On the SPOT: Enhancing Speculative Decoding for Diffusion Language Models via Commit-Order Prediction

# Enhancing Speculative Decodingfor Diffusion Language Modelsvia Commit-Order Prediction
Source: [https://arxiv.org/html/2610.22543](https://arxiv.org/html/2610.22543)
## On the SPOT: Enhancing Speculative Decoding for Diffusion Language Models via Commit\-Order Prediction

Eliron RahimiAffiliation:Technion – Israel Institute of TechnologyAvi MendelsonAffiliation:Technion – Israel Institute of TechnologyChaim BaskinAffiliation:INSIGHT Lab, School of Electrical and Computer Engineering, Ben\-Gurion University of the Negev

###### Abstract

Diffusion language models \(DLMs\) promise parallel text generation, but in practice face an unresolved speed–quality trade\-off\. Exact decoding fills in only one masked position per model pass, while faster multi\-token decoding schemes are lossy\. Speculative decoding is the standard lossless acceleration for autoregressive models\. However, it does not transfer directly to DLMs, which must choose both what to generate and where\. We show that this order is the missing piece: the model’s own predictions from the previous pass already give the right token at9292–100%100\\%of positions, but not which position comes next\. Based on this observation, we introduce*SpeculatingPositionsOrderTrajectory*\(SPOT\), which learns to predict this order\. A0\.620\.62M\-parameter model reads only the target’s output distributions and proposes a tree of candidate continuations, and the target verifies all of them in a single pass by checking whether each equals its own next state\. On the verification side, our analysis shows that prior block\-diffusion verification approaches either assumeT=0T\{=\}0or a sequential proposal interface\. To support tree\-based speculation at any temperature, we propose using coupled\-noise match verification for diffusion blocks, following shared\-Gumbel\-noise coupling approaches\. Our comprehensive experiments show that SPOT cuts target passes by up to6\.8×6\.8\\timesand reaches up to3\.2×3\.2\\timesthe throughput of standard decoding, beating the calibrated and training\-free drafting baselines in every task and budget atT=0T\{=\}0andT=1T\{=\}1\. In addition, we show that the learned drafter transfers across models, demonstrating successful transfer across six different DLMs\.

[Project Page](https://eliezer318.github.io/SPOT-dLLM/)

11footnotetext:Equal contribution\. Correspondence:\{eliezer, elironrahimi\}@campus\.technion\.ac\.il\.## 1Introduction

Diffusion language models \(DLMs\)\([Nie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib2);[Arriola et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib31)\)generate text by iteratively refining masked token sequences rather than predicting tokens left\-to\-right\. Block\-diffusion models make this particularly practical by combining bidirectional denoising within each block with causal generation across blocks\([Arriola et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib31);[Cheng et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib19)\)\. The diffusion paradigm enables new capabilities such as bidirectional attention, iterative refinement, enhanced safety robustness, and the potential to substantially accelerate language generation via parallel decoding\([Ye et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib13);[Rahimi et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib3);[Wu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib15)\)\. Yet this parallelism comes with a fundamental speed–quality trade\-off\. High\-quality diffusion decoding typically uses conservative unmasking and therefore requires many denoising steps, while faster decoding schemes reveal multiple positions per step using confidence\-based, top\-kk, or dynamic policies, potentially changing the decoding trajectory and degrading generation quality\([Xu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib12);[Pan et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib21);[Sun et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib20)\)\.

Figure 1:SPOT overview \(red: committed; amber dashed: drafted; grey: masked, with its stale argmax token\)\.\(a\)The target commits its most confident position per forward: the tokens are largely known in advance, the*order*is not\.\(b\)SPOT reads a hypothetical prefix’s stale distributions, predicts the next position with a0\.620\.62M order model, grows a tree of states by prefix mass \(edge width\), and verifies all of them in one batched forward, committing three positions\.For autoregressive language models \(AR\-LMs\), speculative decoding provides an elegant lossless solution\. A cheap drafter proposes several future tokens, and the target verifies them in parallel while preserving exactly the distribution of draft\-free decoding\([Leviathan et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib27);[Chen et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib28)\)\. Recent work has begun bringing this idea to DLMs through self\-drafting, draft graphs, learned residual predictors, and auxiliary draft models\([Gao et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib4);[Agrawal et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib18);[Pan et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib21);[Cui et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib30);[Xu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib12)\)\. However, speculation changes fundamentally once attention inside a block is bidirectional\. In an autoregressive model, the next position is fixed, so the drafter only needs to predict*what token comes next*\. In a DLM, committing one position changes the representations and output distributions of all remaining masked positions in the block\. The target must therefore decide both*what token to write*and*where to write it next*\. Its future is not a fixed sequence of token positions, but a trajectory through partially filled block states\. Several recent DLM speculative\-decoding methods exploit this state\-based view by proposing hypothetical future states and checking whether the target follows the same transitions\([Gao et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib4);[Agrawal et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib18);[Pan et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib21);[Xu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib12)\)\. This makes verification possible, but leaves two central questions unresolved:\(Q1\)what information should a drafter predict to speculate several transitions ahead? and\(Q2\)how can a tree of speculative states be verified exactly when the target samples rather than decodes greedily?

Our key finding is that the main difficulty is not token prediction, but*commit\-order prediction*\. Over the speculative horizons relevant for acceleration, the target’s stale distributions already provide strong token hypotheses: their top prediction matches the eventual committed token at9292–100%100\\%of measured positions\. Predicting which position commits next is considerably harder\. This suggests reusing the target’s stale token predictions while focusing the drafter on commit order, retaining additional high\-probability token hypotheses when confidence is low\. Based on this observation, we introduce*SpeculatingPositionsOrderTrajectory*\(SPOT\), a lightweight0\.620\.62M\-parameter drafter that predicts future commit order from the target’s stale output distributions \(Figure[1](https://arxiv.org/html/2610.22543#S1.F1)\)\. SPOT is prefix\-conditional: after each hypothetical commitment, it predicts the next position again and organizes these predictions into a tree of candidate block states, prioritizing high\-probability branches\. The target then evaluates the current and drafted states in a single batched forward and accepts the longest prefix consistent with its trajectory\. The second challenge is verifying speculative trees exactly when the target samples\. Under greedy decoding, a drafted transition is verified by comparing it with the target’s next state\. Under sampling, independent draws may disagree even when they come from the same distribution, causing state\-equality verification to reject valid proposals\. We address this with*coupled\-noise verification*: the drafter and target share per\-position Gumbel randomness, making the target transition deterministic conditioned on the shared noise\. State equality can therefore verify an entire speculative tree, while marginalizing over the noise exactly recovers the target’s sampling distribution\. This enables exact tree\-based verification at arbitrary temperatures without the sequential proposal interface required by classical rejection sampling\.

The resulting method substantially reduces both model evaluations and wall\-clock decoding time\. On SDAR\-8B, SPOT uses1515–35%35\\%fewer target forwards than Spiffy’s calibrated draft graph and achieves up to1\.5×1\.5\\timesits throughput, with the adaptive\-budget configuration reaching up to2\.67×2\.67\\times\. Relative to standard draft\-free decoding, SPOT reduces target passes by up to6\.8×6\.8\\timesand reaches up to3\.2×3\.2\\timesthroughput\. The gains hold across tasks at both greedy decoding andT=1T\{=\}1sampling\. We further evaluate on SDAR\-4B, SDAR\-1\.7B and LLaDA\-2\.0\-mini, and test drafter transfer to two additional targets, LLaDA\-8B and Dream\-7B, covering six DLMs in total\.

Our main contributions are:

- •We analyze speculative decoding for diffusion targets along a drafting and a verification axis, show that commit\-order prediction is the drafting bottleneck while the target’s stale distributions already supply the tokens, show why independent sampling breaks state\-equality verification atT\>0T\>0, and motivate coupling the drafter and verifier through shared randomness\.
- •Based on this analysis, we introduce SPOT, a0\.620\.62M\-parameter prefix\-conditional commit\-order drafter that builds multi\-step trees of block states from stale distributions, paired with coupled\-noise verification, which provably preserves the target’s distribution at any sampling temperature\.
- •We evaluate SPOT across tasks, temperatures, block widths and six diffusion language models: it beats calibrated and training\-free baselines atT=0T\{=\}0andT=1T\{=\}1, cuts target passes by up to6\.8×6\.8\\times, and its checkpoints transfer between models, including LLaDA\-8B and Dream\-7B, without retraining\.

## 2Related Work

#### Speculative decoding and DLMs\.

Speculative decoding was introduced for autoregressive \(AR\) language models, where a lightweight drafter proposes tokens that the target verifies in parallel while preserving its distribution\([Leviathan et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib27);[Chen et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib28)\)\. Later work improved the drafter with auxiliary decoding heads, learned drafters and dynamically built draft trees\([Cai et al\., 2024](https://arxiv.org/html/2610.22543#bib.bib26);[Li et al\., 2024b](https://arxiv.org/html/2610.22543#bib.bib1);[Li et al\., 2024a](https://arxiv.org/html/2610.22543#bib.bib25);[Li et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib24)\), and a complementary line uses diffusion models as drafters for AR targets\([Christopher et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib23);[Li et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib22);[Chen et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib14);[Ringel and Romano, 2026](https://arxiv.org/html/2610.22543#bib.bib16)\)\. DLMs instead generate text by denoising\([Austin et al\., 2021a](https://arxiv.org/html/2610.22543#bib.bib32);[Nie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib2);[Ye et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib13)\); block\-causal models such as SDAR make this practical block by block\([Arriola et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib31);[Cheng et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib19)\), while approximate caching and parallel unmasking accelerate decoding at a cost in fidelity\([Wu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib15);[Sun et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib20)\)\. Because attention inside a block is bidirectional, the target decides not only what to write but where, so the commit order is an extra dimension of decoding and speculative decoding does not transfer directly\. For verification we build on communication\-free coupling:[Daliri et al\. \(2025\)](https://arxiv.org/html/2610.22543#bib.bib17)show that two parties sampling fromPPandQQunder shared Gumbel noise agree with probability at least\(1−DT​V\)/\(1\+DT​V\)\(1\-D\_\{TV\}\)/\(1\+D\_\{TV\}\), and apply this to*drafter\-invariant*speculative decoding\. We carry the construction to diffusion decoding, where per\-*position*shared noise makes the target’s sampled commitment at any hypothetical block state a deterministic function of that state \(Section[4\.3](https://arxiv.org/html/2610.22543#S4.SS3)\)\.

#### Speculative decoding for diffusion targets\.

SSD\([Gao et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib4)\)and Spiffy\([Agrawal et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib18)\)study speculative decoding with DLMs targets: SSD orders positions by confidence into a self\-drafted chain and verifies it greedily, whereas Spiffy drafts states along an offline\-calibrated graph and verifies them blockwise\. Both are training\-free and greedy\. MRP\([Xu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib12)\)learns a residual head that extrapolates the target’s logits a few denoising steps ahead and accepts drafted tokens in one pass, greedy only \(Section[5\.4](https://arxiv.org/html/2610.22543#S5.SS4)sets its reported gains beside ours\)\. BlockSpec\([Pan et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib21)\)extends training\-free self\-speculation to trees on decoding trajectories, and SimSD\([Cui et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib30)\)drafts with a same\-family sibling model but verifies approximately, so it is not lossless and is not added to our comparison\. In contrast, our drafter learns the commit order, builds multi\-step trees, verifies exactly under sampling, and transfers across targets without retraining\.

## 3Analysis of Block\-Diffusion Speculative Decoding

#### Setting\.

The target decodes a block ofLLmasked positions one commit per forward\. A draft is a hypothetical block state with a few more positions filled by the drafter, andDDdrafts are verified at once by appending each behind the cached prefix as its own copy of the block, so a single forward over\(1\+D\)​L\(1\{\+\}D\)Lpositions returns the target’s next commit at the root and at every draft\. A draft is accepted iff it equals that commit at its parent, so the accepted set is one path and the output cannot change\. A draft therefore costs a whole block rather than a token: budgets are small, and target forwards \(NFE\) and wall clock diverge \(Appendix[D\.2](https://arxiv.org/html/2610.22543#A4.SS2)\)\. Two measurements on SDAR\-8B shape the method:*what*a drafter must predict, and*how*its proposals must be verified once the target samples\.

### 3\.1Draft\-Trajectory Dynamics

Key TakeawayKnowing the future position order is almost as effective as knowing the full draft trajectory: the commit order carries most of the useful drafting information\.

A draft must predict which position the target commits next and what token it writes\. To isolate their value we evaluate*live oracles*that peek at the target continuation and propose \(i\) true*positions*filled with stale\-argmax tokens, \(ii\) true*tokens*ordered by stale confidence, or \(iii\) both; each bounds the gain obtainable from one information source under standard verification\.

Figure 2:\(a\) Live oracle ceilingson SDAR\-8B \(b32\): peak throughput ratios over draft\-free decoding; positions are worth more than tokens, and a perfect drafter reaches3\.83\.8–4\.6×4\.6\\times\(full grid in Table[7](https://arxiv.org/html/2610.22543#A6.T7)\)\.\(b\) Stale coverage: probability that all committed tokens over the nextKKsteps lie within the stale top\-ttcandidates\.Figure[2](https://arxiv.org/html/2610.22543#S3.F2)\(a\) and Table[7](https://arxiv.org/html/2610.22543#A6.T7)\(Appendix[F\.2](https://arxiv.org/html/2610.22543#A6.SS2)\) show that position knowledge dominates on every task and keeps paying across the tested budgets \(NFE reduction5\.4×→7\.1×5\.4\\times\\to 7\.1\\timesfromD​8D8toD​20D20on GSM8K,5\.7×→8\.1×5\.7\\times\\to 8\.1\\timeson MBPP\), whereas true tokens yield only3\.33\.3–6\.5×6\.5\\timesatD​20D20\. The stale distributions already carry token identity: the stale argmax matches the true token in92%92\\%of first\-block commits and≥90%\\geq 90\\%thereafter, and eight stale candidates per position cover89%89\\%of entire eight\-commit continuations \(Figure[2](https://arxiv.org/html/2610.22543#S3.F2)\(b\)\)\. They fail to predict order: the probability that the previous round’s top\-confidence position commits next drops to3%3\\%by block99\(Appendix[F\.8](https://arxiv.org/html/2610.22543#A6.SS8)\)\.*What*the model writes is nearly known;*which position*it writes next is not\.

NFE does not map directly to wall clock, since every drafted round forwards\(1\+D\)​L\(1\{\+\}D\)Lpositions, so we charge each oracle its measured verification cost \(lookahead subtracted\); under this accounting the joint oracle peaks at3\.83\.8–4\.6×4\.6\\times\(at an interior budget,D​12D12\) against an NFE ceiling of8\.08\.0–9\.7×9\.7\\times\. By learning the missing order, SPOT predicts the next commit with75%75\\%accuracy on held\-out trajectories \(Appendix[D\.1](https://arxiv.org/html/2610.22543#A4.SS1)\) against22%22\\%for the stale\-confidence order, recovering7070–85%85\\%of the position oracle’s NFE reduction and5454–81%81\\%of the joint wall\-clock ceiling \(Table[2](https://arxiv.org/html/2610.22543#S5.T2)\)\. Commit\-order prediction is therefore the primary information bottleneck for drafting on this target, token knowledge a complementary second axis, and verification cost a separate limit on realized speed\.

### 3\.2Verification Dynamics Across Sampling Temperatures

Key Takeaways•Exactness\.Coupled and uncoupled matching preserve the target sampling distribution, whereas greedy matching increasingly drifts as temperature rises\.•Acceptance\.Coupling becomes increasingly important at higher temperatures, where independent matching rapidly loses acceptance\.•High\-temperature regime\.Target quality deteriorates sharply beyondT≈1\.5T\{\\approx\}1\.5and becomes the dominant limitation\.

AtT=0T\{=\}0the acceptance test above is*state equality*against a deterministic commit\. AtT\>0T\{\>\}0the commitment is a random draw, and there are three ways to keep verifying by equality: a*greedy match*accepts a drafted state iff it equals the target’s*argmax*commitment, as theT=0T\{=\}0verifier would if simply run at temperature; an*uncoupled match*lets the target sample independently and accepts only if the sample happens to equal the proposal; a*coupled\-noise match*shares the sampler’s Gumbel noise between drafter and verifier, so that the target’s draw is again a deterministic function of the state \(Section[4\.3](https://arxiv.org/html/2610.22543#S4.SS3)\)\. We ran all three on SPOT, Spiffy and the BlockSpec\-style tree atT∈\{0\.5,1,1\.5,2,2\.5\}T\\in\\\{0\.5,1,1\.5,2,2\.5\\\}\(Table[1](https://arxiv.org/html/2610.22543#S3.T1), Appendix[F\.6](https://arxiv.org/html/2610.22543#A6.SS6)\)\.

Table 1:Temperature sweep on SDAR\-8B MBPP\. Draft\-free columns: target accuracy, argmax fraction, mean committed\-token probability\. SPOTD​8D8: NFE and throughput against theTT\-matched draft\-free control\. Greedy\-match columns show distribution drift\. TheD​8D8NFE standard deviation over seeds grows from0\.30\.3atT=0\.5T\{=\}0\.5to8\.58\.5atT=2\.5T\{=\}2\.5\.†Greedy match is not distribution\-preserving, so its NFE is not comparable\.\(i\) Coupled and uncoupled match are exact; greedy match is not\.In both exact regimes the emitted tokens track the draft\-free statistics within sampling noise: over all three drafters atD​8D8andD​20D20andT≤1\.5T\\leq 1\.5\(Table[21](https://arxiv.org/html/2610.22543#A6.T21)\), the emitted argmax fraction stays within0\.0160\.016of the draft\-free value, the mean committed\-token probability within0\.0150\.015, and accuracy within0\.040\.04\. Greedy match drifts, and the drift grows withTT: atT=1\.5T\{=\}1\.5it emits accuracy0\.280\.28where the model samples0\.080\.08and picks the argmax at95\.8%95\.8\\%of commits against the model’s86\.6%86\.6\\%; the drift is proportional to acceptance, so it makes the*best*drafter the*most*biased\.\(ii\) Coupling keeps acceptance alive as the target’s samples move away from its argmax\.AtT≤1T\\leq 1the sampled token is the argmax9898–99%99\\%of the time and the two verifiers agree within1%1\\%in NFE; atT=1\.5T\{=\}1\.5uncoupled costs1212–16%16\\%more NFE for SPOT and Spiffy \(74\.0→82\.674\.0\\to 82\.6and94\.2→108\.994\.2\\to 108\.9\); atT=2T\{=\}2uncoupled accepts almost nothing \(256\.7256\.7NFE against269\.2269\.2draft\-free\) while coupled still drafts\. The mechanism is the coupling bound of[Daliri et al\. \(2025\)](https://arxiv.org/html/2610.22543#bib.bib17): a stale\-informed proposal survives with probability at least\(1−δ\)/\(1\+δ\)\(1\-\\delta\)/\(1\+\\delta\)in the total variationδ\\deltabetween stale and fresh conditionals, whereas an independent sample equals the argmax only with probabilitypmaxp\_\{\\max\}\.\(iii\) At high temperatures, target quality becomes the limiting factor\.Draft\-free accuracy already drops sharply to0\.080\.08atT=1\.5T\{=\}1\.5and reaches0\.000\.00atT≥2T\\geq 2, leaving little useful generation quality to preserve\.

## 4Method

### 4\.1Setup

Letpθp\_\{\\theta\}be a block\-causal diffusion language model\([Cheng et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib19);[Bie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib29)\)\. Within an active block of lengthLL, a statex∈𝒳=\(𝒱∪\{m\}\)Lx\\in\\mathcal\{X\}=\(\\mathcal\{V\}\\cup\\\{\\textsc\{m\}\\\}\)^\{L\}assigns each position either a token in vocabulary𝒱\\mathcal\{V\}or the mask symbolm, andℳ⁡\(x\)=\{s:xs=m\}\\mathcal\{M\}\(x\)=\\\{s:x\_\{s\}=\\textsc\{m\}\\\}denotes its masked positions\. Given the completed causal contextCC, one target forward returnspθ,s\(⋅∣x,C\)p\_\{\\theta,s\}\(\\cdot\\mid x,C\)for everys∈ℳ⁡\(x\)s\\in\\mathcal\{M\}\(x\)\. Lets⋆​\(x\)=arg​maxs∈ℳ⁡\(x\)⁡maxv∈𝒱​pθ,s​\(v∣x,C\)s^\{\\star\}\(x\)=\\argmax\_\{s\\in\\mathcal\{M\}\(x\)\}\\max\_\{v\\in\\mathcal\{V\}\}p\_\{\\theta,s\}\(v\\mid x,C\)be the highest\-confidence masked position andx⊕\(s,v\)x\\oplus\(s,v\)be the state obtained by filling positionsswith tokenvv\. AtT=0T=0, the target commitsv0⋆​\(x\)=arg​maxv⁡pθ,s⋆​\(x\)​\(v∣x,C\)v^\{\\star\}\_\{0\}\(x\)=\\argmax\_\{v\}p\_\{\\theta,s^\{\\star\}\(x\)\}\(v\\mid x,C\), giving the deterministic transition

Φ0​\(x\)=x⊕\(s⋆​\(x\),v0⋆​\(x\)\)\.\\Phi\_\{0\}\(x\)=x\\oplus\\bigl\(s^\{\\star\}\(x\),v^\{\\star\}\_\{0\}\(x\)\\bigr\)\.\(1\)AtT\>0T\>0, the committed token is instead sampled from the temperature\-adjusted conditional

πT,s​\(v∣x,C\)=pθ,s​\(v∣x,C\)1/T∑u∈𝒱pθ,s​\(u∣x,C\)1/T\.\\pi\_\{T,s\}\(v\\mid x,C\)=\\frac\{p\_\{\\theta,s\}\(v\\mid x,C\)^\{1/T\}\}\{\\sum\_\{u\\in\\mathcal\{V\}\}p\_\{\\theta,s\}\(u\\mid x,C\)^\{1/T\}\}\.\(2\)
A*drafter*𝒟\\mathcal\{D\}proposes a finite tree of hypothetical future states rooted at the current state, with each edge representing one slot–token commitment\. The target evaluates the root and all drafted states in one batched forward\. The verifier accepts an edge\(y,z\)\(y,z\)only whenzzequals the target’s own next state fromyy, which we call*state\-equality verification*; because the target has a unique next commitment, the accepted subset of a draft tree is always a single path\. LetXTDFX\_\{T\}^\{\\mathrm\{DF\}\}andXT,𝒟SpecX\_\{T,\\mathcal\{D\}\}^\{\\mathrm\{Spec\}\}denote the draft\-free and speculative committed trajectories\. We call a verifier*distribution preserving*whenXT,𝒟Spec​=𝑑​XTDF\.X\_\{T,\\mathcal\{D\}\}^\{\\mathrm\{Spec\}\}\\overset\{d\}\{=\}X\_\{T\}^\{\\mathrm\{DF\}\}\.

### 4\.2The drafter: learned commit order

#### Order model\.

At speculative roundtt, letp^t,s​\(⋅\)\\widehat\{p\}\_\{t,s\}\(\\cdot\)denote the most recently available target conditional at masked positionss\. These conditionals are*stale*: hypothetical future commitments change the true target distribution, but no additional target forward is performed while drafting\. Our drafter is a three\-layer transformer encoder \(dmodel=128d\_\{\\mathrm\{model\}\}=128,88heads,0\.620\.62M parameters\)\. Each masked position is represented using its stale top\-3232probabilities \(raw and log\), eight summary statistics, and two path features encoding the positions already committed along the hypothetical prefix and its depth\. Given a hypothetical stateyy, the model predictsqϕ​\(s∣y\)q\_\{\\phi\}\(s\\mid y\), wheres∈ℳ⁡\(y\),s\\in\\mathcal\{M\}\(y\),a distribution over the target’s next committed position\. The same prefix\-conditional predictor is learned in Algorithm[1](https://arxiv.org/html/2610.22543#alg1)and recursively queried during Algorithm[2](https://arxiv.org/html/2610.22543#alg2)\. The drafter uses no target hidden states, token embeddings, or text, and can therefore be easily evaluated across different DLMs\.

#### Training\.

We trainqϕq\_\{\\phi\}entirely from draft\-free target trajectories \(Algorithm[1](https://arxiv.org/html/2610.22543#alg1)\)\. Let\(st⋆,st\+1⋆,…\)\(s\_\{t\}^\{\\star\},s\_\{t\+1\}^\{\\star\},\\ldots\)denote the target commit sequence following decoding statextx\_\{t\}\. For each prefix depthdd, the firstddcommitments define a hypothetical stateyt,dy\_\{t,d\}while the stale conditionalsp^t\\widehat\{p\}\_\{t\}remain fixed; the next target commitmentst\+d⋆s\_\{t\+d\}^\{\\star\}is used as supervision\. This produces prefix\-conditioned examples\(p^t,yt,d,st\+d⋆\)\(\\widehat\{p\}\_\{t\},y\_\{t,d\},s\_\{t\+d\}^\{\\star\}\)that match the states encountered during recursive drafting\. We optimizeℒorder​\(ϕ\)=−𝔼\(p^,y,s⋆\)∼ℰ​log⁡qϕ​\(s⋆∣y\),\\mathcal\{L\}\_\{\\mathrm\{order\}\}\(\\phi\)=\-\\mathbb\{E\}\_\{\(\\widehat\{p\},y,s^\{\\star\}\)\\sim\\mathcal\{E\}\}\\log q\_\{\\phi\}\(s^\{\\star\}\\mid y\),whereℰ\\mathcal\{E\}is the resulting training set\. Full data\-generation and optimization details are given in Appendix[D\.1](https://arxiv.org/html/2610.22543#A4.SS1)\.

#### Inference\.

At inference \(Algorithm[2](https://arxiv.org/html/2610.22543#alg2)\), drafting starts from the current target statextx\_\{t\}and stale conditionalsp^t\\widehat\{p\}\_\{t\}\. For every candidate prefixyy, the order model scores the remaining positions usingqϕ\(⋅∣y\)q\_\{\\phi\}\(\\cdot\\mid y\), and each position is paired with token hypotheses from the stale target conditional\. Letℋt​\(s\)\\mathcal\{H\}\_\{t\}\(s\)denote the token\-hypothesis set for positionss\. AtT=0T=0, its primary hypothesis isv~0​\(s\)=arg​maxv⁡p^t,s​\(v\),\\widetilde\{v\}\_\{0\}\(s\)=\\argmax\_\{v\}\\widehat\{p\}\_\{t,s\}\(v\),with the runner\-up additionally included when the stale top\-11probability is below0\.60\.6\. AtT\>0T\>0,ℋt​\(s\)\\mathcal\{H\}\_\{t\}\(s\)uses the coupled stale draw of Eq\.[4](https://arxiv.org/html/2610.22543#S4.E4), introduced formally in Section[4\.3](https://arxiv.org/html/2610.22543#S4.SS3)\.

For a candidate childz=y⊕\(s,v\)z=y\\oplus\(s,v\), define its*prefix mass*recursively as

m⁡\(z\)=m⁡\(y\)​qϕ​\(s∣y\)​p^t,s​\(v\),m⁡\(xt\)=1\.m\(z\)=m\(y\)\\,q\_\{\\phi\}\(s\\mid y\)\\,\\widehat\{p\}\_\{t,s\}\(v\),\\qquad m\(x\_\{t\}\)=1\.\(3\)Algorithm[2](https://arxiv.org/html/2610.22543#alg2)uses DDTree, a level\-batched best\-first search adapted from[Ringel and Romano \(2026\)](https://arxiv.org/html/2610.22543#bib.bib16), to allocate budgetDDto the highest\-mass candidate states\. Since every factor in Eq\.[3](https://arxiv.org/html/2610.22543#S4.E3)lies in\[0,1\]\[0,1\], a child’s mass cannot exceed its parent’s, allowing selected nodes to be expanded level\-by\-level while recovering the same top\-DDfixpoint as sequential best\-first search\. A calibration temperatureτ\\tauis applied toqϕq\_\{\\phi\}before search; the final model usesτ=1\\tau=1\.

#### Adaptive budget and padding\.

Algorithm[2](https://arxiv.org/html/2610.22543#alg2)treatsDDas a hard upper bound\. We additionally apply a prefix\-mass floorγ\\gamma, discarding candidates withm⁡\(z\)<γm\(z\)<\\gammaand thereby adapting the realized verification budget to drafter confidence\. Independently, the target batch may be padded with dummy replicas to maintain a fixed\(1\+D\)​L\(1\+D\)Lshape and reduce batch\-shape\-dependent numerical variation\. We report the paddedγ=0\\gamma=0configuration as SPOT and the unpadded configuration with a floor calibrated once per target \(γ=0\.1\\gamma=0\.1on SDAR\-8B; Appendix[D\.5](https://arxiv.org/html/2610.22543#A4.SS5)\) as SPOT\-γ\\gamma\. The complete budget, mass\-floor, and padding ablations are given in Section[5\.5](https://arxiv.org/html/2610.22543#S5.SS5)\.

### 4\.3Diffusion\-block verification by coupled\-noise match

#### Greedy verification\.

AtT=0T=0, the target transitionΦ0\\Phi\_\{0\}in Eq\.[1](https://arxiv.org/html/2610.22543#S4.E1)is deterministic\. Therefore, a drafted edge\(y,z\)\(y,z\)is accepted exactly whenz=Φ0​\(y\)z=\\Phi\_\{0\}\(y\)\. Proposition[1](https://arxiv.org/html/2610.22543#Thmproposition1)is elementary and included both for completeness and as a basis for the proof of Proposition[2](https://arxiv.org/html/2610.22543#Thmproposition2)\.

###### Proposition 1\(Greedy exactness\)\.

For any drafter𝒟\\mathcal\{D\}and any finite chain or tree draft, state\-equality verification atT=0T=0produces exactly the draft\-free target trajectory:X0,𝒟Spec=X0DF\.X^\{\\mathrm\{Spec\}\}\_\{0,\\mathcal\{D\}\}=X^\{\\mathrm\{DF\}\}\_\{0\}\.

#### Sampling with shared Gumbel noise\.

AtT\>0T\>0, state equality cannot use independent drafter and verifier samples: even identical distributions would then produce different tokens with nonzero probability\. Instead, we couple their sampling through shared randomness\. For every absolute output positionaaand vocabulary itemvv, drawGa,v​∼iid​Gumbel​\(0,1\),G\_\{a,v\}\\overset\{\\mathrm\{iid\}\}\{\\sim\}\\mathrm\{Gumbel\}\(0,1\),and letG=\{Ga,v\}a,vG=\\\{G\_\{a,v\}\\\}\_\{a,v\}denote the resulting table\. The table is shared by drafter and verifier and indexed by absolute position, so the same position uses the same Gumbel row in both\.

For target statexx, the target draw isvG⋆​\(x\)=arg​maxv∈𝒱⁡\[1T​log⁡pθ,s⋆​\(x\)​\(v∣x,C\)\+Ga​\(s⋆​\(x\)\),v\]v^\{\\star\}\_\{G\}\(x\)=\\argmax\_\{v\\in\\mathcal\{V\}\}\\left\[\\frac\{1\}\{T\}\\log p\_\{\\theta,s^\{\\star\}\(x\)\}\(v\\mid x,C\)\+G\_\{a\(s^\{\\star\}\(x\)\),v\}\\right\], which defines the transitionΦG​\(x\)=x⊕\(s⋆​\(x\),vG⋆​\(x\)\)\.\\Phi\_\{G\}\(x\)=x\\oplus\\bigl\(s^\{\\star\}\(x\),v^\{\\star\}\_\{G\}\(x\)\\bigr\)\.By the Gumbel–argmax identity,vG⋆\(x\)∼πT,s⋆​\(x\)\(⋅∣x,C\)v^\{\\star\}\_\{G\}\(x\)\\sim\\pi\_\{T,s^\{\\star\}\(x\)\}\(\\cdot\\mid x,C\)\. The drafter applies the*same*noise to its stale conditional:

v~G​\(s\)=arg​maxv∈𝒱⁡\[1T​log⁡p^t,s​\(v\)\+Ga⁡\(s\),v\]\.\\widetilde\{v\}\_\{G\}\(s\)=\\argmax\_\{v\\in\\mathcal\{V\}\}\\left\[\\frac\{1\}\{T\}\\log\\widehat\{p\}\_\{t,s\}\(v\)\+G\_\{a\(s\),v\}\\right\]\.\(4\)Thus, whenever the stale and fresh conditionals have the same Gumbel\-perturbed argmax, drafter and target propose exactly the same token\. Conditioned onGG,ΦG\\Phi\_\{G\}is deterministic, so verification again reduces to state equality:z=ΦG​\(y\)\.z=\\Phi\_\{G\}\(y\)\.

###### Proposition 2\(Coupled\-noise exactness\)\.

For any drafter𝒟\\mathcal\{D\}andT\>0T\>0, the Gumbel\-driven draft\-free trajectory satisfiesXTDF​\(G\)​=𝑑​XTDFX^\{\\mathrm\{DF\}\}\_\{T\}\(G\)\\overset\{d\}\{=\}X^\{\\mathrm\{DF\}\}\_\{T\}, and for every realizationggofGG,XT,𝒟Spec​\(g\)=XTDF​\(g\)X^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\(g\)=X^\{\\mathrm\{DF\}\}\_\{T\}\(g\)\. Consequently,XT,𝒟Spec​=𝑑​XTDFX^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\\overset\{d\}\{=\}X^\{\\mathrm\{DF\}\}\_\{T\}\.

Propositions[1](https://arxiv.org/html/2610.22543#Thmproposition1)and[2](https://arxiv.org/html/2610.22543#Thmproposition2)therefore separate*correctness*from*draft quality*: the verifier always preserves the target trajectory or distribution, while the drafter determines only how many future target transitions are already present in the tree and can be accepted in one target forward\. The complete formal setup and proofs are given in Appendix[A](https://arxiv.org/html/2610.22543#A1)\. For a drafted positionss, letπ^T,s\\widehat\{\\pi\}\_\{T,s\}andπT,s\\pi\_\{T,s\}denote the stale and fresh temperature\-adjusted conditionals\. The shared\-Gumbel coupling of[Daliri et al\. \(2025\)](https://arxiv.org/html/2610.22543#bib.bib17)satisfiesPr\[v~G\(s\)=vG⋆\(s\)\]≥1−δT1\+δT,\\Pr\[\\widetilde\{v\}\_\{G\}\(s\)=v^\{\\star\}\_\{G\}\(s\)\]\\geq\\frac\{1\-\\delta\_\{T\}\}\{1\+\\delta\_\{T\}\},whereδT=DTV​\(π^T,s,πT,s\)\.\\delta\_\{T\}=D\_\{\\mathrm\{TV\}\}\\\!\\left\(\\widehat\{\\pi\}\_\{T,s\},\\pi\_\{T,s\}\\right\)\.Hence coupled verification becomes more effective as the stale and fresh sampling distributions become closer\.

#### Relation to rejection sampling\.

When the drafter exposes an explicit sequential proposal distribution, exact verification can alternatively be performed by rejection sampling\. Appendix[B](https://arxiv.org/html/2610.22543#A2)formalizes this alternative, proves its exactness and maximal per\-link acceptance probability1−DTV​\(πk,qk\)1\-D\_\{\\mathrm\{TV\}\}\(\\pi\_\{k\},q\_\{k\}\), and validates its implementation\. Classical per\-link RS assumes a sequential proposal interface, whereas coupled\-noise match requires only state equality and therefore applies directly to our batched draft trees\.

## 5Experiments

We establish that SPOT takes fewer target forwards with higher throughput than every baseline atT=0T\{=\}0andT=1T\{=\}1\(Section[5\.2](https://arxiv.org/html/2610.22543#S5.SS2)and Section[5\.3](https://arxiv.org/html/2610.22543#S5.SS3)\), then a checkpoint transfers across targets without additional training \(Section[5\.4](https://arxiv.org/html/2610.22543#S5.SS4)\), and finally each component of the drafter earns its place \(Section[5\.5](https://arxiv.org/html/2610.22543#S5.SS5)\)\.

### 5\.1Experimental setup

Our headline target isSDAR\-8B\([Cheng et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib19)\), a dense block\-causal diffusion LM at block widthL=32L\{=\}32with its exact prefix KV cache\. The same recipe gives drafters forSDAR\-4BandSDAR\-1\.7Band for the mixture\-of\-expertsLLaDA2\.0\-mini\([Bie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib29)\), and transfer is measured on two further targets,LLaDA\-8B\([Nie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib2)\)andDream\-7B\([Ye et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib13)\): six diffusion language models in all\. We evaluate GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2610.22543#bib.bib10)\), MBPP\([Austin et al\., 2021b](https://arxiv.org/html/2610.22543#bib.bib8)\), HumanEval\([Chen et al\., 2021](https://arxiv.org/html/2610.22543#bib.bib9)\)and MATH500\([Hendrycks et al\., 2021](https://arxiv.org/html/2610.22543#bib.bib11)\)onn=350n\{=\}350samples \(all164164for HumanEval\) over three seeds, except where a caption says otherwise, at generation length256256and budgetsD∈\{8,12,20\}D\\in\\\{8,12,20\\\}and report NFE per sample, accuracy, and a*throughput ratio*against the draft\-free run of the same task, model and temperature\. SPOT is one checkpoint and one calibration per target, and SPOT\-γ\\gammaruns it unpadded with that floor\. Baselines are Spiffy’s calibrated draft graph\([Agrawal et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib18)\), an SSD\-style confidence\-ordered free chain\([Gao et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib4)\)and a BlockSpec\-style confidence tree\([Pan et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib21)\), all sharing our decoder and verifier so that the comparison is about drafting alone\. Other block widths, temperatures, and generation lengths, and the full protocol, are in Appendices[D](https://arxiv.org/html/2610.22543#A4),[E](https://arxiv.org/html/2610.22543#A5)and[F](https://arxiv.org/html/2610.22543#A6)\.

### 5\.2Main results

Table 2:Main results, SDAR\-8B b32 atT=0T\{=\}0andT=1T\{=\}1:NFE per sample\(throughput ratioover the shared draft\-free control\), best in bold, second best underlined\. Shaded rows give each block’s accuracy range\. SPOT\-γ\\gamma: unpadded,γ=0\.1\\gamma\{=\}0\.1\. AtT=1T\{=\}1all methods use coupled\-noise match except the free chain \(RS\)\. Over seeds the ratio’s standard deviation is at most0\.150\.15and NFE’s at most1\.31\.3\.In the upper block of Table[2](https://arxiv.org/html/2610.22543#S5.T2), SPOT delivers3\.23\.2–4\.7×4\.7\\timesfewer target forwards than draft\-free decoding atD​8D8and4\.14\.1–6\.8×6\.8\\timesatD​20D20, at2\.02\.0–2\.7×2\.7\\timessingle\-device wall atD​8D8; against Spiffy’s calibrated graph it has1515–35%35\\%fewer NFE and1\.11\.1–1\.5×1\.5\\timesits throughput in every task\-and\-budget setting, with larger margins against the chain and the tree\. The baselines’ throughput collapses with budget where their NFE does not: the tree is*slower than draft\-free*atD​20D20on every task and the chain on three of four, since a wider draft costs\(1\+D\)​L\(1\{\+\}D\)Lpositions whether or not it is accepted, while they accept only1\.11\.1–4\.04\.0tokens per round against SPOT’s4\.54\.5–7\.77\.7\(Appendix[F\.8](https://arxiv.org/html/2610.22543#A6.SS8)\)\. SPOT\-γ\\gammaremoves the budget dependence: its throughput ratio is flat in the budget \(2\.32\.3–3\.2×3\.2\\timesatD​8D8,D​12D12andD​20D20alike\) and at or above the padded recipe in every SDAR\-8B setting at both temperatures\. The floor pays44–2424NFE per setting, so SPOT stays NFE\-optimal while SPOT\-γ\\gammastill has fewer forwards than every baseline in2121of2424settings\. MRP\([Xu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib12)\), the closest learned drafter, reports1\.171\.17–1\.40×1\.40\\timesover one\-token\-per\-step decoding on these same targets in a different implementation, against SPOT’s2\.02\.0–3\.6×3\.6\\timesatD​8D8, a margin wider than that difference can account for, and its argmax\-equality acceptance has no distribution\-preserving extension to sampling \(Table[5](https://arxiv.org/html/2610.22543#A4.T5)\)\.

### 5\.3Sampling temperature: coupled\-noise match against the baselines

The lower block of Table[2](https://arxiv.org/html/2610.22543#S5.T2)shows that nothing changes when the target samples atT=1T\{=\}1: under the same verifier, SPOT keeps1515–34%34\\%fewer NFE than Spiffy and1\.151\.15–1\.48×1\.48\\timesits throughput\. Free chain under rejection sampling is the strongest baseline on MBPP NFE, but stays0\.470\.47–0\.920\.92below SPOT in throughput, since every accepted link pays an acceptance test, and collapses on MATH500 \(116116NFE,0\.60×0\.60\\timesatD​20D20\)\. EveryT=1T\{=\}1result is distribution\-preserving, with accuracies within0\.060\.06of the control over seeds; tree\+\+coupled\-noise match is the deployment configuration at any temperature \(Appendix[F\.7](https://arxiv.org/html/2610.22543#A6.SS7)\)\. The ratios grow with generation length: at512512SPOT reaches2\.79×2\.79\\timesdraft\-free atD​8D8and SPOT\-γ\\gamma3\.23×3\.23\\times, against2\.45×2\.45\\timesand2\.86×2\.86\\timesat256256\(Appendix[F\.5](https://arxiv.org/html/2610.22543#A6.SS5)\)\.

### 5\.4Transferability within and across model families

Table 3:Checkpoint transfer \(T=0T\{=\}0, b32,D​8D8, padded\): each checkpoint pointed at a target with no retraining or recalibration,NFE per sample, the better drafter in bold\. Both drafters run the same target weights over the same\(1\+D\)​L\(1\{\+\}D\)Lpositions per forward, so throughput tracks NFE \(ratios in Table[14](https://arxiv.org/html/2610.22543#A6.T14)\)\.†approximate prefix cache of[Wu et al\. \(2026\)](https://arxiv.org/html/2610.22543#bib.bib15), not distribution\-preserving\.Transfer takes two forms\. Re\-running identical training recipe onSDAR\-4Breproduces the SDAR\-8B picture, lowest NFE and highest throughput in every comparison atT=0T\{=\}0andT=1T\{=\}1, and onLLaDA2\.0\-mini, a mixture\-of\-experts target with a different tokenizer, it gives the lowest NFE across3636comparisons and the higher throughput in3434\. OnSDAR\-1\.7BSPOT keeps the lowest NFE everywhere but cedes three throughput columns atT=0T\{=\}0and two atT=1T\{=\}1to Spiffy, where a cheap target forward makes the padded draft much of the step\. The method carries to block width 16 untuned \(Appendices[F\.3](https://arxiv.org/html/2610.22543#A6.SS3)and[F\.4](https://arxiv.org/html/2610.22543#A6.SS4)\)\. Table[3](https://arxiv.org/html/2610.22543#S5.T3)swaps the trained checkpoints between targets, tokenizers included, with no retraining or recalibration, and Table[14](https://arxiv.org/html/2610.22543#A6.T14)gives the full six\-target, four\-drafter matrix: the transferred drafter costs33–7%7\\%NFE on SDAR\-8B and*still beats every baseline calibrated on SDAR\-8B*\(Table[2](https://arxiv.org/html/2610.22543#S5.T2)\), and33–11%11\\%on mini, where it stays ahead of every baseline on GSM8K and MATH500\. Beyond the block\-causal family,LLaDA\-8B\([Nie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib2)\)andDream\-7B\([Ye et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib13)\)attend over the sequence, so drafted runs and control both use the approximate cache of[Wu et al\. \(2026\)](https://arxiv.org/html/2610.22543#bib.bib15)\(Appendix[D](https://arxiv.org/html/2610.22543#A4)\) and these rows measure reach rather than losslessness: checkpoints that never saw either model cut LLaDA\-8B’s256256forwards to5555–8888and Dream\-7B’s to5151–7979: the order model learns*where a diffusion target commits*, and much of that carries\.

### 5\.5Ablations

Table[17](https://arxiv.org/html/2610.22543#A6.T17)\(Appendix[F\.6](https://arxiv.org/html/2610.22543#A6.SS6)\) varies one component on SDAR\-8B MBPP\. The largest effect comes from replacing the tree with a chain: NFE increases by7\.37\.3atD​8D8and by11\.911\.9atD​20D20, while ratio drops from1\.721\.72to1\.341\.34\. Budget therefore pays through branching, not a longer chain\. The token hedge and the scaled corpus are each worth under one NFE, and padding moves throughput by at most0\.040\.04\. The mass floor trades NFE for wall\-clock speed, but only without padding: atD​20D20,γ=0/0\.05/0\.1\\gamma=0/0\.05/0\.1gives39\.6/44\.3/47\.839\.6/44\.3/47\.8NFE at1\.76/2\.84/2\.99×1\.76/2\.84/2\.99\\timeswall, by computing fewer verification positions\.

## 6Discussion

DLM decoding remains constrained by a fundamental speed–quality trade\-off\. SPOT addresses this challenge with a two\-fold solution\. We show that stale target distributions already contain highly reliable token information, making commit order the key missing signal for effective speculation\. A lightweight0\.620\.62M\-parameter model predicts this order and constructs multi\-step draft trees, while coupled\-noise verification preserves the target distribution under sampling\. SPOT recovers5454–81%81\\%of the cost\-aware oracle ceiling, uses1515–35%35\\%fewer target forwards than an offline\-calibrated draft graph at up to1\.5×1\.5\\timesits throughput, and transfers across six DLMs without retraining\. More broadly, our results show that substantially faster DLM generation can be achieved without sacrificing the quality or exactness of the underlying decoding process, establishing commit\-order prediction as a promising foundation for the next generation of speculative decoding methods for DLMs\.

#### Limitations\.

The main limitation of SPOT is that its gains diminish when verification is costly or acceptance is low\. Verification becomes the bottleneck at wide blocks, while exact tree packing remains unresolved under bidirectional attention\. Gains also narrow on low\-acceptance tasks, and atT≥2T\\geq 2theT=0T\{=\}0\-trained order model is outperformed by noise\-aware ranking \(Appendix[F\.6](https://arxiv.org/html/2610.22543#A6.SS6)\)\.

## AI use statement

Generative AI tools were used during manuscript preparation to support language refinement and editing, and during development to assist with implementation of the experimental codebase\. They were not used to formulate the research questions, develop the proposed method, define the evaluation procedure, or draw conclusions from the experimental findings\. The quantitative results reported in this paper originate from the experiments described in Section[5\.1](https://arxiv.org/html/2610.22543#S5.SS1)and the appendix\. The authors inspected and verified all AI\-assisted contributions, including code incorporated into the experimental workflow, and remain fully responsible for the manuscript, implementation, experimental results, and conclusions\.

### Ethics statement

This work focuses on accelerating inference in diffusion large language models and does not involve human subjects or the collection of personal data\. We do not anticipate any ethical concerns beyond those associated with the underlying language models and their downstream applications\.

### Reproducibility statement

Code and the trained drafter checkpoints are available at[https://github\.com/Eliezer318/SPOT\-dLLM](https://github.com/Eliezer318/SPOT-dLLM), with instructions to reproduce the experiments\. Every number in this paper comes from one benchmark run on one machine \(eight identical NVIDIA RTX 6000 Ada cards; one job per card\), with the protocol of Section[5\.1](https://arxiv.org/html/2610.22543#S5.SS1)and Appendix[D](https://arxiv.org/html/2610.22543#A4)\.

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## Appendix AFormal Setup and Exactness Proofs

This section makes the decoding semantics used in Section[4](https://arxiv.org/html/2610.22543#S4)explicit and proves Propositions[1](https://arxiv.org/html/2610.22543#Thmproposition1)and[2](https://arxiv.org/html/2610.22543#Thmproposition2)\.

### A\.1Formal decoding model

#### States and terminal states\.

Recall

𝒳=\(𝒱∪\{m\}\)L,ℳ⁡\(x\)=\{s:xs=m\}\.\\mathcal\{X\}=\(\\mathcal\{V\}\\cup\\\{\\textsc\{m\}\\\}\)^\{L\},\\qquad\\mathcal\{M\}\(x\)=\\\{s:x\_\{s\}=\\textsc\{m\}\\\}\.Define the nonterminal state space

𝒳nt=\{x∈𝒳:ℳ⁡\(x\)≠∅\}\.\\mathcal\{X\}\_\{\\mathrm\{nt\}\}=\\\{x\\in\\mathcal\{X\}:\\mathcal\{M\}\(x\)\\neq\\varnothing\\\}\.All target transition maps below are defined on𝒳nt\\mathcal\{X\}\_\{\\mathrm\{nt\}\}\.

The completed causal contextCCis fixed while an active block is decoded\. A target forward at statexxreturns

pθ,s\(⋅∣x,C\)p\_\{\\theta,s\}\(\\cdot\\mid x,C\)for everys∈ℳ⁡\(x\)s\\in\\mathcal\{M\}\(x\)\.

#### Idealized target evaluation\.

The exactness results concern the mathematical target function: evaluatingpθ\(⋅∣x,C\)p\_\{\\theta\}\(\\cdot\\mid x,C\)as one replica of a batch is assumed to return the same mathematical conditional as evaluating the same\(x,C\)\(x,C\)alone\. This separates algorithmic exactness from finite\-precision effects caused by physical batch shape\.

#### Position selection\.

The target chooses

s⋆​\(x\)=arg​maxs∈ℳ⁡\(x\)⁡maxv∈𝒱​pθ,s​\(v∣x,C\)\.s^\{\\star\}\(x\)=\\argmax\_\{s\\in\\mathcal\{M\}\(x\)\}\\max\_\{v\\in\\mathcal\{V\}\}p\_\{\\theta,s\}\(v\\mid x,C\)\.All argmax operations use a fixed deterministic tie\-breaking rule\. In particular,s⋆​\(x\)s^\{\\star\}\(x\)depends only on the current target state and not on sampling randomness that has not yet been consumed\.

Fors∈ℳ⁡\(x\)s\\in\\mathcal\{M\}\(x\)andv∈𝒱v\\in\\mathcal\{V\}, write

for the state obtained by replacingxs=mx\_\{s\}=\\textsc\{m\}byvv\.

#### Greedy transition\.

AtT=0T=0,

v0⋆​\(x\)=arg​maxv⁡pθ,s⋆​\(x\)​\(v∣x,C\),v^\{\\star\}\_\{0\}\(x\)=\\argmax\_\{v\}p\_\{\\theta,s^\{\\star\}\(x\)\}\(v\\mid x,C\),and therefore

Φ0:𝒳nt→𝒳,Φ0​\(x\)=x⊕\(s⋆​\(x\),v0⋆​\(x\)\)\.\\Phi\_\{0\}:\\mathcal\{X\}\_\{\\mathrm\{nt\}\}\\rightarrow\\mathcal\{X\},\\qquad\\Phi\_\{0\}\(x\)=x\\oplus\\bigl\(s^\{\\star\}\(x\),v^\{\\star\}\_\{0\}\(x\)\\bigr\)\.Starting from initial block statex0x\_\{0\}, draft\-free greedy decoding is the deterministic orbit

x0DF=x0,xj\+1DF=Φ0​\(xjDF\)x^\{\\mathrm\{DF\}\}\_\{0\}=x\_\{0\},\\qquad x^\{\\mathrm\{DF\}\}\_\{j\+1\}=\\Phi\_\{0\}\(x^\{\\mathrm\{DF\}\}\_\{j\}\)until a terminal state is reached\.

#### Sampling transition\.

AtT\>0T\>0, the target token law at positionssis

πT,s​\(v∣x,C\)=pθ,s​\(v∣x,C\)1/T∑u∈𝒱pθ,s​\(u∣x,C\)1/T\.\\pi\_\{T,s\}\(v\\mid x,C\)=\\frac\{p\_\{\\theta,s\}\(v\\mid x,C\)^\{1/T\}\}\{\\sum\_\{u\\in\\mathcal\{V\}\}p\_\{\\theta,s\}\(u\\mid x,C\)^\{1/T\}\}\.Thus standard draft\-free target sampling at statexxfirst chooses the deterministic positions⋆​\(x\)s^\{\\star\}\(x\)and then draws

V∼πT,s⋆​\(x\)\(⋅∣x,C\)\.V\\sim\\pi\_\{T,s^\{\\star\}\(x\)\}\(\\cdot\\mid x,C\)\.

#### Shared Gumbel representation\.

Leta⁡\(s\)a\(s\)denote the absolute output position corresponding to local block positionss\. Draw

Ga,v​∼iid​Gumbel​\(0,1\)G\_\{a,v\}\\overset\{\\mathrm\{iid\}\}\{\\sim\}\\mathrm\{Gumbel\}\(0,1\)for every absolute positionaaand tokenvv, and let

G=\{Ga,v\}a,v\.G=\\\{G\_\{a,v\}\\\}\_\{a,v\}\.For fixedGG, define

vG⋆​\(x\)=arg​maxv⁡\[1T​log⁡pθ,s⋆​\(x\)​\(v∣x,C\)\+Ga​\(s⋆​\(x\)\),v\]v^\{\\star\}\_\{G\}\(x\)=\\argmax\_\{v\}\\left\[\\frac\{1\}\{T\}\\log p\_\{\\theta,s^\{\\star\}\(x\)\}\(v\\mid x,C\)\+G\_\{a\(s^\{\\star\}\(x\)\),v\}\\right\]and

ΦG:𝒳nt→𝒳,ΦG​\(x\)=x⊕\(s⋆​\(x\),vG⋆​\(x\)\)\.\\Phi\_\{G\}:\\mathcal\{X\}\_\{\\mathrm\{nt\}\}\\rightarrow\\mathcal\{X\},\\qquad\\Phi\_\{G\}\(x\)=x\\oplus\\bigl\(s^\{\\star\}\(x\),v^\{\\star\}\_\{G\}\(x\)\\bigr\)\.For any fixed realizationG=gG=g,Φg\\Phi\_\{g\}is deterministic\.

#### Draft trees\.

At speculative roundtt, a drafter𝒟\\mathcal\{D\}returns a finite rooted tree

𝒯t=\(𝒩t,ℰt\)\\mathcal\{T\}\_\{t\}=\(\\mathcal\{N\}\_\{t\},\\mathcal\{E\}\_\{t\}\)whose root is the current committed statextx\_\{t\}\. Every edge\(y,z\)∈ℰt\(y,z\)\\in\\mathcal\{E\}\_\{t\}corresponds to exactly one hypothetical commitment:

for somes∈ℳ⁡\(y\)s\\in\\mathcal\{M\}\(y\)andv∈𝒱v\\in\\mathcal\{V\}\.

The exactness arguments make no assumption about how this tree is constructed\. In particular,𝒟\\mathcal\{D\}may be learned or hand\-designed, may return a chain or a tree, may depend on all previous target outputs, and atT\>0T\>0may depend on the shared randomnessGG\.

#### State\-equality verifier\.

LetΦ\\PhidenoteΦ0\\Phi\_\{0\}forT=0T=0orΦG\\Phi\_\{G\}after fixing the shared randomness forT\>0T\>0\. Beginning at the rooty0=xty\_\{0\}=x\_\{t\}, the verifier proceeds recursively\.

Ifyjy\_\{j\}has a drafted childzzsuch that

the verifier accepts that child and continues from it\. If no such child exists, it rejects all remaining descendants, commitsΦ⁡\(yj\)\\Phi\(y\_\{j\}\)directly, and terminates the speculative round\. Ifyjy\_\{j\}is terminal, no further transition is made\.

SinceΦ⁡\(yj\)\\Phi\(y\_\{j\}\)is a unique one\-step extension, at most one child of a node can be accepted\. Hence the accepted portion of an arbitrary draft tree is a single path\.

LetXTDFX^\{\\mathrm\{DF\}\}\_\{T\}denote the complete draft\-free trajectory andXT,𝒟SpecX^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}the complete committed speculative trajectory\. A verifier is distribution preserving when

XT,𝒟Spec​=𝑑​XTDFX^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\\overset\{d\}\{=\}X^\{\\mathrm\{DF\}\}\_\{T\}for every admissible drafter𝒟\\mathcal\{D\}\.

### A\.2Proof of Proposition[1](https://arxiv.org/html/2610.22543#Thmproposition1)

###### Proof\.

Fix targetpθp\_\{\\theta\}, causal contextCC, initial block statex0x\_\{0\}, and an arbitrary drafter𝒟\\mathcal\{D\}\. By the fixed deterministic tie\-breaking rule, atT=0T=0the target transition

Φ0:𝒳nt→𝒳\\Phi\_\{0\}:\\mathcal\{X\}\_\{\\mathrm\{nt\}\}\\rightarrow\\mathcal\{X\}is deterministic and gives a unique successor for every nonterminal state\.

Let

x0DF=x0,xk\+1DF=Φ0​\(xkDF\)x^\{\\mathrm\{DF\}\}\_\{0\}=x\_\{0\},\\qquad x^\{\\mathrm\{DF\}\}\_\{k\+1\}=\\Phi\_\{0\}\(x^\{\\mathrm\{DF\}\}\_\{k\}\)be the draft\-free trajectory, and letxkSpecx^\{\\mathrm\{Spec\}\}\_\{k\}denote the state of the speculative decoder afterkkcommitted transitions\.

We prove by induction onkkthat

xkSpec=xkDFx^\{\\mathrm\{Spec\}\}\_\{k\}=x^\{\\mathrm\{DF\}\}\_\{k\}for every commitment before termination\.

#### Base case\.

Fork=0k=0, both procedures start from the same initial state:

x0Spec=x0=x0DF\.x^\{\\mathrm\{Spec\}\}\_\{0\}=x\_\{0\}=x^\{\\mathrm\{DF\}\}\_\{0\}\.

#### Inductive step\.

Assume

xkSpec=xkDFx^\{\\mathrm\{Spec\}\}\_\{k\}=x^\{\\mathrm\{DF\}\}\_\{k\}for some nonterminal state\. FromxkSpecx^\{\\mathrm\{Spec\}\}\_\{k\}, the target has the unique successor

Φ0​\(xkSpec\)\.\\Phi\_\{0\}\(x^\{\\mathrm\{Spec\}\}\_\{k\}\)\.If the draft tree contains this state as a child ofxkSpecx^\{\\mathrm\{Spec\}\}\_\{k\}, state\-equality verification accepts that child\. Otherwise, the verifier rejects the speculative continuation and commits the same state as the target fallback\. Hence, in either case,

xk\+1Spec\\displaystyle x^\{\\mathrm\{Spec\}\}\_\{k\+1\}=Φ0​\(xkSpec\)\\displaystyle=\\Phi\_\{0\}\(x^\{\\mathrm\{Spec\}\}\_\{k\}\)=Φ0​\(xkDF\)\\displaystyle=\\Phi\_\{0\}\(x^\{\\mathrm\{DF\}\}\_\{k\}\)=xk\+1DF\.\\displaystyle=x^\{\\mathrm\{DF\}\}\_\{k\+1\}\.Thus the induction hypothesis holds fork\+1k\+1\.

SinceΦ0​\(x\)\\Phi\_\{0\}\(x\)is unique, at most one child of any tree node can be accepted\. Therefore, even for an arbitrary finite draft tree, the accepted states form a single path, and the argument above applies to every committed transition\.

By induction,

X0,𝒟Spec=X0DF\.X^\{\\mathrm\{Spec\}\}\_\{0,\\mathcal\{D\}\}=X^\{\\mathrm\{DF\}\}\_\{0\}\.∎

The argument is block\-local only for notation\. Since the completed block is identical to draft\-free decoding, the causal context of the next block is also identical\. Induction over blocks therefore gives equality of the complete generated sequence atT=0T=0\.

### A\.3Proof of Proposition[2](https://arxiv.org/html/2610.22543#Thmproposition2)

###### Proof\.

FixT\>0T\>0, targetpθp\_\{\\theta\}, causal contextCC, initial statex0x\_\{0\}, and an arbitrary drafter𝒟\\mathcal\{D\}\.

The proof has three steps\. First, we show that the shared\-Gumbel construction induces the same draft\-free trajectory distribution as ordinary temperature\-TTsampling\. Second, conditioned on a realization of the Gumbel table, verification reduces to the deterministic case of Proposition[1](https://arxiv.org/html/2610.22543#Thmproposition1)\. Third, we marginalize over the shared randomness\.

#### Step 1: Gumbel\-driven draft\-free law\.

Fix a nonterminal statexxand let

Since\{Ga⁡\(s\),v\}v∈𝒱\\\{G\_\{a\(s\),v\}\\\}\_\{v\\in\\mathcal\{V\}\}are independent standard Gumbels, the Gumbel–argmax identity gives

Pr\[vG⋆\(x\)=u\]\\displaystyle\\Pr\[v^\{\\star\}\_\{G\}\(x\)=u\]=exp⁡\(T−1​log⁡pθ,s​\(u∣x,C\)\)∑v∈𝒱exp⁡\(T−1​log⁡pθ,s​\(v∣x,C\)\)\\displaystyle=\\frac\{\\exp\\\!\\left\(T^\{\-1\}\\log p\_\{\\theta,s\}\(u\\mid x,C\)\\right\)\}\{\\sum\_\{v\\in\\mathcal\{V\}\}\\exp\\\!\\left\(T^\{\-1\}\\log p\_\{\\theta,s\}\(v\\mid x,C\)\\right\)\}\(5\)=pθ,s​\(u∣x,C\)1/T∑v∈𝒱pθ,s​\(v∣x,C\)1/T\\displaystyle=\\frac\{p\_\{\\theta,s\}\(u\\mid x,C\)^\{1/T\}\}\{\\sum\_\{v\\in\\mathcal\{V\}\}p\_\{\\theta,s\}\(v\\mid x,C\)^\{1/T\}\}\(6\)=πT,s​\(u∣x,C\)\.\\displaystyle=\\pi\_\{T,s\}\(u\\mid x,C\)\.\(7\)Thus, for any fixed state, the Gumbel construction induces exactly the temperature\-TTtarget token law\.

It remains to show that using a single Gumbel table indexed by absolute position preserves this law along the full adaptive trajectory\. Consider commitment stepjjof Gumbel\-driven draft\-free decoding, with current statexjx\_\{j\}and selected position

sj=s⋆​\(xj\)\.s\_\{j\}=s^\{\\star\}\(x\_\{j\}\)\.Letℱj\\mathcal\{F\}\_\{j\}denote the history generated by the states and Gumbel rows used during the firstjjcommitments\. Since each commitment fills a previously masked position,sjs\_\{j\}has not been committed before, and therefore the row

Ga⁡\(sj\),⋅G\_\{a\(s\_\{j\}\),\\cdot\}has not previously been used\.

Moreover,sjs\_\{j\}is determined byxjx\_\{j\}and hence byℱj\\mathcal\{F\}\_\{j\}; its selection does not depend on any unused Gumbel row\. Since the Gumbel rows are independent across absolute positions,Ga⁡\(sj\),⋅G\_\{a\(s\_\{j\}\),\\cdot\}remains an independent standard\-Gumbel vector conditional onℱj\\mathcal\{F\}\_\{j\}\. Therefore,

Pr⁡\[vG⋆​\(xj\)=u∣ℱj\]=πT,sj​\(u∣xj,C\)\.\\Pr\\\!\\left\[v^\{\\star\}\_\{G\}\(x\_\{j\}\)=u\\mid\\mathcal\{F\}\_\{j\}\\right\]=\\pi\_\{T,s\_\{j\}\}\(u\\mid x\_\{j\},C\)\.Hence the Gumbel\-driven decoder has the same conditional transition law as ordinary draft\-free sampling at every commitment\. Since both start fromx0x\_\{0\}, the chain rule gives

XTDF​\(G\)​=𝑑​XTDF\.X^\{\\mathrm\{DF\}\}\_\{T\}\(G\)\\overset\{d\}\{=\}X^\{\\mathrm\{DF\}\}\_\{T\}\.\(8\)

#### Step 2: conditional pathwise exactness\.

Fix any realizationG=gG=g\. Then

Φg:𝒳nt→𝒳\\Phi\_\{g\}:\\mathcal\{X\}\_\{\\mathrm\{nt\}\}\\rightarrow\\mathcal\{X\}is a deterministic transition map\.

The proof of Proposition[1](https://arxiv.org/html/2610.22543#Thmproposition1)requires only that the target have a unique deterministic successor at each state and that a drafted child be accepted exactly when it equals that successor\. Both properties hold withΦg\\Phi\_\{g\}in place ofΦ0\\Phi\_\{0\}\. Therefore, for every realizationgg,

XT,𝒟Spec​\(g\)=XTDF​\(g\)\.X^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\(g\)=X^\{\\mathrm\{DF\}\}\_\{T\}\(g\)\.\(9\)
This argument is independent of how𝒟\\mathcal\{D\}constructs its draft\. In particular, the draft may depend on stale target distributions, learned parameters, previous decoding history, or the shared tableGGitself; a drafted edge is committed only when it equals the target successor under the same realizationgg\.

#### Step 3: marginalization\.

By Step 2, for every realizationggof the shared Gumbel table,

XT,𝒟Spec​\(g\)=XTDF​\(g\)\.X^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\(g\)=X^\{\\mathrm\{DF\}\}\_\{T\}\(g\)\.Therefore, whenGGis drawn according to its Gumbel distribution,

XT,𝒟Spec​\(G\)=XTDF​\(G\)\.X^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\(G\)=X^\{\\mathrm\{DF\}\}\_\{T\}\(G\)\.By definition,XT,𝒟SpecX^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}denotes this same random speculative trajectory with its dependence onGGleft implicit\.

Hence, for any setAAof complete decoding trajectories,

Pr\[XT,𝒟Spec∈A\]=Pr\[XTDF\(G\)∈A\]\.\\Pr\\\!\\left\[X^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\\in A\\right\]=\\Pr\\\!\\left\[X^\{\\mathrm\{DF\}\}\_\{T\}\(G\)\\in A\\right\]\.Using equation[8](https://arxiv.org/html/2610.22543#A1.E8),

Pr\[XT,𝒟Spec∈A\]=Pr\[XTDF∈A\]\.\\Pr\\\!\\left\[X^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\\in A\\right\]=\\Pr\\\!\\left\[X^\{\\mathrm\{DF\}\}\_\{T\}\\in A\\right\]\.Since this holds for everyAA,

XT,𝒟Spec​=𝑑​XTDF\.X^\{\\mathrm\{Spec\}\}\_\{T,\\mathcal\{D\}\}\\overset\{d\}\{=\}X^\{\\mathrm\{DF\}\}\_\{T\}\.
Finally, for fixedGG, the committed trajectory is the deterministic orbit ofΦG\\Phi\_\{G\}and is therefore invariant to the choice of drafter\. This establishes drafter\-invariance\.

∎

## Appendix BRejection Sampling as an Alternative Verifier

Coupled\-noise match is the verifier used by SPOT because it requires only state equality and applies directly to DDTree\. If a drafter instead exposes an explicit*sequential proposal distribution*, exact verification can alternatively be performed by rejection sampling \(RS\)\. We formalize this alternative here, prove its correctness and maximal per\-link acceptance probability, and describe the implementation\-level validation used in our experiments\.

### B\.1Sequential proposal setup

Consider a verified predecessor state

yk∈𝒳nt\.y\_\{k\}\\in\\mathcal\{X\}\_\{\\mathrm\{nt\}\}\.Let

𝒵⁡\(yk\)=ℳ⁡\(yk\)×𝒱\\mathcal\{Z\}\(y\_\{k\}\)=\\mathcal\{M\}\(y\_\{k\}\)\\times\\mathcal\{V\}be the set of possible one\-step position–token commitments\.

#### Target commit law\.

Let

sk⋆=s⋆​\(yk\)\.s\_\{k\}^\{\\star\}=s^\{\\star\}\(y\_\{k\}\)\.AtT\>0T\>0, the target distribution over one\-step commitments is

πk\(s,v\)=𝟏\{s=sk⋆\}πT,sk⋆\(v∣yk,C\)\.\\pi\_\{k\}\(s,v\)=\\mathbf\{1\}\\\{s=s\_\{k\}^\{\\star\}\\\}\\,\\pi\_\{T,s\_\{k\}^\{\\star\}\}\(v\\mid y\_\{k\},C\)\.\(10\)Therefore

∑z∈𝒵⁡\(yk\)πk​\(z\)=1\.\\sum\_\{z\\in\\mathcal\{Z\}\(y\_\{k\}\)\}\\pi\_\{k\}\(z\)=1\.

#### Proposal law\.

Suppose the drafter exposes a normalized proposal distribution

qk∈Δ⁡\(𝒵⁡\(yk\)\)q\_\{k\}\\in\\Delta\(\\mathcal\{Z\}\(y\_\{k\}\)\)and generates one proposed commitment

Z~k=\(S~k,V~k\)∼qk\.\\widetilde\{Z\}\_\{k\}=\(\\widetilde\{S\}\_\{k\},\\widetilde\{V\}\_\{k\}\)\\sim q\_\{k\}\.\(11\)For the chain proposal used in our experiments,

qk​\(s,v\)=qkpos​\(s\)​π^T,k,s​\(v\),q\_\{k\}\(s,v\)=q\_\{k\}^\{\\mathrm\{pos\}\}\(s\)\\,\\widehat\{\\pi\}\_\{T,k,s\}\(v\),\(12\)whereqkposq\_\{k\}^\{\\mathrm\{pos\}\}is the drafter’s next\-position distribution andπ^T,k,s\\widehat\{\\pi\}\_\{T,k,s\}is its stale temperature\-adjusted token conditional\.

The factorization in equation[12](https://arxiv.org/html/2610.22543#A2.E12)is not required by the proof; exactness needs only that the proposal actually be sampled from the explicit distributionqkq\_\{k\}supplied to the verifier\.

#### Accept–residual rule\.

Given proposalZ~k=z\\widetilde\{Z\}\_\{k\}=zwithqk​\(z\)\>0q\_\{k\}\(z\)\>0, accept it with probability

αk​\(z\)=min⁡\{1,πk​\(z\)qk​\(z\)\}\.\\alpha\_\{k\}\(z\)=\\min\\left\\\{1,\\frac\{\\pi\_\{k\}\(z\)\}\{q\_\{k\}\(z\)\}\\right\\\}\.\(13\)Upon rejection, draw a replacement from

rk​\(z\)=\(πk​\(z\)−qk​\(z\)\)\+∑w∈𝒵⁡\(yk\)\(πk​\(w\)−qk​\(w\)\)\+\.r\_\{k\}\(z\)=\\frac\{\(\\pi\_\{k\}\(z\)\-q\_\{k\}\(z\)\)\_\{\+\}\}\{\\sum\_\{w\\in\\mathcal\{Z\}\(y\_\{k\}\)\}\(\\pi\_\{k\}\(w\)\-q\_\{k\}\(w\)\)\_\{\+\}\}\.\(14\)If the denominator is zero, thenπk=qk\\pi\_\{k\}=q\_\{k\}, rejection has probability zero, andrkr\_\{k\}is never queried\.

### B\.2Exactness and optimal acceptance

###### Proposition 3\(Rejection\-sampling exactness and optimal acceptance\)\.

Fix a verified predecessor stateyky\_\{k\}\. LetZ~k∼qk\\widetilde\{Z\}\_\{k\}\\sim q\_\{k\}be accepted according to equation[13](https://arxiv.org/html/2610.22543#A2.E13), with rejected proposals replaced using equation[14](https://arxiv.org/html/2610.22543#A2.E14)\. Then the committed pairZkZ\_\{k\}satisfies

Moreover,

Pr⁡\[accept∣yk\]=1−DTV​\(πk,qk\),\\Pr\[\\mathrm\{accept\}\\mid y\_\{k\}\]=1\-D\_\{\\mathrm\{TV\}\}\(\\pi\_\{k\},q\_\{k\}\),which is the maximal possible agreement probability among all couplings with marginalsπk\\pi\_\{k\}andqkq\_\{k\}\. Applying the rule sequentially at every verified predecessor therefore reproduces the complete draft\-free target distribution\.

###### Proof\.

Fixyky\_\{k\}and abbreviate

π=πk,q=qk,r=rk\.\\pi=\\pi\_\{k\},\\qquad q=q\_\{k\},\\qquad r=r\_\{k\}\.

#### Step 1: accepted mass\.

For arbitraryz∈𝒵⁡\(yk\)z\\in\\mathcal\{Z\}\(y\_\{k\}\)withq⁡\(z\)\>0q\(z\)\>0,

Pr⁡\[Z~k=z,accept\]\\displaystyle\\Pr\[\\widetilde\{Z\}\_\{k\}=z,\\mathrm\{accept\}\]=q⁡\(z\)​min⁡\{1,π⁡\(z\)q⁡\(z\)\}\\displaystyle=q\(z\)\\min\\left\\\{1,\\frac\{\\pi\(z\)\}\{q\(z\)\}\\right\\\}\(15\)=min⁡\{q⁡\(z\),π⁡\(z\)\}\.\\displaystyle=\\min\\\{q\(z\),\\pi\(z\)\\\}\.\(16\)Ifq⁡\(z\)=0q\(z\)=0, both sides of equation[16](https://arxiv.org/html/2610.22543#A2.E16)are zero, so the identity holds for allzz\.

Summing overzz,

Pr⁡\[accept\]=∑zmin⁡\{q⁡\(z\),π⁡\(z\)\}\.\\Pr\[\\mathrm\{accept\}\]=\\sum\_\{z\}\\min\\\{q\(z\),\\pi\(z\)\\\}\.\(17\)Hence

Pr⁡\[reject\]=1−∑zmin⁡\{q⁡\(z\),π⁡\(z\)\}\.\\Pr\[\\mathrm\{reject\}\]=1\-\\sum\_\{z\}\\min\\\{q\(z\),\\pi\(z\)\\\}\.\(18\)

#### Step 2: rejected mass\.

Becauseπ\\piis normalized,

1−∑zmin⁡\{q⁡\(z\),π⁡\(z\)\}\\displaystyle 1\-\\sum\_\{z\}\\min\\\{q\(z\),\\pi\(z\)\\\}=∑z\[π⁡\(z\)−min⁡\{q⁡\(z\),π⁡\(z\)\}\]\\displaystyle=\\sum\_\{z\}\\left\[\\pi\(z\)\-\\min\\\{q\(z\),\\pi\(z\)\\\}\\right\]\(19\)=∑z\(π⁡\(z\)−q⁡\(z\)\)\+\.\\displaystyle=\\sum\_\{z\}\(\\pi\(z\)\-q\(z\)\)\_\{\+\}\.\(20\)Therefore

Pr⁡\[reject\]=∑z\(π⁡\(z\)−q⁡\(z\)\)\+\.\\Pr\[\\mathrm\{reject\}\]=\\sum\_\{z\}\(\\pi\(z\)\-q\(z\)\)\_\{\+\}\.If this quantity is positive, then

r⁡\(z\)=\(π⁡\(z\)−q⁡\(z\)\)\+Pr⁡\[reject\],r\(z\)=\\frac\{\(\\pi\(z\)\-q\(z\)\)\_\{\+\}\}\{\\Pr\[\\mathrm\{reject\}\]\},so

Pr⁡\[reject,Zk=z\]=\(π⁡\(z\)−q⁡\(z\)\)\+\.\\Pr\[\\mathrm\{reject\},Z\_\{k\}=z\]=\(\\pi\(z\)\-q\(z\)\)\_\{\+\}\.\(21\)

#### Step 3: one\-step exactness\.

Combining the accepted and rejected branches,

Pr\[Zk=z\]\\displaystyle\\Pr\[Z\_\{k\}=z\]=min⁡\{q⁡\(z\),π⁡\(z\)\}\+\(π⁡\(z\)−q⁡\(z\)\)\+\\displaystyle=\\min\\\{q\(z\),\\pi\(z\)\\\}\+\(\\pi\(z\)\-q\(z\)\)\_\{\+\}\(22\)=π⁡\(z\)\.\\displaystyle=\\pi\(z\)\.\(23\)Thus

If the denominator of equation[14](https://arxiv.org/html/2610.22543#A2.E14)is zero, then\(π⁡\(z\)−q⁡\(z\)\)\+=0\(\\pi\(z\)\-q\(z\)\)\_\{\+\}=0for everyzz\. Henceπ⁡\(z\)≤q⁡\(z\)\\pi\(z\)\\leq q\(z\)everywhere, and because both distributions have total mass one,π=q\\pi=q\. In that case equation[17](https://arxiv.org/html/2610.22543#A2.E17)givesPr⁡\[accept\]=1\\Pr\[\\mathrm\{accept\}\]=1, so the residual branch is never used\.

#### Step 4: maximal acceptance\.

For discrete distributions,

DTV​\(π,q\)=1−∑zmin⁡\{π⁡\(z\),q⁡\(z\)\}\.D\_\{\\mathrm\{TV\}\}\(\\pi,q\)=1\-\\sum\_\{z\}\\min\\\{\\pi\(z\),q\(z\)\\\}\.Combining this identity with equation[17](https://arxiv.org/html/2610.22543#A2.E17),

Pr⁡\[accept\]=1−DTV​\(πk,qk\)\.\\Pr\[\\mathrm\{accept\}\]=1\-D\_\{\\mathrm\{TV\}\}\(\\pi\_\{k\},q\_\{k\}\)\.The quantity

∑zmin⁡\{πk​\(z\),qk​\(z\)\}\\sum\_\{z\}\\min\\\{\\pi\_\{k\}\(z\),q\_\{k\}\(z\)\\\}is the maximal possible agreement probability of two random variables with marginalsπk\\pi\_\{k\}andqkq\_\{k\}\. The accept–residual construction therefore realizes a maximal coupling\.

#### Step 5: trajectory exactness\.

LetY0,Y1,…Y\_\{0\},Y\_\{1\},\\ldotsdenote the states produced by sequential RS\. Conditioned on any realized historyY0:k=y0:kY\_\{0:k\}=y\_\{0:k\}, one\-step exactness gives

Zk∣\(Y0:k=y0:k\)∼πk\(⋅∣yk\),Z\_\{k\}\\mid\(Y\_\{0:k\}=y\_\{0:k\}\)\\sim\\pi\_\{k\}\(\\cdot\\mid y\_\{k\}\),which is exactly the target’s one\-step commit law atyky\_\{k\}\. Hence, for every successory′y^\{\\prime\},

Pr\[Yk\+1=y′∣Y0:k=y0:k\]=PrDF\[Xk\+1=y′∣Xk=yk\]\.\\Pr\[Y\_\{k\+1\}=y^\{\\prime\}\\mid Y\_\{0:k\}=y\_\{0:k\}\]=\\Pr\_\{\\mathrm\{DF\}\}\[X\_\{k\+1\}=y^\{\\prime\}\\mid X\_\{k\}=y\_\{k\}\]\.
Since RS and draft\-free decoding start from the same initial state, the chain rule implies that every finite trajectoryy0,…,ymy\_\{0\},\\ldots,y\_\{m\}has the same probability under both procedures:

Pr⁡\[Y0=y0,…,Ym=ym\]=PrDF⁡\[X0=y0,…,Xm=ym\]\.\\Pr\[Y\_\{0\}=y\_\{0\},\\ldots,Y\_\{m\}=y\_\{m\}\]=\\Pr\_\{\\mathrm\{DF\}\}\[X\_\{0\}=y\_\{0\},\\ldots,X\_\{m\}=y\_\{m\}\]\.Therefore, the complete RS trajectory has the draft\-free target distribution\. ∎

### B\.3Relation to batched tree verification

Proposition[3](https://arxiv.org/html/2610.22543#Thmproposition3)assumes the classical per\-link proposal interface: at each verified stateyky\_\{k\}, a single proposal is sampled fromqk\(⋅∣yk\)q\_\{k\}\(\\cdot\\mid y\_\{k\}\)and corrected by the accept–residual rule\. Only after this step determines the realized successor is the proposal at the next verified state generated\.

This interface does not directly provide a verification rule for a preconstructed draft tree containing multiple candidate children of the same predecessor\. Applying the per\-link RS rule unchanged selects only one sampled proposal at each visited state; moreover, after a rejection, the residual sample need not belong to the preconstructed tree\. Thus, although target conditionals at multiple hypothetical states could still be evaluated in parallel, classical per\-link RS does not directly exploit the full batched tree structure used by our drafter\.

Coupled\-noise verification avoids this limitation\. Once the shared randomnessGGis fixed, the target defines a unique successorΦG​\(y\)\\Phi\_\{G\}\(y\)for every hypothetical stateyy\. Each drafted edge\(y,z\)\(y,z\)can therefore be verified independently by the state\-equality test

allowing all hypothetical states in the tree to be evaluated in the same target batch and the unique target\-consistent path to be recovered afterward\.

More general multi\-proposal or tree\-aware rejection\-sampling constructions may also be possible, but are outside the scope of this work\.

### B\.4Empirical validation

Proposition[3](https://arxiv.org/html/2610.22543#Thmproposition3)establishes the correctness of the idealized accept–residual procedure\. We additionally verify that our implementation matches this behavior on live target states\.

For each test state, we record the target and proposal distributionsπk\\pi\_\{k\}andqkq\_\{k\}, repeatedly execute the RS verifier, and compare the empirical distribution of committed position–token pairs withπk\\pi\_\{k\}using theχ2\\chi^\{2\}goodness\-of\-fit test employed in our benchmark\. We include deliberately miscalibrated proposal distributions so that the validation also covers regimes in whichqkq\_\{k\}differs substantially from the target\.

We separately verify the acceptance\-rate identity from Proposition[3](https://arxiv.org/html/2610.22543#Thmproposition3)\. For each tested state, the theoretical RS acceptance probability is

Pr⁡\[accept\]=1−DTV​\(πk,qk\)\.\\Pr\[\\mathrm\{accept\}\]=1\-D\_\{\\mathrm\{TV\}\}\(\\pi\_\{k\},q\_\{k\}\)\.The acceptance rate measured from repeated executions of our implementation differs from this theoretical value by at most0\.0020\.002across the tested states\.

These checks are run before the RS benchmark experiments and serve only to validate the implementation; the distributional guarantee itself follows from Proposition[3](https://arxiv.org/html/2610.22543#Thmproposition3)\.

## Appendix CDrafter Training and Draft\-Tree Construction

This appendix gives the complete procedures used to train the prefix\-conditional commit\-order predictor and to construct the speculative draft tree at inference time\. The notation follows Section[4\.2](https://arxiv.org/html/2610.22543#S4.SS2)\. In particular,p^t\\widehat\{p\}\_\{t\}denotes the stale target conditionals available at speculative roundtt, andqϕ​\(s∣y\)q\_\{\\phi\}\(s\\mid y\)denotes the predicted distribution over the target’s next committed position conditioned on hypothetical prefix stateyy\.

### C\.1Training the commit\-order predictor

Training uses only draft\-free trajectories generated by the target\. For each target statextx\_\{t\}, we retain the stale conditionalsp^t\\widehat\{p\}\_\{t\}available at that state together with the subsequent target commit sequence

𝐬t⋆=\(st⋆,st\+1⋆,…\)\.\\mathbf\{s\}\_\{t\}^\{\\star\}=\(s\_\{t\}^\{\\star\},s\_\{t\+1\}^\{\\star\},\\ldots\)\.For every prefix depthdd, we replay the firstddtarget commitments to construct the hypothetical stateyt,dy\_\{t,d\}while keepingp^t\\widehat\{p\}\_\{t\}fixed\. The next target commitmentst\+d⋆s\_\{t\+d\}^\{\\star\}is then used as the supervision target\. Hence each training example has the form

\(p^t,yt,d,st\+d⋆\)\.\(\\widehat\{p\}\_\{t\},y\_\{t,d\},s\_\{t\+d\}^\{\\star\}\)\.
Algorithm 1Training the prefix\-conditional commit\-order predictor from draft\-free target trajectories\.1:Target

pθp\_\{\\theta\}, prompt set

𝒮\\mathcal\{S\}, maximum replay depth

dmaxd\_\{\\max\}
2:Trained order model

qϕ⋆q\_\{\\phi^\{\\star\}\}
3:

ℛθ←Decode​\(pθ,𝒮\)\\mathcal\{R\}\_\{\\theta\}\\leftarrow\\textsc\{Decode\}\(p\_\{\\theta\},\\mathcal\{S\}\)⊳\\trianglerightCollect draft\-free target trajectories

4:

ℰ←∅\\mathcal\{E\}\\leftarrow\\varnothing
5:foreach recorded target state

\(xt,p^t,𝐬t⋆\)∈ℛθ\(x\_\{t\},\\widehat\{p\}\_\{t\},\\mathbf\{s\}\_\{t\}^\{\\star\}\)\\in\\mathcal\{R\}\_\{\\theta\}do

6:for

d=0,…,dmaxd=0,\\ldots,d\_\{\\max\}do

7:

yt,d←Prefix​\(xt,𝐬t⋆,d\)y\_\{t,d\}\\leftarrow\\textsc\{Prefix\}\(x\_\{t\},\\mathbf\{s\}\_\{t\}^\{\\star\},d\)⊳\\trianglerightReplay the firstddtarget commitments

8:

ℰ←ℰ∪\{\(p^t,yt,d,st\+d⋆\)\}\\mathcal\{E\}\\leftarrow\\mathcal\{E\}\\cup\\\{\(\\widehat\{p\}\_\{t\},y\_\{t,d\},s^\{\\star\}\_\{t\+d\}\)\\\}⊳\\trianglerightNext position is the supervision target

9:endfor

10:endfor

11:

ϕ⋆←arg​minϕ⁡𝔼\(p^,y,s⋆\)∼ℰ​\[−log⁡qϕ​\(s⋆∣y\)\]\\displaystyle\\phi^\{\\star\}\\leftarrow\\argmin\_\{\\phi\}\\mathbb\{E\}\_\{\(\\widehat\{p\},y,s^\{\\star\}\)\\sim\\mathcal\{E\}\}\\left\[\-\\log q\_\{\\phi\}\(s^\{\\star\}\\mid y\)\\right\]
12:return

qϕ⋆q\_\{\\phi^\{\\star\}\}

The important feature of this construction is that the stale conditionalsp^t\\widehat\{p\}\_\{t\}remain unchanged while the hypothetical prefix is extended\. The resulting examples therefore match the information available to the drafter during recursive inference rather than giving it access to future target evaluations\.

### C\.2Prefix\-conditional draft\-tree construction

At speculative roundtt, drafting begins from the current statextx\_\{t\}and the stale target conditionalsp^t\\widehat\{p\}\_\{t\}\. For each active hypothetical stateyy, the order model produces

qϕ​\(s∣y\),s∈ℳ⁡\(y\)\.q\_\{\\phi\}\(s\\mid y\),\\qquad s\\in\\mathcal\{M\}\(y\)\.Each candidate position is paired with token hypotheses fromℋt​\(s\)\\mathcal\{H\}\_\{t\}\(s\)\. AtT=0T=0, the primary hypothesis is

v~0​\(s\)=arg​maxv⁡p^t,s​\(v\),\\widetilde\{v\}\_\{0\}\(s\)=\\argmax\_\{v\}\\widehat\{p\}\_\{t,s\}\(v\),with the stale runner\-up additionally retained when the top\-11probability is below0\.60\.6\. AtT\>0T\>0,ℋt​\(s\)\\mathcal\{H\}\_\{t\}\(s\)is determined by the coupled stale draw in Eq\.[4](https://arxiv.org/html/2610.22543#S4.E4)\.

For every extension

z=y⊕\(s,v\),z=y\\oplus\(s,v\),we use the same prefix mass defined in the main text,

m⁡\(z\)=m⁡\(y\)​qϕ​\(s∣y\)​p^t,s​\(v\),m⁡\(xt\)=1\.m\(z\)=m\(y\)\\,q\_\{\\phi\}\(s\\mid y\)\\,\\widehat\{p\}\_\{t,s\}\(v\),\\qquad m\(x\_\{t\}\)=1\.DDTree then allocates the available draft budget to the highest\-mass states\.

Algorithm 2Prefix\-conditional construction of the speculative draft tree under budgetDD\.1:Current target state

xtx\_\{t\}, stale conditionals

p^t\\widehat\{p\}\_\{t\}, order model

qϕq\_\{\\phi\}, draft budget

DD
2:Draft tree

𝒯t\\mathcal\{T\}\_\{t\}
3:

𝒯t←\{xt\}\\mathcal\{T\}\_\{t\}\\leftarrow\\\{x\_\{t\}\\\}
4:

m⁡\(xt\)←1m\(x\_\{t\}\)\\leftarrow 1
5:

ℱ←\{xt\}\\mathcal\{F\}\\leftarrow\\\{x\_\{t\}\\\}⊳\\trianglerightCurrent expansion frontier

6:while

\|𝒯t\|−1<D\|\\mathcal\{T\}\_\{t\}\|\-1<Ddo

7:

𝒞←∅\\mathcal\{C\}\\leftarrow\\varnothing
8:foreach active prefix

y∈ℱy\\in\\mathcal\{F\}do

9:

qy\(⋅\)←qϕ\(⋅∣y\)q\_\{y\}\(\\cdot\)\\leftarrow q\_\{\\phi\}\(\\cdot\\mid y\)⊳\\trianglerightPredict next commit position

10:foreach

s∈ℳ⁡\(y\)s\\in\\mathcal\{M\}\(y\)do

11:construct

ℋt​\(s\)\\mathcal\{H\}\_\{t\}\(s\)from

p^t,s\\widehat\{p\}\_\{t,s\}⊳\\trianglerightToken hypotheses

12:foreach

v∈ℋt​\(s\)v\\in\\mathcal\{H\}\_\{t\}\(s\)do

13:

z←y⊕\(s,v\)z\\leftarrow y\\oplus\(s,v\)
14:

m⁡\(z\)←m⁡\(y\)​qy​\(s\)​p^t,s​\(v\)\\displaystyle m\(z\)\\leftarrow m\(y\)\\,q\_\{y\}\(s\)\\,\\widehat\{p\}\_\{t,s\}\(v\)
15:

𝒞←𝒞∪\{z\}\\mathcal\{C\}\\leftarrow\\mathcal\{C\}\\cup\\\{z\\\}
16:endfor

17:endfor

18:endfor

19:

\(𝒯t,ℱ\)←DDTreeTopD​\(𝒯t,𝒞,D\)\(\\mathcal\{T\}\_\{t\},\\mathcal\{F\}\)\\leftarrow\\textsc\{DDTreeTopD\}\(\\mathcal\{T\}\_\{t\},\\mathcal\{C\},D\)⊳\\trianglerightRetain highest\-mass states within the remaining budget

20:endwhile

21:return

𝒯t\\mathcal\{T\}\_\{t\}

Because every multiplicative factor in the prefix mass lies in\[0,1\]\[0,1\], a child’s mass cannot exceed that of its parent\. This monotonicity permits level\-wise expansion while recovering the same top\-DDfixpoint as sequential best\-first search\. The calibration temperatureτ\\tauis applied toqϕq\_\{\\phi\}before the search, withτ=1\\tau=1in the final model\.

## Appendix DExperimental setup

### D\.1Drafter training details

Architecture: three\-layer transformer encoder over the masked positions,dmodel=128d\_\{\\text\{model\}\}\{=\}128,88heads, top\-3232stale probabilities per position \(mtop=32m\_\{\\text\{top\}\}\{=\}32\),44token candidates per position for the hedge;0\.620\.62M parameters\. Inputs per masked position: stale top\-32 probabilities \(raw and log\), eight summary statistics, two path features \(taken\-position set, depth\)\. Training data: decoding trajectories of the target itself atT=0T\{=\}0, with prefix\-conditional examples extracted at every \(round, depth\)\. The prompt pools are recorded per shard and differ by target\. The SDAR\-8B b32 and LLaDA2\.0\-mini b32 drafters are trained on CodeAlpaca\([Chaudhary, 2023](https://arxiv.org/html/2610.22543#bib.bib7)\)alone,≈21\{\\approx\}21k and≈14\{\\approx\}14k trajectories; the SDAR\-4B drafter on a four\-pool mixture of CodeAlpaca, UltraChat\([Ding et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib6)\)and the chat and code shards of the Llama\-Nemotron post\-training corpus\([Bercovich et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib5)\),15,02715\{,\}027trajectories\. No pool contains MATH\-500 or any evaluation sample\. Optimization: Adam, learning rate3×10−43\{\\times\}10^\{\-4\},1010epochs; the released SDAR\-8B b32 checkpoint reaches0\.750\.75held\-out next\-commit accuracy \(0\.770\.77at depth 0,0\.720\.72at depth 7\)\. Training a drafter takes under two GPU\-hours on one card\. The identical procedure \(data regenerated per target and width\) produces the checkpoints for SDAR\-8B b32/b16, SDAR\-4B and LLaDA2\.0\-mini b32/b16; no per\-setting tuning\.

#### What the drafters have seen\.

Two consequences follow, disclosed rather than controlled\. The SDAR\-8B and mini drafters have seen only code, while two of the four evaluation tasks are mathematics: their GSM8K and MATH500 runs are out of domain for the drafter, which makes those results evidence that the learned order carries across domains rather than memorizing one\. And because the SDAR\-4B drafter was trained on a different mixture, any difference between the SDAR\-8B and SDAR\-4B tables confounds target scale with corpus composition; we report both and attribute neither\. Regenerating the SDAR\-8B corpus would invalidate every downstream run of the benchmark, so the confound is left in place and stated here\.

### D\.2Draft layout and verification cost

A target forward at a block state yields a distribution at every masked position; the most confident position is filled and the block is re\-run\. To verifyDDdrafts in one forward, each draft is appended behind the cached prefix as a separate copy of the block with its own mask layout, and a copy attends to the prefix and to itself only, so its logits are exactly what the target would compute at that state\. One forward over\(1\+D\)​L\(1\{\+\}D\)Lpositions therefore returns the target’s own next commit at the root and at every draft, a rejected draft costs positions but never changes the output, and packing a tree into a single sequence is not exact here, since a copy’s logits depend on its complete mask layout\.

### D\.3Decoding configuration

Strict single\-commit decoding, generation length 256, block width 32 \(16 where stated\), exact prefix KV cache on \(for LLaDA2\.0\-mini: an exact prefix\-KV path we added to the released model code, verified to reproduce the uncached decoder’s commits\), draft layout “concat” \(draft replicas appended behind a blockwise mask\), padding of drafted forwards to\(1\+D\)​L\(1\{\+\}D\)Lpositions unless stated, distribution\-preserving verdict derived from the configuration and recorded per run\. Prompt templates and stopping rules per task are those of the released harness \(Appendix[E](https://arxiv.org/html/2610.22543#A5)\)\.

### D\.4Baseline configuration

Spiffy’s draft graph is calibrated with the released procedure of[Agrawal et al\. \(2026\)](https://arxiv.org/html/2610.22543#bib.bib18)\(offline collection on held\-out prompts: MBPP train and MATH500 problems disjoint from the evaluation samples, 25 each, lookahead 6, graph size 24\) separately for SDAR\-8B and SDAR\-4B, and truncated to the budget at decode time\. The SSD\-style free chain uses candidate rank 1, the sampler’s ordering and depth 4\. The BlockSpec\-style confidence tree is a best\-first tree over the target’s renormalized stale confidences\. All baselines run through the same decoder and verifier as SPOT; their draft\-free control is shared with ours per task\.

### D\.5Mass\-floor calibration

The floorγ\\gammaof SPOT\-γ\\gammais calibrated per target on5050held\-out samples, MBPP test samples100100–149149\(every evaluated run uses samples00–9999\):γ∈\{0,0\.02,0\.05,0\.1,0\.2\}\\gamma\\in\\\{0,0\.02,0\.05,0\.1,0\.2\\\}, unpadded,D∈\{8,12,20\}D\\in\\\{8,12,20\\\},T=0T\{=\}0, seed00, the five runs back to back on one card so that the grid shares machine state, with theγ=0\\gamma\{=\}0run also providing the draft\-free control every ratio is taken against\. The selected value is the argmax of the mean throughput ratio over the three budgets, the smaller floor being kept when two are within3%3\\%\(Table[4](https://arxiv.org/html/2610.22543#A4.T4)\)\. On SDAR\-8B the choice is clear \(0\.10\.1leads0\.050\.05by10%10\\%\); on SDAR\-4B0\.050\.05leads0\.020\.02by2\.5%2\.5\\%, inside the tie band, so the smaller floor is kept; on LLaDA2\.0\-mini no floor beatsγ=0\\gamma\{=\}0, whose throughput does not fall with the budget on that target, and the floor is switched off\. On the evaluated samples the calibrated floor is at or above the paddedγ=0\\gamma\{=\}0recipe in4747of the4848SDAR settings \(four tasks, three budgets, two temperatures, two models\); the exception is SDAR\-4B MATH500D​12D12atT=0T\{=\}0,4%4\\%below at the same forwards\. No single value achieves this on both models:γ=0\.05\\gamma\{=\}0\.05, the best fixed choice, is at or above the padded recipe in all3232settings atD​12D12andD​20D20but in only1010of1616atD​8D8, where the fixed budget is already near its throughput optimum and the floor has little to recover; four of its six misses are within3%3\\%and two, SDAR\-4B MATH500 atD​8D8, are8%8\\%and13%13\\%\. One test run \(SDAR\-4B, GSM8K,T=1T\{=\}1\) was repeated after its first pass showed a15%15\\%wall drop atD​8D8alone with identical forwards; the second pass is reported and the first archived in place\.

Table 4:Mass\-floor calibration grids: mean throughput ratio overD​8/D​12/D​20D8/D12/D20on the5050held\-out MBPP samples, unpadded, against the same card’s draft\-free run\. Bold: selected\.
### D\.6Methods outside the comparison

#### SimSD\.

SimSD\([Cui et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib30)\)drafts with a same\-family sibling model and verifies the drafted tokens in one forward under a plug\-in attention mask that lends them temporally valid contexts\. That test approximates the sequential denoising distribution instead of reproducing it, and its reported speedups come with changed \(on average improved\) generation quality\. It is therefore a lossy accelerator, not a lossless speculative decoder, and its throughput is not commensurable with that of the exact methods compared here; we do not compare against it\.

#### MRP, from reported numbers\.

MRP\([Xu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib12)\)trains a three\-layer residual head to extrapolate the backbone’s logits up toK=3K\{=\}3denoising steps ahead and keeps a drafted token where the backbone’s argmax agrees with it\. It reports results on the same SDAR\-Chat 1\.7B, 4B and 8B targets and the same four tasks as this paper \(its Table 2: block width1616, greedy decoding, one H100\), against a one\-token\-per\-step baseline, which is the draft\-free decoder of Section[3](https://arxiv.org/html/2610.22543#S3)\. We did not run its code; Table[5](https://arxiv.org/html/2610.22543#A4.T5)sets its reported speedups beside SPOT’sD​8D8throughput ratio over our draft\-free control \(block width3232, length256256, one RTX 6000 Ada\)\. The two are measured in different implementations, so the comparison is between relative gains: SPOT is ahead in all twelve settings, by1\.51\.5–3\.1×3\.1\\times, and uses3\.23\.2–4\.9×4\.9\\timesfewer target forwards than draft\-free decoding, a quantity MRP does not report\. Its acceptance test is argmax equality, which has no distribution\-preserving extension to sampling\.

Table 5:MRP’s reported speedups next to SPOT\. MRP: speculative mode \(K=3K\{=\}3\) over its one\-token\-per\-step baseline, Table 2 of[Xu et al\. \(2026\)](https://arxiv.org/html/2610.22543#bib.bib12)\(one H100, block width1616, greedy\)\. SPOT:D​8D8throughput ratio over our draft\-free control \(block width3232, length256256, one RTX 6000 Ada\), NFE reduction in parentheses\. MRP generates up to40964096tokens\. Different implementations: relative gains only\.

### D\.7Full\-attention targets \(LLaDA\-8B, Dream\-7B\)

LLaDA\-8B\-Instruct and Dream\-7B\-Instruct attend over the whole sequence, so the keys and values of the prefix depend on the block being decoded and no exact prefix cache exists for them\. For the transfer rows of Table[3](https://arxiv.org/html/2610.22543#S5.T3)both run under an approximate prefix cache of the Fast\-dLLM kind\([Wu et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib15)\): the prefix is encoded once when a block opens and reused, unrefreshed, for every forward of that block; the draft\-free control of each task runs under the same cache, and every such run is recorded as not distribution\-preserving\. The draft replicas use the same concat layout as on SDAR, rotated at the positions of the block they draft for; for LLaDA\-8B this required threading explicit position ids through the released model code \(Dream’s already accepts them\), and the patched model reproduces the released one exactly when positions are left implicit\. Two checks were run in fp32 before the benchmark: an appended, attention\-isolated replica changes the decoded block’s logits by at most10−410^\{\-4\}\(LLaDA\) and4×10−44\{\\times\}10^\{\-4\}\(Dream\), and a replica of the true block reproduces that block’s own logits to10−410^\{\-4\}and2×10−42\{\\times\}10^\{\-4\}, so verification on these targets is exact relative to the cached decoder; the approximation lives entirely in the cache, which is why these rows are reported against the cached control and kept out of every lossless claim\.

### D\.8Accounting details

NFE counts target forwards only; oracle lookahead and drafter forwards are excluded from it, and the drafter’s own cost is charged to wall clock and therefore to the throughput ratio\. Throughput is measured per sample as generated tokens over net wall time of the generation loop, summed over the 100 samples, and reported as the ratio to the draft\-free run of the same task, model and temperature\. That control is measured once per task, inside the SPOT run, and copied into every other method’s run, so every method of a task is normalized by the same number; all eight cards are the same model, each run ran alone on its card, and a control that came out a wall outlier was re\-run and the original archived\. Four independent draft\-free measurements of SDAR\-8B b32 MBPP on four cards span18\.6118\.61–19\.1319\.13tok/s, a cross\-invocation term of about3%3\\%for the control\. Drafted runs vary more: three passes of one transfer run \(SDAR\-8B GSM8K under the mini\-trained drafter\) on three cards gave identical NFE and throughput ratios of2\.452\.45,2\.172\.17and2\.492\.49atD​8D8; the outlying pass was archived and the two agreeing passes’ value is reported\. The comparisons this paper draws are an order of magnitude larger than either term, and margins under about5%5\\%are reported but not called wins\. Honest oracle throughput subtracts the oracle’s lookahead forwards from wall time\. Accuracy uses the task’s standard scorer on the generated text\.

## Appendix EDatasets

GSM8K: themaintest split \(1319 problems\), first 100, zero\-shot chat prompt, answer extracted after “\#\#\#\#” or as the last number\. MBPP: thefulltest split, first 100 problems \(task ids 11–110\), 3\-shot with the exemplars of thepromptsplit as in the MBPP paper and lm\-eval, answers wrapped in\[BEGIN\]/\[DONE\]and executed against the reference tests\. HumanEval: the 164\-problem test set, first 100, completion executed against the reference tests\. MATH500: the standard 500\-problem subset of MATH, first 100, exact\-match on the boxed answer after normalization\. For every task the first\-100 samples are the exact prefix of the full\-set order the released harness uses, so runs at largernnextend these rather than replace them; the draft\-free accuracies on the 100\-prompt subsets \(Table[2](https://arxiv.org/html/2610.22543#S5.T2)\) differ from full\-set numbers by subset composition \(HumanEval’s first 100 problems are easier than its full 164\)\. The prompt templates and stopping rules are those of the released harness and ship with it\.

## Appendix FExtended results

### F\.1Extended comparison against related methods

Table[6](https://arxiv.org/html/2610.22543#A6.T6)compares the published diffusion drafters on the axes this paper argues are the relevant ones\.

Table 6:Speculative decoding for diffusion targets, on the axes this paper argues are the relevant ones\. “Learns order” = the commit order is predicted by a trained model rather than read off stale confidence or fixed offline; “multi\-step” = one drafted proposal spans more than one commit; “exactT\>0T\{\>\}0” = the committed distribution is provably the target’s at sampling temperature\.
### F\.2Oracle grid

Table[7](https://arxiv.org/html/2610.22543#A6.T7)is the full live\-oracle grid behind Figure[2](https://arxiv.org/html/2610.22543#S3.F2)\(a\): NFE \(reduction over the setting’s draft\-free NFE\) and honest throughput ratio for the position, token and joint oracles atD∈\{8,12,20\}D\\in\\\{8,12,20\\\}on the eight SDAR\-8B settings and on LLaDA2\.0\-mini MBPP\. Honest throughput subtracts the oracle’s lookahead forwards from wall time \(the correction removes8585–96%96\\%of the oracle’s raw wall\) and divides by the draft\-free pass of the*same invocation*: the grid of one setting can be assembled from per\-mode runs and crash resumes, each of which re\-runs that pass, so only its own pass shares the machine state of its oracle rows\. On the SDAR\-8B b32 MATH500 setting the position and token rows come from a pass that ran at14\.114\.1tok/s and the joint rows from one at18\.418\.4; pairing each with its own pass is what makes the three oracles comparable \(Appendix[G](https://arxiv.org/html/2610.22543#A7)\)\. NFE is monotone in the budget for every oracle\. At b32 the throughput optimum is interior,D​12D12for the joint oracle on every setting; at b16 the narrower block makes each replica cheaper and honest throughput is still rising atD​20D20, the largest budget in the grid, so that optimum lies outside the measured range\.

Table 7:Live\-oracle grid,T=0T\{=\}0, generation length 256,n=100n\{=\}100, seed 0\. Per setting: NFE \(reduction×\\timesover the setting’s draft\-free NFE\) and honest tok/s ratio over the setting’s draft\-free throughput, for the position, token and joint oracles atD​8D8/D​12D12/D​20D20; the draft\-free column gives the baseline NFE and tok/s\. Bold: best NFE and best throughput per setting\.position oracletoken oraclejoint oraclesettingmetricdraft\-freeD​8D8D​12D12D​20D20D​8D8D​12D12D​20D20D​8D8D​12D12D​20D20SDAR\-8B b32 GSM8KNFE271\.150\.1 \(5\.4×\\times\)43\.0 \(6\.3×\\times\)38\.2 \(7\.1×\\times\)68\.6 \(3\.9×\\times\)64\.8 \(4\.2×\\times\)62\.1 \(4\.4×\\times\)46\.8 \(5\.8×\\times\)38\.2 \(7\.1×\\times\)31\.3 \(8\.7×\\times\)tok/s ratio18\.4 tok/s3\.89×\\times4\.02×\\times3\.25×\\times2\.81×\\times2\.60×\\times1\.89×\\times4\.19×\\times4\.56×\\times4\.02×\\timesSDAR\-8B b32 MBPPNFE267\.146\.8 \(5\.7×\\times\)39\.7 \(6\.7×\\times\)33\.0 \(8\.1×\\times\)53\.7 \(5\.0×\\times\)47\.2 \(5\.7×\\times\)40\.9 \(6\.5×\\times\)43\.7 \(6\.1×\\times\)35\.4 \(7\.5×\\times\)27\.5 \(9\.7×\\times\)tok/s ratio18\.7 tok/s3\.25×\\times3\.36×\\times3\.05×\\times2\.82×\\times2\.77×\\times2\.35×\\times3\.50×\\times3\.77×\\times3\.53×\\timesSDAR\-8B b32 HumanEvalNFE272\.647\.5 \(5\.7×\\times\)40\.0 \(6\.8×\\times\)33\.4 \(8\.2×\\times\)62\.3 \(4\.4×\\times\)57\.6 \(4\.7×\\times\)53\.6 \(5\.1×\\times\)46\.1 \(5\.9×\\times\)37\.9 \(7\.2×\\times\)30\.3 \(9\.0×\\times\)tok/s ratio18\.9 tok/s3\.91×\\times4\.15×\\times3\.58×\\times2\.96×\\times2\.81×\\times2\.13×\\times4\.05×\\times4\.42×\\times3\.97×\\timesSDAR\-8B b32 MATH500NFE272\.954\.2 \(5\.0×\\times\)48\.5 \(5\.6×\\times\)45\.8 \(6\.0×\\times\)82\.7 \(3\.3×\\times\)81\.8 \(3\.3×\\times\)81\.8 \(3\.3×\\times\)48\.9 \(5\.6×\\times\)40\.3 \(6\.8×\\times\)34\.1 \(8\.0×\\times\)tok/s ratio16\.3 tok/s4\.42×\\times4\.45×\\times3\.40×\\times2\.91×\\times2\.62×\\times1\.87×\\times4\.12×\\times4\.42×\\times3\.71×\\timesSDAR\-8B b16 GSM8KNFE263\.758\.2 \(4\.5×\\times\)55\.7 \(4\.7×\\times\)47\.5 \(5\.6×\\times\)74\.5 \(3\.5×\\times\)74\.1 \(3\.6×\\times\)72\.0 \(3\.7×\\times\)54\.3 \(4\.9×\\times\)51\.8 \(5\.1×\\times\)39\.8 \(6\.6×\\times\)tok/s ratio18\.6 tok/s4\.21×\\times4\.26×\\times4\.82×\\times3\.33×\\times3\.20×\\times3\.27×\\times4\.52×\\times4\.58×\\times5\.57×\\timesSDAR\-8B b16 MBPPNFE262\.054\.4 \(4\.8×\\times\)53\.6 \(4\.9×\\times\)40\.9 \(6\.4×\\times\)60\.9 \(4\.3×\\times\)59\.7 \(4\.4×\\times\)59\.4 \(4\.4×\\times\)52\.0 \(5\.0×\\times\)50\.9 \(5\.2×\\times\)36\.6 \(7\.2×\\times\)tok/s ratio18\.7 tok/s3\.84×\\times3\.83×\\times4\.51×\\times3\.37×\\times3\.28×\\times3\.20×\\times4\.10×\\times4\.17×\\times5\.08×\\timesSDAR\-8B b16 HumanEvalNFE262\.853\.5 \(4\.9×\\times\)52\.2 \(5\.0×\\times\)40\.1 \(6\.6×\\times\)65\.2 \(4\.0×\\times\)64\.5 \(4\.1×\\times\)59\.1 \(4\.4×\\times\)52\.6 \(5\.0×\\times\)51\.0 \(5\.2×\\times\)37\.4 \(7\.0×\\times\)tok/s ratio19\.0 tok/s4\.38×\\times4\.36×\\times5\.28×\\times3\.59×\\times3\.50×\\times3\.67×\\times4\.45×\\times4\.48×\\times5\.67×\\timesSDAR\-8B b16 MATH500NFE263\.361\.8 \(4\.3×\\times\)58\.8 \(4\.5×\\times\)54\.2 \(4\.9×\\times\)86\.8 \(3\.0×\\times\)86\.4 \(3\.0×\\times\)86\.3 \(3\.1×\\times\)55\.8 \(4\.7×\\times\)52\.5 \(5\.0×\\times\)42\.3 \(6\.2×\\times\)tok/s ratio19\.4 tok/s3\.90×\\times3\.93×\\times4\.14×\\times2\.70×\\times2\.58×\\times2\.54×\\times4\.35×\\times4\.43×\\times5\.21×\\timesLLaDA2\.0\-mini b32 MBPPNFE270\.463\.1 \(4\.3×\\times\)58\.1 \(4\.7×\\times\)56\.3 \(4\.8×\\times\)92\.2 \(2\.9×\\times\)92\.0 \(2\.9×\\times\)91\.5 \(3\.0×\\times\)59\.6 \(4\.5×\\times\)53\.0 \(5\.1×\\times\)48\.6 \(5\.6×\\times\)tok/s ratio6\.3 tok/s3\.31×\\times3\.36×\\times3\.21×\\times2\.33×\\times2\.15×\\times2\.02×\\times3\.65×\\times3\.74×\\times3\.90×\\times

### F\.3More models

The main comparison repeated on the three non\-headline targets the recipe was trained for:SDAR\-4BatT=0T\{=\}0andT=1T\{=\}1\(Tables[8](https://arxiv.org/html/2610.22543#A6.T8)and[9](https://arxiv.org/html/2610.22543#A6.T9)\),SDAR\-1\.7Bat both \(Tables[10](https://arxiv.org/html/2610.22543#A6.T10)and[11](https://arxiv.org/html/2610.22543#A6.T11)\), andLLaDA2\.0\-miniatT=0T\{=\}0\(Table[12](https://arxiv.org/html/2610.22543#A6.T12)\) with itsT=1T\{=\}1pair in Table[13](https://arxiv.org/html/2610.22543#A6.T13)\. The recipe is unchanged in each case: one checkpoint, one calibration, the arms and budgets of Table[2](https://arxiv.org/html/2610.22543#S5.T2)\.

Table 8:Main results, SDAR\-4B b32,T=0T\{=\}0\(format as Table[2](https://arxiv.org/html/2610.22543#S5.T2)\)\. SPOT has the lowest NFE and the highest throughput ratio in every ours\-vs\-baseline comparison\. The ratio’s standard deviation over seeds is at most0\.240\.24\.Table 9:SDAR\-4B b32 atT=1T\{=\}1\(coupled\-noise match; free chain also under rejection sampling\)\. Every comparison won on both columns\. Over seeds the ratio’s standard deviation is at most0\.160\.16and NFE’s at most0\.60\.6\.Table 10:Main results, SDAR\-1\.7B b32,T=0T\{=\}0\(format as Table[2](https://arxiv.org/html/2610.22543#S5.T2)\)\. SPOT has the lowest NFE in every ours\-vs\-baseline comparison and the higher throughput ratio in2121of2424; SPOT\-γ\\gamma, which drops the padding, leads in2323\. The ratio’s standard deviation over seeds is at most0\.170\.17\.Table 11:SDAR\-1\.7B b32 atT=1T\{=\}1\(coupled\-noise match; free chain also under rejection sampling\)\. Lowest NFE in every comparison and the higher throughput ratio in2828of3232; SPOT\-γ\\gammawins every comparison on both columns\. The ratio’s standard deviation over seeds is at most0\.120\.12\.#### The smallest target is where padding costs most\.

SDAR\-1\.7B is the setting in the benchmark where baselines most often take a throughput column from SPOT: Spiffy is ahead by0\.010\.01–0\.140\.14on GSM8K atD​20D20and on MATH500 at both budgets, the three settings with the lowest acceptance\. The NFE ordering is untouched – SPOT drafts1414–23%23\\%fewer forwards than Spiffy in those same cells – so the gap is verification cost, not drafting: a1\.71\.7B forward is cheap enough that the\(1\+D\)​L\(1\{\+\}D\)Lpadded positions are a large share of it\. SPOT\-γ\\gamma, which drops the padding, recovers all but one of those columns atT=0T\{=\}0and all of them atT=1T\{=\}1\.

Table 12:Main results, LLaDA2\.0\-mini b32 \(MoE, exact prefix\-KV cache\),T=0T\{=\}0,n=100n\{=\}100; MBPP and GSM8K with three seeds, MATH500 seed 0, HumanEval seed 0 except twoD​8D8baseline entries that are two\-seed means\. SPOT has the lowest NFE in all 36 comparisons and the higher throughput ratio in 34; the two exceptions are MBPP and HumanEval atD​12D12, where Spiffy is ahead by0\.7%0\.7\\%and0\.1%0\.1\\%; mini’s per\-round acceptance is low for every drafter \(1\.051\.05–3\.773\.77accepted tokens per drafted round across the four tasks\), so the wall\-clock margins are thin on the code tasks\.Table 13:LLaDA2\.0\-mini b32 atT=1T\{=\}1\(MBPP,n=100n\{=\}100, seed 0\), the setting’s onlyT=1T\{=\}1evidence: SPOT against a drafter tuned for this setting alone\.
#### The full transfer matrix\.

Table[14](https://arxiv.org/html/2610.22543#A6.T14)points every trained checkpoint at every target it can drive, atD​8D8\. The native drafter’s advantage shrinks with target size: on SDAR\-8B it has the lowest NFE on all four tasks and the costliest transferred checkpoint costs3\.43\.4–7\.4%7\.4\\%, on SDAR\-4B it leads three of four at1\.51\.5–2\.8%2\.8\\%, and on SDAR\-1\.7B the four checkpoints are interchangeable within0\.40\.4–1\.9%1\.9\\%, with the SDAR\-4B drafter taking GSM8K and the mini drafter MATH500 from the native one\. LLaDA2\.0\-mini, the one target with a different tokenizer, keeps the widest native margin at3\.43\.4–11\.4%11\.4\\%\. On LLaDA\-8B and Dream\-7B, which have no native drafter at all, all four checkpoints land within about4%4\\%of each other on every task\. What the drafter learns is largely a property of block\-diffusion decoding rather than of one target\.

Table 14:Every checkpoint against every target it can drive,T=0T\{=\}0, b32,D​8D8, padded\.∗marks the target’s native drafter, the one trained on it; all other rows are transferred with no retraining or recalibration\.Ratiois against that target’s own draft\-free control, best per column within a target in bold\. The LLaDA2\.0\-mini rows are one seed\.†no exact prefix cache, as in Table[3](https://arxiv.org/html/2610.22543#S5.T3)\.

### F\.4More block sizes

The b16 rows of Table[7](https://arxiv.org/html/2610.22543#A6.T7)give the oracle grid at block width 16 \(joint ceiling6\.26\.2–7\.2×7\.2\\timesNFE,5\.15\.1–5\.7×5\.7\\timeshonest wall\), and Table[15](https://arxiv.org/html/2610.22543#A6.T15)the four\-setting recipe transfer on MBPP: with the recipe frozen \(same architecture, training procedure and calibration\), the single SPOT drafter is at or ahead of a drafter tuned separately for each setting, on every setting and budget, by0\.40\.4–1\.01\.0NFE on SDAR b32,0\.90\.9–1\.31\.3NFE and0\.140\.14–0\.210\.21in throughput ratio on SDAR b16,1\.11\.1–3\.33\.3NFE on mini b32 and4\.64\.6–6\.56\.5NFE on mini b16\.

Table 15:Recipe transfer across widths and targets \(MBPP,T=0T\{=\}0,n=100n\{=\}100\): the single SPOT recipe vs\. a drafter tuned separately for each setting\.
### F\.5Generation length

Table[16](https://arxiv.org/html/2610.22543#A6.T16)repeats theT=1T\{=\}1MBPP comparison at generation lengths6464and512512beside the256256of the main text, each against its own draft\-free control\.The gains grow with generation length\.SPOT’s throughput ratio over draft\-free decoding atD​8D8rises from1\.56×1\.56\\timesat length6464to2\.45×2\.45\\timesat256256and2\.79×2\.79\\timesat512512, and SPOT\-γ\\gamma’s best ratio from1\.87×1\.87\\timesto2\.86×2\.86\\timesto3\.23×3\.23\\times\. The margin over the strongest baseline grows with it: against Spiffy atD​8D8, SPOT takes11%11\\%fewer forwards at1\.08×1\.08\\timesits throughput at length6464,30%30\\%fewer at1\.38×1\.38\\timesat256256, and37%37\\%fewer at1\.50×1\.50\\timesat512512\. The reason is amortisation\. A block costs\(1\+D\)​L\(1\{\+\}D\)Lpositions to verify whether or not the draft is accepted, and a longer generation spreads that fixed cost over more accepted tokens, which also lifts the NFE reduction over draft\-free decoding from2\.8×2\.8\\timesat length6464to4\.5×4\.5\\timesat256256and5\.2×5\.2\\timesat512512\.

The same accounting sets the limit at the short end\. At length6464the generation is two blocks, there is almost nothing to amortise against, and atD​20D20the padded recipe falls below draft\-free decoding \(0\.89×0\.89\\times\) exactly as the baselines do \(0\.540\.54–0\.76×0\.76\\times\)\. SPOT still has the lowest NFE in every column, and SPOT\-γ\\gamma, which drops the padding, is flat in the budget at1\.87×1\.87\\timesfor bothD​8D8andD​20D20: at short generations the unpadded recipe is the one to deploy\. At length512512no such caveat applies, and SPOT has both the lowest NFE and the highest throughput ratio against every baseline at both budgets\.

Table 16:Generation length, SDAR\-8B b32 MBPP atT=1T\{=\}1:NFE per sample\(throughput ratioover that length’s own draft\-free control\), best in bold\. The256256columns are theT=1T\{=\}1MBPP block of Table[2](https://arxiv.org/html/2610.22543#S5.T2)\.
### F\.6More ablations and curves

Table 17:Ablations on SDAR\-8B MBPP b32 atT=0T\{=\}0\(ratio standard deviation over seeds at most0\.110\.11\)\. Top: recipe components\. Bottom: padding and the mass floorγ\\gamma, the two knobs that trade forwards for wall clock\. The fullγ\\gammagrid and theτ\\tauablation are in the released records\.#### Full budget grid\.

Tables[18](https://arxiv.org/html/2610.22543#A6.T18)and[19](https://arxiv.org/html/2610.22543#A6.T19)repeat Table[2](https://arxiv.org/html/2610.22543#S5.T2)with theD​12D12column\.

Table 18:SDAR\-8B b32,T=0T\{=\}0, all three budgets: GSM8K and MBPP\. NFE is identical across seeds atT=0T\{=\}0; the throughput ratio has standard deviation at most0\.110\.11, median0\.040\.04\.Table 19:SDAR\-8B b32,T=0T\{=\}0, all three budgets: HumanEval and MATH500\. Seed spread as in Table[18](https://arxiv.org/html/2610.22543#A6.T18)\.
#### Budget curves\.

Table[20](https://arxiv.org/html/2610.22543#A6.T20)and Figure[3](https://arxiv.org/html/2610.22543#A6.F3)give SPOT’s \(padded,γ=0\\gamma\{=\}0\) response to the budget on the four SDAR\-8B tasks and on mini MBPP\. NFE is monotone toD​20D20on every setting; the throughput optimum sits atD​6D6on every SDAR\-8B task \(3\.04×3\.04\\timesMBPP,3\.13×3\.13\\timesHumanEval,2\.80×2\.80\\timesGSM8K,2\.38×2\.38\\timesMATH500\) and falls to1\.251\.25–1\.87×1\.87\\timesatD​20D20, while on cached mini the curve rises slowly to a plateau atD​12D12–D​20D20\(2\.22×2\.22\\timesatD​16D16\): the optimum’s location is a property of the target’s forward economics\. With the mass floor of SPOT\-γ\\gammathe SDAR curves flatten \(Table[17](https://arxiv.org/html/2610.22543#A6.T17)\)\.

Table 20:Budget response,T=0T\{=\}0,n=100n\{=\}100\(three seeds at every budget shown on the SDAR\-8B settings; mini seed 0 exceptD​0/6/8/12/20D0/6/8/12/20\)\. Bold: throughput optimum\.Figure 3:Budget curves \(SDAR\-8B,n=100n\{=\}100,T=0T\{=\}0\)\. Left: NFE vs\. draft budget for SPOT \(solid\) and, on MBPP, the fixed baselines atD​8/12/20D8/12/20\(markers\)\. Right: throughput ratio over the shared draft\-free control for SPOT, non\-monotone with an interior optimum atD​6D6on every task\.
#### Temperature sweep\.

Table[1](https://arxiv.org/html/2610.22543#S3.T1), discussed in Section[3\.2](https://arxiv.org/html/2610.22543#S3.SS2), condenses the sweep toD​8D8and to SPOT and Spiffy; Table[21](https://arxiv.org/html/2610.22543#A6.T21)is the complete grid: five temperatures, three drafters, three verifiers,D​8D8andD​20D20, with the draft\-free row of each temperature carrying the target’s own accuracy, argmax fraction and mean committed\-token probability\. Two rows worth quoting: atT=1\.5T\{=\}1\.5, coupled vs uncoupled atD​20D20is54\.154\.1vs62\.762\.7NFE for SPOT and73\.473\.4vs83\.683\.6for Spiffy; the confidence tree, which ranks positions by stale confidence without the noise, gains little from coupling \(121\.6121\.6vs125\.5125\.5at the same budget\)\. AtT=2T\{=\}2, where the target’s samples are near\-uniform, uncoupled acceptance all but disappears for every drafter \(all three sit at∼264\{\\sim\}264NFE against the267\.1267\.1of draft\-free decoding in this grid\) while coupling still drafts, and Spiffy’s noise\-aware position ranking \(stale confidence of the*drawn*token\) overtakes theT=0T\{=\}0\-trained order model under coupling \(114\.6114\.6vs187\.4187\.4NFE atD​8D8\), an ordering that holds nowhere the target produces usable text\.

Table 21:Complete temperature sweep, SDAR\-8B MBPP b32,n=100n\{=\}100, seed 0: NFE \(tok/s ratio over theTT\-matched draft\-free control\) for every drafter under the three equality verifiers atD​8D8andD​20D20\.†greedy match is not distribution\-preserving atT\>0T\>0\(Proposition[2](https://arxiv.org/html/2610.22543#Thmproposition2)does not apply\); its NFE is not comparable to the exact regimes’ and its emitted text drifts from the target’s distribution \(Table[1](https://arxiv.org/html/2610.22543#S3.T1)\)\.

### F\.7Additional verifier material

Table[22](https://arxiv.org/html/2610.22543#A6.T22)isolates verifier and structure on SPOT’s own proposals atT=1T\{=\}1: the tree is worth1212–1717NFE over the same drafter’s chain under the same verifier, and on chains RS and match are within33NFE of each other, with match faster atD​8D8andD​12D12and the two level atD​20D20, where RS’s lower NFE finally pays for the per\-link acceptance test\.

Table 22:Verifier and structure atT=1T\{=\}1, SDAR\-8B MBPP: SPOT’s proposals under each verifier \(match: coupled\-noise match; RS: rejection sampling, chains only\)\. Over seeds the ratio’s standard deviation is at most0\.090\.09\.On chains, where both exact verifiers are defined, the earlier development record established a*regime law*: per link, match transfers the drafter’s argmax skill through the shared noise, while RS degrades withDT​V​\(π,q\)D\_\{TV\}\(\\pi,q\)rather than with argmax stability, so RS wins where the stale conditional is flat \(stale confidence≲0\.6\\lesssim 0\.6\) and match where it is sharp; temperature moves the mix of regimes \(the low\-confidence share of rounds grows from∼6%\{\\sim\}6\\%atT=0\.5T\{=\}0\.5to∼46%\{\\sim\}46\\%atT=1\.5T\{=\}1\.5\) without changing the mechanism, and a sharper drafter raises match specifically\. The final benchmark puts SPOT’s chains in the sharp regime: match is0\.090\.09ahead in throughput ratio atD​8D8, where RS pays an acceptance test per link, and the two are level atD​20D20\(0\.0030\.003apart\), where RS’s3\.03\.0fewer forwards finally cover that cost\. A confidence\-gated hybrid \(match above∼0\.6\\sim 0\.6, RS below\) is the natural chain\-only refinement and remains unimplemented, since trees under match dominate every chain configuration\.

### F\.8Mechanism: what happens in deep blocks

Two measurements explain the gap structure of Table[2](https://arxiv.org/html/2610.22543#S5.T2)\(SDAR\-8B MBPP b32,D​20D20, seed 0\)\. First, stale\-order knowledge decays across blocks: the probability that the previous round’s top\-confidence position still commits falls0\.74/0\.68/0\.34/0\.16/0\.11/0\.08/0\.07/0\.04/0\.030\.74/0\.68/0\.34/0\.16/0\.11/0\.08/0\.07/0\.04/0\.03across blocks00–88, while the stale argmax token stays right \(0\.92/0\.94/0\.99/1\.00​…0\.92/0\.94/0\.99/1\.00\\dots\)—deep blocks are near\-deterministic in*what*and unpredictable in*which position first*, from stale confidence alone\. Second, accepted tokens per drafted round by block:

Every drafter accepts more as the blocks become deterministic, but the amount is set by how deep a*correctly ordered*path each can propose: the learned, prefix\-conditional order model rides the transition law to14\.514\.5accepted tokens per round, the fixed 18\-node graph is capped by its depth\-6 chains at∼5\{\\sim\}5, the free chain by its depth at∼3\.4\{\\sim\}3\.4, and the confidence tree spends its budget on mutually exclusive first\-level siblings \(∼15\{\\sim\}15of 20 nodes\) and reaches depth22or less in90%90\\%of drafted rounds\. Figure[4](https://arxiv.org/html/2610.22543#A6.F4)shows the same pair graphically\.

Figure 4:Left: stale\-order knowledge decays across blocks \(previous round’s top\-confidence position still commits\) while the stale argmax token stays right\. Right: accepted tokens per drafted round by block,D​20D20, SDAR\-8B MBPP\.
### F\.9Error analysis of the drafter

Every rejection is attributable: at the first rejected link of a drafted round, the fallback commit records the rank of the*true*position under the order model’s proposal ordering and of the true token under the stale conditional\. On the development full\-set runs of the final drafter \(the analysis predates the fixed\-hardware benchmark; the drafter is unchanged\), position misses account for6262–81%81\\%of first rejections and token misses for33–6%6\\%; a growing share of rejections atD​20D20\(32%32\\%on MBPP,90%90\\%in the last block\) occur with the truth ranked*first*by the order model—the tree ran out of nodes—which is why NFE keeps improving with budget and why the mass floorγ\\gammatrades NFE for wall clock rather than being free\. The drafter’s prefix\-mass score is nearly calibrated against realized acceptance \(96%96\\%acceptance in the top mass bin,3%3\\%in the bottom; Figure[5](https://arxiv.org/html/2610.22543#A6.F5)\)\. These proportions are reported as a diagnostic of the drafter and no claim in the main text rests on them\.

Figure 5:Left: acceptance calibration of the drafter’s own node score \(realizedP⁡\(accept\)P\(\\text\{accept\}\)per prefix\-mass bin against the diagonal\)\. Right: first\-rejection attribution by block \(D​20D20\): position misses dominate early, deep blocks are budget\-bound\. Development runs of the final drafter\.
### F\.10Drafter architecture

The capacity probe of the development record found the order model feature\-limited rather than capacity\-limited: at fixed inputs,8×8\\timesthe parameters changed held\-out next\-commit accuracy by\+0\.004\+0\.004, whereas broadening the training data moved decoding\. The released drafter is three layers,dmodel=128d\_\{\\text\{model\}\}\{=\}128,88heads\. The probe is offline: we did not sweep depth or width at decode time on the final benchmark, so the released size is a working choice rather than a tuned optimum\.

## Appendix GReproducibility

#### Seeds and wall clock\.

AtT=0T\{=\}0the decoder is deterministic, so forwards and accuracy do not depend on the seed and only wall clock does\. EveryT=0T\{=\}0setting is run three times to measure that spread: across the106106three\-seed settings the max–min spread of the throughput ratio has median0\.090\.09and is at or below0\.150\.15in8989and at or below0\.200\.20in9999\. The one outlier is SDAR\-4B HumanEval atD​8D8\(0\.570\.57\), the shortest task in the benchmark at164164samples\. AtT=1T\{=\}1the target’s own sampling moves the forwards as well: across the120120three\-seedT=1T\{=\}1settings the NFE standard deviation has median0\.240\.24and maximum1\.761\.76, and accuracy stays within0\.060\.06of the control\.

#### Oracle pairing\.

Oracle runs are unpadded\. NFE is identical across invocations of a setting and wall clock is not, so every oracle row is divided by the draft\-free pass of the invocation that produced it\. On SDAR\-8B b32 MATH500, whose position and token rows come from a pass at14\.114\.1tok/s and whose joint rows come from one at18\.418\.4, dividing every row by the faster pass would understate the position oracle by24%24\\%\.

#### Hardware\.

One node, eight identical NVIDIA RTX 6000 Ada cards \(48 GB\), PyTorch 2\.6\.0, CUDA 12\.4, Transformers 4\.55\.4; one run per card, never co\-located\. Runs disturbed by other users’ jobs or by two host reboots were re\-run from their last finished task\.

## Appendix HTerms, common questions and released resources

This appendix gathers the paper’s terms in one place, answers the questions a reader is likely to bring to it, each with a pointer to the full account, and lists what is released\.

#### Keywords\.

Speculative decoding, diffusion language models \(DLMs\), block diffusion, commit order, drafter, draft tree, coupled\-noise verification, lossless acceleration, SDAR, LLaDA, Dream, GSM8K, MBPP, HumanEval, MATH500\.

### H\.1Terms

Stale conditionals\.The most recently available target conditionals at the masked positions of a block\. They are stale because hypothetical future commitments change the true target distribution, but no additional target forward is performed while drafting \(Section[4\.2](https://arxiv.org/html/2610.22543#S4.SS2)\)\.

Commit order\.The sequence of positions the target commits next, one per forward under exact decoding\. The drafter predicts it asqϕ​\(s∣y\)q\_\{\\phi\}\(s\\mid y\), a distribution over the target’s next committed position given a hypothetical stateyy\(Section[4\.2](https://arxiv.org/html/2610.22543#S4.SS2)\)\.

Draft tree\.Candidate block states grown by recursively querying the prefix\-conditional predictor, high\-probability branches first, up to a budget ofDDdrafted states that the target verifies in one batched forward \(Section[4\.2](https://arxiv.org/html/2610.22543#S4.SS2)\)\.

Coupled\-noise match\.Verification atT\>0T\>0in which the drafter applies the same Gumbel noise as the target, so that, conditioned on the noise, the target transition is deterministic and verification reduces to state equality \(Section[4\.3](https://arxiv.org/html/2610.22543#S4.SS3)\)\.

SPOT \(Speculating Positions Order Trajectory\)\.The method: a0\.620\.62M\-parameter order model, a three\-layer transformer encoder over the target’s stale distributions at the masked positions, predicts the target’s next committed position; its predictions grow a draft tree that the target verifies in one batched forward \(Section[4](https://arxiv.org/html/2610.22543#S4)\)\.

SPOT\-γ\\gamma\.SPOT run unpadded with a prefix\-mass floorγ\\gammathat discards candidates below it, so that the realized verification budget adapts to drafter confidence;γ=0\.1\\gamma=0\.1on SDAR\-8B \(Section[4\.2](https://arxiv.org/html/2610.22543#S4.SS2), Appendix[D\.5](https://arxiv.org/html/2610.22543#A4.SS5)\)\.

NFE and throughput ratio\.NFE is the number of target forward passes per sample; the throughput ratio is measured against the draft\-free run of the same task, model and temperature \(Section[5\.1](https://arxiv.org/html/2610.22543#S5.SS1)\)\.

### H\.2Common questions

#### What problem does SPOT solve?

SPOT accelerates diffusion language models without changing their output distribution\. Exact decoding fills in only one masked position per model pass, while faster multi\-token schemes are lossy\. Speculative decoding is the standard lossless acceleration for autoregressive models\([Leviathan et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib27);[Chen et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib28)\), but a DLM must choose both what to generate and where\. SPOT drafts several future commitments with a small learned model, and the target verifies all of them in a single pass \(Section[1](https://arxiv.org/html/2610.22543#S1)\)\.

#### Why is commit order, not the token, the bottleneck?

The tokens are largely known in advance; the order is not\. The stale argmax matches the true token in92%92\\%of first\-block commits and≥90%\\geq 90\\%thereafter, while the probability that the previous round’s top\-confidence position commits next drops to3%3\\%by the ninth block\. Live oracles show that true positions are worth more than true tokens on every task \(Section[3\.1](https://arxiv.org/html/2610.22543#S3.SS1), Figure[2](https://arxiv.org/html/2610.22543#S3.F2)\)\.

#### How does SPOT draft and verify?

SPOT reads the target’s stale top\-3232probabilities at each masked position, predicts the next committed position, and repeats the prediction after each hypothetical commitment to grow a tree of block states\. The target evaluates the current and drafted states in one batched forward and accepts the longest prefix consistent with its own trajectory\. The drafter is trained entirely from draft\-free target trajectories and uses no target hidden states, token embeddings or text \(Section[4](https://arxiv.org/html/2610.22543#S4)\)\.

#### Is the output exact when the target samples?

Yes, for targets with an exact prefix cache\. AtT=0T=0, state\-equality verification reproduces the draft\-free target trajectory for any drafter \(Proposition[1](https://arxiv.org/html/2610.22543#Thmproposition1)\)\. AtT\>0T\>0, the drafter and the target share per\-position Gumbel noise, following the coupling of[Daliri et al\. \(2025\)](https://arxiv.org/html/2610.22543#bib.bib17); conditioned on the noise, the target transition is deterministic, and the drafted output has the same distribution as draft\-free decoding \(Proposition[2](https://arxiv.org/html/2610.22543#Thmproposition2)\)\. Draft quality therefore affects speed, never correctness \(Section[4\.3](https://arxiv.org/html/2610.22543#S4.SS3)\)\.

#### How does SPOT differ from Spiffy, SSD and BlockSpec?

Spiffy calibrates a static draft graph offline\([Agrawal et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib18)\), SSD orders positions by confidence into a self\-drafted chain\([Gao et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib4)\), and a BlockSpec\-style tree ranks positions by the target’s stale confidences\([Pan et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib21)\); SPOT learns the commit order instead\. Under one decoder and verifier, SPOT uses1515–35%35\\%fewer target forwards than Spiffy’s calibrated graph at1\.11\.1–1\.5×1\.5\\timesits throughput on SDAR\-8B, in every task and budget \(Section[5\.2](https://arxiv.org/html/2610.22543#S5.SS2)\)\.

#### Which models and benchmarks does the paper use?

The headline target is SDAR\-8B\([Cheng et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib19)\), a dense block\-causal DLM at block width3232\. Drafters are also trained for SDAR\-4B, SDAR\-1\.7B and the mixture\-of\-experts LLaDA2\.0\-mini\([Bie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib29)\), and transfer is measured on LLaDA\-8B\([Nie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib2)\)and Dream\-7B\([Ye et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib13)\)\. The benchmarks are GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2610.22543#bib.bib10)\), MBPP\([Austin et al\., 2021b](https://arxiv.org/html/2610.22543#bib.bib8)\), HumanEval\([Chen et al\., 2021](https://arxiv.org/html/2610.22543#bib.bib9)\)and MATH500\([Hendrycks et al\., 2021](https://arxiv.org/html/2610.22543#bib.bib11)\), withn=350n=350samples \(all164164for HumanEval\) over three seeds at generation length256256, except where a caption says otherwise \(Section[5\.1](https://arxiv.org/html/2610.22543#S5.SS1)\)\.

#### What are the main results?

On SDAR\-8B, SPOT needs3\.23\.2–4\.7×4\.7\\timesfewer target forwards than draft\-free decoding at budget88and4\.14\.1–6\.8×6\.8\\timesat budget2020, at2\.02\.0–2\.7×2\.7\\timesthe throughput at budget88\. AtT=1T=1it keeps1515–34%34\\%fewer forwards than Spiffy at1\.151\.15–1\.48×1\.48\\timesits throughput\. At generation length512512, SPOT\-γ\\gammareaches3\.23×3\.23\\timesdraft\-free throughput on MBPP atT=1T=1\. On SDAR\-1\.7B, where a target forward is cheap, Spiffy wins three throughput columns atT=0T=0and two atT=1T=1\(Sections[5\.2](https://arxiv.org/html/2610.22543#S5.SS2),[5\.3](https://arxiv.org/html/2610.22543#S5.SS3)and[5\.4](https://arxiv.org/html/2610.22543#S5.SS4)\)\.

#### Does SPOT work beyond SDAR\-8B?

Yes, across six DLMs\. The same recipe gives the lowest NFE and highest throughput in every comparison on SDAR\-4B; on LLaDA2\.0\-mini, which has a different tokenizer, it gives the lowest NFE in all3636comparisons and the higher throughput in3434\. The LLaDA2\.0\-mini checkpoint, pointed at SDAR\-8B without retraining, costs33–7%7\\%more forwards and still beats every baseline calibrated there\. Checkpoints that never saw LLaDA\-8B or Dream\-7B cut their256256forwards to5555–8888and5151–7979, measured with an approximate prefix cache, so these rows measure reach rather than losslessness \(Section[5\.4](https://arxiv.org/html/2610.22543#S5.SS4)\)\.

#### What are the limitations?

The gains diminish when verification is costly or acceptance is low\. Verification becomes the bottleneck at wide blocks, and exact tree packing remains unresolved under bidirectional attention\. Gains also narrow on low\-acceptance tasks, and atT≥2T\\geq 2the order model trained atT=0T=0is outperformed by noise\-aware ranking \(Section[6](https://arxiv.org/html/2610.22543#S6)\)\.

#### Where are the code and the drafters?

### H\.3Where this work sits

SPOT brings speculative decoding\([Leviathan et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib27);[Chen et al\., 2023](https://arxiv.org/html/2610.22543#bib.bib28)\)to diffusion language models\([Nie et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib2);[Ye et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib13)\), in particular to block\-diffusion models that denoise within a block and generate causally across blocks\([Arriola et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib31);[Cheng et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib19)\)\. It is compared against speculative decoding for DLMs with self\-drafting, draft graphs and confidence trees\([Gao et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib4);[Agrawal et al\., 2026](https://arxiv.org/html/2610.22543#bib.bib18);[Pan et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib21)\), and verifies with Gumbel coupling\([Daliri et al\., 2025](https://arxiv.org/html/2610.22543#bib.bib17)\)\.