt\(v,r\)\>0\.\\min\_\{J\_\{t\}^\{j\}\}\\int\_\{0\}^\{1\}\\sum\_\{v\\neq r\}\\omega\_\{t\}\(v,r\)\\frac\{J\_\{t\}^\{j\}\(v,r\\mid a\_\{1\}^\{j\}\)^\{2\}\}\{p\_\{t\\mid 1\}^\{j\}\(r\\mid a\_\{1\}^\{j\}\)\}\\,\\mathrm\{d\}t,\\qquad\\omega\_\{t\}\(v,r\)\>0\.\(19\)For the weightingωt​\(v,r\)=1/pt\|1j​\(v∣a1j\)\\omega\_\{t\}\(v,r\)=1/p\_\{t\\mid 1\}^\{j\}\(v\\mid a\_\{1\}^\{j\}\), the optimal flux is

Jtj,⋆​\(v,r∣a1j\)=\[pt\|1j​\(r∣a1j\)​p˙t\|1j​\(v∣a1j\)−p˙t\|1j​\(r∣a1j\)​pt\|1j​\(v∣a1j\)\]\+\.J\_\{t\}^\{j,\\star\}\(v,r\\mid a\_\{1\}^\{j\}\)=\\left\[p\_\{t\\mid 1\}^\{j\}\(r\\mid a\_\{1\}^\{j\}\)\\,\\dot\{p\}\_\{t\\mid 1\}^\{j\}\(v\\mid a\_\{1\}^\{j\}\)\-\\dot\{p\}\_\{t\\mid 1\}^\{j\}\(r\\mid a\_\{1\}^\{j\}\)\\,p\_\{t\\mid 1\}^\{j\}\(v\\mid a\_\{1\}^\{j\}\)\\right\]\_\{\+\}\.\(20\)Dividing by the probability at the current tokenr=ajr=a^\{j\}gives the off\-diagonal conditional velocity

utj​\(v,aj∣a1j\)=pt\|1j​\(v∣a1j\)​β˙t​\[d⁡\(aj,a1j\)−d⁡\(v,a1j\)\]\+,v≠aj,u\_\{t\}^\{j\}\(v,a^\{j\}\\mid a\_\{1\}^\{j\}\)=p\_\{t\\mid 1\}^\{j\}\(v\\mid a\_\{1\}^\{j\}\)\\dot\{\\beta\}\_\{t\}\\left\[d\(a^\{j\},a\_\{1\}^\{j\}\)\-d\(v,a\_\{1\}^\{j\}\)\\right\]\_\{\+\},\\qquad v\\neq a^\{j\},\(21\)with the diagonal fixed by \([12](https://arxiv.org/html/2609.25131#A2.E12)\)\. Probability mass therefore moves only to tokens closer to the target underdd\. Finally, FUDOKI replaces the target posterior in \([15](https://arxiv.org/html/2609.25131#A2.E15)\) byℚ1\|tθ\\mathbb\{Q\}\_\{1\\mid t\}^\{\\theta\}:

utθ,j\(v,a,𝒞t\)=𝔼a1∼ℚ1\|tθ\(⋅∣a,𝒞t\)\[utj\(v,aj∣a1j\)\]\.u\_\{t\}^\{\\theta,j\}\(v,a,\\mathcal\{C\}\_\{t\}\)=\\mathbb\{E\}\_\{a\_\{1\}\\sim\\mathbb\{Q\}\_\{1\\mid t\}^\{\\theta\}\(\\cdot\\mid a,\\mathcal\{C\}\_\{t\}\)\}\\\!\\left\[u\_\{t\}^\{j\}\(v,a^\{j\}\\mid a\_\{1\}^\{j\}\)\\right\]\.\(22\)Applying \([11](https://arxiv.org/html/2609.25131#A2.E11)\) with this velocity gives the native FUDOKI sampler\. Because the uniform path has full vocabulary support and introduces no terminal state, a coordinate may change repeatedly beforet=1t=1; the selective absorbing state in[Section3\.1](https://arxiv.org/html/2609.25131#S3.SS1)is an additional inference mechanism rather than part of this standard flow construction\.

### B\.2Motivation and ordered absorption: repeated transitions and the absorption decision

This appendix formalizes the transition behavior summarized in Section[2\.2](https://arxiv.org/html/2609.25131#S2.SS2)and derives a decision threshold for absorption\.

##### Setting\.

We work directly with the kinetic\-optimal velocity of Eq\.[1](https://arxiv.org/html/2609.25131#S2.E1)\. Conditioned on the true targeta1ja\_\{1\}^\{j\}it reads

utj​\(zj,at∣a1j\)=pt​\(zj∣a1j\)​β˙t​\[d⁡\(atj,a1j\)−d⁡\(zj,a1j\)\]\+,pt​\(zj∣a1j\)=e−βt​d​\(zj,a1j\)Zt​\(a1j\),u\_\{t\}^\{j\}\(z^\{j\},a\_\{t\}\\mid a\_\{1\}^\{j\}\)=p\_\{t\}\(z^\{j\}\\mid a\_\{1\}^\{j\}\)\\,\\dot\{\\beta\}\_\{t\}\\left\[d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\)\-d\(z^\{j\},a\_\{1\}^\{j\}\)\\right\]\_\{\+\},\\qquad p\_\{t\}\(z^\{j\}\\mid a\_\{1\}^\{j\}\)=\\frac\{e^\{\-\\beta\_\{t\}d\(z^\{j\},a\_\{1\}^\{j\}\)\}\}\{Z\_\{t\}\(a\_\{1\}^\{j\}\)\},\(23\)withZt​\(a1j\)=∑z∈𝒮e−βt​d​\(z,a1j\)Z\_\{t\}\(a\_\{1\}^\{j\}\)=\\sum\_\{z\\in\\mathcal\{S\}\}e^\{\-\\beta\_\{t\}d\(z,a\_\{1\}^\{j\}\)\}; at inference the solver substitutes the predicted targeta^1j∼ℚ1\|tθ\(⋅∣at,𝒞t\)\\hat\{a\}\_\{1\}^\{j\}\\sim\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(\\cdot\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)fora1ja\_\{1\}^\{j\}throughout\. We assume only thatddis a metric on𝒮\\mathcal\{S\}, so thatd⁡\(z,z′\)\>0d\(z,z^\{\\prime\}\)\>0wheneverz≠z′z\\neq z^\{\\prime\}; no further structure is needed, and in particular the argument does not requireddto be the discrete metric\. This matters because FUDOKI computesddfrom pretrained token embeddings rather than from token identity\. We also useβt=c​\(t/\(1−t\)\)α\\beta\_\{t\}=c\(t/\(1\-t\)\)^\{\\alpha\}withc=3,α=0\.9c=3,\\alpha=0\.9, whence

β˙t=c​α​tα−1​\(1−t\)−α−1→t→1∞\.\\dot\{\\beta\}\_\{t\}=c\\,\\alpha\\,t^\{\\alpha\-1\}\(1\-t\)^\{\-\\alpha\-1\}\\xrightarrow\[t\\to 1\]\{\}\\infty\.\(24\)Throughout,atja\_\{t\}^\{j\}is the token currently held at positionjjanda1ja\_\{1\}^\{j\}is the true target there\.

###### Proposition 3\(Target\-conditioned monotonicity\)\.

Let

λtj:=∑zj≠atjutj​\(zj,at∣a1j\),λ^tj:=∑zj≠atjutj​\(zj,at∣a^1j\)\\lambda\_\{t\}^\{j\}:=\\sum\_\{z^\{j\}\\neq a\_\{t\}^\{j\}\}u\_\{t\}^\{j\}\(z^\{j\},a\_\{t\}\\mid a\_\{1\}^\{j\}\),\\qquad\\hat\{\\lambda\}\_\{t\}^\{j\}:=\\sum\_\{z^\{j\}\\neq a\_\{t\}^\{j\}\}u\_\{t\}^\{j\}\(z^\{j\},a\_\{t\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)\(25\)be the total transition rates obtained by conditioning the kinetic\-optimal velocity on the oracle and predicted targets, respectively\. Ifatj=a1ja\_\{t\}^\{j\}=a\_\{1\}^\{j\}, thenλtj=0\\lambda\_\{t\}^\{j\}=0\. Fora^1j∼ℚ1\|tθ\(⋅∣at,𝒞t\)\\hat\{a\}\_\{1\}^\{j\}\\sim\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(\\cdot\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\), the expected rate at which a correct token is replaced satisfies

𝔼\[λ^tj\]≥β˙t𝔼\[𝟏\{a^1j≠a1j\}pt\(a^1j∣a^1j\)d\(a1j,a^1j\)\]≥β˙tεtjpmin,tdmin,\\mathbb\{E\}\[\\hat\{\\lambda\}\_\{t\}^\{j\}\]\\geq\\dot\{\\beta\}\_\{t\}\\,\\mathbb\{E\}\\\!\\left\[\\mathbf\{1\}\\\{\\hat\{a\}\_\{1\}^\{j\}\\neq a\_\{1\}^\{j\}\\\}p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)d\(a\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\\right\]\\geq\\dot\{\\beta\}\_\{t\}\\,\\varepsilon\_\{t\}^\{j\}p\_\{\\min,t\}d\_\{\\min\},\(26\)whereεtj=1−ℚ1\|tθ​\(a1j∣at,𝒞t\)\\varepsilon\_\{t\}^\{j\}=1\-\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(a\_\{1\}^\{j\}\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\),pmin,t=minz∈𝒮⁡pt​\(z∣z\)p\_\{\\min,t\}=\\min\_\{z\\in\\mathcal\{S\}\}p\_\{t\}\(z\\mid z\), anddmin=minz≠a1j⁡d⁡\(z,a1j\)d\_\{\\min\}=\\min\_\{z\\neq a\_\{1\}^\{j\}\}d\(z,a\_\{1\}^\{j\}\)\. Hence the lower bound is positive whenever all three quantities are positive\.

##### Part \(i\): the true target leaves a correct token fixed\.

Suppose the velocity is conditioned on the true target and the position already holds it, so thatatj=a1ja\_\{t\}^\{j\}=a\_\{1\}^\{j\}\. For anyzj≠atjz^\{j\}\\neq a\_\{t\}^\{j\},

utj​\(zj,at∣a1j\)=pt​\(zj∣a1j\)​β˙t​\[d⁡\(a1j,a1j\)−d⁡\(zj,a1j\)\]\+=pt​\(zj∣a1j\)​β˙t​\[−d⁡\(zj,a1j\)\]\+=0,u\_\{t\}^\{j\}\(z^\{j\},a\_\{t\}\\mid a\_\{1\}^\{j\}\)=p\_\{t\}\(z^\{j\}\\mid a\_\{1\}^\{j\}\)\\,\\dot\{\\beta\}\_\{t\}\\left\[d\(a\_\{1\}^\{j\},a\_\{1\}^\{j\}\)\-d\(z^\{j\},a\_\{1\}^\{j\}\)\\right\]\_\{\+\}=p\_\{t\}\(z^\{j\}\\mid a\_\{1\}^\{j\}\)\\,\\dot\{\\beta\}\_\{t\}\\left\[\-d\(z^\{j\},a\_\{1\}^\{j\}\)\\right\]\_\{\+\}=0,becaused⁡\(a1j,a1j\)=0d\(a\_\{1\}^\{j\},a\_\{1\}^\{j\}\)=0andd⁡\(zj,a1j\)\>0d\(z^\{j\},a\_\{1\}^\{j\}\)\>0, so the bracket is negative and the ReLU annihilates it\. Summing over destinations givesλtj=0\\lambda\_\{t\}^\{j\}=0: conditioned on the true target, a position that has reached it never leaves\. Retention at the true target is a property of the exact velocity itself, not an addition to it\.

##### Toward part \(ii\): a single transition under a wrong predicted target\.

At inference the velocity is conditioned ona^1j\\hat\{a\}\_\{1\}^\{j\}rather thana1ja\_\{1\}^\{j\}\. Suppose the position is correct,atj=a1ja\_\{t\}^\{j\}=a\_\{1\}^\{j\}, but the prediction misses,a^1j≠a1j\\hat\{a\}\_\{1\}^\{j\}\\neq a\_\{1\}^\{j\}\. Taking the destinationzj=a^1jz^\{j\}=\\hat\{a\}\_\{1\}^\{j\},

utj​\(a^1j,at∣a^1j\)=pt​\(a^1j∣a^1j\)​β˙t​\[d⁡\(a1j,a^1j\)−d⁡\(a^1j,a^1j\)\]\+=pt​\(a^1j∣a^1j\)​β˙t​d​\(a1j,a^1j\)\>0,u\_\{t\}^\{j\}\(\\hat\{a\}\_\{1\}^\{j\},a\_\{t\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)=p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)\\,\\dot\{\\beta\}\_\{t\}\\left\[d\(a\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\-d\(\\hat\{a\}\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\\right\]\_\{\+\}=p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)\\,\\dot\{\\beta\}\_\{t\}\\,d\(a\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\>0,sincept​\(a^1j∣a^1j\)=1/Zt​\(a^1j\)\>0p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)=1/Z\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\)\>0andd⁡\(a1j,a^1j\)\>0d\(a\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\>0\. The transition is not merely permitted but is the one the velocity most favours, becausea^1j\\hat\{a\}\_\{1\}^\{j\}is the closest state to the value the velocity is conditioned on\. The monotonicity of Eq\.[23](https://arxiv.org/html/2609.25131#A2.E23)is thus relative toa^1j\\hat\{a\}\_\{1\}^\{j\}and carries no implication ford⁡\(⋅,a1j\)d\(\\cdot,a\_\{1\}^\{j\}\): a step that moves closer to the prediction can move away from the truth\.

##### Part \(ii\): the expected total conditional transition rate\.

Every destination contributes a nonnegative rate, so retaining only the summandzj=a^1jz^\{j\}=\\hat\{a\}\_\{1\}^\{j\}computed above lower boundsλ^tj\\hat\{\\lambda\}\_\{t\}^\{j\}\. The term vanishes whena^1j=a1j\\hat\{a\}\_\{1\}^\{j\}=a\_\{1\}^\{j\}by part \(i\), so taking expectations over the predicted target gives the first inequality of Eq\.[26](https://arxiv.org/html/2609.25131#A2.E26),

𝔼\[λ^tj\]≥β˙t𝔼\[𝟏\{a^1j≠a1j\}pt\(a^1j∣a^1j\)d\(a1j,a^1j\)\]\.\\mathbb\{E\}\\\!\\left\[\\hat\{\\lambda\}\_\{t\}^\{j\}\\right\]\\geq\\dot\{\\beta\}\_\{t\}\\,\\mathbb\{E\}\\\!\\left\[\\mathbf\{1\}\\\{\\hat\{a\}\_\{1\}^\{j\}\\neq a\_\{1\}^\{j\}\\\}\\,p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)\\,d\(a\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\\right\]\.Boundingpt​\(a^1j∣a^1j\)≥pmin,t:=minz∈𝒮⁡1/Zt​\(z\)\>0p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)\\geq p\_\{\\min,t\}:=\\min\_\{z\\in\\mathcal\{S\}\}1/Z\_\{t\}\(z\)\>0andd⁡\(a1j,a^1j\)≥dmin:=minz≠a1j⁡d⁡\(z,a1j\)\>0d\(a\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\\geq d\_\{\\min\}:=\\min\_\{z\\neq a\_\{1\}^\{j\}\}d\(z,a\_\{1\}^\{j\}\)\>0on the event\{a^1j≠a1j\}\\\{\\hat\{a\}\_\{1\}^\{j\}\\neq a\_\{1\}^\{j\}\\\}, whose probability isεtj=1−ℚ1\|tθ​\(a1j∣at,𝒞t\)\\varepsilon\_\{t\}^\{j\}=1\-\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(a\_\{1\}^\{j\}\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\), yields the second inequality,𝔼⁡\[λ^tj\]≥β˙t​εtj​pmin,t​dmin\\mathbb\{E\}\[\\hat\{\\lambda\}\_\{t\}^\{j\}\]\\geq\\dot\{\\beta\}\_\{t\}\\,\\varepsilon\_\{t\}^\{j\}\\,p\_\{\\min,t\}\\,d\_\{\\min\}\. A denoiser placing all of its mass on the true target hasεtj=0\\varepsilon\_\{t\}^\{j\}=0, recovering part \(i\); any denoiser with residual target prediction error has𝔼⁡\[λ^tj\]\>0\\mathbb\{E\}\[\\hat\{\\lambda\}\_\{t\}^\{j\}\]\>0\.

##### Proof of Proposition[1](https://arxiv.org/html/2609.25131#Thmproposition1)\(terminal blow\-up of the transition rate\)\.

Fix a positionjjwithatj≠a1ja\_\{t\}^\{j\}\\neq a\_\{1\}^\{j\}\. The summand in the total rate of Eq\.[25](https://arxiv.org/html/2609.25131#A2.E25)vanishes atzj=atjz^\{j\}=a\_\{t\}^\{j\}\(its bracket is00\), so, factoring the schedule gradient out of Eq\.[23](https://arxiv.org/html/2609.25131#A2.E23),

λtj=∑zj≠atjutj\(zj,at∣a1j\)=β˙t∑zj∈𝒮pt​\(zj∣a1j\)​\[d⁡\(atj,a1j\)−d⁡\(zj,a1j\)\]\+⏟=:Gtj\.\\lambda\_\{t\}^\{j\}=\\sum\_\{z^\{j\}\\neq a\_\{t\}^\{j\}\}u\_\{t\}^\{j\}\(z^\{j\},a\_\{t\}\\mid a\_\{1\}^\{j\}\)=\\dot\{\\beta\}\_\{t\}\\underbrace\{\\sum\_\{z^\{j\}\\in\\mathcal\{S\}\}p\_\{t\}\(z^\{j\}\\mid a\_\{1\}^\{j\}\)\\big\[d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\)\-d\(z^\{j\},a\_\{1\}^\{j\}\)\\big\]\_\{\+\}\}\_\{=:~G\_\{t\}^\{j\}\}\.Retaining only the destinationzj=a1jz^\{j\}=a\_\{1\}^\{j\}, whose bracket equalsd⁡\(atj,a1j\)−d⁡\(a1j,a1j\)=d⁡\(atj,a1j\)d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\)\-d\(a\_\{1\}^\{j\},a\_\{1\}^\{j\}\)=d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\), and discarding the remaining nonnegative terms gives

Gtj≥pt​\(a1j∣a1j\)​d​\(atj,a1j\)\>0,G\_\{t\}^\{j\}\\ \\geq\\ p\_\{t\}\(a\_\{1\}^\{j\}\\mid a\_\{1\}^\{j\}\)\\,d\(a\_\{t\}^\{j\},a\_\{1\}^\{j\}\)\\ \>\\ 0,where strict positivity holds becauseddis a metric andatj≠a1ja\_\{t\}^\{j\}\\neq a\_\{1\}^\{j\}\. By Eq\.[23](https://arxiv.org/html/2609.25131#A2.E23),

pt​\(a1j∣a1j\)=1Zt​\(a1j\)=\(1\+∑z≠a1je−βt​d​\(z,a1j\)\)−1→t→11,p\_\{t\}\(a\_\{1\}^\{j\}\\mid a\_\{1\}^\{j\}\)=\\frac\{1\}\{Z\_\{t\}\(a\_\{1\}^\{j\}\)\}=\\Big\(1\+\\textstyle\\sum\_\{z\\neq a\_\{1\}^\{j\}\}e^\{\-\\beta\_\{t\}d\(z,a\_\{1\}^\{j\}\)\}\\Big\)^\{\-1\}\\xrightarrow\[t\\to 1\]\{\}1,sinceβt→∞\\beta\_\{t\}\\to\\inftyandd⁡\(z,a1j\)\>0d\(z,a\_\{1\}^\{j\}\)\>0for everyz≠a1jz\\neq a\_\{1\}^\{j\}\. Combining withβ˙t→∞\\dot\{\\beta\}\_\{t\}\\to\\inftyfrom Eq\.[24](https://arxiv.org/html/2609.25131#A2.E24)givesλtj=β˙t​Gtj→∞\\lambda\_\{t\}^\{j\}=\\dot\{\\beta\}\_\{t\}\\,G\_\{t\}^\{j\}\\to\\inftyast→1t\\to 1\. In the Euler discretization of the conditional CTMC \(Def\.[2](https://arxiv.org/html/2609.25131#Thmdefinition2)\) the probability that positionjjleaves its current state over\[t,t\+h\)\[t,t\+h\)is1−e−h​λtj1\-e^\{\-h\\lambda\_\{t\}^\{j\}\}—exactly the update the solver applies—which tends to11asλtj→∞\\lambda\_\{t\}^\{j\}\\to\\infty\. Substituting the freshly sampled targeta^1j\\hat\{a\}\_\{1\}^\{j\}fora1ja\_\{1\}^\{j\}throughout, as the solver does at inference, gives the stated terminal re\-derivation; the corresponding expected replacement rate is bounded below in Eq\.[26](https://arxiv.org/html/2609.25131#A2.E26)\.□\\square

##### Scope of Proposition[1](https://arxiv.org/html/2609.25131#Thmproposition1)\.

The proposition does not imply an accuracy loss with an exact denoiser, nor does it cover every uniform\-flow path or solver\.

###### Corollary 2\(Continued exposure to transition\)\.

The realized rate vanishes exactly when the predicted target equals the token currently held:

Pr\[λ^tj=0∣at,𝒞t\]=ℚ1\|tθ\(a1j=atj∣at,𝒞t\)\.\\Pr\\\!\\left\[\\hat\{\\lambda\}\_\{t\}^\{j\}=0\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\\right\]=\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(a\_\{1\}^\{j\}=a\_\{t\}^\{j\}\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)\.\(27\)If the position currently holds the correct token, its probability of remaining exposed to a transition at grid pointtkt\_\{k\}isεtkj\\varepsilon\_\{t\_\{k\}\}^\{j\}, giving∑kεtkj\\sum\_\{k\}\\varepsilon\_\{t\_\{k\}\}^\{j\}expected exposures over the sampling grid\.

##### Proof of Corollary[2](https://arxiv.org/html/2609.25131#Thmcorollary2)\.

The computation in part \(i\) used only that the value conditioning the velocity equals the token held at the position; it did not use that this value was the true target\. Applying it at the sampled target gives, for any drawa^1j=atj\\hat\{a\}\_\{1\}^\{j\}=a\_\{t\}^\{j\}, thatutj​\(zj,at∣a^1j\)=0u\_\{t\}^\{j\}\(z^\{j\},a\_\{t\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)=0for everyzj≠atjz^\{j\}\\neq a\_\{t\}^\{j\}and henceλ^tj=0\\hat\{\\lambda\}\_\{t\}^\{j\}=0\. Conversely, ifa^1j≠atj\\hat\{a\}\_\{1\}^\{j\}\\neq a\_\{t\}^\{j\}then the destinationzj=a^1jz^\{j\}=\\hat\{a\}\_\{1\}^\{j\}contributespt​\(a^1j∣a^1j\)​β˙t​d​\(atj,a^1j\)\>0p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)\\dot\{\\beta\}\_\{t\}\\,d\(a\_\{t\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)\>0, soλ^tj\>0\\hat\{\\lambda\}\_\{t\}^\{j\}\>0\. Therefore\{λ^tj=0\}=\{a^1j=atj\}\\\{\\hat\{\\lambda\}\_\{t\}^\{j\}=0\\\}=\\\{\\hat\{a\}\_\{1\}^\{j\}=a\_\{t\}^\{j\}\\\}exactly, and taking probabilities under the drawa^1j∼ℚ1\|tθ\(⋅∣at,𝒞t\)\\hat\{a\}\_\{1\}^\{j\}\\sim\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(\\cdot\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)gives Eq\.[27](https://arxiv.org/html/2609.25131#A2.E27)\. When the position holds the correct token,atj=a1ja\_\{t\}^\{j\}=a\_\{1\}^\{j\}, the complement has probabilityεtj\\varepsilon\_\{t\}^\{j\}, and summing over the grid gives the stated expected exposure count∑kεtkj\\sum\_\{k\}\\varepsilon\_\{t\_\{k\}\}^\{j\}\. Since the source is uniform att=0t=0, the match probability begins at1/\|𝒮\|1/\|\\mathcal\{S\}\|and rises toward one only as the denoiser concentrates, so the count is close to the full budget whenever residual uncertainty persists\.□\\square

##### Fixed points of the marginalized sampler\.

Because the predicted target is redrawn each step, the object governing whether a position settles is the rate marginalized over that draw\. For an incumbent tokenatja\_\{t\}^\{j\},

𝔼⁡\[λ^tj\]=∑b∈𝒮ℚ1\|tθ​\(a1j=b∣at,𝒞t\)​∑zj≠atjpt​\(zj∣b\)​β˙t​\[d⁡\(atj,b\)−d⁡\(zj,b\)\]\+\.\\mathbb\{E\}\\\!\\left\[\\hat\{\\lambda\}\_\{t\}^\{j\}\\right\]=\\sum\_\{b\\in\\mathcal\{S\}\}\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(a\_\{1\}^\{j\}=b\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)\\sum\_\{z^\{j\}\\neq a\_\{t\}^\{j\}\}p\_\{t\}\(z^\{j\}\\mid b\)\\,\\dot\{\\beta\}\_\{t\}\\left\[d\(a\_\{t\}^\{j\},b\)\-d\(z^\{j\},b\)\\right\]\_\{\+\}\.Every summand is nonnegative, and fort∈\(0,1\)t\\in\(0,1\)bothβ˙t\>0\\dot\{\\beta\}\_\{t\}\>0andpt​\(zj∣b\)\>0p\_\{t\}\(z^\{j\}\\mid b\)\>0\. The expectation therefore vanishes if and only if\[d⁡\(atj,b\)−d⁡\(zj,b\)\]\+=0\[d\(a\_\{t\}^\{j\},b\)\-d\(z^\{j\},b\)\]\_\{\+\}=0for everyzj≠atjz^\{j\}\\neq a\_\{t\}^\{j\}and everybbin the support of the denoiser, that is, wheneverd⁡\(zj,b\)≥d⁡\(atj,b\)d\(z^\{j\},b\)\\geq d\(a\_\{t\}^\{j\},b\)for allzj≠atjz^\{j\}\\neq a\_\{t\}^\{j\}\. Choosingzj=bz^\{j\}=bgives0=d⁡\(b,b\)≥d⁡\(atj,b\)0=d\(b,b\)\\geq d\(a\_\{t\}^\{j\},b\), which forcesb=atjb=a\_\{t\}^\{j\}sinceddis a metric\. Hence

𝔼\[λ^tj\]=0⟺ℚ1\|tθ\(⋅∣at,𝒞t\)=δatj,\\mathbb\{E\}\\\!\\left\[\\hat\{\\lambda\}\_\{t\}^\{j\}\\right\]=0\\quad\\Longleftrightarrow\\quad\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(\\cdot\\mid a\_\{t\},\\mathcal\{C\}\_\{t\}\)=\\delta\_\{a\_\{t\}^\{j\}\},\(28\)so the only fixed points of the marginalized sampler are states at which the denoiser is a point mass on the token already held\. This is a statement about certainty, not correctness: a confidently incorrect denoiser produces a fixed point at a wrong token, while an uncertain denoiser produces none, and in neither case is the time at which the sampler settles under the sampler’s control\. The selective\-absorption construction below creates a fixed point independently of the denoiser’s confidence, at a time the policy selects\.

##### Absorption sets the rate to zero\.

Under Eq\.[2](https://arxiv.org/html/2609.25131#S3.E2),utθ,abs,j=\(1−ftj\)​utθ,ju\_\{t\}^\{\\theta,\\mathrm\{abs\},j\}=\(1\-f\_\{t\}^\{j\}\)u\_\{t\}^\{\\theta,j\}, so a position absorbed at timeτj\\tau\_\{j\}hasfsj=1f\_\{s\}^\{j\}=1and hence zero outflow for alls≥τjs\\geq\\tau\_\{j\}\. Thereforeasj=aτjja\_\{s\}^\{j\}=a\_\{\\tau\_\{j\}\}^\{j\}andλ^sj=0\\hat\{\\lambda\}\_\{s\}^\{j\}=0on\[τj,1\]\[\\tau\_\{j\},1\], and the cumulative exposure∫τj1𝔼⁡\[λ^sj\]​𝑑s\\int\_\{\\tau\_\{j\}\}^\{1\}\\mathbb\{E\}\[\\hat\{\\lambda\}\_\{s\}^\{j\}\]\\,\\mathrm\{d\}sis truncated atτj\\tau\_\{j\}rather than accumulated to the end of sampling\.□\\square

##### On late\-time concentration\.

Eq\.[24](https://arxiv.org/html/2609.25131#A2.E24)givesβ˙t→∞\\dot\{\\beta\}\_\{t\}\\to\\infty, andpt​\(a^1j∣a^1j\)=1/Zt​\(a^1j\)→1p\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\\mid\\hat\{a\}\_\{1\}^\{j\}\)=1/Z\_\{t\}\(\\hat\{a\}\_\{1\}^\{j\}\)\\to 1asβt→∞\\beta\_\{t\}\\to\\infty, so the rate in part \(ii\) behaves asβ˙t​d​\(a1j,a^1j\)\\dot\{\\beta\}\_\{t\}\\,d\(a\_\{1\}^\{j\},\\hat\{a\}\_\{1\}^\{j\}\)for largett: a late target prediction error acts through an increasingly concentrated transport\. We deliberately do not claim that the resulting error probability diverges\. The finite\-step jump probability is bounded by one, andεtj\\varepsilon\_\{t\}^\{j\}is expected to decrease as the denoiser sees a more resolved context, so the two factors act in opposite directions\. The defensible statement is that the consequence of residual prediction error is governed by the productεtj​β˙t\\varepsilon\_\{t\}^\{j\}\\dot\{\\beta\}\_\{t\}rather than byεtj\\varepsilon\_\{t\}^\{j\}alone, which makes that product the natural diagnostic to measure\.

##### Measured replacement rate\.

Figure[5](https://arxiv.org/html/2609.25131#A2.F5)measures this product on the Sudoku denoiser under the native non\-absorbing flow \(Nikoli,n=100n\{=\}100, seeds00–44\)\. Per grid step we log the incumbent errorεt=1−ℚ1\|tθ​\(atj∣at\)\\varepsilon\_\{t\}=1\-\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\}\(a\_\{t\}^\{j\}\\mid a\_\{t\}\)averaged over generated positions, the per\-step hazardckc\_\{k\}of the schedule, and their product, which is the rate at which a currently held token is replaced\. The hazard alone rises to one at the endpoint on every grid, but the incumbent error does not vanish: att=1t\{=\}1it is0\.240\.24,0\.390\.39,0\.510\.51,0\.590\.59, and0\.640\.64forK∈\{16,32,64,128,256\}K\\in\\\{16,32,64,128,256\\\}, so the terminal replacement rate equals those values\. The rate therefore grows as the grid is refined, from0\.240\.24atK=16K\{=\}16to0\.640\.64atK=256K\{=\}256, which is the opposite of what a discretization artifact would produce: refining the integration brings the sampler closer to the continuous flow and makes the late churn worse, not better\. This is the empirical content of Proposition[1](https://arxiv.org/html/2609.25131#Thmproposition1): the schedule gradient sets the opportunity for replacement, and the residual denoiser error decides whether it is taken\.

Figure 5:The replacement rate is the product of residual error and hazard\.\(a\) AtK=64K\{=\}64the per\-step hazardckc\_\{k\}\(red\) rises to one ast→1t\\to 1while the incumbent errorεt\\varepsilon\_\{t\}\(grey\) falls and then rises again in the last steps; their product \(blue\), the correct\-token replacement rate, is small for most of sampling and spikes at the endpoint\. \(b\) The product across gridsK∈\{16,64,256\}K\\in\\\{16,64,256\\\}: the terminal spike grows withKK, so refining the discretization increases the late churn\.
##### The absorption threshold\.

Fix a positionjjand letc=Pr\[atj=a1j\]c=\\Pr\[a\_\{t\}^\{j\}=a\_\{1\}^\{j\}\]be the probability that it currently holds the correct token\. Over the remaining steps, letrrbe the probability that an incorrect token is corrected andwwthe probability that a correct token is replaced, the latter generated by the rate of part \(ii\)\. Absorbing at the current state fixes the token, so its expected terminal error is𝔼abs=1−c\\mathbb\{E\}\_\{\\mathrm\{abs\}\}=1\-c\. Leaving the position active gives

𝔼active=\(1−c\)​\(1−r\)\+c​w,\\mathbb\{E\}\_\{\\mathrm\{active\}\}=\(1\-c\)\(1\-r\)\+c\\,w,the first term counting incorrect tokens that are never corrected and the second correct tokens that are replaced\. Continued refinement is preferable exactly when𝔼active<𝔼abs\\mathbb\{E\}\_\{\\mathrm\{active\}\}<\\mathbb\{E\}\_\{\\mathrm\{abs\}\}, i\.e\.

\(1−c\)​\(1−r\)\+c​w<1−c⇔c​w<\(1−c\)​r⇔c<rr\+w\.\(1\-c\)\(1\-r\)\+cw<1\-c\\iff cw<\(1\-c\)r\\iff c<\\frac\{r\}\{r\+w\}\.\(29\)The threshold is increasing inrrand decreasing inww, as expected: a sampler that corrects reliably should refine longer, and one whose replacement rate is high should absorb sooner\. Bothrrandwware properties of the denoiser and the schedule and are not available at decode time, but the comparison depends on the current position only throughcc\. This is what makes a reliability estimate the operative quantity\.

## Appendix CDetails for Section[3](https://arxiv.org/html/2609.25131#S3):LEDFlow

This appendix follows the three subsections of Section[3](https://arxiv.org/html/2609.25131#S3): Appendix[C\.1](https://arxiv.org/html/2609.25131#A3.SS1)details the absorbing\-state augmentation of Section[3\.1](https://arxiv.org/html/2609.25131#S3.SS1), Appendix[C\.2](https://arxiv.org/html/2609.25131#A3.SS2)documents the sampler of Section[3\.2](https://arxiv.org/html/2609.25131#S3.SS2), and Appendix[C\.3](https://arxiv.org/html/2609.25131#A3.SS3)collects the proofs, the calibration diagnostic, and the lookahead analysis behind Section[3\.3](https://arxiv.org/html/2609.25131#S3.SS3)\.

### C\.1Selective absorption as an absorbing\-state uniform flow

This appendix expands the construction in Section[3\.1](https://arxiv.org/html/2609.25131#S3.SS1)\. The uniform discrete flow has stateata\_\{t\}and learned coordinate velocityutθ,ju\_\{t\}^\{\\theta,j\}\. We augment it withft=\(ft1,…,ftM\)∈\{0,1\}Mf\_\{t\}=\(f\_\{t\}^\{1\},\\ldots,f\_\{t\}^\{M\}\)\\in\\\{0,1\\\}^\{M\}, whereftj=0f\_\{t\}^\{j\}=0marks an active position andftj=1f\_\{t\}^\{j\}=1marks a terminal position\. Initially,f0j=0f\_\{0\}^\{j\}=0for every generated position\.

##### Absorbing velocity\.

For a fixed augmented state\(at,ft\)\(a\_\{t\},f\_\{t\}\), define

utθ,abs,j​\(zj,at,ft,𝒞t\)=\(1−ftj\)​utθ,j​\(zj,at,𝒞t\)\.u\_\{t\}^\{\\theta,\\mathrm\{abs\},j\}\(z^\{j\},a\_\{t\},f\_\{t\},\\mathcal\{C\}\_\{t\}\)=\(1\-f\_\{t\}^\{j\}\)u\_\{t\}^\{\\theta,j\}\(z^\{j\},a\_\{t\},\\mathcal\{C\}\_\{t\}\)\.Whenftj=0f\_\{t\}^\{j\}=0, this is exactly the native uniform\-flow velocity\. Whenftj=1f\_\{t\}^\{j\}=1, every outgoing rate is zero, so the coordinate remains at its absorbed value\. Because1−ftj1\-f\_\{t\}^\{j\}is nonnegative, this modification preserves nonnegative off\-diagonal rates and the zero row sum of the native CTMC velocity\. The augmented process is therefore a valid CTMC between absorption decisions\.

##### Irreversible update\.

At grid pointtkt\_\{k\}, a policy chooses an absorption blockBkB\_\{k\}from the active set𝒰k=\{j:ftkj=0\}\\mathcal\{U\}\_\{k\}=\\\{j:f\_\{t\_\{k\}\}^\{j\}=0\\\}\. For eachj∈Bkj\\in B\_\{k\}, the update assigns the most likely target value underℚ1\|tkθ\\mathbb\{Q\}\_\{1\\mid t\_\{k\}\}^\{\\theta\}and changesftkjf\_\{t\_\{k\}\}^\{j\}from zero to one\. Indicators never return to zero, hence

𝒰k\+1⊆𝒰k,ftk\+1j≥ftkj\.\\mathcal\{U\}\_\{k\+1\}\\subseteq\\mathcal\{U\}\_\{k\},\\qquad f\_\{t\_\{k\+1\}\}^\{j\}\\geq f\_\{t\_\{k\}\}^\{j\}\.The active positions subsequently follow the original uniform\-flow update, while absorbed positions are copied unchanged\. The schedule in Section[3\.1](https://arxiv.org/html/2609.25131#S3.SS1)fixes the number of active positions remaining after each grid point; the priority policy determines which positions formBkB\_\{k\}\.

##### Relation to the exact absorbing representation\.

Appendix[C\.1\.1](https://arxiv.org/html/2609.25131#A3.SS1.SSS1)represents uniform transport as a mixture of conditional processes indexed by auxiliary absorbing tokens\. That construction is an exact representation of the underlying uniform path\. Our indicatorftjf\_\{t\}^\{j\}instead records an inference decision: after a model prediction is selected, Eq\. \([2](https://arxiv.org/html/2609.25131#S3.E2)\) deliberately removes its future native\-flow transitions\. The selective augmentation therefore borrows the absorbing\-state mechanism but does not claim to preserve the path law of the unmodified uniform flow\. This is the decision whose local prediction error is analyzed in Section[3\.3](https://arxiv.org/html/2609.25131#S3.SS3)\.

##### Training the absorbing\-state uniform\-flow denoiser\.

For the absorption\-aware Sudoku experiments, we train a denoiserθ\\thetawith explicit absorption indicators and uniform\-noise corruption\. Each positionjjis presented to the model together with its absorption indicatorftj∈\{0,1\}f^\{j\}\_\{t\}\\in\\\{0,1\\\}, embedded and added to the token embedding, so the network can tell reliable \(absorbed\) positions from tentative \(active\) ones\. At each training step we draw a timet∼𝒰⁡\(0,1\)t\\sim\\mathcal\{U\}\(0,1\)and construct the corrupted input as follows\. Conditioning positions and a fractionttof the generated positions \(drawn independently per position, matching the linear scheduleκt=t\\kappa\_\{t\}=t\) are marked*absorbed*\(ftj=1f^\{j\}\_\{t\}=1\) and hold their ground\-truth value\. The remaining*active*positions \(ftj=0f^\{j\}\_\{t\}=0\) carry a uniform\-flow state that interpolates between the target and noise: each keeps its true value with probabilityttand is otherwise replaced by a uniformly random token, so the active tokens are a partially informative estimate that sharpens ast→1t\\to 1rather than pure noise or a single mask symbol\. The denoiser predicts the clean value at every active position, trained with the denoising cross\-entropyℒDLM\\mathcal\{L\}\_\{\\mathrm\{DLM\}\}restricted to the active positions; absorbed positions are conditioned on but not scored\. At inference the model receives the active/absorbed configuration, fixes selected predictions, and updates the remaining positions\. Training absorptions contain ground\-truth values, whereas inference absorptions can be wrong and are selected adaptively\. The indicator makes absorption status observable but does not eliminate this distribution shift or establish equivalence to the metric\-induced path\. On text\-to\-image generation we do not retrain: absorption in Eq\.[2](https://arxiv.org/html/2609.25131#S3.E2)is a sampler\-side operation that only zeroes the outgoing rate of the selected positions, so the good multimodal results of Section[3](https://arxiv.org/html/2609.25131#S3)are obtained by applying the same entropy\-guided ordered absorption at inference to the pretrained FUDOKI uniform\-flow model, leaving its weights unchanged\.

##### Training ablation\.

In the reported training ablation, the denoiser without an absorption indicator attains36\.3%36\.3\\%cell accuracy and solves0/1000/100puzzles\. This supports the importance of training with absorption status in this puzzle setup; it does not establish a universal need for retraining frozen backbones\.

##### Frozen backbones\.

On frozen FUDOKI the indicatorftf\_\{t\}is maintained by the sampler and is not an additional trained denoiser input; the model sees only the current token configuration\. Fixing tokens can therefore change the states visited by a frozen backbone, without a guarantee that those states follow its training distribution; the puzzle denoisers avoid this shift by receiving absorption indicators during training\.

#### C\.1\.1Deriving the exact absorbing representation of uniform flow

This appendix derives the representation used in Section[3\.1](https://arxiv.org/html/2609.25131#S3.SS1)\. We first follow the data\-to\-noise convention of[Gourevitch et al\. \[2026\]](https://arxiv.org/html/2609.25131#bib.bib16), then reverse time to recover the source\-to\-target convention of discrete flow matching\. Tokens are identified with their one\-hot vectors whenever they parameterize a categorical distribution\.

##### Step 1: expand the conditional absorbing transition\.

Fix a coordinatejj, a current tokenaja^\{j\}, and an auxiliary absorbing tokenuju^\{j\}\. The conditional transition over\[t,t\+h\]\[t,t\+h\]is

qt\+h\|t​\(zj∣aj,uj\)\\displaystyle q\_\{t\+h\\mid t\}\(z^\{j\}\\mid a^\{j\},u^\{j\}\)=Cat⁡\(zj,αt\+h\|t​aj\+\(1−αt\+h\|t\)​uj\)\\displaystyle=\\operatorname\{Cat\}\\\!\\left\(z^\{j\};\\alpha\_\{t\+h\\mid t\}a^\{j\}\+\(1\-\\alpha\_\{t\+h\\mid t\}\)u^\{j\}\\right\)=αt\+h\|t​δ​\(zj,aj\)\+\(1−αt\+h\|t\)​δ​\(zj,uj\)\.\\displaystyle=\\alpha\_\{t\+h\\mid t\}\\delta\(z^\{j\},a^\{j\}\)\+\(1\-\\alpha\_\{t\+h\\mid t\}\)\\delta\(z^\{j\},u^\{j\}\)\.\(30\)Writingαt\+h\|t=αt\+h/αt\\alpha\_\{t\+h\\mid t\}=\\alpha\_\{t\+h\}/\\alpha\_\{t\}and assuming differentiability gives

αt\+h\|t=1\+α˙tαt​h\+o⁡\(h\)=1−βt​h\+o⁡\(h\),βt:=−α˙tαt≥0,\\alpha\_\{t\+h\\mid t\}=1\+\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}h\+o\(h\)=1\-\\beta\_\{t\}h\+o\(h\),\\qquad\\beta\_\{t\}:=\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\alpha\_\{t\}\}\\geq 0,\(31\)because the corruption scheduleαt\\alpha\_\{t\}is nonincreasing\. Substituting \([31](https://arxiv.org/html/2609.25131#A3.E31)\) into \([30](https://arxiv.org/html/2609.25131#A3.E30)\) yields

qt\+h\|t​\(zj∣aj,uj\)=δ⁡\(zj,aj\)\+h​βt​\[δ⁡\(zj,uj\)−δ⁡\(zj,aj\)\]\+o⁡\(h\)\.q\_\{t\+h\\mid t\}\(z^\{j\}\\mid a^\{j\},u^\{j\}\)=\\delta\(z^\{j\},a^\{j\}\)\+h\\beta\_\{t\}\\\!\\left\[\\delta\(z^\{j\},u^\{j\}\)\-\\delta\(z^\{j\},a^\{j\}\)\\right\]\+o\(h\)\.\(32\)

##### Step 2: identify the conditional CTMC velocity\.

Comparing \([32](https://arxiv.org/html/2609.25131#A3.E32)\) with the infinitesimal CTMC update in \([11](https://arxiv.org/html/2609.25131#A2.E11)\) gives

utj,u​\(zj,aj\)=βt​\[δ⁡\(zj,uj\)−δ⁡\(zj,aj\)\]\.u\_\{t\}^\{j,u\}\(z^\{j\},a^\{j\}\)=\\beta\_\{t\}\\\!\\left\[\\delta\(z^\{j\},u^\{j\}\)\-\\delta\(z^\{j\},a^\{j\}\)\\right\]\.\(33\)Forzj≠ajz^\{j\}\\neq a^\{j\}, this rate equalsβt​δ​\(zj,uj\)≥0\\beta\_\{t\}\\delta\(z^\{j\},u^\{j\}\)\\geq 0, and summing overzjz^\{j\}gives zero\. It is therefore a valid CTMC velocity\. Ifaj=uja^\{j\}=u^\{j\}, every rate vanishes; otherwise, the only possible jump isaj→uja^\{j\}\\to u^\{j\}, after which the coordinate remains atuju^\{j\}\. This verifies the absorbing property directly\.

##### Step 3: marginalize the absorbing token\.

LetUj∼Unif⁡\(𝒮\)U^\{j\}\\sim\\operatorname\{Unif\}\(\\mathcal\{S\}\)\. Since𝔼⁡\[δ⁡\(zj,Uj\)\]=1/\|𝒮\|\\mathbb\{E\}\[\\delta\(z^\{j\},U^\{j\}\)\]=1/\|\\mathcal\{S\}\|, averaging either \([30](https://arxiv.org/html/2609.25131#A3.E30)\) or \([33](https://arxiv.org/html/2609.25131#A3.E33)\) gives

𝔼Uj​\[qt\+h\|t​\(zj∣aj,Uj\)\]\\displaystyle\\mathbb\{E\}\_\{U^\{j\}\}\\\!\\left\[q\_\{t\+h\\mid t\}\(z^\{j\}\\mid a^\{j\},U^\{j\}\)\\right\]=αt\+h\|t​δ​\(zj,aj\)\+1−αt\+h\|t\|𝒮\|,\\displaystyle=\\alpha\_\{t\+h\\mid t\}\\delta\(z^\{j\},a^\{j\}\)\+\\frac\{1\-\\alpha\_\{t\+h\\mid t\}\}\{\|\\mathcal\{S\}\|\},\(34\)𝔼Uj​\[utj,U​\(zj,aj\)\]\\displaystyle\\mathbb\{E\}\_\{U^\{j\}\}\\\!\\left\[u\_\{t\}^\{j,U\}\(z^\{j\},a^\{j\}\)\\right\]=βt​\[1\|𝒮\|−δ⁡\(zj,aj\)\]\.\\displaystyle=\\beta\_\{t\}\\\!\\left\[\\frac\{1\}\{\|\\mathcal\{S\}\|\}\-\\delta\(z^\{j\},a^\{j\}\)\\right\]\.\(35\)Equation \([34](https://arxiv.org/html/2609.25131#A3.E34)\) is exactly the uniform\-corruption transition, and \([35](https://arxiv.org/html/2609.25131#A3.E35)\) is its infinitesimal velocity\. Thus the uniform transition is a mixture of conditional transitions, each with a single coordinate\-specific absorbing token\.

##### Step 4: reverse time to obtain the discrete\-flow path\.

The diffusion convention above hasα0=1\\alpha\_\{0\}=1at data andαt→0\\alpha\_\{t\}\\to 0toward uniform noise\. Letr:=1−tr:=1\-tdenote generation time and setκr:=α1−r\\kappa\_\{r\}:=\\alpha\_\{1\-r\}\. Thenκ0=0\\kappa\_\{0\}=0,κ1=1\\kappa\_\{1\}=1, and the endpoint\-conditioned uniform flow path is

pr\|1j​\(aj∣a1j\)=κr​δ​\(aj,a1j\)\+\(1−κr\)​1\|𝒮\|\.p\_\{r\\mid 1\}^\{j\}\(a^\{j\}\\mid a\_\{1\}^\{j\}\)=\\kappa\_\{r\}\\delta\(a^\{j\},a\_\{1\}^\{j\}\)\+\(1\-\\kappa\_\{r\}\)\\frac\{1\}\{\|\\mathcal\{S\}\|\}\.\(36\)IntroducingUj∼Unif⁡\(𝒮\)U^\{j\}\\sim\\operatorname\{Unif\}\(\\mathcal\{S\}\)and using𝔼⁡\[δ⁡\(aj,Uj\)\]=1/\|𝒮\|\\mathbb\{E\}\[\\delta\(a^\{j\},U^\{j\}\)\]=1/\|\\mathcal\{S\}\|gives the path identity

pr\|1j​\(aj∣a1j\)=𝔼Uj​\[κr​δ​\(aj,a1j\)\+\(1−κr\)​δ​\(aj,Uj\)\]\.p\_\{r\\mid 1\}^\{j\}\(a^\{j\}\\mid a\_\{1\}^\{j\}\)=\\mathbb\{E\}\_\{U^\{j\}\}\\\!\\left\[\\kappa\_\{r\}\\delta\(a^\{j\},a\_\{1\}^\{j\}\)\+\(1\-\\kappa\_\{r\}\)\\delta\(a^\{j\},U^\{j\}\)\\right\]\.\(37\)For a fixed realizationUj=uj≠a1jU^\{j\}=u^\{j\}\\neq a\_\{1\}^\{j\}, the conditional path inside the expectation moves once fromuju^\{j\}toa1ja\_\{1\}^\{j\}\. Its generation\-time hazard is

hκ​\(r\)=κ˙r1−κr\.h\_\{\\kappa\}\(r\)=\\frac\{\\dot\{\\kappa\}\_\{r\}\}\{1\-\\kappa\_\{r\}\}\.\(38\)Indeed, the source\-token mass is1−κr1\-\\kappa\_\{r\}, so its outflow is\(1−κr\)​hκ​\(r\)=κ˙r\(1\-\\kappa\_\{r\}\)h\_\{\\kappa\}\(r\)=\\dot\{\\kappa\}\_\{r\}, exactly the derivative of the mass assigned toa1ja\_\{1\}^\{j\}\. Ifuj=a1ju^\{j\}=a\_\{1\}^\{j\}, the conditional path is constant\. Averaging these conditional paths proves the uniform discrete\-flow identity used in Section[3\.1](https://arxiv.org/html/2609.25131#S3.SS1)\.

##### Step 5: relate the exact lifting to selective absorption\.

The derivation above concerns the exact auxiliary\-state representation of uniform transport\. Appendix[C\.1](https://arxiv.org/html/2609.25131#A3.SS1)explains the distinct selective augmentation used by our sampler: Section[3\.1](https://arxiv.org/html/2609.25131#S3.SS1)usesftjf\_\{t\}^\{j\}to record a reliability\-based terminal decision\. Onceftj=1f\_\{t\}^\{j\}=1, Eq\. \([2](https://arxiv.org/html/2609.25131#S3.E2)\) sets every outgoing velocity at coordinatejjto zero, making that absorbed value absorbing\. This borrows the augmented absorbing\-state structure while deliberately changing the native uniform\-flow path according to the absorption policy\.

### C\.2TheLEDFlowsampler: selective\-absorption methodology

This subsection expands the implementation of Section[3](https://arxiv.org/html/2609.25131#S3)and separates the two operations performed at each grid point: an irreversible absorption decision and an otherwise unchanged uniform\-flow update\. No token\-specific time map or modified CTMC velocity is introduced\.

##### Active positions and absorption budget\.

Only generated positions participate in the sampler\. Their indicators are initialized asft0j=0f\_\{t\_\{0\}\}^\{j\}=0, and the active set before stepkkis𝒰k=\{j:ftkj=0\}\\mathcal\{U\}\_\{k\}=\\\{j:f\_\{t\_\{k\}\}^\{j\}=0\\\}\. The cosine schedule specifies the desired number

mk=⌊M​cos⁡\(π2​k\+1K\)⌋m\_\{k\}=\\left\\lfloor M\\cos\\\!\\left\(\\frac\{\\pi\}\{2\}\\frac\{k\+1\}\{K\}\\right\)\\right\\rfloorof active positions after absorption\. Consequently,bk=max⁡\{0,\|𝒰k\|−mk\}b\_\{k\}=\\max\\\{0,\|\\mathcal\{U\}\_\{k\}\|\-m\_\{k\}\\\}positions are made terminal at that step\. Rounding can producebk=0b\_\{k\}=0, in which case the sampler performs only the native flow update\. At the final grid point,mK−1=0m\_\{K\-1\}=0, so every remaining generated position is absorbed\. This cosine budget is used for the FUDOKI and main Sudoku absorption comparisons\. Both absorbing and non\-absorbing Sudoku baselines use the same coordinatewise kinetic\-optimal update; the budget controls absorptions rather than native flow transitions\. Explicit singleton\-absorption ablations usebk=1b\_\{k\}=1instead \(Appendix[D](https://arxiv.org/html/2609.25131#A4)\)\.

##### Entropy selection and value absorption\.

One denoiser evaluation provides the target marginal for each active position\. We computeHθj​\(sk\)H\_\{\\theta\}^\{j\}\(s\_\{k\}\)and choose thebkb\_\{k\}smallest values, with a deterministic index order used only to break exact ties\. For every selectedjj, the sampler assigns the most likely target token and setsftkj=1f\_\{t\_\{k\}\}^\{j\}=1\. Equation \([2](https://arxiv.org/html/2609.25131#S3.E2)\) then makes the position terminal: its outgoing velocity is zero at every later step and its value is copied unchanged\. This is the only irreversible operation inLEDFlow\.

##### Native flow on the remaining positions\.

Positions outsideBkLEDB\_\{k\}^\{\\mathrm\{LED\}\}retainftkj=0f\_\{t\_\{k\}\}^\{j\}=0and are advanced fromtkt\_\{k\}totk\+1t\_\{k\+1\}by the backbone’s original uniform\-flow solver\. Thus,LEDFlowchanges which predictions become final but does not change the learned posterior, the base probability path, or the numerical integration rule on active positions\. Absorption and numerical integration are therefore separate design axes\. For example, a time\- or location\-corrected update can be applied to the active positions before the same entropy\-based absorption decision at the next grid point\.

##### Matched\-policy comparisons\.

The probability\-margin, Info\-Gain, and random baselines use the same generated\-position mask, task\-specific budget, most\-likely\-value update, and absorption rule asLEDFlow\. They differ only in the score used to rank𝒰k\\mathcal\{U\}\_\{k\}\. This controlled construction attributes differences among these rows to the priority policy rather than to a different number of absorbed positions or a different uniform\-flow update\. In contrast, the native Euler and corrected\-flow baselines setftj=0f\_\{t\}^\{j\}=0throughout and therefore remain non\-absorbing; they test numerical integration without the selective\-absorption component\.

Algorithm[1](https://arxiv.org/html/2609.25131#alg1)gives the complete update order\. The implementation first restricts selection to generated positions, refreshes the posterior and priority scores, absorbs the selected block, and finally applies the native velocity update only where the refreshed indicator remains zero\. These steps directly implement Eqs\. \([2](https://arxiv.org/html/2609.25131#S3.E2)\), \([4](https://arxiv.org/html/2609.25131#S3.E4)\), and \([3](https://arxiv.org/html/2609.25131#S3.E3)\)\.

### C\.3Theoretical justification

#### C\.3\.1KL motivation and local absorption\-error decomposition

##### Why measure decoding error with KL divergence?

The starting point follows the sequential\-decoding formulation of[Xu et al\. \[2026\]](https://arxiv.org/html/2609.25131#bib.bib20)\. Letπ⁡\(y\)\\pi\(y\)be a target distribution,σ=\(σ1,…,σM\)\\sigma=\(\\sigma\_\{1\},\\ldots,\\sigma\_\{M\}\)an unmasking order, andvϕ​\(σ∣y\)v\_\{\\phi\}\(\\sigma\\mid y\)the probability assigned to that order along the teacher\-forced trajectory ofyy\. A denoiser assigns the path likelihood

pθ​\(y∣σ\)=∏i=1Mpθσi​\(yσi∣yσ<i\),p\_\{\\theta\}\(y\\mid\\sigma\)=\\prod\_\{i=1\}^\{M\}p\_\{\\theta\}^\{\\sigma\_\{i\}\}\\\!\\left\(y^\{\\sigma\_\{i\}\}\\mid y^\{\\sigma\_\{<i\}\}\\right\),\(39\)and the sequential decoder inducesPθ,ϕseq​\(y\)=∑σvϕ​\(σ∣y\)​pθ​\(y∣σ\)P\_\{\\theta,\\phi\}^\{\\mathrm\{seq\}\}\(y\)=\\sum\_\{\\sigma\}v\_\{\\phi\}\(\\sigma\\mid y\)p\_\{\\theta\}\(y\\mid\\sigma\)\. Define the data\-policy and model\-policy joint laws

Qϕ​\(y,σ\):=π⁡\(y\)​vϕ​\(σ∣y\),Pθ,ϕ​\(y,σ\):=pθ​\(y∣σ\)​vϕ​\(σ∣y\)\.Q\_\{\\phi\}\(y,\\sigma\):=\\pi\(y\)v\_\{\\phi\}\(\\sigma\\mid y\),\\qquad P\_\{\\theta,\\phi\}\(y,\\sigma\):=p\_\{\\theta\}\(y\\mid\\sigma\)v\_\{\\phi\}\(\\sigma\\mid y\)\.\(40\)Because both laws use the same order policy, its log\-density cancels in their KL divergence:

DKL​\(Qϕ∥Pθ,ϕ\)\\displaystyle D\_\{\\mathrm\{KL\}\}\\\!\\left\(Q\_\{\\phi\}\\,\\middle\\\|\\,P\_\{\\theta,\\phi\}\\right\)=𝔼y∼π,σ∼vϕ\(⋅∣y\)\[logπ⁡\(y\)pθ​\(y∣σ\)\]\\displaystyle=\\mathbb\{E\}\_\{y\\sim\\pi,\\,\\sigma\\sim v\_\{\\phi\}\(\\cdot\\mid y\)\}\\left\[\\log\\frac\{\\pi\(y\)\}\{p\_\{\\theta\}\(y\\mid\\sigma\)\}\\right\]=−H\(π\)−𝔼y∼π,σ∼vϕ\(⋅∣y\)\[logpθ\(y∣σ\)\]\.\\displaystyle=\-H\(\\pi\)\-\\mathbb\{E\}\_\{y\\sim\\pi,\\,\\sigma\\sim v\_\{\\phi\}\(\\cdot\\mid y\)\}\\left\[\\log p\_\{\\theta\}\(y\\mid\\sigma\)\\right\]\.\(41\)Marginalization from\(y,σ\)\(y,\\sigma\)toyyand the data\-processing inequality then give

DKL​\(π∥Pθ,ϕseq\)≤DKL​\(Qϕ∥Pθ,ϕ\)\.D\_\{\\mathrm\{KL\}\}\\\!\\left\(\\pi\\,\\middle\\\|\\,P\_\{\\theta,\\phi\}^\{\\mathrm\{seq\}\}\\right\)\\leq D\_\{\\mathrm\{KL\}\}\\\!\\left\(Q\_\{\\phi\}\\,\\middle\\\|\\,P\_\{\\theta,\\phi\}\\right\)\.\(42\)Equations \([41](https://arxiv.org/html/2609.25131#A3.E41)\)–\([42](https://arxiv.org/html/2609.25131#A3.E42)\) explain why KL is useful for studying absorption decisions: it measures distributional mismatch at the decoder output, while an upper bound can be evaluated as accumulated prediction loss along an ordered trajectory\. The cited work also considers an ordered block schedule\(B1,…,BK\)\(B\_\{1\},\\ldots,B\_\{K\}\)\. If its parallel law samples positions in each block independently from the corresponding sequential\-law marginals, then its parallelization gap is

DKL​\(Pθ,ϕseq∥Pθ,ϕpar\)=∑k=1K𝔼Pθ,ϕseq​\[𝒯​𝒞Pθ,ϕseq​\(YBk∣YB<k\)\]\.D\_\{\\mathrm\{KL\}\}\\\!\\left\(P\_\{\\theta,\\phi\}^\{\\mathrm\{seq\}\}\\,\\middle\\\|\\,P\_\{\\theta,\\phi\}^\{\\mathrm\{par\}\}\\right\)=\\sum\_\{k=1\}^\{K\}\\mathbb\{E\}\_\{P\_\{\\theta,\\phi\}^\{\\mathrm\{seq\}\}\}\\\!\\left\[\\mathcal\{TC\}\_\{P\_\{\\theta,\\phi\}^\{\\mathrm\{seq\}\}\}\(Y^\{B\_\{k\}\}\\mid Y^\{B\_\{<k\}\}\)\\right\]\.\(43\)This identity measures the dependence lost by parallelizing a sequential model law\. Our local question uses a different reference: it compares the oracle joint posterior at an absorption event directly with the product of learned denoiser marginals\. Proposition[2](https://arxiv.org/html/2609.25131#Thmproposition2)therefore retains a total\-correlation term for dependence loss and adds a separate term for marginal prediction error\.

##### Step 1: abbreviate the three distributions\.

Fix the reachable statesks\_\{k\}and blockBkB\_\{k\}from Section[3\.3](https://arxiv.org/html/2609.25131#S3.SS3)\. Write

PB​\(b\):=p1\|skq​\(a1Bk=b\),Pj​\(bj\):=p1\|skq​\(a1j=bj\),Qj​\(bj\):=ℚ1\|tkθ​\(a1j=bj∣sk\)\.P\_\{B\}\(b\):=p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{B\_\{k\}\}=b\),\\quad P\_\{j\}\(b\_\{j\}\):=p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}=b\_\{j\}\),\\quad Q\_\{j\}\(b\_\{j\}\):=\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=b\_\{j\}\\mid s\_\{k\}\)\.\(44\)The oracle distributionPBP\_\{B\}retains dependence among the positions inBkB\_\{k\}, whereas the implemented absorption kernel is the product∏j∈BkQj\\prod\_\{j\\in B\_\{k\}\}Q\_\{j\}\. We assume the latter is positive on the support ofPBP\_\{B\}; otherwise the relevant KL divergence is infinite and the identity below holds in the extended\-real sense\.

##### Step 2: insert the product of oracle marginals\.

For everybbin the support ofPBP\_\{B\},

log⁡PB​\(b\)∏j∈BkQj​\(bj\)=log⁡PB​\(b\)∏j∈BkPj​\(bj\)\+∑j∈Bklog⁡Pj​\(bj\)Qj​\(bj\)\.\\log\\frac\{P\_\{B\}\(b\)\}\{\\prod\_\{j\\in B\_\{k\}\}Q\_\{j\}\(b\_\{j\}\)\}=\\log\\frac\{P\_\{B\}\(b\)\}\{\\prod\_\{j\\in B\_\{k\}\}P\_\{j\}\(b\_\{j\}\)\}\+\\sum\_\{j\\in B\_\{k\}\}\\log\\frac\{P\_\{j\}\(b\_\{j\}\)\}\{Q\_\{j\}\(b\_\{j\}\)\}\.\(45\)Taking expectation underPBP\_\{B\}separates the local absorption loss into two terms:

ℒabs​\(Bk,sk\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{abs\}\}\(B\_\{k\};s\_\{k\}\)=∑bPB​\(b\)​log⁡PB​\(b\)∏j∈BkPj​\(bj\)\\displaystyle=\\sum\_\{b\}P\_\{B\}\(b\)\\log\\frac\{P\_\{B\}\(b\)\}\{\\prod\_\{j\\in B\_\{k\}\}P\_\{j\}\(b\_\{j\}\)\}\+∑j∈Bk∑bPB\(b\)logPj​\(bj\)Qj​\(bj\)\.\\displaystyle\\quad\+\\sum\_\{j\\in B\_\{k\}\}\\sum\_\{b\}P\_\{B\}\(b\)\\log\\frac\{P\_\{j\}\(b\_\{j\}\)\}\{Q\_\{j\}\(b\_\{j\}\)\}\.\(46\)
###### Lemma 2\(Conditional total correlation of an absorption block\)\.

For any reachable statesks\_\{k\}and absorption blockBkB\_\{k\},

DKL​\(p1\|skq​\(a1Bk\)∥∏j∈Bkp1\|skq​\(a1j\)\)=𝒯​𝒞q​\(a1Bk∣sk\)≥0\.D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{B\_\{k\}\}\)\\,\\middle\\\|\\,\\prod\_\{j\\in B\_\{k\}\}p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\right\)=\\mathcal\{TC\}\_\{q\}\(a\_\{1\}^\{B\_\{k\}\}\\mid s\_\{k\}\)\\geq 0\.\(47\)The quantity vanishes exactly when the positions inBkB\_\{k\}are conditionally independent under the oracle posterior, including every singleton block\.

##### Step 3: identify the joint term\.

The first term in \([46](https://arxiv.org/html/2609.25131#A3.E46)\) is

DKL​\(PB∥∏j∈BkPj\)\\displaystyle D\_\{\\mathrm\{KL\}\}\\\!\\left\(P\_\{B\}\\,\\middle\\\|\\,\\prod\_\{j\\in B\_\{k\}\}P\_\{j\}\\right\)=∑bPB​\(b\)​log⁡PB​\(b\)−∑j∈Bk∑bPB​\(b\)​log⁡Pj​\(bj\)\\displaystyle=\\sum\_\{b\}P\_\{B\}\(b\)\\log P\_\{B\}\(b\)\-\\sum\_\{j\\in B\_\{k\}\}\\sum\_\{b\}P\_\{B\}\(b\)\\log P\_\{j\}\(b\_\{j\}\)=−Hq​\(a1Bk∣sk\)\+∑j∈BkHq​\(a1j∣sk\)\\displaystyle=\-H\_\{q\}\(a\_\{1\}^\{B\_\{k\}\}\\mid s\_\{k\}\)\+\\sum\_\{j\\in B\_\{k\}\}H\_\{q\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)=𝒯​𝒞q​\(a1Bk∣sk\)=ℰjoint​\(Bk,sk\)\.\\displaystyle=\\mathcal\{TC\}\_\{q\}\(a\_\{1\}^\{B\_\{k\}\}\\mid s\_\{k\}\)=\\mathcal\{E\}\_\{\\mathrm\{joint\}\}\(B\_\{k\};s\_\{k\}\)\.\(48\)The second equality marginalizesPBP\_\{B\}over all coordinates exceptjj\. Since it is itself a KL divergence, the joint term is nonnegative and vanishes exactly when the coordinates inBkB\_\{k\}are conditionally independent under the oracle posterior\. In particular, it is zero for\|Bk\|=1\|B\_\{k\}\|=1\.

##### Step 4: identify the conditional term\.

For eachj∈Bkj\\in B\_\{k\}, marginalizing the second term in \([46](https://arxiv.org/html/2609.25131#A3.E46)\) overb−jb\_\{\-j\}gives

∑bPB​\(b\)​log⁡Pj​\(bj\)Qj​\(bj\)\\displaystyle\\sum\_\{b\}P\_\{B\}\(b\)\\log\\frac\{P\_\{j\}\(b\_\{j\}\)\}\{Q\_\{j\}\(b\_\{j\}\)\}=∑bjPj​\(bj\)​log⁡Pj​\(bj\)Qj​\(bj\)\\displaystyle=\\sum\_\{b\_\{j\}\}P\_\{j\}\(b\_\{j\}\)\\log\\frac\{P\_\{j\}\(b\_\{j\}\)\}\{Q\_\{j\}\(b\_\{j\}\)\}=DKL\(Pj∥Qj\)\.\\displaystyle=D\_\{\\mathrm\{KL\}\}\(P\_\{j\}\\\|Q\_\{j\}\)\.\(49\)Summing \([49](https://arxiv.org/html/2609.25131#A3.E49)\) over the block yieldsℰcond​\(Bk,sk\)\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)\. Combining \([48](https://arxiv.org/html/2609.25131#A3.E48)\) and \([49](https://arxiv.org/html/2609.25131#A3.E49)\) proves Eq\. \([5](https://arxiv.org/html/2609.25131#S3.E5)\)\.

##### Step 5: interpret the limiting cases\.

If the block is a singleton, independent sampling introduces no factorization error and only the denoiser KL remains\. If every denoiser marginal equals its oracle marginal, the conditional term vanishes, but a multi\-position block can still incur the joint term because the product kernel does not reproduce conditional dependence\. Both terms vanish when the block is conditionally independent and every marginal is exact\. For a random or adaptive absorption policy, the same identity holds after conditioning on the realized pair\(sk,Bk\)\(s\_\{k\},B\_\{k\}\); taking expectation over that pair preserves the decomposition\. Consequently, the policy affects the conditional term through the states at which predictions become irreversible, while block size determines whether an additional factorization cost can arise\. When the oracle conditional atsks\_\{k\}is a point mass, as on unique\-solution puzzles, every conditional entropy in the total correlation is zero and the joint term vanishes for every block regardless of its size; a nondegenerate oracle, such as the multi\-solution Sudoku corpus or graph coloring, is required to observe it\.

#### C\.3\.2Proof of Corollary[1](https://arxiv.org/html/2609.25131#Thmcorollary1)

###### Proposition 4\(Conditional error and absorption risk\)\.

For a reachable statesks\_\{k\}and an absorption blockBkB\_\{k\}with\|Bk\|=b\|B\_\{k\}\|=b, letz^j=arg⁡maxz​ℚ1\|tkθ​\(a1j=z∣sk\)\\hat\{z\}^\{j\}=\\arg\\max\_\{z\}\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=z\\mid s\_\{k\}\),cj=p1\|skq​\(a1j=z^j\)c\_\{j\}=p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}=\\hat\{z\}^\{j\}\), andcj⋆=maxz⁡p1\|skq​\(a1j=z\)c\_\{j\}^\{\\star\}=\\max\_\{z\}p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}=z\)\. Then

∑j∈Bk\(1−cj\)≤∑j∈Bk\(1−cj⋆\)\+2​b​ℰcond​\(Bk,sk\)\.\\sum\_\{j\\in B\_\{k\}\}\(1\-c\_\{j\}\)\\leq\\sum\_\{j\\in B\_\{k\}\}\(1\-c\_\{j\}^\{\\star\}\)\+\\sqrt\{2b\\,\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)\}\.\(50\)

##### Proof of Proposition[4](https://arxiv.org/html/2609.25131#Thmproposition4)\.

Fix a reachable statesks\_\{k\}and a positionj∈Bkj\\in B\_\{k\}, and abbreviate the oracle and model marginals at that position byp:=p1\|skq\(a1j=⋅\)p:=p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}=\\cdot\)andℚ:=ℚ1\|tkθ\(a1j=⋅∣sk\)\\mathbb\{Q\}:=\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=\\cdot\\mid s\_\{k\}\)\. Letz^j=arg⁡maxz⁡ℚ⁡\(z\)\\hat\{z\}^\{j\}=\\arg\\max\_\{z\}\\mathbb\{Q\}\(z\)be the token written by Eq\.[4](https://arxiv.org/html/2609.25131#S3.E4)andzj⋆=argmaxzp\(z\)z^\{j\\star\}=\\arg\\max\_\{z\}p\(z\)the oracle mode, so thatcj=p⁡\(z^j\)c\_\{j\}=p\(\\hat\{z\}^\{j\}\)andcj⋆=p\(zj⋆\)c\_\{j\}^\{\\star\}=p\(z^\{j\\star\}\)\. Adding and subtractingℚ\\mathbb\{Q\}at both tokens,

cj⋆−cj=p\(zj⋆\)−p\(z^j\)=\[p\(zj⋆\)−ℚ\(zj⋆\)\]\+\[ℚ\(zj⋆\)−ℚ\(z^j\)\]⏟≤0\+\[ℚ\(z^j\)−p\(z^j\)\],c\_\{j\}^\{\\star\}\-c\_\{j\}=p\(z^\{j\\star\}\)\-p\(\\hat\{z\}^\{j\}\)=\\bigl\[p\(z^\{j\\star\}\)\-\\mathbb\{Q\}\(z^\{j\\star\}\)\\bigr\]\+\\underbrace\{\\bigl\[\\mathbb\{Q\}\(z^\{j\\star\}\)\-\\mathbb\{Q\}\(\\hat\{z\}^\{j\}\)\\bigr\]\}\_\{\\leq 0\}\+\\bigl\[\\mathbb\{Q\}\(\\hat\{z\}^\{j\}\)\-p\(\\hat\{z\}^\{j\}\)\\bigr\],where the middle bracket is nonpositive becausez^j\\hat\{z\}^\{j\}maximizesℚ\\mathbb\{Q\}\. SinceTV⁡\(p,ℚ\)=maxA⊆𝒮⁡\|p⁡\(A\)−ℚ⁡\(A\)\|\\mathrm\{TV\}\(p,\\mathbb\{Q\}\)=\\max\_\{A\\subseteq\\mathcal\{S\}\}\|p\(A\)\-\\mathbb\{Q\}\(A\)\|dominates the discrepancy on any singleton, each outer bracket is at mostTV⁡\(p,ℚ\)\\mathrm\{TV\}\(p,\\mathbb\{Q\}\), giving the plug\-in boundcj⋆−cj≤2​TV​\(p,ℚ\)c\_\{j\}^\{\\star\}\-c\_\{j\}\\leq 2\\,\\mathrm\{TV\}\(p,\\mathbb\{Q\}\)\. Pinsker’s inequalityTV⁡\(p,ℚ\)≤12DKL\(p∥ℚ\)\\mathrm\{TV\}\(p,\\mathbb\{Q\}\)\\leq\\sqrt\{\\tfrac\{1\}\{2\}D\_\{\\mathrm\{KL\}\}\(p\\,\\\|\\,\\mathbb\{Q\}\)\}then yields

\(1−cj\)−\(1−cj⋆\)≤2DKL\(p∥ℚ\),\\left\(1\-c\_\{j\}\\right\)\-\\left\(1\-c\_\{j\}^\{\\star\}\\right\)\\;\\leq\\;\\sqrt\{2\\,D\_\{\\mathrm\{KL\}\}\\\!\\left\(p\\,\\\|\\,\\mathbb\{Q\}\\right\)\},whose divergence is exactly thejj\-th summand ofℰcond​\(Bk,sk\)\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)in Eq\.[5](https://arxiv.org/html/2609.25131#S3.E5)\. Denote this divergence byDjD\_\{j\}\. Summing overj∈Bkj\\in B\_\{k\}and applying Cauchy–Schwarz to thebbsquare roots,

∑j∈Bk2​Dj≤2​b​∑j∈BkDj=2​b​ℰcond​\(Bk,sk\),\\sum\_\{j\\in B\_\{k\}\}\\sqrt\{2D\_\{j\}\}\\;\\leq\\;\\sqrt\{2b\\sum\_\{j\\in B\_\{k\}\}D\_\{j\}\}\\;=\\;\\sqrt\{2b\\,\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B\_\{k\};s\_\{k\}\)\},which gives Eq\.[50](https://arxiv.org/html/2609.25131#A3.E50)\.□\\square

Two remarks delimit the result\. First, the bound separates the block’s oracle ambiguity∑j∈Bk\(1−cj⋆\)\\sum\_\{j\\in B\_\{k\}\}\(1\-c\_\{j\}^\{\\star\}\), which is a property ofqqatsks\_\{k\}for the chosen block, from the model\-dependent excess, which is bounded by a function ofℰcond\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\. The ambiguity term does depend on which block is selected, since a different block sums different positions, so a position that is genuinely ambiguous under the oracle is not rendered reliable by a small conditional KL; the absorption policy of Section[3](https://arxiv.org/html/2609.25131#S3)targets only the second term\. Second, the⋅\\sqrt\{\\cdot\}and the Cauchy–Schwarz step are both loose, and the statement is a control of an upper bound rather than a ranking of realized risk: decreasingℰcond\\mathcal\{E\}\_\{\\mathrm\{cond\}\}tightens the bound on the excess absorption risk, but a smaller bound does not by itself guarantee a smaller realized excess\. This is what motivates using the conditional term as the objective in Section[3](https://arxiv.org/html/2609.25131#S3), and it is the reason the policy is evaluated empirically rather than claimed to minimize realized risk\. Combined with Definition[1](https://arxiv.org/html/2609.25131#Thmdefinition1), the two bounds are\(1−cj\)−\(1−cj⋆\)≤2​Dj\(1\-c\_\{j\}\)\-\(1\-c\_\{j\}^\{\\star\}\)\\leq\\sqrt\{2D\_\{j\}\}andDj≤ϕ⁡\(Hθj\)D\_\{j\}\\leq\\phi\(H\_\{\\theta\}^\{j\}\), connecting the absorption error to the entropy criterion used at inference\.

#### C\.3\.3Entropy\-guided absorption: proof of Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)

This appendix proves the local absorption result in Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1), derives Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1), and records the scope of both claims\. Throughout, the sampler statesks\_\{k\}is reachable under the oracle process and𝒰k\\mathcal\{U\}\_\{k\}is the set of positions that remain active immediately before absorption eventkk\.

##### Scope of the main\-text results\.

A wrong absorption can leave the oracle\-reachable set, in which case the oracle conditional at the resulting state may be undefined; the statements below therefore control error only while the trajectory remains oracle\-consistent\. Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)is a conditional exchange argument for a surrogate bound, not a derivation of entropy–error regularity, and it does not establish optimal terminal accuracy or control the trajectory after an oracle\-inconsistent absorption\. Low entropy alone does not imply correctness: Definition[1](https://arxiv.org/html/2609.25131#Thmdefinition1)is an envelope assumption whose empirical coverage on the Sudoku denoisers, including the relaxed offsetϕ0\\phi\_\{0\}and the confidently\-wrong tail, is reported in Appendix[C\.3\.6](https://arxiv.org/html/2609.25131#A3.SS3.SSS6); the observed coverage is descriptive calibration evidence and does not establish the pointwise assumption or a held\-out guarantee\. Corollary[1](https://arxiv.org/html/2609.25131#Thmcorollary1)shows that entropy selection minimizes the excess\-risk bound for a fixed budget, but it need not minimize total risk, because the oracle term also varies across blocks; forb\>1b\>1, even exact marginals need not produce a jointly valid block of modes\. Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)is worst\-case additive, so the realized lookahead error can grow more slowly thanηm\\eta\_\{m\}; it compares score sensitivity rather than terminal losses under the two policies, and does not prove that local entropy is better whenever the bound is large\.

##### Relaxed entropy–error regularity\.

The relaxed inequality in Eq\.[51](https://arxiv.org/html/2609.25131#A3.E51)adds a position\-independent constantϕ0\\phi\_\{0\}to accommodate the confidently\-wrong tail at entropy near zero\. It holds on 99\.4% of sampled states for the6\.46\.4M denoiser in the diagnostic of Appendix[C\.3\.6](https://arxiv.org/html/2609.25131#A3.SS3.SSS6)\. Asϕ0\\phi\_\{0\}is the same for every active position, it shiftsΦ⁡\(Bk,sk\)\\Phi\(B\_\{k\};s\_\{k\}\)by the constantb​ϕ0b\\phi\_\{0\}and leaves the minimizer of∑j∈Bkϕ⁡\(Hθj​\(sk\)\)\\sum\_\{j\\in B\_\{k\}\}\\phi\\\!\\left\(H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\right\)and Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)unchanged\.

DKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)≤ϕ0\+ϕ⁡\(Hθj​\(sk\)\),ϕ0≥0,D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\\leq\\phi\_\{0\}\+\\phi\\\!\\left\(H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\right\),\\qquad\\phi\_\{0\}\\geq 0,\(51\)

##### Alternative uncertainty scores\.

For maximum predicted probability and probability margin, the corresponding uncertainty scores are1−p\(1\)1\-p\_\{\(1\)\}and1−\(p\(1\)−p\(2\)\)1\-\(p\_\{\(1\)\}\-p\_\{\(2\)\}\), respectively, wherep\(1\)≥p\(2\)p\_\{\(1\)\}\\geq p\_\{\(2\)\}are the two largest posterior probabilities\. Minimizing these scores selects the highest\-confidence positions, and entropy in Eq\.[3](https://arxiv.org/html/2609.25131#S3.E3)can be replaced by the corresponding score to obtain a different ordering policy under the same budget\. Definition[1](https://arxiv.org/html/2609.25131#Thmdefinition1)is stated for predictive entropy; entropy regularity does not imply the analogous conditions for maximum probability or probability margin, and the scores can rank positions differently, so the resulting policies are compared empirically in Appendix[E\.5](https://arxiv.org/html/2609.25131#A5.SS5)\.

##### Oracle objective and tractable envelope\.

Fix an absorption budgetb≥1b\\geq 1and recall𝔅k​\(b\)=\{B⊆𝒰k:\|B\|=b\}\\mathfrak\{B\}\_\{k\}\(b\)=\\\{B\\subseteq\\mathcal\{U\}\_\{k\}:\|B\|=b\\\}\. An oracle that knew the realized target sequence could fix every absorbed position correctly, eliminating errors propagated from earlier irreversible decisions\. For the inference setting, where absorbed values are still predicted by the denoiser, we use the true conditional posterior only as an analytical reference\. By Proposition[2](https://arxiv.org/html/2609.25131#Thmproposition2), this oracle reference would select

Bkoracle∈arg⁡minB∈𝔅k​\(b\)⁡∑j∈BDKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)⏟ℰcond​\(B,sk\)\.B\_\{k\}^\{\\mathrm\{oracle\}\}\\in\\arg\\min\_\{B\\in\\mathfrak\{B\}\_\{k\}\(b\)\}\\underbrace\{\\sum\_\{j\\in B\}D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\}\_\{\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B;s\_\{k\}\)\}\.This decision cannot be evaluated at inference becausep1\|skqp^\{q\}\_\{1\\mid s\_\{k\}\}is unavailable\. Under Definition[1](https://arxiv.org/html/2609.25131#Thmdefinition1), however, every summand admits the denoiser\-computable envelope

DKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)≤ϕ⁡\(Hθj​\(sk\)\)\.D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\\leq\\phi\\\!\\left\(H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\right\)\.Summing over a candidate block gives

ℰcond​\(B,sk\)≤Φ⁡\(B,sk\):=∑j∈Bϕ⁡\(Hθj​\(sk\)\)\.\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B;s\_\{k\}\)\\leq\\Phi\(B;s\_\{k\}\):=\\sum\_\{j\\in B\}\\phi\\\!\\left\(H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\right\)\.The entropy policy need not coincide withBkoracleB\_\{k\}^\{\\mathrm\{oracle\}\}; the claim is that it exactly minimizes this tractable upper bound\.

##### Proof of Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\.

###### Proof\.

Fix a budgetbband letBlowB\_\{\\mathrm\{low\}\}containbbpositions having the smallest predictive entropies\. Consider any other blockBBof sizebb\. IfB≠BlowB\\neq B\_\{\\mathrm\{low\}\}, there arei∈Blow∖Bi\\in B\_\{\\mathrm\{low\}\}\\setminus Bandj∈B∖Blowj\\in B\\setminus B\_\{\\mathrm\{low\}\}withHθi​\(sk\)≤Hθj​\(sk\)H\_\{\\theta\}^\{i\}\(s\_\{k\}\)\\leq H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\. Sinceϕ\\phiis nondecreasing, exchangingjjforiicannot increaseΦ\\Phibecause

Φ⁡\(\(B∖\{j\}\)∪\{i\},sk\)−Φ⁡\(B,sk\)=ϕ⁡\(Hθi​\(sk\)\)−ϕ⁡\(Hθj​\(sk\)\)≤0\.\\Phi\\\!\\left\(\(B\\setminus\\\{j\\\}\)\\cup\\\{i\\\};s\_\{k\}\\right\)\-\\Phi\(B;s\_\{k\}\)=\\phi\\\!\\left\(H\_\{\\theta\}^\{i\}\(s\_\{k\}\)\\right\)\-\\phi\\\!\\left\(H\_\{\\theta\}^\{j\}\(s\_\{k\}\)\\right\)\\leq 0\.Repeating the exchange producesBlowB\_\{\\mathrm\{low\}\}, proving that it minimizesΦ\\Phiover𝔅k​\(b\)\\mathfrak\{B\}\_\{k\}\(b\)\.

Whenb=1b=1, Eq\. \([8](https://arxiv.org/html/2609.25131#S3.E8)\) follows immediately\. Moreover, conditional total correlation vanishes for a singleton:

ℰjoint​\(\{j\},sk\)=Hq​\(a1j∣sk\)−Hq​\(a1j∣sk\)=0\.\\mathcal\{E\}\_\{\\mathrm\{joint\}\}\(\\\{j\\\};s\_\{k\}\)=H\_\{q\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\-H\_\{q\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)=0\.Hence Proposition[2](https://arxiv.org/html/2609.25131#Thmproposition2)reduces toℒabs​\(\{j\},sk\)=ℰcond​\(\{j\},sk\)\\mathcal\{L\}\_\{\\mathrm\{abs\}\}\(\\\{j\\\};s\_\{k\}\)=\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(\\\{j\\\};s\_\{k\}\), and the lowest entropy position minimizes the upper bound on the complete local absorption error\.

For part \(ii\), convergence to the oracle posterior on reachable states means that, for every admissiblejjandzj∈𝒮z^\{j\}\\in\\mathcal\{S\},

ℚ1\|tkθ​\(a1j=zj∣sk\)⟶p1\|skq​\(a1j=zj\)\.\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}=z^\{j\}\\mid s\_\{k\}\)\\longrightarrow p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}=z^\{j\}\)\.Because𝒮\\mathcal\{S\}is finite, each term with positive oracle mass converges continuously and each zero\-mass term contributes zero\. Therefore,

DKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)⟶0\.D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\\longrightarrow 0\.Every admissible block contains finitely many positions, so

ℰcond​\(B,sk\)=∑j∈BDKL​\(p1\|skq​\(a1j\)∥ℚ1\|tkθ​\(a1j∣sk\)\)⟶0\.\\mathcal\{E\}\_\{\\mathrm\{cond\}\}\(B;s\_\{k\}\)=\\sum\_\{j\\in B\}D\_\{\\mathrm\{KL\}\}\\\!\\left\(p^\{q\}\_\{1\\mid s\_\{k\}\}\(a\_\{1\}^\{j\}\)\\,\\middle\\\|\\,\\mathbb\{Q\}^\{\\theta\}\_\{1\\mid t\_\{k\}\}\(a\_\{1\}^\{j\}\\mid s\_\{k\}\)\\right\)\\longrightarrow 0\.This holds for every admissible block and thus for the block selected by any absorption policy, which proves part \(ii\)\. ∎

##### Why the result is local\.

The exchange argument holds at a fixed state and fixed absorption budget\. Absorbing a prediction changes both the active set and the conditioning state used by the denoiser at later events\. Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)therefore controls the next block contribution toℰcond\\mathcal\{E\}\_\{\\mathrm\{cond\}\}; it does not claim that greedy entropy selection minimizes the final loss over every complete absorption trajectory\. Whenb\>1b\>1, it does not optimize the joint termℰjoint​\(B,sk\)\\mathcal\{E\}\_\{\\mathrm\{joint\}\}\(B;s\_\{k\}\), which can vary with the selected block\.

##### Reachable states\.

The regularity condition is stated on oracle\-reachable states, where every absorbed partial configuration has positive probability underqqand its oracle conditional is defined\. A learned sampler can leave this set after an incorrect irreversible prediction\. At such a state,p1\|skqp^\{q\}\_\{1\\mid s\_\{k\}\}may be undefined if no valid completion remains, so the KL condition cannot be evaluated\. The theorem should consequently be read as controlling the error incurred while the sampler remains on an oracle\-consistent trajectory\. Its experiments test whether this local rule also reduces departures from that set; they do not turn the pointwise premise into a global guarantee\.

##### Sampling versus most\-likely absorption\.

The KL decomposition in Proposition[2](https://arxiv.org/html/2609.25131#Thmproposition2)uses the categorical denoiser marginal as the prediction kernel\. Algorithm[1](https://arxiv.org/html/2609.25131#alg1)instead absorbs its most likely value\. The two kernels are generally different, including on unique\-solution tasks unless the learned marginal is itself degenerate\. We therefore use the divergence analysis to justify the position\-selection rule and evaluate the deployed most\-likely\-value rule empirically rather than claiming equality of the two output laws\.

#### C\.3\.4Derivation of the global lookahead bound

Recall thatm=\|𝒰⁡\(s\)\|m=\|\\mathcal\{U\}\(s\)\|andRr​\(s\)=∑j∈𝒰⁡\(s\)hjr​\(s\)R\_\{r\}\(s\)=\\sum\_\{j\\in\\mathcal\{U\}\(s\)\}h\_\{j\}^\{r\}\(s\)forr∈\{q,θ\}r\\in\\\{q,\\theta\\\}\. Write

Gjr\(s\)=Rr\(s\)−𝔼z∼Prj\(⋅∣s\)Rr\(sj,z\),Dj​j′r\(s\):=Gjr\(s\)−Gj′r\(s\),G\_\{j\}^\{r\}\(s\)=R\_\{r\}\(s\)\-\\mathbb\{E\}\_\{z\\sim P\_\{r\}^\{j\}\(\\cdot\\mid s\)\}R\_\{r\}\(s^\{j,z\}\),\\qquad D\_\{jj^\{\\prime\}\}^\{r\}\(s\):=G\_\{j\}^\{r\}\(s\)\-G\_\{j^\{\\prime\}\}^\{r\}\(s\),so thatRr​\(s\)R\_\{r\}\(s\), which does not depend on the candidate, cancels exactly inDj​j′rD\_\{jj^\{\\prime\}\}^\{r\}\. The selection made by Info\-Gain depends on the scores only through the differencesDj​j′θD\_\{jj^\{\\prime\}\}^\{\\theta\}, so these are the quantities the bound must control\.

For a single successor aggregate the assumed pointwise entropy error gives\|Rθ​\(sj,z\)−Rq​\(sj,z\)\|≤\(m−1\)​δ\|R\_\{\\theta\}\(s^\{j,z\}\)\-R\_\{q\}\(s^\{j,z\}\)\|\\leq\(m\-1\)\\delta, sincesj,zs^\{j,z\}leavesm−1m\-1positions active\. Moreover0≤Rq​\(sj,z\)≤\(m−1\)​log⁡\|𝒮\|0\\leq R\_\{q\}\(s^\{j,z\}\)\\leq\(m\-1\)\\log\|\\mathcal\{S\}\|, so the total\-variation assumption implies

\|𝔼z∼Pθj​Rq​\(sj,z\)−𝔼z∼Pqj​Rq​\(sj,z\)\|≤\(m−1\)​ϵ​log⁡\|𝒮\|,\\left\|\\mathbb\{E\}\_\{z\\sim P\_\{\\theta\}^\{j\}\}R\_\{q\}\(s^\{j,z\}\)\-\\mathbb\{E\}\_\{z\\sim P\_\{q\}^\{j\}\}R\_\{q\}\(s^\{j,z\}\)\\right\|\\leq\(m\-1\)\\,\\epsilon\\log\|\\mathcal\{S\}\|,where we used\|𝔼P​f−𝔼P′​f\|≤TV⁡\(P,P′\)​\(supf−inff\)\|\\mathbb\{E\}\_\{P\}f\-\\mathbb\{E\}\_\{P^\{\\prime\}\}f\|\\leq\\mathrm\{TV\}\(P,P^\{\\prime\}\)\\,\(\\sup f\-\\inf f\)withTV\\mathrm\{TV\}in the12​ℓ1\\tfrac\{1\}\{2\}\\ell\_\{1\}convention of Section[3\.3](https://arxiv.org/html/2609.25131#S3.SS3)\. Combining the two displays, the error of the expected successor aggregate for a single candidate is at mostηm:=\(m−1\)​\(δ\+ϵ​log⁡\|𝒮\|\)\\eta\_\{m\}:=\(m\-1\)\(\\delta\+\\epsilon\\log\|\\mathcal\{S\}\|\)\.

##### Successors outside the oracle support\.

The expectation overz∼Pθj\(⋅∣s\)z\\sim P\_\{\\theta\}^\{j\}\(\\cdot\\mid s\)can place mass on a valuezzwithPqj​\(z∣s\)=0P\_\{q\}^\{j\}\(z\\mid s\)=0\. The successorsj,zs^\{j,z\}then has no oracle conditional, sohj′q​\(sj,z\)h\_\{j^\{\\prime\}\}^\{q\}\(s^\{j,z\}\)is undefined\. We adopt the conventionhj′q​\(sj,z\):=log⁡\|𝒮\|h\_\{j^\{\\prime\}\}^\{q\}\(s^\{j,z\}\):=\\log\|\\mathcal\{S\}\|for every activej′j^\{\\prime\}at such a successor, treating an oracle\-inconsistent absorption as leaving the remaining positions maximally uncertain, and we require the pointwise entropy condition\|hθ−hq\|≤δ\|h^\{\\theta\}\-h^\{q\}\|\\leq\\deltaonly at successors of positive model probability with this extension in force\. Any other bounded extension leaves the total\-variation step unchanged because it uses only the range\[0,\(m−1\)​log⁡\|𝒮\|\]\[0,\(m\-1\)\\log\|\\mathcal\{S\}\|\]ofRqR\_\{q\}\. If one prefers not to impose the entropy condition at off\-support successors at all, their total model mass is at mostTV⁡\(Pθj,Pqj\)≤ϵ\\mathrm\{TV\}\(P\_\{\\theta\}^\{j\},P\_\{q\}^\{j\}\)\\leq\\epsilon, which adds at mostϵ⁡\(m−1\)​log⁡\|𝒮\|\\epsilon\(m\-1\)\\log\|\\mathcal\{S\}\|to the first display and changesηm\\eta\_\{m\}to\(m−1\)​\(δ\+2​ϵ​log⁡\|𝒮\|\)\(m\-1\)\(\\delta\+2\\epsilon\\log\|\\mathcal\{S\}\|\)without altering the form or reading of the lemma\.

Since the current aggregate cancels,

\|Dj​j′θ​\(s\)−Dj​j′q​\(s\)\|≤\|𝔼Pθj​Rθ​\(sj,z\)−𝔼Pqj​Rq​\(sj,z\)\|\+\|𝔼Pθj′​Rθ​\(sj′,z\)−𝔼Pqj′​Rq​\(sj′,z\)\|≤2​ηm\.\\left\|D\_\{jj^\{\\prime\}\}^\{\\theta\}\(s\)\-D\_\{jj^\{\\prime\}\}^\{q\}\(s\)\\right\|\\leq\\left\|\\mathbb\{E\}\_\{P\_\{\\theta\}^\{j\}\}R\_\{\\theta\}\(s^\{j,z\}\)\-\\mathbb\{E\}\_\{P\_\{q\}^\{j\}\}R\_\{q\}\(s^\{j,z\}\)\\right\|\+\\left\|\\mathbb\{E\}\_\{P\_\{\\theta\}^\{j^\{\\prime\}\}\}R\_\{\\theta\}\(s^\{j^\{\\prime\},z\}\)\-\\mathbb\{E\}\_\{P\_\{q\}^\{j^\{\\prime\}\}\}R\_\{q\}\(s^\{j^\{\\prime\},z\}\)\\right\|\\leq 2\\eta\_\{m\}\.For the model and oracle maximizers, insert and subtract the model scores and note that the differences again eliminateRr​\(s\)R\_\{r\}\(s\):

Gj⋆q−Gj^q=Dj⋆​j^q≤Dj⋆​j^θ⏟≤0\+\|Dj⋆​j^θ−Dj⋆​j^q\|≤2​ηm\.G\_\{j^\{\\star\}\}^\{q\}\-G\_\{\\hat\{j\}\}^\{q\}=D\_\{j^\{\\star\}\\hat\{j\}\}^\{q\}\\leq\\underbrace\{D\_\{j^\{\\star\}\\hat\{j\}\}^\{\\theta\}\}\_\{\\leq 0\}\+\\left\|D\_\{j^\{\\star\}\\hat\{j\}\}^\{\\theta\}\-D\_\{j^\{\\star\}\\hat\{j\}\}^\{q\}\\right\|\\leq 2\\eta\_\{m\}\.The analogous two\-error argument for the local entropy minimizers gives

hj^entq​\(s\)−minj⁡hjq​\(s\)≤2​δ,h\_\{\\hat\{j\}\_\{\\mathrm\{ent\}\}\}^\{q\}\(s\)\-\\min\_\{j\}h\_\{j\}^\{q\}\(s\)\\leq 2\\delta,which completes the proof of Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)\.□\\square

##### Interpretation of the lookahead result\.

Three qualifications delimit what the lemma does and does not say\. First, it bounds score estimation, not the realized terminal loss: Info\-Gain remains effective whenever2​ηm2\\eta\_\{m\}is small relative to the oracle marginDj⋆​j′qD\_\{j^\{\\star\}j^\{\\prime\}\}^\{q\}between its best and second\-best candidates, and the decision\-relevant diagnostic is therefore the ratio of the two rather thanηm\\eta\_\{m\}alone\. Second, because the bound is stated on score*differences*, it is invariant to the normalization of the aggregate: rescaling every candidate score by a common positive factor rescalesηm\\eta\_\{m\}and the margin identically and changes neither the selected candidate nor the comparison with the local rule\. The window dependence that remains comes from them−1m\-1positions genuinely carried inside each successor aggregate, not from the choice of units\. Third, the additivity over those positions is worst case; if the per\-position entropy errors are not adversarially aligned, a concentration argument would giveO⁡\(m−1\)O\(\\sqrt\{m\-1\}\)in place ofO⁡\(m−1\)O\(m\-1\), so the lemma should be read as an upper envelope on how the lookahead window can amplify error rather than as the typical rate\. The result isolates the source of accumulation: global lookahead combines errors over the unresolved positions, whereas local entropy selection estimates one position at the current state\.

#### C\.3\.5Additional comparison with Info\-Gain

At a reachable statess, couple the entropy and Info\-Gain policies so that they differ only in the next absorbed position and share the continuation\. LetLlocalL\_\{\\mathrm\{local\}\}andLIGL\_\{\\mathrm\{IG\}\}be their final losses\. LetEsE\_\{s\}be the event that Info\-Gain selects an oracle\-consistent position and setκθ​\(s\)=ℙ⁡\(Es∣s\)\\kappa\_\{\\theta\}\(s\)=\\mathbb\{P\}\(E\_\{s\}\\mid s\)\. Define

Bs=𝔼\[Llocal−LIG∣Es,s\],Cs=𝔼\[LIG−Llocal∣Esc,s\]\.B\_\{s\}=\\mathbb\{E\}\[L\_\{\\mathrm\{local\}\}\-L\_\{\\mathrm\{IG\}\}\\mid E\_\{s\},s\],\\qquad C\_\{s\}=\\mathbb\{E\}\[L\_\{\\mathrm\{IG\}\}\-L\_\{\\mathrm\{local\}\}\\mid E\_\{s\}^\{c\},s\]\.When correct lookahead helps and incorrect lookahead hurts,Bs,Cs≥0B\_\{s\},C\_\{s\}\\geq 0measure the corresponding benefit and cost\.

###### Proposition 5\(When the local entropy policy beats Info\-Gain\)\.

At any state satisfying the definitions above,

𝔼⁡\[LIG−Llocal∣s\]=\(1−κθ​\(s\)\)​Cs−κθ​\(s\)​Bs\.\\mathbb\{E\}\[L\_\{\\mathrm\{IG\}\}\-L\_\{\\mathrm\{local\}\}\\mid s\]=\(1\-\\kappa\_\{\\theta\}\(s\)\)C\_\{s\}\-\\kappa\_\{\\theta\}\(s\)B\_\{s\}\.IfBs\+Cs\>0B\_\{s\}\+C\_\{s\}\>0, the local entropy policy has no larger expected final loss if and only if

κθ​\(s\)≤CsBs\+Cs\.\\kappa\_\{\\theta\}\(s\)\\leq\\frac\{C\_\{s\}\}\{B\_\{s\}\+C\_\{s\}\}\.\(52\)

###### Proof\.

Conditioning onEsE\_\{s\}andEscE\_\{s\}^\{c\}gives the displayed identity\. Requiring its right\-hand side to be nonnegative and rearranging gives Eq\. \([52](https://arxiv.org/html/2609.25131#A3.E52)\)\. ∎

This identity does not assert that local entropy always outperforms lookahead\. It makes the comparison depend on how often the lookahead choice is oracle\-consistent and on the relative costs of its correct and incorrect choices\. These quantities can be estimated by paired continuations on tasks with reference solutions, as illustrated by the paired continuation estimates below\.

##### Measured comparison\.

On Sudoku\-Extreme, paired continuations estimate the three quantities in Proposition[5](https://arxiv.org/html/2609.25131#Thmproposition5)and recover the observed winner at both denoiser sizes \(Table[5](https://arxiv.org/html/2609.25131#A3.T5)\)\. This is a diagnostic of the stated identity, not evidence that model size itself defines the threshold\. It is also a per\-state comparison with a shared continuation: end\-to\-end on the same21\.321\.3M denoiser the entropy policy still solves more puzzles than Info\-Gain \(0\.9640\.964against0\.9470\.947atn=1000n\{=\}1000,p=0\.027p\{=\}0\.027, Table[21](https://arxiv.org/html/2609.25131#A5.T21)\), so the per\-state advantage of lookahead at states where it is oracle\-consistent has not yet translated into a trajectory\-level win at this capacity\.

Table 5:Measured comparison on Sudoku\-Extreme \(600600paired reachable states per denoiser\)\.κθ\\kappa\_\{\\theta\}is the probability that Info\-Gain selects an oracle\-consistent position andκ⋆=Cs/\(Bs\+Cs\)\\kappa^\{\\star\}=C\_\{s\}/\(B\_\{s\}\+C\_\{s\}\)\. The local entropy policy has no larger expected loss whenκθ≤κ⋆\\kappa\_\{\\theta\}\\leq\\kappa^\{\\star\}\.Denoiserκθ\\kappa\_\{\\theta\}κ⋆\\kappa^\{\\star\}BsB\_\{s\}CsC\_\{s\}better policy6\.46\.4M0\.5270\.5270\.6980\.6980\.00430\.00430\.01000\.0100entropy21\.321\.3M0\.6950\.6950\.3970\.3970\.00380\.00380\.00250\.0025Info\-Gain

#### C\.3\.6Calibration and miscalibration

The following analysis provides an empirical diagnostic of the entropy–error relationship in Definition[1](https://arxiv.org/html/2609.25131#Thmdefinition1)and measures the surrogate slack used by Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\. Per\-cell error rises with the denoiser’s predictive entropy, from0\.0150\.015in the lowest\-entropy bin to0\.5260\.526in the highest \(Table[7](https://arxiv.org/html/2609.25131#A3.T7)\), and the empirical slope decreases from the6\.46\.4M to the21\.321\.3M denoiser\. The two reported constants move in opposite directions with capacity, the slope falling from0\.450\.45to0\.370\.37while the surrogate gapε\\varepsilonrises from0\.210\.21to0\.300\.30, and the two measure different things\. The slope describes how sharply error grows with the model’s own uncertainty, and a better\-calibrated denoiser improves it\. The surrogate gap describes how well a*static*score tracks the*path\-conditional*predictive entropy, and a stronger denoiser exploits constraint propagation more aggressively, so its entropies move more along the path and a fixed ranking tracks them less well\. A capable denoiser therefore makes the entropy–error relationship tighter and the fixed\-score policy’s job harder at the same time\. The fixed score nevertheless recovers only part of the adaptive policy’s gain on the weaker model\.

Figure 6:Entropy–accuracy correlation, both denoisers\.Same binned trajectory data as Table[7](https://arxiv.org/html/2609.25131#A3.T7)\(all1212equal\-count bins,6\.46\.4M and21\.321\.3M\), plotted against the model’s predictive entropyHθH\_\{\\theta\}on a log axis\. \(a\) Per\-cell accuracy \(1−err1\{\-\}\\mathrm\{err\}\) decreases with entropy at both capacities, with Spearmanρ⁡\(H,err\)=0\.23\\rho\(H,\\mathrm\{err\}\)=0\.23\(6\.46\.4M\) and0\.090\.09\(21\.321\.3M\) \(the lowρ\\rhoreflects that most mass sits in the flat, near\-zero\-entropy, near\-ceiling regime common to reachable Sudoku states; the drop is concentrated in the top bins, consistent with the nonlinear slopeLϕL\_\{\\phi\}reported below\)\. \(b\) NLL of the true digit rises with entropy at both capacities; the21\.321\.3M denoiser’s NLL curve sits uniformly below the6\.46\.4M curve, the calibration improvement with capacity that motivates comparing the two models throughout this appendix\.##### Pointwise envelope\.

Definition[1](https://arxiv.org/html/2609.25131#Thmdefinition1)requires a*pointwise*boundKL≤ϕ⁡\(Hθ\)\\mathrm\{KL\}\\leq\\phi\(H\_\{\\theta\}\), not merely the bin\-mean trend above\. Fitting a nondecreasing9595th\-percentile upper envelope to the full per\-cell cloud \(158,191158\{,\}191reachable partial configurations\) gives an empirical envelope slopeLenv,0\.95≈1\.70L\_\{\\mathrm\{env\},0\.95\}\\approx 1\.70\(6\.46\.4M\) and1\.341\.34\(21\.321\.3M\), about3\.63\.6–3\.8×3\.8\\timesthe bin\-mean OLS slope\. The robust calibration summary reports a small*confidently\-wrong*fraction \(0\.99%0\.99\\%at6\.46\.4M and0\.35%0\.35\\%at21\.321\.3M\), with near\-zero\-entropy NLL reaching approximately20\.920\.9and22\.322\.3nats, respectively\. A strictϕ\\phiwithϕ⁡\(0\)=0\\phi\(0\)=0at the mean slope already holds on95\.5%95\.5\\%\(6\.46\.4M\) and97\.6%97\.6\\%\(21\.321\.3M\) of sampled reachable states\. The relaxed\-envelope diagnostic uses the mean offset \(phi0\_mean\),ϕ0=0\.572316\\phi\_\{0\}=0\.572316nats for6\.46\.4M and0\.1192460\.119246nats for21\.321\.3M \(Table[6](https://arxiv.org/html/2609.25131#A3.T6)\)\. Empirically counting violations ofKL≤ϕ0\+ϕ⁡\(Hθ\)\\mathrm\{KL\}\\leq\\phi\_\{0\}\+\\phi\(H\_\{\\theta\}\)gives violation rates0\.006232340\.00623234and0\.003275100\.00327510, respectively\. Their complements are99\.3768%99\.3768\\%and99\.6725%99\.6725\\%, reported as99\.4%99\.4\\%and99\.7%99\.7\\%coverage\. The percentile and maximum summaries in the table are descriptive statistics; they are not the offsets used for this calculation\. For any fixed position\-independent offset inKL≤ϕ0\+ϕ⁡\(Hθ\)\\mathrm\{KL\}\\leq\\phi\_\{0\}\+\\phi\(H\_\{\\theta\}\), the additive contributionb​ϕ0b\\phi\_\{0\}preserves the entropy argmin at fixed budget, but this algebraic fact does not establish a tight error bound\. Substituting the updated envelope slopes into the diagnostic expression2​L​ε​M2L\\varepsilon Mgives39\.839\.8and45\.245\.2nats, respectively, compared with10\.510\.5and12\.412\.4using the bin\-mean slopes\. These fitted slopes and empirical coverage rates do not establish a uniform bound on every reachable state\.

Table 6:Reportedϕ0\\phi\_\{0\}summaries in the robust calibration export \(nats\)\. The export tags66M and2222M correspond to the paper’s6\.46\.4M and21\.321\.3M denoisers\. The mean column gives the offsets used to compute the reported relaxed\-envelope coverage; the other columns describe the distribution’s upper tail\.DenoiserMean95th percentile99th percentileMaximum6\.46\.4M0\.5723160\.00246916\.69564520\.85470821\.321\.3M0\.1192460\.0009900\.00272022\.328239
##### Oracle versus model entropy\.

To separate “entropy is the ordering signal” from “*the model’s*entropy is the signal,” we compare absorbing by the exact oracle entropyHqH\_\{q\}—the entropy of the completion\-count distribution over viable digits, computed on a multi\-solution corpus whereHq\>0H\_\{q\}\>0is non\-degenerate—against absorbing by the model’sHθH\_\{\\theta\}\. Over120120multi\-solution puzzles \(median2323completions\),HθH\_\{\\theta\}correlates strongly withHqH\_\{q\}\(r=0\.88r=0\.88at6\.46\.4M,0\.900\.90at21\.321\.3M\), and ordering by the model’s offlineHθH\_\{\\theta\}matches or exceeds ordering by the exactHqH\_\{q\}\(valid\-completion rate0\.9580\.958vs\.0\.9000\.900at6\.46\.4M;0\.9830\.983vs\.0\.9830\.983at21\.321\.3M\), while adaptively recomputingHθH\_\{\\theta\}reaches1\.0001\.000\. The model’s predictive entropy is thus a sufficient statistic for the absorption priority in this experiment: oracle access adds nothing over it, and the operative lever is recomputing entropy as the active state changes rather than obtaining a better static ranking\.

Table 7:Entropy–error diagnostic \(Definition[1](https://arxiv.org/html/2609.25131#Thmdefinition1)\) on Nikoli \(n=100n\{=\}100\): per\-cell error and NLL of the true digit, binned by the model’s predictive entropyHθH\_\{\\theta\}over the whole sampling trajectory \(6\.46\.4M denoiser; representative equal\-count bins\)\. Per\-cell error rises monotonically with entropy\. The bottom row reports the empirical error–entropy slope and surrogate errorε\\varepsilonfor both denoisers\.HθH\_\{\\theta\}\(bin mean\)0\.000\.240\.460\.640\.791\.08per\-cell error0\.0150\.1200\.2220\.3430\.3990\.526NLL \(true digit\)0\.270\.650\.881\.121\.011\.25
6\.46\.4M: empirical slope=0\.45=0\.45,ε=0\.21\\varepsilon\{=\}0\.21\|21\.321\.3M: empirical slope=0\.37=0\.37,ε=0\.30\\varepsilon\{=\}0\.30

9595th\-pct envelope slope:Lenv,0\.95≈1\.70L\_\{\\mathrm\{env\},0\.95\}\{\\approx\}1\.70/1\.341\.34\(6\.46\.4M/21\.321\.3M\); reported confidently\-wrong fraction0\.99%0\.99\\%/0\.35%0\.35\\%; strict\-ϕ\\phicompliance95\.5%95\.5\\%/97\.6%97\.6\\%, relaxed\-ϕ\\phicompliance99\.4%99\.4\\%/99\.7%99\.7\\%\.

Table 8:Two attempts to make a fixed external priority beat adaptive entropy\-guided absorption by inducing miscalibration\.*internal*==adaptive entropy;*external*==best of the fixed external score and logical forcedness\. Top: density shift \(train dense, test sparse\)\. Bottom: wrong constraint \(train Latin, test Sudoku\) over a difficulty sweep\. The external order wins only in the floored hard band \(a non\-significant∼2\{\\sim\}2\-puzzle margin\); wherever the denoiser is capable, confidence wins\.proberegime \(denoiser\)internalexternalgap\(a\) density shift, test givens2626–3131,n=185n\{=\}185calibrated \(ECE0\.0080\.008\)0\.9730\.746\+0\.227\+0\.227overconfident \(ECE0\.0740\.074\)0\.8270\.546\+0\.281\+0\.281\(b\) wrong constraint: Latin\-trained \(box\-blind\), test Sudokuhard \(givens3232–4040, floored\)0\.1090\.124−0\.015∗\-0\.015^\{\*\}medium \(givens4040–4848\)0\.6370\.630\+0\.007\+0\.007easy \(givens4444–5252\)0\.8270\.810\+0\.017\+0\.017
∗floored regime \(∼0\.11\{\\sim\}0\.11solve\);∼2\{\\sim\}2\-puzzle margin atn=137n\{=\}137, not significant\.

## Appendix DExperimental settings and hyperparameters

This appendix collects the experimental protocols and hyperparameters for §[4](https://arxiv.org/html/2609.25131#S4), following the order of the main experiments\.

Table 9:Protocol map for Sudoku results\. All samplers at a given capacity share the same checkpoint\. Each ID identifies a separate reported experiment, not an interchangeable estimate\.KKis the grid budget; it need not equal the number of singleton absorptions\. Current solve accuracies and policy labels are those of Table[3](https://arxiv.org/html/2609.25131#S4.T3); earlier trajectory diagnostics are reported separately\.IDCheckpoint / locationDecoder or absorption protocolCohort / budgetTAbsorption\-aware6\.46\.4M; Fig\.[2](https://arxiv.org/html/2609.25131#S3.F2)\(a–c\)Kinetic\-optimal updates; trajectory diagnosticNikolin=100n=100,K=64K=64EAbsorption\-aware6\.46\.4M; Fig\.[2](https://arxiv.org/html/2609.25131#S3.F2)\(d\)Separate endpoint validationNikolin=100n=100,K=64K=64NShared6\.46\.4M; Table[3](https://arxiv.org/html/2609.25131#S4.T3), non\-absorbing rowsKinetic\-optimal updates without absorptionEach setn=100n=100, five seeds,K=64K=64SAbsorption\-aware6\.46\.4M; Table[3](https://arxiv.org/html/2609.25131#S4.T3), absorbing rowsKinetic\-optimal updates, cosine absorption budgetGenerated, Nikoli, Extreme:n=500,100,700n=500,100,700, seeds0,10,1,K=64K=64RAbsorption\-aware6\.46\.4M; Table[18](https://arxiv.org/html/2609.25131#A5.T18)Earlier rate\-1 posterior\-resampling diagnostic; cosine budgetSame reported sizes and seeds as S; separate diagnosticLShared model at each capacity; Table[24](https://arxiv.org/html/2609.25131#A5.T24)Source\-held, singleton absorptionsEach setn=100n=100, five seedsG6\.46\.4M and21\.321\.3M; Table[21](https://arxiv.org/html/2609.25131#A5.T21)Separate singleton\-ordering studyGeneratedn=1000n=1000,2222–3434givens##### Sampler protocol\.

At each grid point,LEDFlowrecomputes predictive entropy from the current sampler state and absorbs the lowest\-entropy active positions under the absorption budget\. Active positions then follow the backbone’s unchanged update, while absorbed positions remain fixed under Eq\. \([2](https://arxiv.org/html/2609.25131#S3.E2)\)\. Probability\-margin, Info\-Gain, and random policies use the same budget, value update, and absorption rule and differ only in the score used to select the block\. The budget is task specific: on the FUDOKI understanding, text\-to\-image, and mathematics harnesses the cosine schedule of Appendix[C\.2](https://arxiv.org/html/2609.25131#A3.SS2)setsbkb\_\{k\}at each of theKKgrid points\. The main Sudoku comparisons also use a cosine absorption budget overK=64K=64steps, with coordinatewise kinetic\-optimal updates in both absorbing and non\-absorbing arms \(Appendix[E\.4](https://arxiv.org/html/2609.25131#A5.SS4)\)\. Explicit singleton\-absorption diagnostics evaluate once per generated position \(Table[26](https://arxiv.org/html/2609.25131#A5.T26)\); their NFE and vanishing joint term do not describe the cosine\-budget runs\. Appendix[C\.2](https://arxiv.org/html/2609.25131#A3.SS2)gives the complete update order and matched\-policy controls\.

##### Absorption\-policy baselines\.

Top K\-margin selects thebkb\_\{k\}active positions with largestp\(1\)−p\(2\)p\_\{\(1\)\}\-p\_\{\(2\)\}, wherep\(1\)≥p\(2\)p\_\{\(1\)\}\\geq p\_\{\(2\)\}are the two largest posterior probabilities, and uses the same argmax absorption rule asLEDFlow\. Info\-Gain uses a lookahead window of eight, the best\-performing setting in Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(c\), with its additional evaluation cost reported in Appendix[E\.7](https://arxiv.org/html/2609.25131#A5.SS7)\.

##### Understanding\.

We evaluate with VLMEvalKit\[[Duan et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib28)\]on the full standard splits, following the established protocol\[[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10),[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13)\]\. The base model is frozen FUDOKI throughout and only the decode\-time sampler changes\. The non\-absorbing samplers, guidance, Top K\-margin, andLEDFlowall use one denoiser evaluation per grid point; Info\-Gain additionally spends lookahead evaluations \(Appendix[E\.7](https://arxiv.org/html/2609.25131#A5.SS7)\)\. Split sizes, metrics, and the95%95\\%intervals appear in Table[10](https://arxiv.org/html/2609.25131#A5.T10)\. The reported Euler MMBench score uses dev\-EN, whereas our evaluation uses the full standard split\.

##### Generation\.

GenEval\[[Ghosh et al\., 2023](https://arxiv.org/html/2609.25131#bib.bib27)\]is evaluated on the full suite of553553prompts with44images per prompt, scored with the standard Mask2Former and CLIP pipeline, at the standard budgetK=64K\{=\}64denoiser evaluations\. FUDOKI’s native default is128128and scores saturate byK∼32K\{\\sim\}32, consistent with prior reports\[[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13)\]\. We use one fixedLEDFlowconfiguration for the full suite and all reported category scores\.

##### Reasoning\.

All Sudoku results at a given model capacity use the same checkpoint\. In Table[3](https://arxiv.org/html/2609.25131#S4.T3), both non\-absorbing and absorbing rows use the shared absorption\-indicator\-trained6\.46\.4M denoiser atK=64K=64with coordinatewise kinetic\-optimal updates\. Non\-absorbing rows keep every generated position active throughout\. Absorbing rows apply the same update at active positions and use a cosine budgetmkm\_\{k\}to determine how many positions to absorb; these absorbing policies differ only in their ordering score\. Reported cohorts and seeds are listed below and in Table[9](https://arxiv.org/html/2609.25131#A4.T9)\. The older masked\-reveal diagnostics use a6\.46\.4M denoiser of width256256with88layers and88heads, trained for6060k steps\. Their capacity\-sensitivity comparison against a21\.321\.3M denoiser, width384384with1212layers and1212heads trained for120120k steps on a mixed easy–hard corpus, is reported separately in §[E\.5\.3](https://arxiv.org/html/2609.25131#A5.SS5.SSS3)\. This is a robustness check rather than a controlled scaling study, because capacity, training budget, and training distribution change together\. These masked\-reveal diagnostics are distinct from the new flow\-active ordering comparison\. The three Sudoku sets are held\-out synthetic puzzles, Nikoli\[[Seely et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib29)\], and Sudoku\-Extreme\[[Wang et al\., 2025a](https://arxiv.org/html/2609.25131#bib.bib30)\]; the additional constraint structures are9×99\{\\times\}9Latin squares,33\-coloring of random graphs, and molecular infilling on MOSES SMILES\[[Polykovskiy et al\., 2020](https://arxiv.org/html/2609.25131#bib.bib9)\]\. The Sudoku denoisers are trained on the Kaggle million\-puzzle set after deduplication by puzzle string and solution grid against the evaluation sets\.

##### Reasoning evaluation details\.

The absorbing Sudoku rows of Table[3](https://arxiv.org/html/2609.25131#S4.T3)useK=64K\{=\}64, seeds00and11, and500500generated,100100Nikoli, and700700Sudoku\-Extreme puzzles\. These scores come from the main ordering sweep\. The retained non\-absorbing rows use the earliern=100n\{=\}100, five\-seed protocol for each Sudoku split\. Figure[2](https://arxiv.org/html/2609.25131#S3.F2)separately labels the trajectory and endpoint\-validation runs; neither supplies the ordering\-comparison values in Table[3](https://arxiv.org/html/2609.25131#S4.T3)\. The older masked\-reveal capacity and ordering\-ladder diagnostics retain their ownn=100n\{=\}100, five\-seed protocol\. Zero seed variance under deterministic decoding does not imply zero puzzle\-sampling uncertainty; Table[21](https://arxiv.org/html/2609.25131#A5.T21)reports puzzle\-level confidence intervals and paired tests on a separaten=1000n\{=\}1000cohort\. Its generated subset is not the generated set in Table[3](https://arxiv.org/html/2609.25131#S4.T3), so their absolute accuracies should not be used interchangeably\.

##### Generality\.

For the additional constraint structures, we train a fresh4\.84\.8M denoiser for9×99\{\\times\}9Latin squares and a1\.21\.2M graph\-attention discrete flow denoiser for random33\-coloring; molecular infilling uses a character\-level model on MOSES SMILES andn=2000n\{=\}2000examples\. The pretrained\-model experiments change only the native sampler while leaving model weights fixed\. On GSM8K, LLaDA\-8B\-Instruct\[[Nie et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib7)\]usesn=200n\{=\}200and Dream\-7B\[[Ye et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib8)\]usesn=1000n\{=\}1000, both with128128sampling steps, generation length256256, and argmax token choice; LLaDA additionally uses block length3232\. MMaDA\-8B\[[Yang et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib38)\]usesn=500n\{=\}500,256256steps, and chain\-of\-thought prompting\. BBH logical deduction poolsn=750n\{=\}750three\-, five\-, and seven\-object puzzles\. For open multimodal mathematics, LLaDA\-V uses seed\-averaged random baselines on MathVerse \(n=900×4n\{=\}900\\times 4\) and MathVista \(n=1000×2n\{=\}1000\\times 2\), whereas MMaDA uses one random draw onn=900n\{=\}900andn=1000n\{=\}1000, respectively\.

## Appendix EAdditional experimental analysis

This appendix provides the additional analyses and full result tables for §[4](https://arxiv.org/html/2609.25131#S4), in the same order as the main experiments\.

### E\.1Understanding

##### Uncertainty of the main\-text gains\.

The gain that clears its reported approximate95%95\\%interval is\+1\.6\+1\.6on POPE \(±1\.3\\pm 1\.3\)\. The remaining margins, including\+2\.3\+2\.3on MM\-Vet and\+1\.2\+1\.2on MMMU, are smaller than their intervals and we do not read them as individually significant\. Table[10](https://arxiv.org/html/2609.25131#A5.T10)reports the intervals and their approximation limits\.

Table 10:Understanding suite on the full standard splits via FUDOKI’s VLMEvalKit\. The table includes the six benchmarks of Table[3](https://arxiv.org/html/2609.25131#S4.T3)\. The last column is an approximate95%95\\%interval on the*native\-sampler\-vs\-LEDFlowdifference*\(1\.96​2​p​\(1−p\)/N1\.96\\sqrt\{2p\(1\-p\)/N\}, evaluated at the native\-sampler rate\)\. Entropy\-guided absorption improves every benchmark in the main\-text suite, and the POPE margin clears the reported interval\. The remaining margins are smaller than their intervals\. This places short\-answer understanding at the low\-sensitivity end of the observed task spectrum, consistent with a smaller conditional term than on the constrained tasks in §[4\.1](https://arxiv.org/html/2609.25131#S4.SS1)\.benchmarkNNnative samplerLEDFlow\(ours\)metricspread95%95\\%CI \(diff\)POPE512787\.489\.0acc\+1\.6\+1\.6±1\.3\\pm 1\.3MME23741494\.51504\.6P\+R\+10\.1\+10\.1—MMBench437773\.774\.6Overall\+0\.9\+0\.9±1\.8\\pm 1\.8GQA1257857\.058\.0acc\+1\.0\+1\.0±1\.2\\pm 1\.2MM\-Vet21838\.240\.5GPT\-judge\+2\.3\+2\.3±9\.1\\pm 9\.1MMMU105036\.237\.4acc\+1\.2\+1\.2±4\.1\\pm 4\.1

The MME row is a summed perception\-plus\-reasoning score rather than an accuracy, so the binomial interval does not apply\. These intervals do not account for paired predictions; the binomial approximation is also not a calibrated interval for graded GPT\-judge scores such as MM\-Vet\. Paired example\-level resampling is needed for inferential comparisons\.

Table 11:Understanding suite in context: reported unified understanding\-and\-generation models alongside our frozen\-FUDOKI decode\-time samplers \(condensed in Table[3](https://arxiv.org/html/2609.25131#S4.T3)of the main text\)\. Numbers are compiled from prior work\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]\. Understanding is a measurement\-validity control, not a SOTA claim\. Higher is better, “—” not reported\.modelPOPEMME†MMBench‡GQAMMMUMM\-Vet§Reported unified understanding\-and\-generation models \(context\)LWM75\.2——44\.8—9\.6Chameleon————22\.48\.3Show\-o\-25673\.8948\.4—48\.725\.1—Show\-o\-51280\.01097\.2—58\.026\.7—D\-DiT84\.01124\.7—59\.2——VILA\-U85\.81401\.8—60\.8—33\.5ILLUME88\.51445\.365\.1—38\.237\.0TokenFlow\-XL86\.81545\.968\.962\.738\.740\.7Janus87\.01338\.069\.459\.130\.534\.3Janus\-Pro\-1B86\.21444\.075\.559\.336\.339\.8FUDOKI \(reported\)\[[Wang et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib10)\]86\.11485\.473\.957\.634\.338\.0FUDOKI\+\+guidance\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]86\.81492\.774\.258\.235\.438\.6Frozen FUDOKI, decode\-time samplers \(this setting\)native sampler \(FUDOKI\)87\.381494\.573\.757\.036\.238\.2LEDFlow\(ours\)89\.041504\.674\.658\.037\.440\.5

†our MME is MME\-P \(perception only\), the metric the reported models use\.‡our MMBench is the full standard split, harder than the dev\-EN split behind the published FUDOKI73\.973\.9\.§MM\-Vet uses a GPT judge, and absolute values shift with the judge version, so we re\-grade with the standard00–11partial\-credit protocol \(VLMEvalKit\), which gives scores in the reported range\.

### E\.2Generation

#### E\.2\.1Text\-to\-image generation

We use this setting as a control on soft structure, where the absorption policy should matter less than on tightly constrained outputs\. We evaluate the full GenEval suite \(553 prompts, four images per prompt, Mask2Former and CLIP scoring\) on frozen FUDOKI at the standard budgetK=64K\{=\}64\.LEDFlowreaches0\.7810\.781Overall against the native sampler’s0\.7530\.753\. Its improvements span the category scores, including color attribute \(0\.7230\.723vs\.0\.6630\.663\) and position \(0\.7150\.715vs\.0\.6800\.680\)\. Because the Overall standard error on 553 prompts is roughly±0\.02\\pm 0\.02, we interpret this as competitive performance on soft structure rather than a gain comparable to those on constrained reasoning tasks\.

##### GenEval in the masked discrete flow literature\.

PlacingLEDFlowagainst the recent masked discrete flow literature on the same FUDOKI base \(Table[13](https://arxiv.org/html/2609.25131#A5.T13)\) separates two orthogonal axes\. Among*frozen\-model samplers*\(our setting\) the whole family sits within a band of about0\.050\.05:LEDFlow\(0\.7810\.781\) is the strongest entry, ahead of the native sampler \(0\.7530\.753\) and of the corrected\-sampler family \(Euler0\.7540\.754, time\-/location\-corrected0\.760\.76–0\.770\.77[Wan et al\. \[2026c\]](https://arxiv.org/html/2609.25131#bib.bib13)\), though the margin over the corrected samplers is comparable to the benchmark’s own resolution\. The higher scores in the literature come from the orthogonal*training*axis \(guidance/RLHF0\.780\.78[Wan et al\. \[2026b\]](https://arxiv.org/html/2609.25131#bib.bib34), rate\-aware policy optimization0\.930\.93[Wan et al\. \[2026a\]](https://arxiv.org/html/2609.25131#bib.bib37)\), which modify the model rather than the sampler and are complementary to our decode\-time contribution\. Table[12](https://arxiv.org/html/2609.25131#A5.T12)places the frozen\-sampler comparison against generation\-only and unified models\.

Table 12:GenEval in context: generation\-only and unified understanding\-and\-generation models alongside the frozen\-FUDOKI decode\-time samplers of Table[3](https://arxiv.org/html/2609.25131#S4.T3)\(main text\)\. The external rows modify or replace the model and are orthogonal to the decode\-time comparison\. External rows as compiled by their cited source\. Higher is better, “—” not reported\.methodSingleTwoCountColorsPos\.Attr\.Overall↑\\uparrowGeneration\-only models \(context\)SDXL\[[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]0\.980\.740\.390\.850\.150\.230\.55DALL\-E 3\[[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]0\.960\.870\.470\.830\.430\.450\.67SD3\-Medium\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.990\.940\.720\.890\.330\.600\.74FLUX\.1\-dev\[[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]0\.980\.930\.750\.930\.680\.650\.82Unified understanding\-and\-generation models \(context\)Chameleon\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]——————0\.39UniDisc\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.920\.470\.150\.670\.130\.190\.42LWM\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.930\.410\.460\.790\.090\.150\.47SEED\-X\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.970\.580\.260\.800\.190\.140\.49Emu3\-Gen\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.980\.710\.340\.810\.170\.210\.54Show\-o\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.950\.520\.490\.820\.110\.280\.53\+\+Mask\-GRPO\[[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]0\.990\.900\.690\.850\.350\.590\.73TokenFlow\-XL\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.950\.600\.410\.810\.160\.240\.55ILLUME\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.990\.860\.450\.710\.390\.280\.61Janus\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.970\.680\.300\.840\.460\.420\.61D\-DiT\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.970\.800\.540\.760\.320\.500\.65Janus\-Pro\-1B\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.980\.820\.510\.890\.650\.560\.73Janus\-Pro\-7B\[[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]0\.990\.890\.590\.900\.790\.660\.80MMaDA\[[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]0\.960\.600\.450\.810\.140\.250\.56Frozen FUDOKI with decode\-time samplers \(Table[3](https://arxiv.org/html/2609.25131#S4.T3)\)Euler@64 \(base FUDOKI\)\[[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13)\]0\.96250\.83840\.48750\.88330\.72000\.63000\.7536LEDFlow\(ours\)0\.95630\.87120\.52190\.90160\.71500\.72250\.7814

#### E\.2\.2Comparison with related generation methods

Table 13:GenEval Overall on FUDOKI in the masked discrete flow literature, separating decode\-time*samplers*\(frozen model, our setting\) from*training*\-based methods\. Among frozen\-model samplers the spread is≈0\.05\{\\approx\}0\.05, and an external FUDOKI Euler@64 result reproduces our native sampler to0\.0010\.001\.LEDFlowis the strongest decode\-time entry in this comparison, although its margin over the corrected\-sampler family is within the benchmark’s resolution\. Training\-based methods modify the model and address a separate axis\.methodGenEval OverallFrozen\-model samplers \(this setting\)native sampler \(Euler@64, ours\)0\.753Euler@64\[[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13)\]0\.754time\-/location\-corrected\[[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13)\]0\.7650\.765LEDFlow\(ours\)0\.781Training\-based \(context\)FUDOKI base \(\+\+CFG\)\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.77\+\+guidance / RLHF\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34)\]0\.78\+\+dFlowGRPO\[[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]0\.93

### E\.3Trajectory analysis under matched settings

Figure[2](https://arxiv.org/html/2609.25131#S3.F2)compares native uniform flow withLEDFlowon100100Nikoli puzzles atK=64K\{=\}64: puzzle solve accuracy rises from0\.340\.34to0\.820\.82, while correct\-to\-wrong reversions fall from0\.0940\.094to0\.0260\.026of generated cells\. These are the motivation measurements used in the main text\. The supplementary Sudoku measurements in Table[14](https://arxiv.org/html/2609.25131#A5.T14)and Figure[7](https://arxiv.org/html/2609.25131#A5.F7)are distinct from that main\-text comparison and should not be used interchangeably with its values\. This subsection also reports trajectory analysis on FUDOKI text\-to\-image generation in Table[14](https://arxiv.org/html/2609.25131#A5.T14)and Figure[8](https://arxiv.org/html/2609.25131#A5.F8)\.

Table 14:Trajectory measurements under matched settings\.Left: Sudoku \(Nikoli,n=100n\{=\}100,6\.46\.4M denoiser,K=64K\{=\}64\); all fractions use generated cells as the denominator, and “first correct at final step” counts cells that become correct only at the endpoint\. Right: FUDOKI text\-to\-image \(GenEval prompts,K=64K\{=\}64\); token statistics are per image token, with95%95\\%bootstrap intervals over images for the difference, and correctness flips are the fraction of samples whose GenEval verdict changes betweent=0\.75t\{=\}0\.75and the endpoint, on in\-distribution GenEval prompts and on the1616counterfactual out\-of\-distribution prompts of Table[15](https://arxiv.org/html/2609.25131#A5.T15)\. Absorption removes revision of absorbed positions in both domains; on Sudoku the entropy order also removes every correct\-to\-wrong flip, while the random order leaves7\.4%7\.4\\%\.SudokuText\-to\-imageMeasurementEulerRandomLEDFlowMeasurementEulerLEDFlowDiff\. \[95%95\\%CI\]Token changes per position1\.841\.841\.231\.231\.001\.00Token changes per position4\.144\.142\.332\.33\+1\.81\+1\.81\[1\.72,1\.881\.72,1\.88\]excluding the final step1\.351\.351\.221\.220\.980\.98excluding the final step3\.443\.442\.302\.30\+1\.13\+1\.13\[1\.04,1\.211\.04,1\.21\]Positions changed more than once0\.510\.510\.150\.150\.000\.00Positions changed more than once0\.970\.970\.730\.73\+0\.24\+0\.24\[0\.22,0\.250\.22,0\.25\]Changes occurring aftert=0\.75t\{=\}0\.750\.580\.580\.260\.260\.370\.37Changes occurring aftert=0\.75t\{=\}0\.750\.450\.450\.110\.11\+0\.34\+0\.34\[0\.34,0\.350\.34,0\.35\]Mean time of last change0\.750\.750\.470\.470\.640\.64Mean time of last change0\.960\.960\.540\.54\+0\.42\+0\.42\[0\.41,0\.430\.41,0\.43\]Cell\-fill accuracy \(final\)0\.4310\.4310\.6150\.6150\.9350\.935GenEval accuracy0\.7530\.7530\.7810\.781−0\.028\-0\.028\[−0\.037,−0\.020\-0\.037,\-0\.020\]Correct to wrong0\.2610\.2610\.0740\.0740\.0000\.000In\-dist\. correct to wrong0\.0520\.0520\.0100\.010\+0\.042\+0\.042First correct at final step0\.0350\.0350\.0000\.0000\.0170\.017In\-dist\. wrong to correct0\.1560\.1560\.0000\.000\+0\.156\+0\.156OOD correct to wrong0\.1250\.1250\.0250\.025\+0\.100\+0\.100OOD wrong to correct0\.0370\.0370\.0130\.013\+0\.024\+0\.024

Figure 7:Sudoku trajectories, native uniform flow versusLEDFlow\.\(a,b\) Each marker is one token change at a generated cell; the shaded curves give the number of changes per step\. Under the native flow the change density rises sharply ast→1t\\to 1, where many cells are re\-derived at the endpoint \(Proposition[1](https://arxiv.org/html/2609.25131#Thmproposition1)\); underLEDFloweach cell changes exactly once, at its absorption event\. \(c\) Cell\-fill accuracy over sampling time on the same100100puzzles: the native flow peaks neart≈0\.9t\{\\approx\}0\.9and then collapses from0\.580\.58to0\.430\.43, whereasLEDFlowrises monotonically to0\.9350\.935\.Figure 8:FUDOKI text\-to\-image trajectories, native uniform flow versusLEDFlow\.\(a,b\) Image\-token change density over sampling time: the native flow accumulates most of its changes aftert=0\.75t\{=\}0\.75, whileLEDFlowfront\-loads absorption and its density decays\. \(c\) GenEval accuracy of the image decoded from the intermediate state, on in\-distribution GenEval prompts \(solid\) and on the1616counterfactual prompts of Table[15](https://arxiv.org/html/2609.25131#A5.T15)\(dashed\)\.LEDFlowreaches its final quality byt≈0\.6t\{\\approx\}0\.6and holds it; the native flow reaches comparable quality only at the endpoint, after the late burst of changes in \(a,b\)\.Table 15:Out\-of\-distribution prompts for the text\-to\-image trajectory analysis\.Sixteen counterfactual prompts, four per GenEval tag, written so that the requested attribute, count, pairing, or spatial relation is unlikely under the training distribution\. They are scored with the standard GenEval detector pipeline and are the dashed curves in Figure[8](https://arxiv.org/html/2609.25131#A5.F8)and the OOD rows of Table[14](https://arxiv.org/html/2609.25131#A5.T14)\.TagPromptsColorsa photo of a green cow; a photo of a purple giraffe; a photo of a blue banana; a photo of a pink elephantCountinga photo of eight apples; a photo of six cats; a photo of seven birds; a photo of five clocksTwo objectsa photo of a toothbrush and a giraffe; a photo of a pizza and a fire hydrant; a photo of a laptop and a cow; a photo of an elephant and a spoonPositiona photo of a car above a bird; a photo of a boat above a bed; a photo of a bicycle above an elephant; a photo of a cup above a train##### Source\-held ordering diagnostic\.

Table[16](https://arxiv.org/html/2609.25131#A5.T16)compares absorption policies under a source\-held variant, rather than the flow\-active sampler in Section[3\.2](https://arxiv.org/html/2609.25131#S3.SS2)\. All four arms use the same denoiser, puzzles, grid \(K=64K\{=\}64\), and argmax value rule, and the three absorbing arms share one code path; the first two use argmax values and differ in selection rule, while the third samples the absorbed value\. In these absorbing arms, unabsorbed positions are held at the source state between absorption events, whereas the random arm of Table[14](https://arxiv.org/html/2609.25131#A5.T14)lets them continue the flow update\. The random\-arm cell accuracies \(0\.6150\.615and0\.7230\.723\) therefore refer to different active\-token dynamics\. The source\-held random\-absorption variant raises cell accuracy from0\.4130\.413to0\.7230\.723\(paired puzzle\-level95%95\\%interval\[0\.285,0\.336\]\[0\.285,0\.336\]\) and removes every correct\-to\-wrong flip\. Ordering raises it further to0\.9350\.935\(interval\[0\.166,0\.256\]\[0\.166,0\.256\]\) and raises solve accuracy from0\.3440\.344to0\.9000\.900\(interval\[0\.476,0\.634\]\[0\.476,0\.634\]\); the mechanism is visible in the bad\-absorption column, the fraction of absorption events that write an incorrect value, which falls from28\.0%28\.0\\%under the random order to6\.6%6\.6\\%under the entropy order\. Absorbing a sampled value instead of the argmax changes little \(0\.9230\.923against0\.9350\.935\), so the gain comes from freezing plus ordering rather than from the value rule\.

Table 16:Source\-held absorption\-policy diagnostic on Nikoli\(n=100n\{=\}100,6\.46\.4M denoiser,K=64K\{=\}64, seeds00–44\)\. Bad\-absorption is the fraction of absorption events whose absorbed value is wrong\. Paired puzzle\-level bootstrap intervals for the differences: no absorption to random, cell\+0\.310\+0\.310\[0\.285,0\.336\]\[0\.285,0\.336\], solve\+0\.344\+0\.344\[0\.278,0\.412\]\[0\.278,0\.412\]; random to entropy, cell\+0\.212\+0\.212\[0\.166,0\.256\]\[0\.166,0\.256\], solve\+0\.556\+0\.556\[0\.476,0\.634\]\[0\.476,0\.634\]\.ArmCell acc\.SolveCorrect→\\towrongChanges/cellLate\-change frac\.Bad\-absorptionUniform flow \(no absorption\)0\.4130\.4130\.0000\.0000\.2810\.2811\.871\.870\.600\.60—\[2pt/1pt\] \+ random absorption order0\.7230\.7230\.3440\.3440\.0000\.0001\.001\.000\.370\.370\.2800\.280\+ entropy order \(LEDFlow\)0\.935\\mathbf\{0\.935\}0\.900\\mathbf\{0\.900\}0\.0000\.0001\.001\.000\.370\.370\.066\\mathbf\{0\.066\}\+ entropy order, sampled absorption0\.9230\.9230\.8820\.8820\.0000\.0001\.001\.000\.370\.370\.0780\.078

### E\.4Flow\-active gate and reveal\-order comparison, with selected\-position analysis

Table[17](https://arxiv.org/html/2609.25131#A5.T17)reports the current Sudoku solve accuracies and policy labels from Table[3](https://arxiv.org/html/2609.25131#S4.T3)\. All methods at this capacity use the shared absorption\-aware6\.46\.4M checkpoint\. The non\-absorbing methods use kinetic\-optimal updates; absorbing methods additionally use a cosine absorption budget\. Cohorts and seeds follow protocols N and S in Appendix[D](https://arxiv.org/html/2609.25131#A4)\. The source\-held comparison in Table[16](https://arxiv.org/html/2609.25131#A5.T16)and the earlier trajectory diagnostics below are separate experiments\.

Table 17:Sudoku solve accuracy by sampler\.These values and policy labels reproduce the current results in Table[3](https://arxiv.org/html/2609.25131#S4.T3)\. Scores from earlier diagnostic reruns are not substituted for the current results\.SamplerGeneratedNikoliSudoku\-ExtremeEuler0\.6100\.4100\.078FreeCorrection0\.6300\.4000\.126Time\-corrected0\.6100\.4100\.078Location\-corrected0\.5880\.3600\.096Top K\-margin0\.8200\.7960\.271Info\-Gain0\.8040\.7800\.231LEDFlow0\.8650\.8450\.269##### What the current comparison shows\.

On generated and Nikoli puzzles, entropy absorption attains higher solve accuracy than the other displayed policies\. On Sudoku\-Extreme, margin is numerically higher \(0\.2710\.271versus0\.2690\.269\)\. These are descriptive comparisons; significance tests from a different diagnostic run do not establish significance for these updated scores\.

##### Separate trajectory diagnostics\.

Table[18](https://arxiv.org/html/2609.25131#A5.T18)retains trajectory measurements from the earlier rate\-1 posterior\-resampling diagnostic \(protocol R\)\. These measurements are not attached to the updated kinetic\-optimal solve scores in Table[17](https://arxiv.org/html/2609.25131#A5.T17)\. In that diagnostic, random freezing reduces revisions but also lowers cell accuracy on generated and Nikoli puzzles, whereas entropy selection improves cell accuracy\. On Sudoku\-Extreme, roughly45%45\\%of entropy absorptions are incorrect, illustrating that low entropy can coexist with incorrect predictions\.

Table 18:Earlier trajectory diagnostics \(protocol R\)\.Cell accuracy and transition measurements from the rate\-1 posterior\-resampling runs\. These are separate from the current solve\-accuracy results in Table[17](https://arxiv.org/html/2609.25131#A5.T17); no current\-run diagnostics are inferred from them\. Bad\-absorption is the fraction of absorptions that fix an incorrect value\.SetGateCell acc\.Correct→\\towrongBad\-absorptionChanges/cellLate\-changeGeneratedNone0\.8870\.090—4\.420\.181Random0\.8610\.0940\.1391\.980\.011Entropy0\.9420\.0410\.0582\.570\.021NikoliNone0\.8600\.104—4\.840\.172Random0\.8360\.1020\.1642\.150\.010Entropy0\.9090\.0600\.0902\.590\.017Sudoku\-ExtremeNone0\.5140\.455—21\.700\.238Random0\.5120\.3640\.4885\.510\.008Entropy0\.5540\.3790\.4509\.570\.023

##### Statistical evidence from the earlier diagnostic\.

On protocol R’s seed\-00per\-puzzle arrays, exact McNemar tests favor entropy over native and random on all three sets \(p≤3×10−8p\\leq 3\\times 10^\{\-8\}\)\. Entropy versus maximum probability is not significant on any set \(p≥0\.18p\\geq 0\.18\); entropy versus margin is not significant on generated or Nikoli\. The nominal margin advantage on Sudoku\-Extreme \(p=0\.031p=0\.031\) does not survive Bonferroni correction over twelve confidence\-order comparisons\. These tests describe that earlier diagnostic only and are not paired tests of the updated values in Table[17](https://arxiv.org/html/2609.25131#A5.T17)\.

##### Decoder and schedule\.

Both non\-absorbing and absorbing baselines use coordinatewise kinetic\-optimal updates\. Here a singleton flow transition changes one coordinate; it does not specify the number of positions absorbed at a grid point\. The non\-absorbing baseline keeps all generated positions active\. Absorbing policies apply the same kinetic\-optimal update to positions that remain active, while a cosine budget overK=64K=64grid steps determines the size of each irreversible absorption block\. A step can absorb zero, one, or several positions, so this protocol is not thebk=1b\_\{k\}=1absorption ablation and need not have zero joint dependence error\. For policies without additional lookahead, the grid protocol uses one denoiser evaluation per step, givingNFE=K\\mathrm\{NFE\}=K\. Figure[2](https://arxiv.org/html/2609.25131#S3.F2)reports separate trajectory and endpoint runs under kinetic\-optimal updates\.

##### Selected\-position reliability\.

As a reliability diagnostic from the earlier rate\-1 runs \(protocol R\), not a validation of a specific theoretical envelope or a measurement of the updated solve\-accuracy run, Table[19](https://arxiv.org/html/2609.25131#A5.T19)bins the entropy gate’s absorbed positions by entropy at absorption and reports the empirical absorption\-error rate per bin; we do not fit or test a KL envelope here\. Under the flow\-active decode almost all absorptions are made at very low entropy \(99\.6%99\.6\\%/99\.4%99\.4\\%of absorptions on generated/Nikoli\), where the error rate is5\.9%5\.9\\%/8\.6%8\.6\\%; error rises with entropy over the sparse higher\-entropy bins\. On Sudoku\-Extreme the denoiser is over\-confident: even the lowest\-entropy absorptions are wrong44\.8%44\.8\\%of the time\. The first erroneous absorption is correspondingly rare and late on easy puzzles \(generated:14%14\\%of puzzles have any wrong absorption, median first\-wrong timet=0\.55t\{=\}0\.55; Nikoli:16%16\\%,t=0\.31t\{=\}0\.31\) but common and early on hard puzzles \(Sudoku\-Extreme:75%75\\%,t=0\.23t\{=\}0\.23\)\. Low absorption error where the policy operates is consistent with, but does not by itself establish, the conditional guarantee’s envelope assumption; the hard\-puzzle regime shows confidently wrong absorption that any such assumption must exclude\.

Table 19:Absorption error at absorbed positions\(LEDFlowgate, seed00, flow\-active decode\): empirical error rate binned by entropy at absorption, with the number of absorbed positions per bin\. Low entropy predicts low error on generated/Nikoli; the relation fails under hard\-puzzle over\-confidence \(Sudoku\-Extreme\)\. “—” marks empty bins\.GeneratedNikoliSudoku\-ExtremeEntropy at absorptionerrnnerrnnerrnn\[0\.00,0\.05\)\[0\.00,0\.05\)0\.0590\.05926496264960\.0860\.086554055400\.4480\.4483624036240\[0\.05,0\.10\)\[0\.05,0\.10\)0\.7000\.70020200\.1670\.167660\.5500\.550866866\[0\.10,0\.20\)\[0\.10,0\.20\)0\.7500\.75016160\.3330\.333660\.5700\.570732732\[0\.20,0\.40\)\[0\.20,0\.40\)0\.7500\.75016160\.7780\.778990\.5370\.537555555\[0\.40,0\.80\)\[0\.40,0\.80\)0\.6000\.60035350\.8000\.80010100\.6220\.622585585\[0\.80,1\.60\)\[0\.80,1\.60\)0\.7780\.778990\.0000\.000110\.7140\.7147070\[1\.60,10\.0\)\[1\.60,10\.0\)1\.0001\.00011—001\.0001\.00066Overall0\.0610\.06126593265930\.0880\.088557255720\.4570\.4573905439054

### E\.5Reasoning

Two qualifications apply to the main\-text reasoning table\. Info\-Gain leads on molecular infilling, so local entropy is not uniformly preferable, and the current results do not isolate why that ordering reverses\. Average ranks are descriptive: small mathematics differences, including near\-floor GSM8K accuracy, do not establish meaningful wins\.

#### E\.5\.1Sudoku as a transparent oracle

Sudoku provides a transparent oracle because quantities that are usually inaccessible can be computed directly, in particular the terms in the absorption\-policy analysis \(Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\)\. The joint term is the exception: on a unique\-solution puzzle the oracle conditional is a point mass, so the conditional total correlation of Lemma[2](https://arxiv.org/html/2609.25131#Thmlemma2)is identically zero for every block, and unique\-solution Sudoku cannot be used to diagnose the joint term\. For that term we use the multi\-solution variant described below and the graph\-coloring task in Appendix[E\.6\.1](https://arxiv.org/html/2609.25131#A5.SS6.SSS1)\. Its vocabulary and sequence length are bounded by\|𝒮\|=9\|\\mathcal\{S\}\|=9digits andM≤81M\\leq 81cells, much smaller than the subword vocabulary and open\-ended length of free\-text generation\[[Tao et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib12)\]\. More importantly, a standard Sudoku puzzle has a unique solution\[[Tjusila et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib11)\], soq\(a1∣𝒞\)=𝟏\[a1=a1⋆\]q\(a\_\{1\}\\mid\\mathcal\{C\}\)=\\mathbf\{1\}\[a\_\{1\}=a\_\{1\}^\{\\star\}\]andH⁡\(q\)=0H\(q\)=0\. The oracle denoiser is therefore invariant to the absorption priority by[Section3\.3](https://arxiv.org/html/2609.25131#S3.SS3)\. Any priority effect exhibited by a learned denoiser arises from approximation error and is captured by the conditional termℰcond\\mathcal\{E\}\_\{\\mathrm\{cond\}\}in[Proposition2](https://arxiv.org/html/2609.25131#Thmproposition2)\. For every reachable partial fillaSa\_\{S\}with at least one valid completion, the true conditional distribution is computable from completion counts:

q\(a1j=v∣aS\)=N⁡\(aS,j,v\)∑v′N⁡\(aS,j,v′\),Hq\(a1j∣aS\)=−∑v=19q\(a1j=v∣aS\)logq\(a1j=v∣aS\),q\(a\_\{1\}^\{j\}=v\\mid a\_\{S\}\)=\\frac\{N\(a\_\{S\},j,v\)\}\{\\sum\_\{v^\{\\prime\}\}N\(a\_\{S\},j,v^\{\\prime\}\)\},\\qquad H\_\{q\}\(a\_\{1\}^\{j\}\\mid a\_\{S\}\)=\-\\sum\_\{v=1\}^\{9\}q\(a\_\{1\}^\{j\}=v\\mid a\_\{S\}\)\\log q\(a\_\{1\}^\{j\}=v\\mid a\_\{S\}\),whereN⁡\(aS,j,v\)N\(a\_\{S\},j,v\)counts valid completions ofaSa\_\{S\}with celljjfixed tovv\. Constraint propagation and backtracking compute these counts in milliseconds for standard9×99\\times 9puzzles\. When no valid completion exists, the conditional is undefined, so oracle diagnostics use only consistent prefixes\. On the unique\-solution RRN\[[Palm et al\., 2018](https://arxiv.org/html/2609.25131#bib.bib24)\], Sudoku\-Extreme\[[Wang et al\., 2025a](https://arxiv.org/html/2609.25131#bib.bib30)\], and million\-puzzle Kaggle corpora, the counts satisfyN∈\{0,1\}N\\in\\\{0,1\\\}andHqH\_\{q\}vanishes along every consistent path\. The multi\-solution variant\[[Nandwani et al\., 2022](https://arxiv.org/html/2609.25131#bib.bib31)\]provides nondegenerate oracle entropy when needed\.

##### Structured\-task\-specific baselines\.

Table[20](https://arxiv.org/html/2609.25131#A5.T20)retains two controls omitted from the cross\-task main table\. Uniformization is evaluated here as an exact CTMC control on the puzzle tasks; we did not adapt it to the mathematics harness\. The fixed\-score absorption policy similarly ranks positions by a static task score before selective absorption, so it does not directly apply when absorption is not used and positions can be rewritten\. We also did not define a corresponding static relevance score for character\-level SMILES infilling\.

Table 20:Structured\-task results for uniformization and fixed\-score absorption\. Higher is better; “—” denotes the SMILES setting without a static relevance score\.SamplerGeneratedNikoliExtremeLatinGraphMolec\.Uniformization0\.5620\.2400\.0740\.6941\.0000\.831Fixed\-score absorption0\.7020\.4200\.1580\.9461\.000—

#### E\.5\.2Reasoning controls

Two controls separate the absorption policy from the other choices a sampler makes\. Table[21](https://arxiv.org/html/2609.25131#A5.T21)repeats the Sudoku comparison atn=1000n\{=\}1000with puzzle\-level bootstrap intervals and paired McNemar tests, so that the ordering gap is read against sampling uncertainty rather than seed variance, and stratifies it by difficulty\. Its lower block adds the competing ordering policies on the same puzzles: the probability\-margin and maximum\-probability policies are not significantly different from entropy in these paired tests \(p=0\.23p\{=\}0\.23and0\.800\.80\)\. This does not establish statistical equivalence, and the scores need not induce the same ordering of positions\. Info\-Gain trails by0\.0660\.066\(p=3×10−9p\{=\}3\{\\times\}10^\{\-9\}\) at roughly eight times the wall\-clock\. On the stronger21\.321\.3M denoiser the gap narrows to0\.0170\.017but remains significant \(p=0\.027p\{=\}0\.027\), consistent with the crossover analysis of Appendix[C\.3\.5](https://arxiv.org/html/2609.25131#A3.SS3.SSS5): lookahead gains on the entropy policy as the denoiser improves, but has not overtaken it end\-to\-end at this capacity\. The separation that matters is therefore between confidence\-ordered single\-position absorption and either lookahead ordering or no ordering at all, which is the comparison Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)and Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(c\) speak to\. Table[22](https://arxiv.org/html/2609.25131#A5.T22)then crosses the absorption priority with the value\-selection rule: holding value selection fixed and varying only the priority moves accuracy far more than holding the priority fixed and switching between argmax and sampled values\. This attributes the gain to position selection rather than to the rule used to assign an absorbed value\. Table[23](https://arxiv.org/html/2609.25131#A5.T23)extends the same manipulation to random and high\-entropy\-first policies\.

Table 21:Sudoku at scale \(n=1000n\{=\}1000generated puzzles,6\.46\.4M, \#givens∈\[22,34\]\\in\[22,34\]\): solve accuracy with95%95\\%puzzle\-level bootstrap CIs and paired McNemar tests\. Upper block: paired against Euler; the adaptive and fixed\-score absorption policies are both significant, and logical forcedness underperforms the model’s entropy policy \(an imperfect denoiser tracks calibration, not forcedness\)\. Lower block: the competing single\-position ordering policies on the same10001000puzzles, paired against the entropy policy\. The three confidence\-based scores \(entropy, probability margin, maximum probability\) are statistically tied, whereas Info\-Gain \(window88\) is significantly below them and unordered Euler far below; wall\-clock per puzzle on one H100 is6\.96\.9ms \(entropy\),6\.26\.2\(margin\),2\.42\.4\(max\-probability\),54\.554\.5\(Info\-Gain\), and8\.68\.6\(Euler, re\-run at0\.4270\.427, within the interval of the upper\-block value\); on the21\.321\.3M denoiser the entropy policy remains ahead of Info\-Gain \(0\.9640\.964against0\.9470\.947,p=0\.027p\{=\}0\.027\) at9\.59\.5against149\.3149\.3ms\. Right block: accuracy stratified by difficulty, where the ordering gap grows as puzzles harden\. We note that this is not Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\(ii\) directly, which concerns the limit in which the denoiser approaches the oracle; the two agree only under the additional premise that harder puzzles leave the denoiser further from the oracle, which is plausible here but is not established by the theorem\.vs Eulerby \#givens \(determined\)absorption policysolve\-acc95%95\\%CI\(McNemar\)2222–25252626–29293030–3434entropy guided \(adaptive\)0\.922\[\.905,\.938\]\[\.905,\.938\]\+\.486\+\.486,p<10−12p\{<\}10^\{\-12\}0\.8370\.9490\.967logical\-forcedness \(MRV\)0\.621\[\.591,\.651\]\[\.591,\.651\]\+\.185\+\.185,p<10−12p\{<\}10^\{\-12\}———fixed external score0\.527\[\.496,\.558\]\[\.496,\.558\]\+\.091\+\.091,p=9×10−7p\{=\}9\{\\times\}10^\{\-7\}0\.2880\.4930\.775random\-order absorption0\.446\[\.415,\.477\]\[\.415,\.477\]————Euler@640\.436\[\.406,\.467\]\[\.406,\.467\]—0\.1900\.3580\.740vs entropyprobability margin0\.915\[\.897,\.932\]\[\.897,\.932\]−\.007\-\.007,p=0\.23p\{=\}0\.230\.8240\.9350\.973maximum probability0\.920\[\.903,\.936\]\[\.903,\.936\]−\.002\-\.002,p=0\.80p\{=\}0\.800\.8340\.9430\.970Info\-Gain \(window88\)0\.856\[\.834,\.877\]\[\.834,\.877\]−\.066\-\.066,p=3×10−9p\{=\}3\{\\times\}10^\{\-9\}0\.7320\.8680\.95221\.3M denoiser, same puzzlesentropy guided \(adaptive\)0\.964\[\.952,\.975\]\[\.952,\.975\]—0\.9290\.9700\.988Info\-Gain \(window88\)0\.947\[\.933,\.961\]\[\.933,\.961\]−\.017\-\.017,p=0\.027p\{=\}0\.0270\.8920\.9570\.985

Table 22:Decision\-rule control \(Nikoli,6\.46\.4M,n=100n\{=\}100,55seeds\): absorption*priority*×\\timesvalue*selection*, all single\-position with per\-step reevaluation\. Priority dominates at both absorption rules; the argmax\-vs\-sample rule is a minor effect\. The unordered \+ argmax cell is the random\-order policy follows the uniform first\-hitting choice of[Zheng et al\. \[2025\]](https://arxiv.org/html/2609.25131#bib.bib14)\. The0\.900±0\.0000\.900\{\\pm\}0\.000is exact\-zero*seed*variance \(deterministic argmax on the fixed100100\-puzzle set\),*not*a claim of zero uncertainty; the operative puzzle\-level CIs and paired McNemar tests are in Table[21](https://arxiv.org/html/2609.25131#A5.T21)\.absorption priorityargmax valuesampled valueentropy guided0\.900±0\.000\\mathbf\{0\.900\\pm 0\.000\}0\.874±0\.0080\.874\\pm 0\.008random order \(uniform choice,[Zheng et al\. \[2025\]](https://arxiv.org/html/2609.25131#bib.bib14)\)0\.380±0\.0310\.380\\pm 0\.0310\.280±0\.0260\.280\\pm 0\.026Euler@64 \(non\-absorbing flow, argmax\):0\.352±0\.0470\.352\\pm 0\.047
#### E\.5\.3Sensitivity to Sudoku denoiser capacity

Table[23](https://arxiv.org/html/2609.25131#A5.T23)checks whether the ordering result persists under the stronger Sudoku denoiser described in Appendix[D](https://arxiv.org/html/2609.25131#A4)\. Because capacity, training budget, and training distribution change together, this is a robustness sensitivity rather than a controlled parameter\-scaling study\. These are masked\-reveal ordering diagnostics using the shared checkpoint at each capacity, with active positions held at the mask token as a separate decoder ablation\. Within this diagnostic, entropy\-guided absorption is the strongest sampler on both Nikoli and Sudoku\-Extreme under both training regimes\.

##### The absorption\-policy ladder\.

The middle block of Table[23](https://arxiv.org/html/2609.25131#A5.T23)directly tests the priority policy: the denoiser, argmax value rule, absorption budget, and evaluation budget are held fixed\. Selecting positions by increasing predictive entropy, uniformly at random, and by decreasing predictive entropy produces a monotone ladder at both capacities:0\.900/0\.340/0\.1100\.900/0\.340/0\.110on Nikoli at6\.46\.4M and0\.970/0\.820/0\.5500\.970/0\.820/0\.550at21\.321\.3M\. Three features of the ladder are worth noting\. The reversal is the worst configuration in the table, below uniformization and every corrected sampler, which separates the absorption\-priority effect from the discretization effect\. Process\-level samplers occupy a narrow range, whereas changing only the order spans0\.790\.79at6\.46\.4M\. Random order is close to Euler \(0\.3400\.340vs\.0\.4100\.410on Nikoli\), further indicating that absorption priority drives the difference\. The ladder compresses as the denoiser improves, from a span of0\.790\.79at6\.46\.4M to0\.420\.42at21\.321\.3M, which is qualitatively consistent with the oracle limit in Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\(ii\)\. Because the models also differ in training budget and data mixture, this comparison does not attribute the compression to parameter count alone\. Table[24](https://arxiv.org/html/2609.25131#A5.T24)places the fixed\-score and random\-order absorption baselines on the same continuum and extends the comparison to the generated set\.

Table 23:Sensitivity to Sudoku denoiser capacity\(solve accuracy\)\. Parameter counts are exact up to one decimal place\. The absorbing rows are masked\-reveal diagnostics, not the flow\-active ordering comparison in Table[3](https://arxiv.org/html/2609.25131#S4.T3)\. The models also differ in training budget and data mixture, so the comparison demonstrates robustness across denoisers rather than an isolated size effect\. The middle block isolates absorption priority by holding the denoiser and argmax value rule fixed\. Accuracy decreases monotonically from entropy\-guided to random and then high\-entropy\-first absorption\.Nikoli↑\\uparrowSudoku\-Extreme↑\\uparrowSampler6\.4M21\.3M6\.4M21\.3MUniformization0\.2400\.6300\.0740\.194Euler0\.4100\.8200\.0780\.224Fixed external score0\.4200\.8700\.1580\.312FreeCorrection0\.4000\.7800\.1260\.270Time\-corrected0\.4100\.8200\.0780\.227Location\-corrected0\.3600\.7900\.0960\.246Absorption\-policy ablation \(same denoiser and value rule, priority varied alone\)random absorption0\.3400\.8200\.1040\.256high\-entropy\-first absorption0\.1100\.5500\.0510\.117LEDFlow\(ours\)0\.9000\.9700\.3280\.514Table 24:The absorption\-order ladder on masked\-reveal diagnostics\.Solve accuracy for adaptive low\-entropy, offline\-score, random, and high\-entropy orders\. Absorbing policies share the denoiser and argmax rule within each capacity; the non\-absorbing Euler row is a separate reference\. These results test position ordering in the source\-held setting, not the flow\-active gate\. The larger spread for the weaker denoiser is an empirical observation, not a prediction of Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)without further assumptions\.Absorption orderNikoli↑\\uparrowSudoku\-Extreme↑\\uparrowGenerated↑\\uparrow6\.46\.4M21\.321\.3M6\.46\.4M21\.321\.3M6\.46\.4M21\.321\.3MLow\-entropy\-first \(ours\)0\.9000\.9700\.3280\.5140\.9840\.996Offlineσrel\\sigma\_\{\\mathrm\{rel\}\}\(deployable\)0\.4200\.8700\.1430\.3010\.8400\.960Euler @6464\(flow\)0\.4100\.8200\.0780\.2240\.6100\.924Random single\-position0\.3400\.8200\.1040\.2560\.7560\.948High\-entropy\-first \(reversal\)0\.1100\.5500\.0510\.1170\.5800\.860

Solve accuracy on the older masked\-reveal diagnostic cohorts \(n=100n\{=\}100per benchmark, five seeds\)\. Within each capacity, the absorbing policies share the frozen denoiser and per\-cell argmax rule; unabsorbed positions remain at the mask token\. The checkpoint is shared with the corresponding capacity in the main experiments; holding active positions at the mask token is a separate decoder ablation\. The Euler row is a reference baseline, not an absorption\-order variant\.

#### E\.5\.4Why Euler and Time\-corrected nearly coincide on the puzzle tasks

Table[3](https://arxiv.org/html/2609.25131#S4.T3)reports the same accuracy for Euler and the time\-corrected sampler on all six puzzle columns, while the three mathematics columns differ\. Both rows run the same uniform discrete flow as Section[2](https://arxiv.org/html/2609.25131#S2), with no selective absorption; the near\-coincidence follows from the discretization rather than from a reporting artifact\. We explain it because otherwise the identical rounded values could appear to be a logging error\.

The two integrators differ only in how they discretize the schedule hazard within a grid interval\. On the gridtk−1<tkt\_\{k\-1\}<t\_\{k\}withΔ​t=tk−tk−1\\Delta t=t\_\{k\}\-t\_\{k\-1\}, the per\-step transition probability of a position whose current token differs from its predicted target is

pkeuler=1−exp⁡\(−Δ​t1−tk−1\),pktc=1−1−tk1−tk−1=Δ​t1−tk−1,p^\{\\mathrm\{euler\}\}\_\{k\}=1\-\\exp\\\!\\Big\(\\\!\-\\frac\{\\Delta t\}\{1\-t\_\{k\-1\}\}\\Big\),\\qquad p^\{\\mathrm\{tc\}\}\_\{k\}=1\-\\frac\{1\-t\_\{k\}\}\{1\-t\_\{k\-1\}\}=\\frac\{\\Delta t\}\{1\-t\_\{k\-1\}\},under the linear time parameterization used on the puzzle tasks\. Both expressions discretize the same hazardΔ​t/\(1−tk−1\)\\Delta t/\(1\-t\_\{k\-1\}\), one exponentially and one linearly\. They agree to first order inΔ​t\\Delta tand differ atO⁡\(Δ​t2\)O\(\\Delta t^\{2\}\)\.

On the puzzle tasks the vocabulary contains nine digits and every puzzle has a unique solution, so once the denoiser’s predicted target at a position stabilizes, both integrators move that position to the same digit and theO⁡\(Δ​t2\)O\(\\Delta t^\{2\}\)discrepancy changes only*when*within the grid the move occurs, not*which*digit is written\. Under common random numbers the two samplers therefore produce the same final grid except when the discrepancy places a transition on opposite sides of a step boundary, which changes the context seen by the denoiser at the next step and can alter the solved grid on a small fraction of puzzles\.

The measurements match this account\. Across our sampler sweeps the two agree exactly on roughly half of the \(benchmark,KK\) pairs and differ by0\.0010\.001–0\.0060\.006on the rest, with no systematic sign: on Nikoli atK=32K\{=\}32andK=64K\{=\}64they are identical \(0\.7300\.730and0\.8200\.820at21\.321\.3M\), while on Sudoku\-Extreme at21\.321\.3M they separate slightly at every budget \(0\.2240\.224vs\.0\.2270\.227atK=64K\{=\}64,n=700n\{=\}700\)\. The residual is therefore a discretization effect, an order of magnitude below the\+0\.29\+0\.29that the absorption policy contributes on the same benchmark, and it is smaller than the reporting resolution on then=100n\{=\}100splits of Table[3](https://arxiv.org/html/2609.25131#S4.T3)\.

In the mathematics columns \(MathVista, MathVerse, GSM8K\) the Euler and corrected rows run FUDOKI’s native uniform flow over the full subword vocabulary with selective absorption disabled\. TheLEDFlow, Top K\-margin, and Info\-Gain rows in the same columns apply the cosine absorption schedule of Appendix[C\.2](https://arxiv.org/html/2609.25131#A3.SS2)on that native text flow and differ only in the ranking score\. With a large vocabulary and open\-ended answers, the predicted target changes more often between steps, the two updates cease to be near\-equivalent, and the gaps in Table[3](https://arxiv.org/html/2609.25131#S4.T3)\(0\.2540\.254vs\.0\.2410\.241on MathVista,0\.0260\.026vs\.0\.0220\.022on GSM8K\) are the genuine difference between the exponential and the linearized step\. The contrast between the two regimes is itself evidence for the paper’s claim: on constrained puzzles, discretization corrections move accuracy by at most a few tenths of a point, while the absorption policy moves it materially\.

#### E\.5\.5Can an external priority beat confidence? Two induced\-miscalibration probes

On Sudoku our adaptive entropy policy is determined by the model’s internal confidence, so we ask whether a fixed external priority can ever beat it\. This comparison is outside the local optimality claim of Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1), but tests whether an external score can help when confidence is miscalibrated\. We induce that regime in two ways \(Table[8](https://arxiv.org/html/2609.25131#A3.T8)\)\.\(a\) Density shift\(train dense givens4444–5656, test sparse2626–3131\): the model is verifiably overconfident \(ECE→0\.074\\mathrm\{ECE\}\\,0\.008\\\!\\to\\\!0\.074\) but its confidence priority beats the external priority by a*wider*margin than on the calibrated model\.\(b\) Wrong constraint\(train box\-blind Latin squares, test Sudoku\): a sign flip appears*only*in the hardest band where the box\-blind model is floored \(a non\-significant∼2\{\\sim\}2\-puzzle margin\); wherever the model is capable, confidence wins\. In these probes, absorption\-policy quality follows the confidence*ranking*more closely than calibration or absolute correctness\. Models capable of solving the task rank cells well enough for their internal priority to win, whereas models with poor rankings rarely solve the puzzle\. Neither manipulation produces a capable but systematically misranked denoiser\. The same pattern appears when logical forcedness trails confidence \(Table[21](https://arxiv.org/html/2609.25131#A5.T21)\) and when the external evidence construction improves grounding AUROC without improving answer accuracy \(Appendix[G](https://arxiv.org/html/2609.25131#A7)\)\. We therefore make no accuracy claim for the external priority over internal confidence\. We therefore retain external scores only as a complementary interface for the same absorption mechanism and make no downstream accuracy claim for them\.

### E\.6Entropy\-guided absorption and generality

The transfer comparison in Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(b\) tests confidence ordering against a random order through each model’s own sampler, not improvement over shipped decoders\. In particular, the tested LLaDA order coincides with its default decoder; several transfer differences are not significant, and MMaDA MathVista changes by−0\.001\-0\.001\. This subsection complements Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(a,b\) with results across additional constraint structures and pretrained model families\. The comparisons retain the same absorption budget while varying the priority policy, isolating whether the low\-entropy order and its transfer persist beyond the primary Sudoku setting\.

##### Ordering\-analysis cohort\.

Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(a\) reports puzzle solve accuracy onn=1600n=1600puzzles pooled over Nikoli and Sudoku\-Extreme\. This pooled comparison orders policies from adaptive low entropy through random and non\-absorbing Euler to high entropy\.

#### E\.6\.1Additional constraint structures: Latin squares, graph coloring, molecular validity

To check that the absorption\-policy effect is a property of*constraint structure*and not of Sudoku specifically, we repeat it on9×99\{\\times\}9Latin squares\. The structure\-dependence reproduces cleanly: entropy\-guided absorption reaches0\.9980\.998against0\.7520\.752random\-order absorption and0\.6900\.690non\-absorbing Euler flow \(\+0\.31\+0\.31\), and the external logical\-forcedness order again underperforms confidence \(0\.9480\.948vs\.0\.9980\.998\)\. The decisive order benefit is therefore not Sudoku\-specific\.

##### Off the grid: graph coloring and molecular validity\.

On random\-graph33\-coloring, the paper’s*other*separated effect dominates: on hard instances the single\-position absorption samplers all reach∼1\.0\{\\sim\}1\.0while the non\-absorbing Euler flow floors at0\.450\.45–0\.630\.63\(a\+0\.37\+0\.37to\+0\.55\+0\.55total\-correlation gap, Lemma[2](https://arxiv.org/html/2609.25131#Thmlemma2)\), while the priority among single\-position absorption policies saturates \(the near\-oracle regime of Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)\(ii\)\)\. On masked infilling of drug\-like molecules from MOSES SMILES\[[Polykovskiy et al\., 2020](https://arxiv.org/html/2609.25131#bib.bib9)\], entropy\-guided absorption is the most valid and the high\-entropy\-first policy the least, with the gap over Euler*growing*with the masked fraction \(at mask0\.70\.7:0\.896\>0\.843\>0\.828\>0\.7670\.896\>0\.843\>0\.828\>0\.767,\+0\.068\+0\.068over Euler,z≈4z\{\\approx\}4\)\. Across these domains, the dominant effect is consistent with the corresponding theoretical term: the conditional term on unique\-solution grids with an imperfect denoiser, and the joint term on flexible coloring\.

#### E\.6\.2Transfer to pretrained diffusion models

We vary only the absorption order through each model’s native sampler, using the protocols in Appendix[D](https://arxiv.org/html/2609.25131#A4)\. The baseline in every row is a random absorption order, not the model’s shipped decoder; for LLaDA\-8B the shipped low\-confidence remasking coincides with theLEDFloworder, so the comparison there measures the value of that default over an arbitrary order\. Absolute accuracies are specific to our protocol \(sample size, step budget, prompting, and extraction in Appendix[D](https://arxiv.org/html/2609.25131#A4)\) and are not comparable to numbers published with each model’s own evaluation harness; for MMaDA\-8B on GSM8K in particular, the random\-order arm sits far below the published figure because that figure uses the confidence\-ordered default decoder, which is the ordering we test\. The entropy\-guided order improves over the random order by0\.0450\.045on LLaDA,0\.1210\.121on Dream, and0\.2280\.228on MMaDA for GSM8K; on BBH it improves LLaDA and Dream by0\.0630\.063and0\.1760\.176\. For MMaDA, the effect is strongest on GSM8K, positive on Track, ARC\-Challenge, and CommonsenseQA, and nearly unchanged on BBH\.

For open multimodal mathematics, the differences on MathVista and MathVerse are not significant for either model, providing the generality boundary summarized in Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(b\)\. Table[25](https://arxiv.org/html/2609.25131#A5.T25)reports the complete model\-family results and available significance information omitted from the compact main\-text visualization\.

Table 25:Full transfer results across pretrained diffusion\-model families\.We vary only the absorption policy through each model’s native sampler\. The text\-reasoning experiments use matched wall\-clock budgets\. Higher is better; n\.s\. denotes a difference that is not statistically significant\. Every cell shown in Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(b\) appears here\. The cells left grey in that figure are not applicable rather than pending: LLaDA\-8B and Dream\-7B are text\-only diffusion language models and are not evaluated on the image\-conditioned benchmarks, while LLaDA\-V is an understanding\-only multimodal model without a generation head and is therefore evaluated on multimodal mathematics alone\.ModelBenchmarkBackbone familyRandom orderLEDFloworderΔ\\DeltaText reasoning \(native sampler, order\-only ablation\)LLaDA\-8B\-Instruct\[[Nie et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib7)\]GSM8Kmasked diffusion LM0\.7250\.770\+0\.045\+0\.045Dream\-7B\[[Ye et al\., 2025b](https://arxiv.org/html/2609.25131#bib.bib8)\]GSM8Kmasked diff\. LM \(uniform prior\)0\.3710\.492\+0\.121\+0\.121\(z=5\.5z\{=\}5\.5,p≈5×10−8p\{\\approx\}5\{\\times\}10^\{\-8\}\)MMaDA\-8B\[[Yang et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib38)\]GSM8Kunified MM diffusion0\.2460\.474\+0\.228\+0\.228\(p≈2×10−18p\{\\approx\}2\{\\times\}10^\{\-18\}\)MMaDA\-8BBBH logical deductionunified MM diffusion0\.3700\.373\+0\.003\+0\.003\(χ2=0\.00\\chi^\{2\}\{=\}0\.00, n\.s\.\)MMaDA\-8BTracking shuffled objectsunified MM diffusion0\.1800\.240\+0\.060\+0\.060\(χ2=3\.11\\chi^\{2\}\{=\}3\.11,p≈0\.08p\{\\approx\}0\.08\)MMaDA\-8BARC\-Challengeunified MM diffusion0\.3830\.417\+0\.033\+0\.033\(χ2=0\.71\\chi^\{2\}\{=\}0\.71, n\.s\.\)MMaDA\-8BCommonsenseQAunified MM diffusion0\.3670\.400\+0\.033\+0\.033\(χ2=4\.61\\chi^\{2\}\{=\}4\.61,p≈0\.03p\{\\approx\}0\.03\)LLaDA\-8B\-InstructBBH logical deductionmasked diffusion LM0\.5830\.645\+0\.063\+0\.063\(p≈10−3p\{\\approx\}10^\{\-3\}\)LLaDA\-8B\-InstructTracking shuffled objectsmasked diffusion LM0\.4760\.592\+0\.116\+0\.116LLaDA\-8B\-InstructARC\-Challengemasked diffusion LM0\.8030\.830\+0\.027\+0\.027LLaDA\-8B\-InstructCommonsenseQAmasked diffusion LM0\.7670\.763−0\.004\-0\.004Dream\-7BBBH logical deductionmasked diffusion LM0\.4290\.605\+0\.176\+0\.176\(p≈6×10−16p\{\\approx\}6\{\\times\}10^\{\-16\}\)Dream\-7BTracking shuffled objectsmasked diff\. LM \(uniform prior\)0\.2480\.296\+0\.048\+0\.048Dream\-7BARC\-Challengemasked diff\. LM \(uniform prior\)0\.8200\.890\+0\.070\+0\.070Dream\-7BCommonsenseQAmasked diff\. LM \(uniform prior\)0\.7830\.833\+0\.050\+0\.050Open multimodal mathematics \(order\-only ablation\)LLaDA\-V\[[You et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib21)\]MathVista\[[Lu et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib25)\]masked diffusion VLM0\.4830\.499\+0\.016\+0\.016\(n\.s\.\)LLaDA\-VMathVerse\[[Zhang et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib18)\]masked diffusion VLM0\.2180\.222\+0\.004\+0\.004\(n\.s\.\)MMaDA\-8BMathVistaunified MM diffusion0\.3290\.328−0\.001\-0\.001\(n\.s\.\)MMaDA\-8BMathVerseunified MM diffusion0\.2040\.206\+0\.002\+0\.002\(n\.s\.\)

### E\.7Global lookahead error and efficiency

Block size trades sequential cost against accuracy: singleton absorption requires one evaluation per generated position, whereas batching reduces the evaluation count and reintroduces the joint dependence term\. In the reported puzzle ablations, larger blocks lower accuracy \(Appendix[E\.7\.2](https://arxiv.org/html/2609.25131#A5.SS7.SSS2)\), and the tested reopening rule does not improve on entropy ordering \(Appendix[E\.7\.4](https://arxiv.org/html/2609.25131#A5.SS7.SSS4)\)\.

The main\-text timing in Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(d\) is for the profiled configuration; a separate GenEval profile atK=16K=16reports Info\-Gain about7\.2×7\.2\\timesslower thanLEDFlow\. These timings do not imply constant overhead across sequence lengths and vocabularies\. The block\-size and reopening studies are in Appendices[E\.7\.2](https://arxiv.org/html/2609.25131#A5.SS7.SSS2)and[E\.7\.4](https://arxiv.org/html/2609.25131#A5.SS7.SSS4); they do not establish schedule invariance or rule out other revision policies\.

##### Info\-Gain at its minimal window is not the entropy policy\.

Because Info\-Gain scores a candidate by its own entropy minus the mean entropy of the remaining positions after a one\-step lookahead, one might expect it to coincide withLEDFlowat window11\. It does not\. On the6\.46\.4M denoiser, evaluated along a sharedLEDFlowtrajectory, the two select the same position in only21\.8%21\.8\\%of steps on Nikoli and33\.6%33\.6\\%on Sudoku\-Extreme, every puzzle diverges at least once, and the final grids agree on80%80\\%of Nikoli puzzles and24%24\\%of Sudoku\-Extreme puzzles\. Where they diverge,LEDFlowis at least as accurate \(0\.9000\.900against0\.8300\.830on Nikoli;0\.2960\.296against0\.2840\.284on Sudoku\-Extreme\)\. The downstream\-entropy term therefore reorders selections even at the smallest window, and Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(c\) measures how that reordering degrades as the window grows\.

The fixed\-state window experiment underlying Figure[4](https://arxiv.org/html/2609.25131#S4.F4)\(c\) varies only the number of unresolved\-position entropies aggregated by Info\-Gain\. The mean absolute score error increases from0\.170\.17with one term to1\.761\.76under full aggregation, while absorption accuracy peaks at window88and then falls to0\.310\.31\. Because window88is the maximum of that curve, we use it as the Info\-Gain configuration in every accuracy and runtime comparison reported in the main text, so the baseline is never evaluated at a setting we have shown to be suboptimal\. This directly complements the runtime comparison in panel \(d\): local entropy selection avoids both the accumulated counterfactual estimates and their additional denoiser evaluations\.

#### E\.7\.1Wall\-clock comparison

Table 26:Sudoku efficiency \(6\.4M denoiser, Nikolin=100n\{=\}100, mean55\.755\.7blanks/puzzle, five seeds,1×1\\timesH100\)\. The single\-positionLEDFlowpolicy evaluates the denoiser once per absorbed position\. The0\.900±0\.0000\.900\{\\pm\}0\.000entry has zero seed variance because argmax absorption is deterministic on this fixed set; Table[21](https://arxiv.org/html/2609.25131#A5.T21)reports puzzle\-level bootstrap intervals and paired McNemar tests\.samplerdenoiser evals\.ms/puzzlesolve\-acc \(mean±\\pm95% CI\)LEDFlow\(single\-position absorption\)55\.710\.50\.900±0\.000\\mathbf\{0\.900\\pm 0\.000\}Euler@16163\.60\.238±0\.0140\.238\\pm 0\.014Euler@32325\.70\.290±0\.0380\.290\\pm 0\.038Euler@64649\.60\.344±0\.0450\.344\\pm 0\.045Euler@12812817\.50\.334±0\.0220\.334\\pm 0\.022Euler@25625633\.30\.350±0\.0300\.350\\pm 0\.030

Table 27:Profiled wall\-clock \(generated puzzles,n=200n\{=\}200, mean∼52\{\\sim\}52blanks,1×1\\timesH100; ms/puzzle includes each policy’s score computation and denoiser reevaluation\)\.LEDFlow’s accuracy advantage over the non\-absorbing baselines grows with denoiser quality \(to0\.9700\.970at21\.321\.3M\), while uniformization remains less accurate at comparable wall\-clock time\.6\.46\.4M21\.321\.3MsampleraccmsaccmsLEDFlow\(single\-position absorption\)0\.9255\.70\.9709\.7Euler@64 / @2560\.520/0\.5608\.2/28\.20\.830/0\.81012\.8/46\.1uniformization \(exact, unord\.\)0\.4257\.30\.75011\.3For GenEval atK=16K\{=\}16, Flow/Euler,LEDFlow, probability\-margin sampling, and Info\-Gain require2\.962\.96,2\.962\.96,2\.982\.98, and21\.2121\.21seconds per image and rank, respectively; Info\-Gain is therefore7\.2×7\.2\\timesslower thanLEDFlow\.

#### E\.7\.2Absorption budget, block size, and the joint term

Two ablations address the block sizebbthat Theorem[1](https://arxiv.org/html/2609.25131#Thmtheorem1)leaves free and the joint term of Proposition[2](https://arxiv.org/html/2609.25131#Thmproposition2)that it does not optimize\. Both use the6\.46\.4M denoiser, entropy ordering, and argmax absorption, and vary only the absorption schedule\.

##### Schedule shape versus block size\.

Table[28](https://arxiv.org/html/2609.25131#A5.T28)compares single\-position absorption with cosine and linear budgets and with fixed blocks of22,44, and88positions\. The table reports identical outcomes for the cosine, linear, and single\-position settings\. Having fewer generated cells than grid points does not by itself make these schedules identical\. Without verified event\-count traces for the nominal cosine and linear settings, their identical outcomes do not establish schedule insensitivity\. Fixed blocks cut denoiser evaluations from5858to88but lower solve accuracy from0\.9000\.900to0\.7200\.720on Nikoli and from0\.9280\.928to0\.8300\.830on generated puzzles, and the fraction of absorption events that absorb a wrong value rises monotonically withbb\. Absorbing several positions at once from independent marginals ignores their dependence, which is theb\>1b\{\>\}1penalty the decomposition predicts\.

Table 28:Absorption budget and block size\(Sudoku,6\.46\.4M denoiser, entropy ordering\)\. Each cell reports puzzle solve accuracy / cell accuracy / fraction of incorrect absorptions, with the number of denoiser evaluations per puzzle\. The reported cosine, linear, and single\-position outcomes coincide; this does not imply that their nominal schedules are identical\. Larger blocks trade accuracy for fewer evaluations\.ScheduleNikoli \(n=100n\{=\}100\)Generated \(n=500n\{=\}500\)solve / cell / badNFEsolve / cell / badNFESingle \(b=1b\{=\}1\)0\.9000\.900/0\.9360\.936/0\.0650\.06558580\.9280\.928/0\.9690\.969/0\.0330\.0335959Cosine0\.9000\.900/0\.9360\.936/0\.0650\.06558580\.9280\.928/0\.9690\.969/0\.0330\.0335959Linear0\.9000\.900/0\.9360\.936/0\.0650\.06558580\.9280\.928/0\.9690\.969/0\.0330\.0335959\[2pt/1pt\] Fixedb=2b\{=\}20\.8800\.880/0\.9220\.922/0\.0790\.07929290\.9180\.918/0\.9660\.966/0\.0360\.0363030Fixedb=4b\{=\}40\.8500\.850/0\.9180\.918/0\.0830\.08315150\.8920\.892/0\.9590\.959/0\.0430\.0431515Fixedb=8b\{=\}80\.7200\.720/0\.8750\.875/0\.1270\.127880\.8300\.830/0\.9410\.941/0\.0610\.06188
##### Isolating the joint term on multi\-solution puzzles\.

On unique\-solution puzzles the oracle conditional is a point mass and the total correlation of any block is zero \(Lemma[2](https://arxiv.org/html/2609.25131#Thmlemma2)\), so the loss in Table[28](https://arxiv.org/html/2609.25131#A5.T28)mixes the joint term with the denoiser’s own error\. To isolate the joint term we build puzzles with2222,2626, or3030givens by blanking complete grids, which at these densities are almost all multi\-solution, and score grid validity \(every row, column, and box a permutation of11–99\) rather than agreement with one reference solution\. Table[29](https://arxiv.org/html/2609.25131#A5.T29)shows that single\-position absorption stays valid on essentially every puzzle, because re\-conditioning after each absorption collapses the remaining ambiguity onto one consistent branch, whereas absorbing a block of mutually dependent cells from independent marginals mixes digits from different solution branches and breaks the constraints: validity falls to0\.510\.51–0\.710\.71atb=4b\{=\}4and0\.210\.21–0\.460\.46atb=8b\{=\}8, and worsens as puzzles become more under\-determined\. On the few unique\-solution puzzles in the same sweep, block absorption is perfectly valid at every block size\. The penalty therefore appears only when the jointly absorbed cells are dependent, which is the total\-correlation term itself\.

Table 29:The joint term isolated\(Sudoku,6\.46\.4M denoiser, entropy ordering,n=150n\{=\}150puzzles per givens level\)\. Fraction of valid final grids on multi\-solution puzzles as a function of block size; the last column reports the unique\-solution puzzles in the same sweep\.GivensMulti\-solutionSingle \(b=1b\{=\}1\)Blockb=4b\{=\}4Blockb=8b\{=\}8Unique, anybb2222150/150150/1501\.0001\.0000\.5070\.5070\.2670\.267—2626150/150150/1501\.0001\.0000\.6670\.6670\.2070\.207—3030147/150147/1500\.9930\.9930\.7080\.7080\.4600\.4601\.0001\.000

#### E\.7\.3Comparison with entropy\-bounded block unmasking

EB\-Sampler\[[Ben\-Hamu et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib39)\]absorbs, at each step, the lowest\-entropy position together with any further positions whose cumulative entropy stays under a budgetϵ\\epsilon; atϵ=0\\epsilon\{=\}0it reduces to single\-positionLEDFlow\. We reimplement it on the puzzle tasks with the same denoiser, argmax absorption, and ordering \(Table[30](https://arxiv.org/html/2609.25131#A5.T30)\)\. Atϵ=0\.5\\epsilon\{=\}0\.5it cuts denoiser evaluations about thirteen\-fold \(55\.755\.7to4\.24\.2on Nikoli\) and loses about fourteen points of solve accuracy \(0\.9000\.900to0\.7600\.760;0\.9280\.928to0\.7920\.792on generated puzzles\), and both the accuracy loss and the incorrect\-absorption fraction grow monotonically withϵ\\epsilon\. EB\-Sampler’s adaptive batching is slightly more evaluation\-efficient than fixed blocks at matched accuracy, but it sits on the same frontier as Table[28](https://arxiv.org/html/2609.25131#A5.T28): fewer steps are bought by absorbing dependent positions together, and single\-positionLEDFlowoccupies the accuracy\-optimal end\.

Table 30:EB\-Sampler reimplementation on Sudoku\(6\.46\.4M denoiser\)\. Cells report solve accuracy / cell accuracy / fraction of incorrect absorptions and the number of denoiser evaluations per puzzle\.ϵ\\epsilonis the cumulative\-entropy budget in nats;ϵ=0\\epsilon\{=\}0isLEDFlow\.Samplerϵ\\epsilonNikoli \(n=100n\{=\}100\)Generated \(n=500n\{=\}500\)solve / cell / badNFEsolve / cell / badNFELEDFlow\(single\)000\.9000\.900/0\.9360\.936/0\.0650\.06555\.755\.70\.9280\.928/0\.9690\.969/0\.0330\.03353\.253\.2\[2pt/1pt\] EB\-Sampler0\.50\.50\.7600\.760/0\.8980\.898/0\.1040\.1044\.24\.20\.7920\.792/0\.9320\.932/0\.0700\.0703\.73\.7EB\-Sampler1\.01\.00\.6700\.670/0\.8710\.871/0\.1300\.1303\.63\.60\.6640\.664/0\.9030\.903/0\.1000\.1003\.23\.2EB\-Sampler2\.02\.00\.5200\.520/0\.8290\.829/0\.1730\.1733\.03\.00\.5280\.528/0\.8790\.879/0\.1240\.1242\.72\.7EB\-Sampler4\.04\.00\.2700\.270/0\.7910\.791/0\.2110\.2112\.62\.60\.3340\.334/0\.8460\.846/0\.1580\.1582\.32\.3
#### E\.7\.4Absorb\-then\-revise

Irreversibility is the cost of absorption, so we test whether restoring a limited form of revision helps\. The reopen variant lets an absorbed position return to the active set if the denoiser’s posterior mass on its absorbed value later falls below0\.50\.5, capped at10%10\\%of the generated cells per puzzle, after which it is re\-absorbed by the same entropy rule\. Table[31](https://arxiv.org/html/2609.25131#A5.T31)shows that the trigger fires on about five cells per puzzle, yet solve accuracy, cell accuracy, and the incorrect\-absorption fraction are unchanged to three decimals: the reopened cells re\-absorb the same values\. Entropy\-guided absorption already absorbs at the moment the denoiser is confident, so there is nothing for revision to repair, and moving back toward revisability does not recover the accuracy that the native flow loses\. The value lies in when the absorption is made rather than in the option to undo it\.

Table 31:Absorb\-then\-revise\(Sudoku,6\.46\.4M denoiser, entropy ordering\)\. Reopening an absorbed cell when its posterior mass drops below0\.50\.5, capped at10%10\\%of cells, leaves every metric unchanged\.DatasetArmSolveCellBad\-absorptionReopens / puzzleNikoliLEDFlow0\.9000\.9000\.9360\.9360\.0650\.0650\.000\.00NikoliLEDFlow\+ reopen0\.9000\.9000\.9360\.9360\.0650\.0654\.984\.98GeneratedLEDFlow0\.9280\.9280\.9690\.9690\.0330\.0330\.000\.00GeneratedLEDFlow\+ reopen0\.9280\.9280\.9690\.9690\.0330\.0334\.834\.83

## Appendix FExtended related work

##### Discrete flow simulation and numerical correction\.

Discrete flow models generate categorical data through continuous\-time probability paths and associated CTMC velocities\[[Campbell et al\., 2024](https://arxiv.org/html/2609.25131#bib.bib1),[Shaul et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib3)\]\. Euler and tau\-leaping approximate these dynamics on a finite grid, while corrected samplers reduce temporal or spatial discretization error and uniformization simulates the learned CTMC more directly\[[Wan et al\., 2026c](https://arxiv.org/html/2609.25131#bib.bib13),[Wan et al\., 2026d](https://arxiv.org/html/2609.25131#bib.bib33)\]\. These methods improve how the native transition process is integrated\. In contrast,LEDFlowaugments inference with an irreversible absorption decision while leaving the learned posterior, base probability path, and native update on active positions unchanged\. The two design axes are complementary: a corrected update may be applied to active positions before the next absorption decision\.

##### Confidence\-guided absorption and revision\.

Confidence\-based easy\-first selection has a long history in iterative masked generation\. MaskGIT fixes high\-confidence image tokens over successive iterations, while later methods learn planners, use model confidence, or reconsider uncertain predictions\[[Chang et al\., 2022](https://arxiv.org/html/2609.25131#bib.bib32),[Kim et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib15),[Kwok et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib41),[Ye et al\., 2025a](https://arxiv.org/html/2609.25131#bib.bib17),[Peng et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib23),[Hong et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib19),[Xu et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib20),[Ouyang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib35)\]\. Info\-Gain instead uses counterfactual lookahead to estimate how one decision changes uncertainty over the remaining positions\[[Yang et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib36)\]\.LEDFlowplaces this policy question inside a uniform discrete flow: it absorbs the lowest\-entropy active positions and suppresses their future velocity, while all other positions continue under the native flow\. Our analysis separates the joint factorization and conditional prediction errors of this local decision, derives entropy from an upper bound on the latter, and quantifies how prediction error accumulates with the global lookahead window\.

##### Entropy\-bounded and gradient\-steered samplers\.

Two recent samplers for masked diffusion apply KL analysis to unmasking, and the gap between what they control and what irreversible absorption costs is the gap this paper closes\. EB\-Sampler\[[Ben\-Hamu et al\., 2025](https://arxiv.org/html/2609.25131#bib.bib39)\]bounds the dependence discarded within one parallel block and spends that bound on speed, enlarging the block to reach22–3×3\\timesfewer function evaluations at matched quality\. Its analysis never asks which position should be made terminal; the ordering criterion is taken from existing local proxies\. Under a revisable sampler that omission is benign, since a poor early choice can be overwritten\. Under absorption it is not, because the block budget is fixed before any ordering is chosen and therefore cannot see the error that ordering determines\. Section[3\.3](https://arxiv.org/html/2609.25131#S3.SS3)supplies exactly that missing quantity, and the two mechanisms are composable: the entropy budget bounds the block, the entropy ranking orders it\. BoE steering\[[Saini et al\., 2026](https://arxiv.org/html/2609.25131#bib.bib40)\]approximates one\-step lookahead by differentiating successor\-state entropy with respect to the injected embedding, which places it on the lookahead side of the contrast drawn in Section[3\.3](https://arxiv.org/html/2609.25131#S3.SS3), and inherits the window sensitivity that Lemma[1](https://arxiv.org/html/2609.25131#Thmlemma1)identifies\. It is also not portable to our setting: its token importance score is a first\-order expansion alongei​\(α\)=em\+α⁡\(e~i−em\)e\_\{i\}\(\\alpha\)=e\_\{m\}\+\\alpha\(\\tilde\{e\}\_\{i\}\-e\_\{m\}\), anchored at the mask embeddingeme\_\{m\}, and a uniform discrete flow has no absorbing mask symbol to supply that basepoint\. Reporting a number for it would require inventing one and would measure our surrogate rather than the published method, so we do not\. Its authors additionally scope the underlying relaxation away from “symbolic reasoning with brittle constraints,” the regime in which our largest gains occur\. Neither method has released code at the time of writing\.

##### Guidance, training, and benchmark scope\.

Guidance methods target reward\- or evidence\-weighted distributions, and reinforcement\-learning methods fine\-tune the denoiser for downstream rewards\[[Wan et al\., 2026b](https://arxiv.org/html/2609.25131#bib.bib34),[Wan et al\., 2026a](https://arxiv.org/html/2609.25131#bib.bib37)\]\. These approaches alter the target, learned model, or training objective;LEDFlowis a training\-free inference policy\. Standard multimodal\-understanding benchmarks often require short answers, leaving limited output structure for an absorption policy to exploit\. We therefore evaluate across a spectrum: understanding measures behavior on weakly constrained outputs, whereas text\-to\-image generation and structured reasoning test outputs with stronger dependencies \(Tables[10](https://arxiv.org/html/2609.25131#A5.T10),[3](https://arxiv.org/html/2609.25131#S4.T3), and[3](https://arxiv.org/html/2609.25131#S4.T3)\)\. Appendix[G](https://arxiv.org/html/2609.25131#A7)further explores a preliminary external grounding signal as an alternative to the denoiser’s online entropy score; it is separate from the core method and accuracy claims\.

## Appendix GExternal evidence signals for offline absorption

*The main experiments use predictive entropy recomputed from the denoiser at each grid point\. This appendix presents a complementary construction in which a frozen external teacher supplies a fixed score before sampling\. It illustrates how the selective\-absorption mechanism can accept information outside the denoiser; it is not required byLEDFlowor by the main accuracy claims\.*

### G\.1Patch grounding field

We first test whether a pretrained multimodal discrete flow model contains a spatial signal that an external score could exploit\. LetAt\(ℓ,h\),θ=softmax⁡\(Qt\(ℓ,h\),θ​\(Kt\(ℓ,h\),θ\)⊤/dh\)A\_\{t\}^\{\(\\ell,h\),\\theta\}=\\operatorname\{softmax\}\(Q\_\{t\}^\{\(\\ell,h\),\\theta\}\(K\_\{t\}^\{\(\\ell,h\),\\theta\}\)^\{\\top\}/\\sqrt\{d\_\{h\}\}\)denote question\-to\-image attention, withAj,i,t\(ℓ,h\),θA\_\{j,i,t\}^\{\(\\ell,h\),\\theta\}the mass from query tokenj∈𝒬j\\in\\mathcal\{Q\}to image patchi∈𝒫i\\in\\mathcal\{P\}\. We aggregate heads and query tokens as

pj,tθ​\(i\)=∑ℓ,hπℓ​h​Aj,i,t\(ℓ,h\),θ,p¯t​\(i\)=∑j∈𝒬uj​pj,tθ​\(i\),p\_\{j,t\}^\{\\theta\}\(i\)=\\sum\_\{\\ell,h\}\\pi\_\{\\ell h\}A\_\{j,i,t\}^\{\(\\ell,h\),\\theta\},\\qquad\\bar\{p\}\_\{t\}\(i\)=\\sum\_\{j\\in\\mathcal\{Q\}\}u\_\{j\}p\_\{j,t\}^\{\\theta\}\(i\),whereπℓ​h=softmaxℓ,h⁡\(π~ℓ​h\)\\pi\_\{\\ell h\}=\\operatorname\{softmax\}\_\{\\ell,h\}\(\\tilde\{\\pi\}\_\{\\ell h\}\)anduj∝exp⁡\(wu⊤​hj\)u\_\{j\}\\propto\\exp\(w\_\{u\}^\{\\top\}h\_\{j\}\)\. Given an annotated regionB⋆⊆𝒫B^\{\\star\}\\subseteq\\mathcal\{P\}, the normalized teacher fieldρ⋆\(i\)∝𝟏\[i∈B⋆\]\\rho^\{\\star\}\(i\)\\propto\\mathbf\{1\}\[i\\in B^\{\\star\}\]supervises this field throughℒgnd=DKL\(ρ⋆∥p¯t\)\\mathcal\{L\}\_\{\\mathrm\{gnd\}\}=D\_\{\\mathrm\{KL\}\}\(\\rho^\{\\star\}\\\|\\bar\{p\}\_\{t\}\)\. A frozen\-model pilot finds that question\-to\-image attention localizes the annotated region above chance, and fixed top\-kkhead selection improves AUROC over uniform pooling\. These observations motivate using grounded evidence as an external priority signal, without making it part of the core denoiser or sampler state\.

### G\.2External offline score

For generated positionjj, an external teacherℚϕ\\mathbb\{Q\}\_\{\\phi\}can estimate how strongly the grounded region determines its target token\. With teacher alignment scorefϕf\_\{\\phi\}and a contrast set of regions or images𝒩\\mathcal\{N\}, define

sjNCE=log⁡exp⁡\(fϕ​\(a1j,B⋆\)/τ\)1\|𝒩\|​∑B′∈𝒩exp⁡\(fϕ​\(a1j,B′\)/τ\)\.s\_\{j\}^\{\\mathrm\{NCE\}\}=\\log\\frac\{\\exp\(f\_\{\\phi\}\(a\_\{1\}^\{j\},B^\{\\star\}\)/\\tau\)\}\{\\frac\{1\}\{\|\\mathcal\{N\}\|\}\\sum\_\{B^\{\\prime\}\\in\\mathcal\{N\}\}\\exp\(f\_\{\\phi\}\(a\_\{1\}^\{j\},B^\{\\prime\}\)/\\tau\)\}\.This InfoNCE\-style quantity estimates token–region dependence and is computed once with the teacher frozen\. A pointwise erasure alternative uses the positive confidence drop

sjerase=\[log⁡ℚϕ​\(a1j∣I,Q\)−log⁡ℚϕ​\(a1j∣I∖B⋆,Q\)\]\+\.s\_\{j\}^\{\\mathrm\{erase\}\}=\\left\[\\log\\mathbb\{Q\}\_\{\\phi\}\(a\_\{1\}^\{j\}\\mid I,Q\)\-\\log\\mathbb\{Q\}\_\{\\phi\}\(a\_\{1\}^\{j\}\\mid I\_\{\\setminus B^\{\\star\}\},Q\)\\right\]\_\{\+\}\.After normalization, either score can replace predictive entropy as the ranking statistic in the otherwise unchanged absorption mechanism of Section[3\.2](https://arxiv.org/html/2609.25131#S3.SS2)\. For an absorption budgetbkb\_\{k\}, the externally guided block is

Bkext∈arg⁡max⁡∑j∈BB⊆𝒰k\|B\|=bk⁡sjext\.B\_\{k\}^\{\\mathrm\{ext\}\}\\in\\arg\\max\_\{\\begin\{subarray\}\{c\}B\\subseteq\\mathcal\{U\}\_\{k\}\\\\ \|B\|=b\_\{k\}\\end\{subarray\}\}\\sum\_\{j\\in B\}s\_\{j\}^\{\\mathrm\{ext\}\}\.The block uses the same value update, absorption rule, and native uniform\-flow transition as Eq\. \([3](https://arxiv.org/html/2609.25131#S3.E3)\); only the priority score differs\. This makes the construction an offline alternative to the online entropy policy rather than a separate probability path\.

### G\.3Preliminary controlled grounding result

We test whether the grounding field can be learned independently of answer accuracy\. Two FUDOKI runs use identical LoRA adapters, data, and optimization, differing only inλgnd∈\{0,0\.5\}\\lambda\_\{\\mathrm\{gnd\}\}\\in\\\{0,0\.5\\\}\. We compare both with the frozen model on held\-out GQA\-val \(n=500n=500\)\.

Table 32:Controlled grounding ablation on held\-out GQA\-val\. Grounding supervision improves where the model attends, while answer accuracy remains unchanged\.field AUROCanswerrununiform poollearned gateaccuracyfrozen base \(no LoRA\)0\.5340\.5770\.418LoRA only \(λgnd=0\\lambda\_\{\\mathrm\{gnd\}\}=0\)0\.5280\.5610\.414grounded \(λgnd=0\.5\\lambda\_\{\\mathrm\{gnd\}\}=0\.5\)0\.9060\.9080\.418The matched LoRA\-only control remains near the frozen model, whereas grounding supervision raises uniform\-pooled field AUROC from0\.5340\.534to0\.9060\.906\. The learned head gate adds little beyond uniform pooling \(0\.9080\.908versus0\.9060\.906\), indicating that the gain is carried by the supervised attention rather than head selection\. Answer accuracy stays near0\.420\.42for all three runs\. We therefore make only the narrow claim that the construction provides a trainable external localization signal; evaluating whether that signal improves selective absorption is left to future work\.