Prefix-Denoising Consistency: Test-Time Verification for Diffusion Language Models

# Prefix-Denoising Consistency: Test-Time Verification for Diffusion Language Models
Source: [https://arxiv.org/html/2608.25311](https://arxiv.org/html/2608.25311)
###### Abstract

Diffusion Language Models \(DLMs\) have recently become increasingly competitive with autoregressive \(AR\) models, and even outperform them on certain tasks\. Unlike AR models, DLMs produce output through iterative denoising without a left\-to\-right order\. To further improve the performance of DLMs, we introducePDC\(*Prefix\-Denoising Consistency*\), a test\-time self\-verification method for DLMs\.PDCexploits a distinctive test\-time signal in DLMs under prefix conditioned regeneration, correct trajectories are more stable and reproducible than incorrect ones\. Concretely, given an initially generated sample,PDCsplits the sentence at an intermediate position and regenerates the remaining tokens conditioned on the fixed prefix\. Across mathematical reasoning and commonsense reasoning benchmarks,PDCconsistently improves upon the initial sample, outperforms independent generations under a computational constrained comparison, and is robust to different unmasking strategies and parameter settings\. These results highlight prefix\-conditioned regeneration as an effective DLM\-specific primitive for test\-time verification\.

Prefix\-Denoising ConsistencyPrefix conditioned inputsRegenerated outputsVoteρ=0\.1\\rho=0\.1ρ=0\.5\\rho=0\.5ρ=0\.9\\rho=0\.9a\(0\.1\)a^\{\(0\.1\)\}a\(0\.5\)a^\{\(0\.5\)\}a\(0\.9\)a^\{\(0\.9\)\}DLM regenerationmajority votinga\(0\.1\),a\(0\.5\),a\(0\.9\)\{\\scriptstyle a^\{\(0\.1\)\},\\,a^\{\(0\.5\)\},\\,a^\{\(0\.9\)\}\}unmasked positionsmasked positions

Figure 1:Overview of Prefix\-Denoising Consistency \(PDC\)\. Given an initial Diffusion Language Model output,PDCkeeps exact prefixes at ratesρ∈\{0\.1,0\.5,0\.9\}\\rho\\in\\\{0\.1,0\.5,0\.9\\\}, remasks the remaining positions within the same length\-LLoutput, and regenerates them to obtain candidate answers\. The final answer is selected by majority vote over the regenerated answers only; the initial answer is recorded for analysis but excluded from the vote\.## 1Introduction

Diffusion Language Models \(DLMs\)\([Nie et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib10);[Zhu et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib20);[Ye et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib19)\)have recently emerged as a compelling alternative to autoregressive \(AR\) Language Models\([OpenAI, 2025](https://arxiv.org/html/2608.25311#bib.bib27);[Qwen Team, 2025](https://arxiv.org/html/2608.25311#bib.bib29);[NVIDIA, 2025](https://arxiv.org/html/2608.25311#bib.bib28)\)\. Unlike AR models, which generate tokens strictly from left to right, DLMs refine outputs through iterative denoising, enabling more global revision and potentially more parallel and efficient inference\. Recent DLMs have shown competitive performance across a range of language and reasoning tasks\([Gong et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib35);[Fu et al\., 2026](https://arxiv.org/html/2608.25311#bib.bib34)\), suggesting that diffusion\-based generation can offer a promising new scaling direction beyond the standard AR paradigm\.

Compared to AR models, state\-of\-the\-art DLMs are decoded at low temperature to obtain strong pass@1 accuracy\. However, this also creates a challenge for test\-time verification: due to the low temperature, repeated samples often have limited diversity, because the DLM can follow similar denoising trajectories and return the same answer\. This makes a naive transfer of self\-consistency from AR models less direct\. In AR reasoning, self\-consistency improves accuracy by sampling multiple reasoning paths and selecting the most frequent answer\([Wang et al\., 2023](https://arxiv.org/html/2608.25311#bib.bib23)\)\. For low\-temperature DLM decoding, however, repeated full generations can result in near\-identical denoising trajectories, so agreement among samples may overstate reliability rather than provide an independent check\. Recent work\([Wang et al\., 2026a](https://arxiv.org/html/2608.25311#bib.bib15)\)addresses this issue by using temporal information \(TiF\) inside the unmasking process\.TiFvotes over answers extracted from the information of intermediate denoising steps, showing that the denoising trajectory contains a useful reasoning signal and can improve accuracy\. However,TiFexploits consistency within a single denoising trajectory: it reuses intermediate states from one run rather than generating multiple alternative reasoning trajectories\. By construction, it therefore cannot directly leverage the multi\-trajectory self\-consistency mechanism that makes majority voting effective in AR reasoning\.TiFcan exploit temporal fluctuations along that run, but it cannot branch into a new trajectory once the run becomes locked into an erroneous reasoning path\.

252550507575100100Dream\-7BGSM8KMATH\-500SVAMP252550507575100100LLaDA\-8B128256512252550507575100100LLaDA\-1\.5128256512128256512R0\.5\+R^\{\+\}\_\{0\.5\}: correct\-answer retentionR0\.5−R^\{\-\}\_\{0\.5\}: same\-wrong persistenceFigure 2:Reproduction rates underρ=0\.5\\rho=0\.5regeneration\. Green denotes correct\-answer preservationR0\.5\+R^\{\+\}\_\{0\.5\}, while red denotes same\-wrong\-answer preservationR0\.5−R^\{\-\}\_\{0\.5\}, across generation lengthsL=128,256,512L=128,256,512on three math benchmarks\. The gapR0\.5\+−R0\.5−\>0R^\{\+\}\_\{0\.5\}\-R^\{\-\}\_\{0\.5\}\>0shows that correct answers are more reproducible than wrong answers\. Full results of reproduction rates, including CSQA and SQA, are reported in Appendix Table[4](https://arxiv.org/html/2608.25311#A2.T4)\.To address this issue, we proposePDC\(*Prefix\-Denoising Consistency*\)\. Starting from a completed output, it keeps an exact prefix, remasks the remaining positions, and re\-denoises them at the same temperature\. Repeating this intervention at several keep rates \(i\.e\., amount of the prefix kept\) produces structured alternative trajectories, whose extracted answers are aggregated by majority vote\. This allows the model to revise errors in the original generation while preserving useful context from the initial solution\.

Our contributions are:

- •Prefix\-conditioned regeneration\.PDCis a test\-time scaling method that holds exact prefixes inside a fixed output window, remasks the unkept positions, and votes only over regenerated answers\. The initial answer is excluded to isolate the effect of prefix\-conditioned denoising\.
- •Initially correct answers are more reproducible\.Figure[2](https://arxiv.org/html/2608.25311#S1.F2)shows that initially correct answers are more reproducible than initially wrong answers\. This observation leads to our algorithm\.
- •Improved performance over the initial sample\.In the full diffusion setting across math and commonsense benchmarks,PDCconsistently improves over the initial sample andTiF\.
- •Compute efficiency and robustness analysis\.We comparePDCto standard self\-consistency \(i\.e\., majority voting\)\. We show that PDC outperforms majority voting with a smaller total denoising\-step budget\. We also conduct validation under different unmasking schemes and settings and observe improvements\.

## 2Prefix\-Denoising Consistency

### 2\.1Standard DLM Inference Time

DLMs generate outputs by iteratively denoising a fixed\-length sequence initialized with mask tokens\. Given a promptxx, a maximum output lengthLL, and a total number of denoising stepsTT, let𝒱≤L\\mathcal\{V\}^\{\\leq L\}denote the set of token sequences over the vocabulary𝒱\\mathcal\{V\}with length at mostLL\.

The denoising process begins with an output window consisting entirely of mask tokens:

z\(0\)\\displaystyle z^\{\(0\)\}=\[x,\[MASK\],…,\[MASK\]⏟L\],\\displaystyle=\\left\[x,\\;\\underbrace\{\\text\{\{\[MASK\]\}\},\\ldots,\\text\{\{\[MASK\]\}\}\}\_\{L\}\\right\],\(1\)
At each denoising stept=1,…,Tt=1,\\ldots,T, the DLM predicts tokens for all currently masked output positions in parallel, conditioned on the partially denoised sequencez\(t−1\)z^\{\(t\-1\)\}\. It then selects a subset of these positions to unmask according to a denoising schedule, often based on the model’s confidence\. The remaining positions stay masked, producing an updated sequencez\(t\)z^\{\(t\)\}, for example,

z\(t\)=\[x,\[MASK\],\[TOKEN\],…,\[TOKEN\],\[MASK\],…⏟L\]\.z^\{\(t\)\}=\\left\[x,\\;\\underbrace\{\\text\{\{\[MASK\]\}\},\\text\{\{\[TOKEN\]\}\},\\ldots,\\text\{\{\[TOKEN\]\}\},\\text\{\{\[MASK\]\}\},\\ldots\}\_\{L\}\\right\]\.Thus, at an intermediate denoising step, the output window contains both positions that remain masked and tokens filled in during earlier steps\. The positions are not necessarily unmasked in left\-to\-right order; instead, they may be resolved in an arbitrary order determined by the denoising schedule\. AfterTTdenoising steps, DLM obtains a fully unmasked output window, output sequencey=z\(T\)y=z^\{\(T\)\}, whose length is at most the output lengthLL\.

Algorithm 1Prefix\-Denoising Consistency \(PDC\)1:Prompt

xx, generation budget

LL, denoising budget

TT, keep rates

ℛ=\{0\.1,0\.5,0\.9\}\\mathcal\{R\}=\\\{0\.1,0\.5,0\.9\\\}, answer extractor

Extract\\mathrm\{Extract\}
2:Regeneration\-vote answer

a^\\hat\{a\}, or no\-vote

3:

y←DLM⁡\(x,L,T\)y\\leftarrow\\mathrm\{DLM\}\(x;L,T\);

ℓy←\|y\|\\ell\_\{y\}\\leftarrow\|y\|
4:Record

a\(0\)←Extract⁡\(y\)a^\{\(0\)\}\\leftarrow\\mathrm\{Extract\}\(y\)for diagnostics only

5:foreach keep rate

ρ∈ℛ\\rho\\in\\mathcal\{R\}do

6:if

ℓy=0\\ell\_\{y\}=0then

7:

Lρprefix←0L\_\{\\rho\}^\{\\text\{prefix\}\}\\leftarrow 0
8:else

9:

Lρprefix←max⁡\(1,⌊ρ​ℓy⌋\)L\_\{\\rho\}^\{\\text\{prefix\}\}\\leftarrow\\max\(1,\\lfloor\\rho\\ell\_\{y\}\\rfloor\)
10:endif

11:

Lρ←L−LρprefixL\_\{\\rho\}\\leftarrow L\-L\_\{\\rho\}^\{\\text\{prefix\}\}
12:

Tρ←LρT\_\{\\rho\}\\leftarrow L\_\{\\rho\}
13:Run the constrained DLM with output positions

1:Lρprefix1\{:\}L\_\{\\rho\}^\{\\text\{prefix\}\}fixed and

LρL\_\{\\rho\}suffix positions masked:

14:

y\(ρ\)←DLM\(x,y1:Lρprefix;Lρ,Tρ\)y^\{\(\\rho\)\}\\leftarrow\\mathrm\{DLM\}\\left\(x,y\_\{1:L\_\{\\rho\}^\{\\text\{prefix\}\}\};L\_\{\\rho\},T\_\{\\rho\}\\right\)
15:

a\(ρ\)←Extract⁡\(y\(ρ\)\)a^\{\(\\rho\)\}\\leftarrow\\mathrm\{Extract\}\(y^\{\(\\rho\)\}\)
16:endfor

17:

ℬ←\{a\(ρ\):ρ∈ℛ,a\(ρ\)≠∅\}\\mathcal\{B\}\\leftarrow\\\{a^\{\(\\rho\)\}:\\rho\\in\\mathcal\{R\},\\;a^\{\(\\rho\)\}\\neq\\varnothing\\\}
18:if

ℬ=\{\}\\mathcal\{B\}=\\\{\\\}then

19:returnno\-vote

20:endif

21:return

Maj⁡\(ℬ\)\\operatorname\{Maj\}\(\\mathcal\{B\}\), breaking ties by the fixed order

ρ=0\.1,0\.5,0\.9\\rho=0\.1,0\.5,0\.9
22:*Note:*the initial answer

a\(0\)a^\{\(0\)\}is excluded from the vote\.

Our method does not rely on the internal form of the DLM update rule\. We therefore treat the DLM as a black\-box generator that, given an inputxx, an output budgetLL, and a number of denoising stepsTT, returns a sequence

y=DLM⁡\(x,L,T\),y∈𝒱≤L\.y=\\mathrm\{DLM\}\(x;L,T\),\\qquad y\\in\\mathcal\{V\}^\{\\leq L\}\.
The output length and initial extracted answer are:

ℓy=\|y\|,a\(0\)=Extract⁡\(y\),\\ell\_\{y\}=\|y\|,\\qquad a^\{\(0\)\}=\\mathrm\{Extract\}\(y\),whereExtract\\mathrm\{Extract\}is a task\-specific canonical answer extractor andExtract⁡\(y\)=∅\\mathrm\{Extract\}\(y\)=\\varnothingdenotes a null extraction that represents a failure to parse the answer\.

In a typical diffusion decoding, the model predicts tokens for the currently masked positions in parallel, and a subset of positions is unmasked at each step according to a denoising schedule, often based on confidence\([Nie et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib10);[Zhu et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib20);[Ye et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib19)\)\.

### 2\.2Prefix\-Denoising Consistency \(PDC\)

Our proposed method,PDC\(Figure[1](https://arxiv.org/html/2608.25311#S0.F1)and Algorithm[1](https://arxiv.org/html/2608.25311#alg1)\) holds a prefix of the initial generated outputyyat a keep rateρ∈ℛ\\rho\\in\\mathcal\{R\}, masks the remaining output positions, regenerates those positions using the same DLM, and then aggregates the regenerated answers by majority vote\.

Throughout the paper, we use a fixed set of three ratesℛ=\{0\.1,0\.5,0\.9\}\\mathcal\{R\}=\\\{0\.1,0\.5,0\.9\\\}, this set covers weak, intermediate, and strong prefix conditioning while requiring only three regenerations\. Given a keep initial outputyyof lengthℓy\\ell\_\{y\}, the number of held tokens for rateρ\\rhois:

Lρprefix=⌊ρ​ℓy⌋\.L\_\{\\rho\}^\{\\text\{prefix\}\}=\\lfloor\\rho\\ell\_\{y\}\\rfloor\.The remaining regeneration length and denoising step are:

Lρ=L−Lρprefix,Tρ=Lρ\.L\_\{\\rho\}=L\-L\_\{\\rho\}^\{\\text\{prefix\}\},\\qquad T\_\{\\rho\}=L\_\{\\rho\}\.Different from Eq\. \([1](https://arxiv.org/html/2608.25311#S2.E1)\), the regeneration input is the following equation:

zρ\(0\)=\[x,y1:Lρprefix,\[MASK\],…,\[MASK\]⏟Lρ\],z^\{\(0\)\}\_\{\\rho\}=\\left\[x,\\;y\_\{1:L\_\{\\rho\}^\{\\text\{prefix\}\}\},\\;\\underbrace\{\\text\{\{\[MASK\]\}\},\\ldots,\\text\{\{\[MASK\]\}\}\}\_\{L\_\{\\rho\}\}\\right\],wherey1:Lρprefixy\_\{1:L\_\{\\rho\}^\{\\text\{prefix\}\}\}is kept output tokens with a rate ofρ\\rhoandLρL\_\{\\rho\}is the number of masked suffix positions to denoise\. Only the remainingLρL\_\{\\rho\}output positions are denoised:

y\(ρ\)=DLM\(x,y1:Lρprefix;Lρ,Tρ\),\|y\(ρ\)\|≤L\.y^\{\(\\rho\)\}=\\mathrm\{DLM\}\\left\(x,y\_\{1:L\_\{\\rho\}^\{\\text\{prefix\}\}\};L\_\{\\rho\},T\_\{\\rho\}\\right\),\\qquad\|y^\{\(\\rho\)\}\|\\leq L\.The regenerated answer is:

a\(ρ\)=Extract⁡\(y\(ρ\)\)\.a^\{\(\\rho\)\}=\\mathrm\{Extract\}\(y^\{\(\\rho\)\}\)\.Thus, the final length ofy\(ρ\)y^\{\(\\rho\)\}may differ from the initial output lengthℓy\\ell\_\{y\}\. However, regeneration does not continue from the end of the initial output\. Instead, it only refills the remasked positions within the same output window of lengthLL\.

Motivated by analogous observations for AR models\([Iwase et al\., 2026](https://arxiv.org/html/2608.25311#bib.bib32)\), we find that the correct reasoning paths are more reproducible under regeneration than incorrect ones in DLMs \(Figure[2](https://arxiv.org/html/2608.25311#S1.F2)\)\. The gold answer is denoted bya⋆a^\{\\star\}\. Reproduction of the correct answer measures whether regeneration preserves a correct initial answer, is defined as:

Rρ\+=Pr⁡\[a\(ρ\)=a⋆∣a\(0\)=a⋆\]\.R^\{\+\}\_\{\\rho\}=\\Pr\\left\[a^\{\(\\rho\)\}=a^\{\\star\}\\mid a^\{\(0\)\}=a^\{\\star\}\\right\]\.Same wrong preservation measures, whether regeneration repeats the same wrong answer when the initial answer is wrong, is defined as:

Rρ−=Pr\[a\(ρ\)=a\(0\)∣a\(0\)≠∅,a\(0\)≠a⋆\]\.R^\{\-\}\_\{\\rho\}=\\Pr\\left\[a^\{\(\\rho\)\}=a^\{\(0\)\}\\mid a^\{\(0\)\}\\neq\\varnothing,\\;a^\{\(0\)\}\\not=a^\{\\star\}\\right\]\.
###### Observation 2\.1\(Reproduction rates\)\. Across the different experiment settings, correct initial answers are more likely to be preserved than incorrect initial answers are to be repeated in DLMs:Rρ\+≥Rρ−\.R^\{\+\}\_\{\\rho\}\\geq R^\{\-\}\_\{\\rho\}\.

Using Observation[2\.1](https://arxiv.org/html/2608.25311#S2.Thmtheorem1),PDCassesses the reliability of the initial answer\.PDCvotes only over regenerated answers, initial answera\(0\)a^\{\(0\)\}is excluded from the final vote\. This isolates the effect of prefix\-conditioned regeneration: the DLMs can improve over the initial answer only if the regenerated candidates support a better answer\. The multiset of non\-null regenerated answers is:

ℬ=\{a\(ρ\):ρ∈ℛ,a\(ρ\)≠∅\}\\mathcal\{B\}=\\left\\\{a^\{\(\\rho\)\}:\\rho\\in\\mathcal\{R\},\\;a^\{\(\\rho\)\}\\neq\\varnothing\\right\\\}Ifℬ=\{\}\\mathcal\{B\}=\\\{\\\}, the example is marked as no\-vote\. Otherwise,PDCreturns the most frequent regenerated answer, denoted byMaj⁡\(ℬ\)\\operatorname\{Maj\}\(\\mathcal\{B\}\):

a^=Maj⁡\(ℬ\)\.\\hat\{a\}=\\operatorname\{Maj\}\(\\mathcal\{B\}\)\.The three keep rates probe different neighborhoods around the same initial output\. The keep rate of0\.10\.1imposes only weak conditioning on the original generation trajectory and allows most of the reasoning path to change\. The keep rate of0\.90\.9largely preserves the answer while perturbing only the final portion of the output\. The keep rate of0\.50\.5provides an intermediate between the two\. Together, these rates evaluate answer stability under varying strengths of prefix conditioning\.

##### When doesPDCimprove accuracy?

To demonstrate the effectiveness of PDC, we consider a stylized setting with only two possible answers: the correct answera⋆a^\{\\star\}and one incorrect answerbb\. The purpose of this analysis is to identify when taking a majority vote over three regenerated answers improves upon the initial answer\.

First, consider three independent Bernoulli variables \(i\.e\., variables that take either one or zero\) whose probabilities of being one arep1,p2,p3p\_\{1\},p\_\{2\},p\_\{3\}, respectively\. The probability that at least two of them are one is

V3​\(𝒑\)\\displaystyle V\_\{3\}\(\\boldsymbol\{p\}\):=Pr⁡\[at least two of the three variables are one\]\\displaystyle:=\\Pr\[\\text\{at least two of the three variables are one\}\]=p1​p2​\(1−p3\)\+p1​\(1−p2\)​p3\+\(1−p1\)​p2​p3\+p1​p2​p3\\displaystyle=p\_\{1\}p\_\{2\}\(1\-p\_\{3\}\)\+p\_\{1\}\(1\-p\_\{2\}\)p\_\{3\}\+\(1\-p\_\{1\}\)p\_\{2\}p\_\{3\}\+p\_\{1\}p\_\{2\}p\_\{3\}=p1​p2\+p1​p3\+p2​p3−2​p1​p2​p3,\\displaystyle=p\_\{1\}p\_\{2\}\+p\_\{1\}p\_\{3\}\+p\_\{2\}p\_\{3\}\-2p\_\{1\}p\_\{2\}p\_\{3\},\(2\)where𝒑=\(p1,p2,p3\)\\boldsymbol\{p\}=\(p\_\{1\},p\_\{2\},p\_\{3\}\)\.

When all three probabilities are equal topp, we write

V3​\(p\):=V3​\(p,p,p\)=3​p2−2​p3\.V\_\{3\}\(p\):=V\_\{3\}\(p,p,p\)=3p^\{2\}\-2p^\{3\}\.Thus,V3V\_\{3\}is the probability that the majority outcome is one\.

For regenerationjj, define

Rρj\+:=Pr⁡\[a\(j\)=a\(0\)∣a\(0\)=a⋆\]R^\{\+\}\_\{\\rho\_\{j\}\}:=\\Pr\[a^\{\(j\)\}=a^\{\(0\)\}\\mid a^\{\(0\)\}=a^\{\\star\}\]and

Rρj−:=Pr\[a\(j\)=a\(0\)∣a\(0\)≠∅,a\(0\)=b\]\.R^\{\-\}\_\{\\rho\_\{j\}\}:=\\Pr\[a^\{\(j\)\}=a^\{\(0\)\}\\mid a^\{\(0\)\}\\neq\\varnothing,\\;a^\{\(0\)\}=b\]\.In words,Rρj\+R^\{\+\}\_\{\\rho\_\{j\}\}is the correct\-answer retention probability, whileRρj−R^\{\-\}\_\{\\rho\_\{j\}\}is the same\-wrong\-answer persistence probability\.

###### Theorem 2\.2\(Benefit of the reproduction gap\)\. Assume a binary answer space𝒜=\{a⋆,b\}\\mathcal\{A\}=\\\{a^\{\\star\},b\\\}with no null answers, and letπ:=Pr\[a\(0\)=a⋆\]\.\\pi:=\\Pr\[a^\{\(0\)\}=a^\{\\star\}\]\.Suppose that, conditional ona\(0\)a^\{\(0\)\}, the three regenerated answers are independent\. Define𝑹±:=\(Rρ1±,Rρ2±,Rρ3±\)\.\\boldsymbol\{R\}^\{\\pm\}:=\\left\(R^\{\\pm\}\_\{\\rho\_\{1\}\},R^\{\\pm\}\_\{\\rho\_\{2\}\},R^\{\\pm\}\_\{\\rho\_\{3\}\}\\right\)\.ThenAcc⁡\(PDC\)=π​V3​\(𝑹\+\)\+\(1−π\)​\[1−V3​\(𝑹−\)\]\.\\operatorname\{Acc\}\(\\textsc\{PDC\}\{\}\)=\\pi V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\+\(1\-\\pi\)\\left\[1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\\right\]\.\(3\)In the non\-degenerate case,PDCimproves uponInitif and only ifπ<1−V3​\(𝑹−\)2−V3​\(𝑹−\)−V3​\(𝑹\+\)\.\\pi<\\frac\{1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\}\{2\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\-V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\}\.\(4\)In particular, ifπ=1/2\\pi=1/2, thenAcc⁡\(PDC\)−Acc⁡\(Init\)=12​\[V3​\(𝑹\+\)−V3​\(𝑹−\)\]\.\\operatorname\{Acc\}\(\\textsc\{PDC\}\{\}\)\-\\operatorname\{Acc\}\(\\textsc\{Init\}\{\}\)=\\frac\{1\}\{2\}\\left\[V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\\right\]\.\(5\)Consequently,V3\(𝑹\+\)\>V3\(𝑹−\)⟺Acc\(PDC\)\>Acc\(Init\)\.V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\>V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\\quad\\Longleftrightarrow\\quad\\operatorname\{Acc\}\(\\textsc\{PDC\}\{\}\)\>\\operatorname\{Acc\}\(\\textsc\{Init\}\{\}\)\.Moreover, becauseV3V\_\{3\}is increasing on each coordinate, a sufficient condition for improvement isR\+ρj≥R−ρj,j=1,2,3,R^\{\+\}\_\{\\rho\_\{j\}\}\\geq R^\{\-\}\_\{\\rho\_\{j\}\},\\qquad j=1,2,3,with strict inequality for at least onejj\.

###### Proof\.

We separate the analysis into two cases\.

##### Case 1: the initial answer is correct\.

Supposea\(0\)=a⋆a^\{\(0\)\}=a^\{\\star\}\. Regenerationjjreproduces the initial answer, which is also the correct answer, with probabilityRρj\+R^\{\+\}\_\{\\rho\_\{j\}\}\. By conditional independence, the probability that at least two of the three regenerated answers are correct is therefore

Pr⁡\[a^=a⋆∣a\(0\)=a⋆\]=V3​\(𝑹\+\)\.\\Pr\[\\hat\{a\}=a^\{\\star\}\\mid a^\{\(0\)\}=a^\{\\star\}\]=V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\.

##### Case 2: the initial answer is wrong\.

Supposea\(0\)=ba^\{\(0\)\}=b\. Because there are only two possible answers, a regeneration is correct exactly when it does not reproduce the initial answer\.

The final majority vote is wrong exactly when at least two regenerations reproducebb\. The probability of this event isV3​\(𝑹−\)V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\. Hence,

Pr⁡\[a^=a⋆∣a\(0\)=b\]=1−V3​\(𝑹−\)\.\\Pr\[\\hat\{a\}=a^\{\\star\}\\mid a^\{\(0\)\}=b\]=1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\.
The initial answer is correct with probabilityπ\\piand wrong with probability1−π1\-\\pi\. Combining the two cases gives

Acc⁡\(PDC\)=π​V3​\(𝑹\+\)\+\(1−π\)​\[1−V3​\(𝑹−\)\],\\operatorname\{Acc\}\(\\textsc\{PDC\}\{\}\)=\\pi V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\+\(1\-\\pi\)\\left\[1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\\right\],which proves Eq\. \([3](https://arxiv.org/html/2608.25311#S2.E3)\)\.

Since the accuracy of the initial answer isAcc⁡\(Init\)=π\\operatorname\{Acc\}\(\\textsc\{Init\}\{\}\)=\\pi, the change in accuracy is

Acc⁡\(PDC\)−Acc⁡\(Init\)\\displaystyle\\operatorname\{Acc\}\(\\textsc\{PDC\}\{\}\)\-\\operatorname\{Acc\}\(\\textsc\{Init\}\{\}\)=\(1−π\)​\[1−V3​\(𝑹−\)\]\\displaystyle=\(1\-\\pi\)\\left\[1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\\right\]−π⁡\[1−V3​\(𝑹\+\)\]\.\\displaystyle\\quad\-\\pi\\left\[1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\\right\]\.\(6\)
This expression has a direct interpretation:

\(1−π\)​\[1−V3​\(𝑹−\)\]⏟wrong initial answers that are corrected−π⁡\[1−V3​\(𝑹\+\)\]⏟correct initial answers that are spoiled\.\\underbrace\{\(1\-\\pi\)\\left\[1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\\right\]\}\_\{\\text\{wrong initial answers that are corrected\}\}\-\\underbrace\{\\pi\\left\[1\-V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\\right\]\}\_\{\\text\{correct initial answers that are spoiled\}\}\.Thus, the method improves accuracy precisely when the benefit from correcting initially wrong answers exceeds the loss from spoiling initially correct answers\. RearrangingAcc⁡\(PDC\)−Acc⁡\(Init\)\>0\\operatorname\{Acc\}\(\\textsc\{PDC\}\{\}\)\-\\operatorname\{Acc\}\(\\textsc\{Init\}\{\}\)\>0gives Eq\. \([4](https://arxiv.org/html/2608.25311#S2.E4)\)\. Finally, suppose thatπ=1/2\\pi=1/2\. Then

Acc⁡\(PDC\)−Acc⁡\(Init\)=12​\[V3​\(𝑹\+\)−V3​\(𝑹−\)\],\\operatorname\{Acc\}\(\\textsc\{PDC\}\{\}\)\-\\operatorname\{Acc\}\(\\textsc\{Init\}\{\}\)=\\frac\{1\}\{2\}\\left\[V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\-V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\)\\right\],which gives Eq\. \([5](https://arxiv.org/html/2608.25311#S2.E5)\) and the stated equivalence\. It remains to establish the sufficient condition\. From Eq\. \([2](https://arxiv.org/html/2608.25311#S2.E2)\),

∂V3∂p1\\displaystyle\\frac\{\\partial V\_\{3\}\}\{\\partial p\_\{1\}\}=p2\+p3−2​p2​p3\\displaystyle=p\_\{2\}\+p\_\{3\}\-2p\_\{2\}p\_\{3\}=p2​\(1−p3\)\+p3​\(1−p2\)\>0\\displaystyle=p\_\{2\}\(1\-p\_\{3\}\)\+p\_\{3\}\(1\-p\_\{2\}\)\>0forp2,p3∈\(0,1\)p\_\{2\},p\_\{3\}\\in\(0,1\)\. The same argument applies to the other two coordinates, soV3V\_\{3\}is strictly increasing in each coordinate on\(0,1\)3\(0,1\)^\{3\}\. Therefore, ifRρj\+≥Rρj−R^\{\+\}\_\{\\rho\_\{j\}\}\\geq R^\{\-\}\_\{\\rho\_\{j\}\}for everyjj, with strict inequality for at least onejj, then

V3​\(𝑹\+\)\>V3​\(𝑹−\),V\_\{3\}\(\\boldsymbol\{R\}^\{\+\}\)\>V\_\{3\}\(\\boldsymbol\{R\}^\{\-\}\),∎

## 3Experiments

The experiments first examine the diagnostic signal behindPDC: whether prefix regeneration preserves initially correct answers more often than it repeats the same wrong answer\. We then evaluatePDCagainst the initial sample and temporal voting \(TiF\([Wang et al\., 2026a](https://arxiv.org/html/2608.25311#bib.bib15)\)\) baselines in full diffusion setting111In the LLaDA family, the block length controls the generation\. When the block length is11, the model reduces to an AR model; when the block length isLL, it corresponds to the full diffusion setting; and when the block length is between11andLL, it yields a semi\-AR setting\.\. In the ablation study, we comparePDCwith standard majority voting that spends more token budget than it\. Furthermore, we had several robustness checks, including several alternative unmasking strategies\.

##### Models and datasets\.

We study Dream\-7B \(Dream\-org/Dream\-v0\-Instruct\-7B,[Ye et al\. \(2025\)](https://arxiv.org/html/2608.25311#bib.bib19)\) and LLaDA\-family models \(GSAI\-ML/LLaDA\-8B\-Instruct,[Nie et al\. \(2025\)](https://arxiv.org/html/2608.25311#bib.bib10);GSAI\-ML/LLaDA\-1\.5,[Zhu et al\. \(2025\)](https://arxiv.org/html/2608.25311#bib.bib20)\)\. The main results use the full\-diffusion setting\. For LLaDA\-family models, we set the block length equal to the generation length\. All models are decoded at temperature0\.20\.2\. The datasets we tested are GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2608.25311#bib.bib3)\), MATH\-500\([Hendrycks et al\., 2021](https://arxiv.org/html/2608.25311#bib.bib5);[Lightman et al\., 2024](https://arxiv.org/html/2608.25311#bib.bib26)\), SVAMP\([Patel et al\., 2021](https://arxiv.org/html/2608.25311#bib.bib11)\), CSQA \(CommonsenseQA,[Talmor et al\. \(2019\)](https://arxiv.org/html/2608.25311#bib.bib21)\), and SQA \(StrategyQA,[Geva et al\. \(2021\)](https://arxiv.org/html/2608.25311#bib.bib33)\)\.

Table 1:For each pair, we select the generation length that yields the highestInitaccuracy; this selection is based only onInitand is not optimized forPDC\.PDCattains the best or tied\-best accuracy in all 15 settings and improves over the selectedInitbaseline in 14 of them, with gains of up to\+5\.68\+5\.68accuracy points\. The full result is reported in Appendix[C](https://arxiv.org/html/2608.25311#A3)\. Signed green/red values denote the advantage of the method relative to the selectedInitbaseline\. Bold indicates the best accuracy in each row, and an underline indicates the second best accuracy, with ties marked together\.ModelDatasetLengthInitTiFFixedTiFLinearTiFExp\.PDCΔ\\DeltaTiFavg\.Δ\\DeltaPDCDream\-7BGSM8K51282\.0381\.9682\.0382\.0384\.08\-0\.02\+2\.05MATH\-50051246\.0046\.0046\.0046\.2047\.00\+0\.07\+1\.00SVAMP25687\.0087\.0087\.0087\.0088\.00\+0\.00\+1\.00CSQA12873\.2273\.7973\.9673\.5574\.20\+0\.55\+0\.98SQA12870\.4570\.7470\.7470\.7471\.76\+0\.29\+1\.31LLaDA\-8BGSM8K25660\.7360\.1260\.8060\.8065\.88\-0\.15\+5\.15MATH\-50025626\.4526\.2526\.2526\.2527\.86\-0\.20\+1\.41SVAMP12883\.6783\.6783\.6783\.6784\.33\+0\.00\+0\.66CSQA12880\.5180\.5180\.5180\.5180\.59\+0\.00\+0\.08SQA12865\.0765\.0765\.0765\.0766\.67\+0\.00\+1\.60LLaDA\-1\.5GSM8K25660\.5860\.0560\.2760\.5066\.26\-0\.30\+5\.68MATH\-50025625\.6525\.6525\.4525\.6526\.65\-0\.07\+1\.00SVAMP12883\.6783\.3383\.6783\.6784\.67\-0\.11\+1\.00CSQA12880\.1080\.1080\.1080\.1080\.10\+0\.00\+0\.00SQA12866\.0866\.0866\.0866\.0866\.81\+0\.00\+0\.73

### 3\.1Correct Answers Are More Reproducible Than Wrong Answers

We first examine the reproduction rates that motivatePDC\. Figure[2](https://arxiv.org/html/2608.25311#S1.F2)shows reproduction rates at keep rate0\.50\.5; the results for keep rates0\.10\.1or0\.90\.9across all datasets are reported in Appendix Table[4](https://arxiv.org/html/2608.25311#A2.T4)\. The figure shows a significant gap between correct\-answer retention and the same wrong answer persistence: initially correct answers are typically reproduced under prefix\-conditioned regeneration, whereas initially, wrong answers are less likely to be regenerated as the same wrong answer\.

### 3\.2PDCImproves the Initial Sample

We next examine whether PDC actually improves the accuracy of the answer\. This is non\-trivial: regenerations can move away from an initial wrong answer, but the new answers may still be wrong, and the final vote may still result in an incorrect answer\.

##### Baselines\.

Initindicates the initial answeryygenerated by the standard denoising process\.PDCvotes over 0\.1, 0\.5, and 0\.9 keep\-rate regenerations\. Ties are broken by the fixed keep\-rate order0\.1,0\.5,0\.90\.1,0\.5,0\.9\.TiF, the state\-of\-the\-art method for improving DLLM accuracy, extracts answers from intermediate steps\. To aggregate answers on these steps,TiFLinear andTiFExp use linearly and exponentially increasing temporal weights, respectively\. The exponential setting usesα=5\\alpha=5, following the settings of temporal voting in DLMs\([Wang et al\., 2026a](https://arxiv.org/html/2608.25311#bib.bib15)\)\.

##### Results

Table[1](https://arxiv.org/html/2608.25311#S3.T1)compares the accuracy of the methods\. For each model and dataset pair, we optimize the generation lengthLLto achieve the highestInitaccuracy\. This comparesPDCagainst the strongest available initial sample baseline for that pair, rather than against a favorable length chosen post hoc forPDC\. Results across several values ofLLare reported in Appendix[C](https://arxiv.org/html/2608.25311#A3)\.

On Dream\-7B,PDCconsistently improves overIniton all five datasets\. The advantage ofPDCoverInitranges from \+0\.98 to \+2\.05 points, with an unweighted mean gain of \+1\.27 points\. On LLaDA\-family models,PDCimproves overInitin nine of ten model–dataset pairs and ties in the remaining pair, with an unweighted mean gain of \+1\.73 points\. Appendix[D](https://arxiv.org/html/2608.25311#A4)reports additional LLaDA\-family runs with block length3232, a semi\-AR setting\. It shows that the same prefix\-regeneration signal remains useful\.

### 3\.3Ablation Study

#### 3\.3\.1PDCOutperforms Self\-Consistency at Lower Inference Cost

We comparePDCwith Self\-Consistency at generation lengthL=128L=128\. TheInitx4 baseline conducts a majority vote over four independent generations and has a total denoising\-step budget of4​T4T\. This budget is approximately2\.5​T2\.5Twhen the initial output fills the length\-LLwindow, can exceed3​T3Tfor shorter outputs, and remains below4​T4Tfor every nonempty initial output\. Thus,Initx4 is a conservative higher\-budget baseline rather than an exactly cost\-matched baseline\. Improvements overInitx4 suggest that the method is using regeneration more effectively rather than relying merely on self\-consistency\. Table[2](https://arxiv.org/html/2608.25311#S3.T2)shows the completedL=128L=128runs\. In this setting,PDCimproves overInitx4 in 13 of the 15 completed model and dataset pairs, ties in one pair, and underperforms in one pair, indicating that the gains stem from a more effective regeneration strategy that utilizes the consistency of the reasoning process, rather than merely drawing additional full generations\.

Table 2:We comparePDCwith Self\-Consistency under different total denoising\-step budgets at generation lengthL=128L=128\. For bothInitandTiF,×4\\times 4denotes four independent initial denoising trajectories\.Initx4 votes over their final answers with budget4​T4T, whilePDCuses one initial generation followed by three regenerations at keep rates0\.10\.1,0\.50\.5, and0\.90\.9\.PDCachieves the best or tied\-best accuracy in 14 of 15 model–dataset settings and improves overInitx4 in 13 of them, with gains of up to\+9\.25\+9\.25accuracy points\.DatasetModelInitInitx4TiFFixed x4TiFLinear x4TiFExp\. x4PDCΔ\\Deltavs\.Initx4GSM8KDream\-7B64\.0664\.2262\.4064\.5265\.8166\.03\+1\.81LLaDA\-1\.556\.4856\.6356\.4156\.5656\.7165\.88\+9\.25LLaDA\-8B58\.3058\.9158\.3858\.6158\.9165\.88\+6\.97MATH\-500Dream\-7B35\.8036\.0034\.2034\.8035\.6037\.20\+1\.20LLaDA\-1\.523\.4023\.2023\.4023\.4023\.0026\.40\+3\.20LLaDA\-8B23\.8023\.6023\.8023\.8023\.6025\.40\+1\.80SVAMPDream\-7B83\.0083\.0084\.0083\.6784\.0084\.33\+1\.33LLaDA\-1\.583\.6783\.6783\.3383\.6783\.6784\.67\+1\.00LLaDA\-8B83\.6784\.3384\.3384\.0084\.3384\.33\+0\.00CSQADream\-7B73\.2273\.2273\.7973\.9673\.5574\.20\+0\.98LLaDA\-1\.580\.1080\.5180\.6780\.6780\.6780\.10\-0\.41LLaDA\-8B80\.5180\.3480\.5980\.5980\.5980\.59\+0\.25SQADream\-7B70\.4570\.4570\.7470\.7470\.7471\.76\+1\.31LLaDA\-1\.566\.0865\.9465\.9465\.7965\.7966\.81\+0\.87LLaDA\-8B65\.0764\.6365\.0765\.0765\.0766\.67\+2\.04

#### 3\.3\.2Robustness

We now conduct robustness checks\. First, we tested several variants of the unmasking strategy\. Up to this point, the results have used entropy unmasking for Dream and low\-confidence unmasking for LLaDA\-family models; Table[3](https://arxiv.org/html/2608.25311#S3.T3)repeats the MATH\-500 evaluation with alternative unmasking rules, including random and origin unmasking\. Table[3](https://arxiv.org/html/2608.25311#S3.T3)shows that larger keep rates also retain more initially correct answers, while smaller keep rates more often break exact repetition of the initial wrong answer\. Accuracy remains comparable to or above the corresponding initial generation in these settings\.

In addition, we conduct two other robustness checks\. Appendix[D](https://arxiv.org/html/2608.25311#A4)reports LLaDA\-family experiments with a block length of 32, corresponding to semi\-AR inference\. The results show thatPDCremains effective under this semi\-AR setting\. Moreover, we test whether the advantage ofPDCis specific to the low\-temperature decoding regime by repeating theL=128L=128comparison under more stochastic decoding atτ=1\.0\\tau=1\.0\. At this temperature,PDCattains the highest accuracy in 12 of the 15 model and dataset settings and exceedsInit×4\\times 4by4\.984\.98accuracy points on average \(see Appendix[G](https://arxiv.org/html/2608.25311#A7)\)\. These runs are not used for the main full\-diffusion claims, but they check whether the prefix\-conditioning signal also appears under a different inference time schedule\. Together, these experiments do not makePDCindependent of the DLM, but they indicate that the signal is not specific to one unmasking heuristic or one block schedule\.

Table 3:Robustness to alternative unmasking rules on MATH\-500 across models and generation lengths\. Dream uses origin unmasking, while the LLaDA\-family models use random remasking\. All rows use the full\-diffusion setting with block length equal to generation length\. We show answer accuracy for the initial generation,PDCwith keep ratesρ∈\{0\.1,0\.5,0\.9\}\\rho\\in\\\{0\.1,0\.5,0\.9\\\}, andTiFvariants\.PDCachieves the best accuracy in all models and length settings\.ModelLengthAccuracy \(%\)Reproduction rates \(%\)TiFvariantsρ=0\.1\\rho=0\.1ρ=0\.5\\rho=0\.5ρ=0\.9\\rho=0\.9InitPDCFixedLinearExp\.Rρ\+R^\{\+\}\_\{\\rho\}Rρ−R^\{\-\}\_\{\\rho\}Rρ\+R^\{\+\}\_\{\\rho\}Rρ−R^\{\-\}\_\{\\rho\}Rρ\+R^\{\+\}\_\{\\rho\}Rρ−R^\{\-\}\_\{\\rho\}Dream\-7B12817\.6017\.6023\.4017\.6017\.6017\.8017\.6017\.6051\.1451\.1412\.6612\.6664\.7764\.7717\.5317\.5395\.4595\.4564\.6164\.6125614\.2014\.2020\.4014\.8014\.8014\.8014\.8015\.0060\.5660\.5610\.7410\.7456\.3456\.3417\.1817\.1878\.8778\.8757\.6757\.6751214\.8014\.8020\.4015\.2015\.2015\.0015\.0054\.0554\.0514\.7814\.7854\.0554\.0519\.8119\.8183\.7883\.7858\.1858\.18LLaDA\-1\.512827\.2027\.2028\.6027\.8027\.8028\.2027\.2027\.2068\.3868\.3821\.2521\.2573\.5373\.5336\.2636\.2694\.8594\.8581\.3081\.3025631\.4631\.4633\.0731\.6631\.6631\.8631\.8632\.0670\.0670\.0618\.3418\.3477\.0777\.0730\.7730\.7792\.9992\.9980\.4780\.4751231\.4531\.4536\.2931\.2531\.2532\.6632\.6632\.8671\.7971\.7911\.6811\.6873\.0873\.0823\.6523\.6590\.3890\.3873\.3573\.35LLaDA\-8B12825\.2025\.2028\.0025\.0025\.0025\.4025\.4064\.2964\.2918\.3118\.3177\.7877\.7834\.3734\.3793\.6593\.6576\.3476\.3425631\.0631\.0633\.2729\.4629\.4630\.8630\.8631\.6669\.0369\.0322\.7122\.7176\.7776\.7733\.9233\.9294\.1994\.1979\.6579\.6551230\.8530\.8534\.6831\.6531\.0531\.0531\.4531\.4565\.3665\.3614\.0814\.0880\.3980\.3923\.1723\.1795\.4295\.4265\.9865\.98

## 4Conclusion

In this work, we presentedPDC\(*Prefix\-Denoising Consistency*\), a test\-time self\-verification method\.PDCexploits a distinctive consistency signal from prefix\-conditioned regeneration, correct reasoning trajectories tend to be more stable and reproducible than incorrect ones\. Across math and commonsense benchmarks,PDCconsistently improved over the initial sample, compared favorably to independent generations under a compute\-constrained setting, and remained robust across unmasking strategies and hyperparameter choices\. These results suggest that prefix\-conditioned regeneration is an effective DLM\-specific primitive for test\-time verification, and point to the broader potential of exploiting denoising\-based regeneration signals to improve the reliability of diffusion language models\.

## Acknowledgments

J\. Komiyama was supported by the MBZUAI Start\-up Fund \[BF0121\]\.

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## Appendix ARelated Work

##### Diffusion language models\.

Diffusion models for text generation succeeded by[Austin et al\. \(2021\)](https://arxiv.org/html/2608.25311#bib.bib36), advanced through the masked token framework\. Recent DLM work has improved both the probabilistic formulation and the scale of masked/discrete diffusion\. Score\-entropy discrete diffusion, masked diffusion language modeling, and simplified masked diffusion objectives improve training and likelihood modeling for token sequences\([Lou et al\., 2024](https://arxiv.org/html/2608.25311#bib.bib8);[Sahoo et al\., 2024](https://arxiv.org/html/2608.25311#bib.bib12);[Shi et al\., 2024](https://arxiv.org/html/2608.25311#bib.bib13)\)\. At a larger scale, LLaDA shows that a masked diffusion model can be trained from scratch and instruction\-tuned as a large language model\([Nie et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib10)\), while LLaDA 1\.5 studies preference optimization for such models\([Zhu et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib20)\)\. Dream further demonstrates a strong open diffusion LLM with parallel iterative refinement and flexible generation orders\([Ye et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib19)\)\. Our work is complementary to these models and training advances: given a base model, we study how its conditional denoising behavior can be used at inference time\.

##### Blockwise and semi\-autoregressive diffusion inference time\.

Several DLMs introduce left\-to\-right or blockwise structure to improve length flexibility and efficiency\. SSD\-LM and AR\-Diffusion use semi\-autoregressive or position\-dependent denoising to combine diffusion with sequential dependencies\([Han et al\., 2023](https://arxiv.org/html/2608.25311#bib.bib4);[Wu et al\., 2023](https://arxiv.org/html/2608.25311#bib.bib17)\)\. Block diffusion interpolates between autoregressive and discrete diffusion models, enabling arbitrary\-length generation and KV\-cache reuse\([Arriola et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib1)\)\. Discrete diffusion forcing similarly turns pretrained dLLMs into an AR\-diffusion hybrid for faster inference\([Wang et al\., 2026b](https://arxiv.org/html/2608.25311#bib.bib14)\)\. These methods modify the model, DLM, or inference time schedule\. In contrast,PDCis a black\-box test\-time procedure that probes the existing DLM by changing which parts of one completed output are held fixed\.

##### Adaptive unmasking and early termination\.

A closely related acceleration direction treats DLM inference as a dynamic unmasking, token\-commitment, or stopping problem\. Fast\-dLLM selectively unmasks tokens whose confidence exceeds a threshold, while using approximate KV caching to reduce per\-step cost\([Wu et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib18)\)\. SlowFast Sampling adapts the inference time pace using token certainty, convergence, and positional structure, alternating between exploratory and accelerated phases\([Wei et al\., 2026](https://arxiv.org/html/2608.25311#bib.bib16)\)\. Learning Unmasking Policies formulates masked diffusion sampling as a Markov decision process and learns token\-unmasking decisions from model confidences\([Jazbec et al\., 2025](https://arxiv.org/html/2608.25311#bib.bib6)\)\. DAWN instead uses dependency graphs to avoid simultaneously unmasking strongly coupled uncertain tokens\([Luo et al\., 2026](https://arxiv.org/html/2608.25311#bib.bib9)\)\. Learn2PD trains a lightweight filter that predicts whether each current token prediction matches the final output, and combines this with End\-of\-Text Prediction to terminate inference after the sequence is complete\([Bao et al\., 2026](https://arxiv.org/html/2608.25311#bib.bib2)\)\. Prophet observes early answer convergence and commits the remaining tokens in one step when the top\-2 confidence gap indicates sufficient stability\([Li et al\., 2026](https://arxiv.org/html/2608.25311#bib.bib7)\)\. These methods decide when to unmask, commit, or stop in order to reduce inference cost\.PDChas a different goal: it starts from a completed sample and uses fresh prefix\-conditioned regenerations to test answer reproducibility and improve accuracy, rather than shortening the original denoising run\.

##### Self\-consistency method\.

Self\-consistency \(majority\-voting\) has been widely used as a decoding\-time strategy for improving chain\-of\-thought reasoning\. Rather than relying on a single reasoning path,[Wang et al\. \(2023\)](https://arxiv.org/html/2608.25311#bib.bib23)samples multiple reasoning traces and aggregates the final answers by majority vote\. Subsequent work has investigated how to reduce the sampling cost of this procedure through early termination\. Adaptive Consistency\([Aggarwal et al\., 2023](https://arxiv.org/html/2608.25311#bib.bib24)\)formulates stopping as a posterior decision problem, using a Beta\-binomial model over the leading answer counts and terminating once the estimated margin of the current top answer is sufficiently large\. Early\-Stopping Self\-Consistency\([Li et al\., 2024](https://arxiv.org/html/2608.25311#bib.bib30)\)instead adopts a simpler window\-based criterion, stopping when all answers within a fixed\-size recent window agree\. More recently,[Sharma and Chopra \(2025\)](https://arxiv.org/html/2608.25311#bib.bib22)demonstrated that sequential, entropy\-aware voting can yield stronger cost\-matched performance than parallel self\-consistency, emphasizing the need for compute\-equivalent comparisons\.[Jindal et al\. \(2026\)](https://arxiv.org/html/2608.25311#bib.bib31)proposed an inference time method that clusters short reasoning prefixes and discards prefixes on non\-dominant clusters to save computation\. In a related theoretical direction,[Komiyama et al\. \(2026\)](https://arxiv.org/html/2608.25311#bib.bib25)studied the asymptotic behavior of majority voting, or best\-of\-∞\\infty, and proposed a Bayesian nonparametric stopping rule\.

## Appendix BFull reproduction rates Results

Due to space limitations, the main paper reports only the results with a keep rate of0\.50\.5on GSM8K, MATH\-500, and SVAMP\. Table[4](https://arxiv.org/html/2608.25311#A2.T4)reports the full conditional stability diagnostics behind Figure[2](https://arxiv.org/html/2608.25311#S1.F2)\.

The full table includes CSQA and SQA across keep rates0\.1,0\.50\.1,0\.5, and0\.90\.9\. Across these additional datasets and rates, correct\-answer retention is generally higher than same\-wrong persistence that is consistent with the results in the main paper\.

Table 4:Reproducing rate across the full model and benchmarks\. We report correct\-answer retentionRρ\+R^\{\+\}\_\{\\rho\}, same\-wrong persistenceRρ−R^\{\-\}\_\{\\rho\}, and their separationΔ=Rρ\+−Rρ−\\Delta=R^\{\+\}\_\{\\rho\}\-R^\{\-\}\_\{\\rho\}\. Bold marks the largestΔ\\Deltawithin each row; negativeΔ\\Deltavalues are shown in red\.DatasetModelLengthRρ\+R^\{\+\}\_\{\\rho\}Rρ−R^\{\-\}\_\{\\rho\}Δ\\Delta0\.10\.50\.90\.10\.50\.90\.10\.50\.9GSM8KDream\-7B12897\.997\.998\.398\.399\.299\.282\.582\.582\.582\.593\.393\.315\.315\.85\.825697\.897\.899\.399\.399\.899\.883\.583\.584\.484\.491\.391\.314\.315\.08\.551297\.597\.599\.399\.399\.599\.586\.886\.882\.882\.890\.790\.710\.716\.48\.9LLaDA\-1\.512892\.692\.694\.494\.499\.299\.242\.842\.836\.036\.086\.486\.449\.858\.412\.825695\.295\.295\.095\.099\.699\.665\.965\.922\.022\.084\.784\.729\.473\.014\.951288\.888\.893\.093\.098\.498\.444\.844\.826\.126\.175\.975\.944\.066\.922\.5LLaDA\-8B12893\.293\.294\.494\.499\.199\.143\.943\.937\.137\.190\.190\.149\.457\.39\.025693\.093\.095\.095\.099\.699\.669\.569\.528\.328\.387\.987\.923\.566\.711\.851280\.380\.387\.687\.699\.299\.248\.548\.526\.326\.377\.877\.831\.861\.321\.4MATH\-500Dream\-7B12890\.590\.592\.292\.297\.897\.878\.378\.374\.174\.191\.591\.512\.218\.16\.325693\.593\.593\.593\.598\.698\.672\.072\.073\.073\.083\.183\.121\.520\.615\.551295\.795\.795\.295\.297\.897\.875\.275\.278\.678\.679\.579\.520\.416\.718\.3LLaDA\-1\.512883\.883\.883\.883\.896\.696\.655\.655\.648\.048\.091\.191\.128\.235\.75\.525688\.388\.384\.484\.499\.299\.269\.369\.346\.646\.694\.194\.119\.037\.85\.251286\.286\.285\.385\.394\.594\.561\.861\.831\.031\.081\.681\.624\.554\.312\.9LLaDA\-8B12879\.879\.880\.780\.797\.597\.554\.254\.249\.249\.288\.788\.725\.631\.58\.825688\.688\.682\.682\.697\.797\.773\.073\.044\.844\.892\.692\.615\.737\.85\.151281\.681\.674\.774\.795\.495\.463\.563\.534\.734\.781\.781\.718\.140\.013\.7SVAMPDream\-7B12898\.898\.898\.898\.899\.299\.274\.474\.471\.871\.879\.579\.524\.427\.019\.725698\.998\.999\.299\.2100\.0100\.081\.681\.668\.468\.486\.886\.817\.330\.813\.251298\.898\.899\.699\.6100\.0100\.082\.982\.980\.080\.082\.982\.916\.019\.617\.1LLaDA\-1\.512897\.297\.298\.898\.899\.699\.673\.573\.549\.049\.095\.995\.923\.749\.83\.725693\.493\.494\.794\.799\.199\.179\.279\.233\.333\.383\.383\.314\.261\.415\.851294\.194\.198\.098\.0100\.0100\.078\.678\.664\.364\.392\.992\.915\.633\.87\.1LLaDA\-8B12898\.098\.096\.896\.8100\.0100\.075\.075\.062\.562\.595\.895\.823\.034\.34\.225689\.489\.495\.695\.699\.499\.475\.075\.038\.638\.695\.595\.514\.457\.03\.951296\.296\.294\.394\.3100\.0100\.0100\.0100\.066\.766\.766\.766\.7\-3\.827\.733\.3CSQADream\-7B12898\.998\.996\.296\.297\.597\.596\.896\.887\.587\.591\.491\.42\.18\.76\.225698\.198\.195\.995\.997\.097\.095\.995\.987\.687\.693\.093\.02\.28\.34\.051298\.898\.895\.495\.498\.598\.595\.195\.187\.587\.593\.193\.13\.77\.95\.5LLaDA\-1\.512897\.997\.996\.996\.999\.399\.386\.086\.089\.789\.797\.197\.111\.87\.22\.225696\.696\.698\.398\.399\.899\.888\.188\.189\.089\.099\.699\.68\.59\.30\.251297\.597\.597\.797\.799\.999\.988\.588\.588\.988\.9100\.0100\.09\.18\.8\-0\.1LLaDA\-8B12897\.697\.698\.698\.699\.799\.788\.788\.790\.890\.897\.597\.58\.97\.82\.225695\.295\.297\.797\.799\.899\.885\.585\.588\.088\.0100\.0100\.09\.79\.7\-0\.251298\.198\.197\.597\.5100\.0100\.081\.681\.685\.185\.1100\.0100\.016\.612\.40\.0SQADream\-7B12898\.698\.695\.795\.796\.996\.992\.492\.491\.991\.997\.897\.86\.13\.8\-0\.925697\.497\.493\.893\.896\.096\.091\.791\.785\.485\.494\.394\.35\.88\.41\.751298\.198\.192\.592\.596\.696\.692\.792\.787\.587\.597\.497\.45\.45\.0\-0\.8LLaDA\-1\.512893\.693\.697\.197\.199\.399\.388\.088\.092\.792\.798\.798\.75\.64\.40\.625694\.394\.396\.496\.498\.598\.587\.187\.189\.189\.195\.095\.07\.37\.43\.551296\.896\.899\.199\.1100\.0100\.086\.286\.291\.991\.997\.697\.610\.67\.22\.4LLaDA\-8B12894\.694\.696\.496\.499\.699\.687\.187\.186\.386\.397\.597\.57\.510\.22\.125687\.587\.596\.796\.799\.299\.290\.890\.884\.284\.293\.493\.4\-3\.312\.55\.851293\.893\.8100\.0100\.0100\.0100\.084\.684\.692\.392\.3100\.0100\.09\.17\.70\.0

## Appendix CFull\-Diffusion Accuracy Sweep

As shown in Table[5](https://arxiv.org/html/2608.25311#A3.T5), the proposed method consistently improvesInitaccuracy across the completed generation lengthsL=128,256,512L=128,256,512\. This indicates that the gains reported in Table[5](https://arxiv.org/html/2608.25311#A3.T5)are not an artifact of selecting a favorable block length, but rather reflect a robust improvement over the baseline\. These results demonstrate the effectiveness of the proposed approach for improving initial\-generation quality in full diffusion settings\.

Table 5:Full accuracy results for Dream and LLaDA\-family diffusion language models\.PDCdenotes majority voting over the three suffix\-regenerated completions obtained with 0\.1, 0\.5, and 0\.9 keep rates\. Signed green/red values report accuracy\-point changes relative to theInitbaseline\. ForTiF, the reported change is averaged over Fixed, Linear, and Exp\.α=5\\alpha=5variants\. The table includes all completed lengths in the current full\-diffusion sweep\.Method / LengthGSM8KMATH\-500SVAMPCSQASQA128256512128256512128256512128256512128256512Dream\-7BInitbaseline64\.0680\.0682\.0335\.8043\.2046\.0083\.0087\.0086\.3373\.2271\.7472\.4070\.4568\.5667\.98\+TiFFixed62\.4079\.6181\.9634\.2043\.0046\.0084\.0087\.0086\.3373\.7971\.6672\.4870\.7469\.2968\.56\+TiFLinear64\.5280\.1482\.0334\.8043\.4046\.0083\.6787\.0086\.3373\.9671\.6672\.4870\.7469\.2968\.56\+TiFExp\.α=5\\alpha=565\.8180\.3682\.0335\.6044\.0046\.2084\.0087\.0086\.3373\.5571\.7472\.4870\.7469\.2968\.56Δ\\DeltaTiFavg\. vs\.Init\+0\.18\-0\.02\-0\.02\-0\.93\+0\.27\+0\.07\+0\.89\+0\.00\+0\.00\+0\.55\-0\.05\+0\.08\+0\.29\+0\.73\+0\.58\+PDC66\.0381\.0584\.0837\.2044\.6047\.0084\.3388\.0088\.0074\.2072\.2473\.3071\.7671\.6270\.60Δ\\DeltaPDCvs\.Init\+1\.97\+0\.99\+2\.05\+1\.40\+1\.40\+1\.00\+1\.33\+1\.00\+1\.67\+0\.98\+0\.50\+0\.90\+1\.31\+3\.06\+2\.62LLaDA\-8BInitbaseline58\.3060\.7318\.8823\.8026\.4517\.5483\.6753\.3317\.6780\.5142\.5148\.3265\.0717\.474\.66\+TiFFixed57\.7760\.1219\.2624\.0026\.2518\.1583\.6756\.3318\.0080\.5145\.2951\.7665\.0728\.389\.46\+TiFLinear58\.0060\.8019\.4823\.8026\.2517\.9483\.6756\.3318\.3380\.5145\.2151\.6065\.0728\.389\.46\+TiFExp\.α=5\\alpha=558\.3060\.8019\.4124\.0026\.2518\.1583\.6756\.3318\.3380\.5145\.2151\.6065\.0728\.389\.46Δ\\DeltaTiFavg\. vs\.Init\-0\.28\-0\.15\+0\.50\+0\.13\-0\.20\+0\.54\+0\.00\+3\.00\+0\.55\+0\.00\+2\.73\+3\.33\+0\.00\+10\.91\+4\.80\+PDC65\.8865\.8833\.3625\.4027\.8622\.3884\.3362\.6731\.0080\.5945\.5450\.7866\.6725\.629\.61Δ\\DeltaPDCvs\.Init\+7\.58\+5\.15\+14\.48\+1\.60\+1\.41\+4\.84\+0\.66\+9\.34\+13\.33\+0\.08\+3\.03\+2\.46\+1\.60\+8\.15\+4\.95LLaDA\-1\.5Initbaseline56\.4860\.5839\.1223\.4025\.6521\.9883\.6775\.6751\.0080\.1073\.2275\.5166\.0851\.5332\.02\+TiFFixed56\.4160\.0539\.8023\.4025\.6521\.9883\.3375\.0051\.3380\.1074\.3775\.8466\.0856\.7733\.33\+TiFLinear56\.5660\.2739\.9523\.4025\.4522\.3883\.6775\.3351\.3380\.1074\.2875\.7666\.0857\.0633\.19\+TiFExp\.α=5\\alpha=556\.5660\.5040\.0323\.4025\.6522\.3883\.6775\.6751\.3380\.1074\.2875\.7666\.0857\.0633\.19Δ\\DeltaTiFavg\. vs\.Init\+0\.03\-0\.30\+0\.81\+0\.00\-0\.07\+0\.27\-0\.11\-0\.34\+0\.33\+0\.00\+1\.09\+0\.28\+0\.00\+5\.43\+1\.22\+PDC65\.8866\.2656\.7926\.4026\.6526\.0184\.6779\.0062\.6780\.1074\.2876\.0966\.8152\.9840\.32Δ\\DeltaPDCvs\.Init\+9\.40\+5\.68\+17\.67\+3\.00\+1\.00\+4\.03\+1\.00\+3\.33\+11\.67\+0\.00\+1\.06\+0\.58\+0\.73\+1\.45\+8\.30

## Appendix DSemi\-AR LLaDA\-Family Results

Table[6](https://arxiv.org/html/2608.25311#A4.T6)and Table[7](https://arxiv.org/html/2608.25311#A4.T7)report the LLaDA\-family results with block length 32\.222Note that all results reported in the main paper correspond to the setting in which the block length equals the generation length, referred to as the full diffusion setting\.When the block length is smaller than the generation length, the sequence is generated block by block, with diffusion performed independently within each block, referred to as the semi\-autoregressive blockwise setting\.

Overall, the results in Table[6](https://arxiv.org/html/2608.25311#A4.T6)and Table[7](https://arxiv.org/html/2608.25311#A4.T7)show that the proposed method remains effective even in this semi\-autoregressive blockwise setting\. Across generation lengths, the proposed method all\-in\-all improvesInitaccuracy for the LLaDA\-family models, indicating that its benefits are not limited to the full\-diffusion setting used for the main claims\.

Table 6:Semi\-AR LLaDA\-family prefix\-denoising consistency results\. These runs use block length 32 with generation lengths 128, 256, and 512\. Values are percentages\. Bold marks the largestΔ\\Deltawithin each row; negativeΔ\\Deltavalues are shown in red\.DatasetModelLengthRρ\+R^\{\+\}\_\{\\rho\}Rρ−R^\{\-\}\_\{\\rho\}Δ\\Delta0\.10\.50\.90\.10\.50\.90\.10\.50\.9GSM8KLLaDA\-1\.512895\.695\.699\.599\.599\.099\.078\.078\.091\.591\.589\.789\.717\.68\.09\.225696\.196\.198\.798\.799\.799\.772\.872\.890\.690\.696\.796\.723\.38\.23\.051297\.797\.799\.299\.299\.999\.967\.867\.886\.886\.898\.798\.729\.912\.41\.2LLaDA\-8B12895\.795\.798\.398\.399\.499\.474\.974\.987\.287\.291\.791\.720\.811\.27\.625697\.397\.399\.499\.499\.699\.670\.170\.187\.887\.895\.695\.627\.211\.64\.051298\.198\.199\.599\.599\.999\.967\.167\.182\.382\.395\.495\.431\.017\.34\.6MATH\-500LLaDA\-1\.512889\.489\.495\.995\.997\.697\.663\.063\.085\.385\.390\.990\.926\.310\.66\.725688\.688\.695\.195\.199\.599\.554\.354\.380\.780\.794\.294\.234\.314\.45\.351288\.588\.593\.893\.899\.599\.545\.645\.666\.666\.693\.493\.442\.827\.26\.1LLaDA\-8B12888\.888\.897\.597\.597\.597\.559\.459\.484\.084\.089\.989\.929\.413\.57\.725690\.390\.396\.696\.699\.499\.455\.055\.077\.777\.794\.294\.235\.318\.95\.351282\.782\.791\.191\.198\.698\.643\.143\.170\.770\.793\.893\.839\.620\.54\.8SVAMPLLaDA\-1\.512898\.998\.999\.699\.699\.699\.668\.668\.688\.688\.6100\.0100\.030\.311\.1\-0\.425698\.598\.599\.699\.6100\.0100\.070\.070\.087\.587\.597\.597\.528\.512\.12\.551298\.198\.199\.399\.3100\.0100\.074\.374\.391\.491\.497\.197\.123\.87\.82\.9LLaDA\-8B12898\.598\.599\.699\.699\.299\.274\.374\.388\.688\.697\.197\.124\.211\.02\.125698\.998\.9100\.0100\.0100\.0100\.060\.560\.584\.284\.294\.794\.738\.315\.85\.351298\.598\.599\.399\.3100\.0100\.073\.573\.591\.291\.297\.197\.125\.08\.12\.9CSQALLaDA\-1\.512895\.795\.798\.398\.399\.399\.374\.574\.590\.790\.795\.795\.721\.37\.63\.625693\.893\.897\.897\.899\.499\.471\.671\.689\.489\.498\.698\.622\.28\.40\.851294\.394\.397\.697\.699\.699\.669\.069\.084\.584\.597\.597\.525\.313\.12\.0LLaDA\-8B12894\.894\.898\.798\.798\.898\.874\.174\.190\.790\.796\.496\.420\.78\.12\.425692\.392\.397\.397\.399\.299\.270\.670\.688\.688\.697\.897\.821\.88\.71\.451293\.993\.996\.896\.899\.299\.265\.065\.084\.384\.396\.796\.729\.012\.52\.4

Table 7:Semi\-AR LLaDA\-family main accuracy results\. These runs use block length 32 with generation lengths 128, 256, and 512\.Method / LengthGSM8KMATH\-500SVAMPCSQA128256512128256512128256512128256512LLaDA\-8BInitbaseline72\.2574\.9181\.8032\.2035\.2743\.1586\.0087\.0088\.6776\.5876\.9977\.07\+TiFFixed71\.8769\.3780\.1429\.6031\.0639\.3184\.6786\.6788\.3378\.7977\.0777\.40\+TiFLinear73\.5474\.7581\.6530\.8034\.6741\.5387\.6786\.3388\.3377\.3177\.0777\.40\+TiFExp\.α=5\\alpha=573\.6276\.2781\.9631\.8035\.2742\.5488\.0086\.6788\.3376\.9977\.3177\.40Δ\\DeltaTiFavg\. vs\.Init\+0\.76\-1\.45\-0\.55\-1\.47\-1\.60\-2\.02\+0\.78\-0\.44\-0\.34\+1\.12\+0\.16\+0\.33\+PDC73\.7776\.0482\.4933\.2036\.4742\.3487\.0088\.0088\.3377\.3177\.3177\.40Δ\\DeltaPDCvs\.Init\+1\.52\+1\.13\+0\.69\+1\.00\+1\.20\-0\.81\+1\.00\+1\.00\-0\.34\+0\.73\+0\.32\+0\.33LLaDA\-1\.5Initbaseline72\.3378\.0182\.7134\.0036\.8741\.9487\.6786\.6788\.3376\.6676\.5875\.92\+TiFFixed71\.9572\.4880\.7432\.8033\.2738\.9184\.3386\.3387\.3378\.3077\.2375\.92\+TiFLinear73\.3177\.5682\.5633\.4036\.6741\.1387\.6786\.3387\.6777\.3177\.1576\.09\+TiFExp\.α=5\\alpha=573\.4679\.0083\.0233\.8037\.0741\.9488\.3386\.3388\.3376\.8276\.5876\.17Δ\\DeltaTiFavg\. vs\.Init\+0\.58\-1\.66\-0\.60\-0\.67\-1\.20\-1\.28\-0\.89\-0\.34\-0\.55\+0\.82\+0\.41\+0\.14\+PDC73\.9278\.4782\.7134\.4137\.2741\.5388\.0087\.6789\.0077\.2377\.3177\.64Δ\\DeltaPDCvs\.Init\+1\.59\+0\.46\+0\.00\+0\.41\+0\.40\-0\.41\+0\.33\+1\.00\+0\.67\+0\.57\+0\.73\+0\.72

## Appendix ECase Study: How Prefix Regeneration Repairs Errors

Problem 97Problem\.*A figure skater is facing north when she begins to spin to her right\. She spins 2250 degrees\. Which direction \(north, south, east or west\) is she facing when she finishes her spin?*Initial generationThe figure skater is facing north when she begins to spin to her right\. She spins 2250 degrees\.…We can do this by subtracting multiples of 360 degrees from 2250 degrees until we get an angle between 0 and 360 degrees\.2250 degrees divided by 360 degrees is 6 with a remainder of 150 degrees\.Therefore, she is facing\\boxed​\{south\}\\backslash\\mathrm\{boxed\}\\\{\\mathrm\{south\}\\\}\.Prefix\-conditioned regenerationThe figure skater is facing north when she begins to spin to her right\. She spins 2250 degrees\.…We can do this by subtracting multiples of 360 degrees from 2250 degrees\.2250 \- 6\*360 = 2250 \- 2160 = 90 degrees\.So, when she finishes her spin, she is facing\\boxed​\{east\}\\backslash\\mathrm\{boxed\}\\\{\\text\{east\}\\\}\.

Problem 144Problem\.*A curve is parameterized by*\(x,y\)=\(t3\+7,−3​t2−6​t−5\)\.\(x,y\)=\(t^\{3\}\+7,\-3t^\{2\}\-6t\-5\)\.*Find the point the curve passes through att=2t=2\.*Initial generationWhent=2,t=2,we have\(x,y\)\\displaystyle\(x,y\)=\(23\+7,−3​\(2\)2−6​\(2\)−5\)\\displaystyle=\(2^\{3\}\+7,\-3\(2\)^\{2\}\-6\(2\)\-5\)=\(15,−27\)\.\\displaystyle=\{\\color\[rgb\]\{0\.7,0,0\}\(15,\-27\)\}\.So the curve passes through the point\\boxed​\{\(15,−27\)\}\.\\backslash\\mathrm\{boxed\}\\\{\(15,\-27\)\\\}\.Prefix\-conditioned regenerationWhent=2,t=2,we have\(x,y\)\\displaystyle\(x,y\)=\(23\+7,−3​\(2\)2−6​\(2\)−5\)\\displaystyle=\(2^\{3\}\+7,\-3\(2\)^\{2\}\-6\(2\)\-5\)=\(8\+7,−12−12−5\)\\displaystyle=\{\\color\[rgb\]\{0,0\.45,0\}\(8\+7,\-12\-12\-5\)\}=\(15,−29\)\.\\displaystyle=\{\\color\[rgb\]\{0,0\.45,0\}\(15,\-29\)\}\.Therefore, the point the curve passes through att=2t=2is\\boxed​\{\(15,−29\)\}\.\\backslash\\mathrm\{boxed\}\\\{\(15,\-29\)\\\}\.The answer is: \(15,\-29\)

Table 8:Two examples in which the wrong answer is corrected by regeneration\. Red text highlights the incorrect continuation in the initial sample; green text highlights the corrected continuation produced by regeneration\. When the wrong intermediate step is remasked, prefix\-conditioned regeneration avoids the same mistake\.Section[2\.2](https://arxiv.org/html/2608.25311#S2.SS2)shows that prefix\-conditioned regeneration has high correct\-answer preservation \(R\+R^\{\+\}\) and low same\-wrong preservation \(R−R^\{\-\}\)\. In words, correct initial answers tend to remain correct after regeneration, while wrong initial answers often do not reproduce the same wrong answer\. This is a useful regime forPDC\. HighR\+R^\{\+\}means regeneration is unlikely to disturb a correct trajectory, while a lowR−R^\{\-\}means an incorrect trajectory often has a chance to escape its original mistake\. Escaping the same wrong answer does not by itself guarantee correctness; the regenerated answer may still be wrong\. However, when the kept prefix contains the right setup and the erroneous intermediate step lies in the regenerated suffix, regeneration can cut away the local mistake and replace it with a corrected continuation\. Table[8](https://arxiv.org/html/2608.25311#A5.T8)shows two MATH\-500 with Dream\-v0\-Instruct\-7B examples of this behavior\.

In Problem 97, the initial sample follows the correct modulo\-reduction strategy but makes a local arithmetic error, treating the remainder of22502250degrees modulo360360degrees as150150degrees and concluding*south*\. The regenerated sample recomputes the suffix as2250−6⋅360=902250\-6\\cdot 360=90degrees and recovers*east*\. Similarly, in Problem 144, the initial sample substitutes the correct valuet=2t=2, but evaluates the resulting expression as−27\-27, whereas regeneration expands the same expression as−12−12−5=−29\-12\-12\-5=\-29and recovers the correct point\.

## Appendix FModel and Dataset Summary

This appendix summarizes the language models and datasets used in the experiments\. Detailed inference time hyperparameters, prompt templates, answer extraction, and voting conventions are given in Appendix[H](https://arxiv.org/html/2608.25311#A8)\.

Table 9:Language models used in the experiments\. Dream\-7B is exclusively for full diffusion, while LLaDA\-family models support both full diffusion and semi\-autoregressive diffusion\. The main paper exclusively considers the full diffusion setting\. The semi\-autoregressive runs are reported separately in Appendix[D](https://arxiv.org/html/2608.25311#A4)\.ModelCheckpointFamilyMain settingDream\-7BDream\-org/Dream\-v0\-Instruct\-7BMasked Diffusion LLMFull\-window diffusion inference timeLLaDA\-1\.5GSAI\-ML/LLaDA\-1\.5Masked Diffusion LLM with Block LengthBlock length matched to generation lengthLLaDA\-8BGSAI\-ML/LLaDA\-8B\-InstructMasked Diffusion LLM with Block LengthBlock length matched to generation length

Table 10:Datasets used in the experiments\. Math\-style datasets are evaluated by canonical answer extraction and symbolic/numeric equivalence when applicable\. Multiple\-choice and binary reasoning datasets are evaluated by canonicalized option extraction\.DatasetNameTask typePrimary answer formatGSM8KGrade\-school math word problems\([Cobbe et al\., 2021](https://arxiv.org/html/2608.25311#bib.bib3)\)Arithmetic reasoningBoxed final answerMATH\-500MATH subset / verification benchmark\([Hendrycks et al\., 2021](https://arxiv.org/html/2608.25311#bib.bib5);[Lightman et al\., 2024](https://arxiv.org/html/2608.25311#bib.bib26)\)Mathematical reasoningBoxed final answerSVAMPArithmetic word\-problem challenge\([Patel et al\., 2021](https://arxiv.org/html/2608.25311#bib.bib11)\)Arithmetic reasoningBoxed final answerCSQACommonsenseQA\([Talmor et al\., 2019](https://arxiv.org/html/2608.25311#bib.bib21)\)Multiple\-choice commonsense reasoningBoxed option letterSQAStrategyQA\([Geva et al\., 2021](https://arxiv.org/html/2608.25311#bib.bib33)\)Binary commonsense reasoningBoxed option letter

## Appendix GResults at Different Sampling Temperatures

To examine whetherPDCremains effective as sampling becomes more stochastic, we repeat theL=128L=128evaluation at token\-sampling temperatureτ=1\.0\\tau=1\.0\.

Table 11:Accuracy at generation lengthL=128L=128under low\- and high\-temperature decoding\. We compare the main setting \(τ=0\.2\\tau=0\.2\) with more stochastic decoding \(τ=1\.0\\tau=1\.0\)\.Init×4\\times 4denotes majority voting over four independent full generations\. Boldface marks the best result in each model–temperature row\. Signed green/red values report the accuracy\-point change ofPDCrelative toInit×4\\times 4at the same temperature\.DatasetModel𝝉\\boldsymbol\{\\tau\}InitInit×4\\times 4TiFFixed×4\\times 4TiFLinear×4\\times 4TiFExp\.×4\\times 4PDCΔ\\Deltavs\.Init×4\\times 4GSM8KDream\-7B0\.264\.0664\.2262\.4064\.5265\.8166\.03\+1\.811\.041\.2450\.1945\.1147\.0849\.1365\.96\+15\.77LLaDA\-1\.50\.256\.4856\.6356\.4156\.5656\.7165\.88\+9\.251\.056\.9458\.8358\.1558\.1558\.2368\.69\+9\.86LLaDA\-8B0\.258\.3058\.9158\.3858\.6158\.9165\.88\+6\.971\.057\.5460\.7359\.1459\.7460\.0569\.45\+8\.72MATH\-500Dream\-7B0\.235\.8036\.0034\.2034\.8035\.6037\.20\+1\.201\.022\.4025\.2024\.2024\.6025\.0037\.60\+12\.40LLaDA\-1\.50\.223\.4023\.2023\.4023\.4023\.0026\.40\+3\.201\.024\.0023\.2023\.6023\.6023\.4030\.20\+7\.00LLaDA\-8B0\.223\.8023\.6023\.8023\.8023\.6025\.40\+1\.801\.024\.4525\.0523\.8523\.8524\.2528\.06\+3\.01SVAMPDream\-7B0\.283\.0083\.0084\.0083\.6784\.0084\.33\+1\.331\.070\.3375\.6775\.0075\.6776\.0082\.67\+7\.00LLaDA\-1\.50\.283\.6783\.6783\.3383\.6783\.6784\.67\+1\.001\.085\.0084\.6785\.0084\.6784\.6789\.00\+4\.33LLaDA\-8B0\.283\.6784\.3384\.3384\.0084\.3384\.33\+0\.001\.084\.0084\.0085\.0084\.3384\.3386\.33\+2\.33CSQADream\-7B0\.273\.2273\.2273\.7973\.9673\.5574\.20\+0\.981\.064\.7876\.4978\.0577\.8977\.8174\.37\-2\.12LLaDA\-1\.50\.280\.1080\.5180\.6780\.6780\.6780\.10\-0\.411\.080\.1080\.8480\.6780\.7580\.7579\.44\-1\.40LLaDA\-8B0\.280\.5180\.3480\.5980\.5980\.5980\.59\+0\.251\.079\.6980\.5980\.7580\.9280\.6779\.85\-0\.74SQADream\-7B0\.270\.4570\.4570\.7470\.7470\.7471\.76\+1\.311\.058\.6663\.1764\.4864\.7764\.7767\.10\+3\.93LLaDA\-1\.50\.266\.0865\.9465\.9465\.7965\.7966\.81\+0\.871\.065\.0764\.7764\.9265\.2164\.9266\.96\+2\.19LLaDA\-8B0\.265\.0764\.6365\.0765\.0765\.0766\.67\+2\.041\.064\.7764\.1964\.6364\.6364\.9266\.67\+2\.48

##### Results\.

Table[11](https://arxiv.org/html/2608.25311#A7.T11)shows that, atτ=1\.0\\tau=1\.0,PDCachieves the highest numerical accuracy in 12 of the 15 model and dataset settings and outperformsInit×4\\times 4in the same 12 settings\. Averaged uniformly over the 15 settings, it exceedsInit×4\\times 4by4\.984\.98accuracy points; the largest dataset\-level mean gains occur on GSM8K and MATH\-500, at11\.4511\.45and7\.477\.47points, respectively\. Across the two tested temperatures, the unweighted mean accuracy ofPDCchanges from65\.3565\.35atτ=0\.2\\tau=0\.2to66\.1666\.16atτ=1\.0\\tau=1\.0\(\+0\.81\+0\.81\), whereasInitandInit×4\\times 4decrease by4\.584\.58and2\.072\.07points, respectively\. Consequently, the mean margin ofPDCoverInit×4\\times 4increases from2\.112\.11to4\.984\.98points\. This larger relative margin partly reflects a weaker full\-generation baselines at the higher temperature and are not uniform across tasks: on all three CSQA settings,PDCtrailsInit×4\\times 4by0\.740\.74–2\.122\.12points\. The results therefore show thatPDCremains effective in the tested higher\-temperature setting, particularly on the mathematical reasoning tasks\.

## Appendix HReproducibility Details

##### Models and inference time hyperparameters\.

We evaluate Dream\-v0\-Instruct\-7B, LLaDA\-1\.5, and LLaDA\-8B\-Instruct\. We use entropy\-based and low confidence unmasking, token temperature0\.20\.2\. The generation lengthLLis set equal to the denoising step budgetT∈\{128,256,512\}T\\in\\\{128,256,512\\\}\.

The main tables use the block\-length\-matched setting, with block length set equal to the generation length\. Appendix[D](https://arxiv.org/html/2608.25311#A4)reports semi\-AR LLaDA\-family runs with block length 32, as follows theTiFsettings\.TiFare captured from the same initial denoising run with a stride of 1\.

##### Regeneration denoising budget\.

During regeneration, the effective denoising budget is reduced according to the kept prefix length\. In the reported full\-diffusion setting,T=LT=L, so the effective regeneration step budget equals the remasked suffix length:

Tρ=Lρ=L−Lρprefix\.T\_\{\\rho\}=L\_\{\\rho\}=L\-L\_\{\\rho\}^\{\\text\{prefix\}\}\.The experiment scripts also support a general nominal step budgetTT, using the proportional rule:

Tρ=max⁡\{1,round⁡\(T​Lρ/L\)\}\.T\_\{\\rho\}=\\max\\\{1,\\operatorname\{round\}\(TL\_\{\\rho\}/L\)\\\}\.This reduces toTρ=LρT\_\{\\rho\}=L\_\{\\rho\}whenT=LT=L\. Dream regenerates the suffix with a reduced number of denoising steps using the floor version of the same proportional rule:

Tρ=max⁡\{1,⌊T​Lρ/L⌋\}\.T\_\{\\rho\}=\\max\\\{1,\\lfloor TL\_\{\\rho\}/L\\rfloor\\\}\.This differs from the rounded rule by at most one denoising step, and all reported tables are computed from the saved records for the corresponding run\.

##### Prompt template\.

For math\-style tasks, all prompts use the same instruction suffix:

> Please reason step by step, and put your final answer within \\boxed\{\}\.

For multiple\-choice tasks, we explicitly require a chain\-of\-thought style reasoning before the final choice\. The prompt lists the answer options and includes the following mandatory format instruction:

> Mandatory format \(do not skip\): 1\) Write your reasoning across several sentences \(facts, definitions, or everyday knowledge you use\)\. 2\) Only after that reasoning, end with exactly one line that contains only one capital option letter \(A\-\-E\) inside \\boxed\{\}, e\.g\. \\boxed\{C\}\.

Thus, the multiple\-choice setting is not a direct\-answer\-only setting; the model is instructed to produce reasoning and then end with a single boxed option letter\.

##### Answer extraction and equivalence\.

For math datasets, we extract the last boxed expression when present\. If no boxed expression is found, it applies final\-answer fallback patterns such as “The answer is”, “answer:”, and “final answer is” in the last lines of the output\. Extracted answers are normalized using Hendrycks\-MATH\-style string normalization, including removal of superficial LaTeX wrappers, unit text, spacing, percent signs, and simple fraction normalization\. If normalized strings do not match exactly, we use a SymPy fallback: both expressions are parsed with the LaTeX parser when possible, otherwise with sympify, and are judged equivalent when the symbolic simplification of their difference is zero\. Numeric decimal strings are compared with tolerance10−610^\{\-6\}\.

For multiple\-choice tasks, the extracted answer is canonicalized to a choice letter\. Exact option\-text matches are mapped to their corresponding letter before loose letter\-form matching, to avoid treating option text such as formulas as a choice letter\. Common letter forms such asC,\(C\),C\., and\\text\{C\}are accepted\.

##### Null predictions, no\-vote cases, and missing triplets\.

If no answer can be extracted from an initial output, the example is counted as incorrect forInitaccuracy\. Conditional quantities that require a non\-null initial answer, such as same\-wrong persistence and the probability that an initially wrong answer becomes correct, exclude such examples from the conditioning set\.