领先一次调用：采用扩散草拟的打包式自推测解码

# Packed Self-SpeculativeDecoding with Diffusion Drafting
Source: [https://arxiv.org/html/2610.33379](https://arxiv.org/html/2610.33379)
## One Call Ahead: Packed Self\-Speculative Decoding with Diffusion Drafting

Yuxiao Chen††thanks:Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, USA;Email:[yuxiaoc@umich\.edu](mailto:[email protected]),[cxcai@umich\.edu](mailto:[email protected])\.

###### Abstract

Autoregressive \(AR\) language models generate tokens sequentially, so generating long outputs is costly\. Speculative decoding mitigates this by using a lightweight drafter to propose multiple tokens for the target AR model to verify in parallel, while a diffusion\-based drafter can produce an entire draft block in one forward pass\. In models with AR and diffusion modes, one network both drafts and verifies, but drafting still waits for verification, so the standard schedule needs two sequential model calls per iteration\. Existing one\-call schedules avoid the wait but incur input size quadratic in the speculative width\. Motivated by this efficiency bottleneck, we introduce Packed\-SS, a self\-speculative decoding schedule that verifies and drafts in a single model call with input size linear in the speculative width, while preserving the target model’s distribution\. The key idea is to let the diffusion drafter run*one call ahead*: while verifying the current draft, the model simultaneously constructs the next draft from an older verified prefix and keeps its unresolved suffix for the next iteration\. Under a mild monotonicity assumption on the drafter, we further show that Packed\-SS requires no more model calls than the standard two\-call decoding in the greedy setting\. Experiments on Nemotron 3B/8B and SetDLM 1\.7B show higher throughput and lower mean wall time than both baselines under greedy decoding\.

###### Contents

1. [1Introduction](https://arxiv.org/html/2610.33379#S1)
2. [2Background](https://arxiv.org/html/2610.33379#S2)1. [2\.1Notation](https://arxiv.org/html/2610.33379#S2.SS1) 2. [2\.2Preliminaries](https://arxiv.org/html/2610.33379#S2.SS2) 3. [2\.3Related work](https://arxiv.org/html/2610.33379#S2.SS3)
3. [3Our Method](https://arxiv.org/html/2610.33379#S3)1. [3\.1Packed self\-speculative procedure under greedy decoding](https://arxiv.org/html/2610.33379#S3.SS1) 2. [3\.2General packed self\-speculative decoding](https://arxiv.org/html/2610.33379#S3.SS2)
4. [4Experiments](https://arxiv.org/html/2610.33379#S4)1. [4\.1Greedy decoding](https://arxiv.org/html/2610.33379#S4.SS1) 2. [4\.2Decoding at temperature 1](https://arxiv.org/html/2610.33379#S4.SS2)
5. [5Discussion and Limitations](https://arxiv.org/html/2610.33379#S5)
6. [References](https://arxiv.org/html/2610.33379#bib)
7. [ANotation](https://arxiv.org/html/2610.33379#A1)
8. [BProofs](https://arxiv.org/html/2610.33379#A2)1. [B\.1Pathwise dominance](https://arxiv.org/html/2610.33379#A2.SS1) 2. [B\.2Exact sampling](https://arxiv.org/html/2610.33379#A2.SS2)
9. [CAny\-Order Extension](https://arxiv.org/html/2610.33379#A3)
10. [DAttention Mask Layouts](https://arxiv.org/html/2610.33379#A4)1. [D\.1Nemotron: sequence packing and padding](https://arxiv.org/html/2610.33379#A4.SS1) 2. [D\.2SetDLM: position\-aligned queries and ordered context](https://arxiv.org/html/2610.33379#A4.SS2) 3. [D\.3Input\-size comparison](https://arxiv.org/html/2610.33379#A4.SS3)
11. [EExperimental Details](https://arxiv.org/html/2610.33379#A5)1. [E\.1Hardware, models, and tasks](https://arxiv.org/html/2610.33379#A5.SS1) 2. [E\.2Initialization and stopping](https://arxiv.org/html/2610.33379#A5.SS2) 3. [E\.3Prompts and scoring](https://arxiv.org/html/2610.33379#A5.SS3) 4. [E\.4Sampling and repetitions](https://arxiv.org/html/2610.33379#A5.SS4) 5. [E\.5Timing and confidence intervals](https://arxiv.org/html/2610.33379#A5.SS5) 6. [E\.6SetDLM decoding settings](https://arxiv.org/html/2610.33379#A5.SS6)
12. [FAdditional Experiments and Tables](https://arxiv.org/html/2610.33379#A6)1. [F\.1Nemotron block size and execution cost](https://arxiv.org/html/2610.33379#A6.SS1) 2. [F\.2Proposal\-only LoRA accounting](https://arxiv.org/html/2610.33379#A6.SS2) 3. [F\.3Model execution and non\-model decoder time](https://arxiv.org/html/2610.33379#A6.SS3) 4. [F\.4Absolute measurements](https://arxiv.org/html/2610.33379#A6.SS4) 5. [F\.5Confidence\-ordered verification on SetDLM](https://arxiv.org/html/2610.33379#A6.SS5) 6. [F\.6Accuracy and output trajectories](https://arxiv.org/html/2610.33379#A6.SS6) 7. [F\.7Paired efficiency comparisons](https://arxiv.org/html/2610.33379#A6.SS7)

## 1Introduction

Modern language models \(LMs\) increasingly generate long sequences of intermediate reasoning before producing a final answer, making inference latency a central bottleneck\. Standard autoregressive \(AR\) models\([Brown et al\., 2020](https://arxiv.org/html/2610.33379#bib.bib34)\)generate one token at a time from left to right\. Producing anLL\-length sequence therefore requires at leastLLsequential model calls, where the computation for each new token depends on all previously generated tokens\.

Speculative decoding\([Leviathan et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib12);[Chen et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib35);[Xia et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib53)\)accelerates AR inference by using a lightweightdraft modelthat proposes multiple tokens for the full\-sizetarget modelto verifyin parallel\. A rejection\-sampling correction step ensures that the resulting sequence follows exactly the target model’s distribution\. The target model can thus output several tokens from a single verification call\. However, when the drafter is AR, creating a draft block remains sequential\. Therefore, speculative decoding shortens the expensive serial chain, but does not eliminate it\.

Masked diffusion language models \(MDLMs\) provide a natural way to parallelize drafting\([Sahoo et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib14);[Shi et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib49);[Gong et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib50);[Nie et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib51);[Ye et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib52)\)\. By learning to reverse a corruption process that transforms data into noise, an MDLM can predict multiple tokens simultaneously\. Therefore, when used as a speculative drafter, it can draft an entire block in one forward pass\([Christopher et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib26);[Li et al\., 2025a](https://arxiv.org/html/2610.33379#bib.bib27);[Chen et al\., 2026a](https://arxiv.org/html/2610.33379#bib.bib20);[Zhang et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib21)\)\. Some LMs support both causal AR prediction and parallel masked diffusion prediction in the same network\([Liu et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib32);[Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)\. They provide a natural form of self\-speculative decoding, with the drafter and verifier sharing the same model weights\.

However, shared weights do not eliminate the sequential dependency between drafting and verification\. The standard self\-speculative schedule, referred to asLinear\-SS, uses two sequential calls: one AR call verifies the current draft and one diffusion call constructs the next draft from the newly verified prefix\. Drafting cannot begin until verification finishes to determine where the verified prefix ends\.

Quadratic\-SS\([Liu et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib32)\)eliminates this dependency by preparing a continuation for every possible verification outcome\. With speculative widthWW, it pre\-drafts separately for the cases in which0,1,…,W0,1,\\ldots,Wdraft tokens are accepted\. Whichever outcome occurs, the next draft is already available\. This reduces each iteration to one call, but the number of evaluated positions grows quadratically withWW\. Thus, Linear\-SS evaluatesO⁡\(W\)O\(W\)positions in two calls, whereas Quadratic\-SS evaluatesO⁡\(W2\)O\(W^\{2\}\)in one\.

At small batch sizes, a substantial fraction of a call’s latency can come from loading model weights from memory, making it attractive to evaluate more positions per call\([Cai et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib37);[Fu et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib15)\)\. To illustrate this tradeoff, suppose a call evaluatingnnpositions costsτ\+δ​n\\tau\+\\delta n, whereτ\\tauis a fixed overhead, mostly weight loading, andδ\\deltais the marginal cost per position\. An iteration costs approximately2​τ\+2​δ​W2\\tau\+2\\delta Wfor Linear\-SS andτ\+δ​W2\\tau\+\\delta W^\{2\}for Quadratic\-SS\. Verifying and drafting in one call with approximately2​W2Wevaluated positions would cost onlyτ\+2​δ​W\\tau\+2\\delta Wper iteration, avoiding both the second\-call overhead and the quadratic branching cost\.

This raises the central question of this work:

Can we verify and draft in one call, while only evaluating positions linear in the speculative width?

#### Contributions\.

We answer affirmatively withPacked\-SS, a plug\-and\-play self\-speculative decoding schedule for such models\. The idea is simple: the next draft does not need to wait for the newest verified prefix\. Instead, we let the diffusion drafter run*one call ahead*\. At iterationtt, one forward pass simultaneously \(i\) verifies the draft, and \(ii\) proposes a new draft block using the prefix available at the beginning of iterationtt\. If verification advances byAApositions, the firstAApositions of this new draft have already been resolved and are discarded\. The remaining suffix becomes the draft for the next iteration\. Drafting therefore no longer waits for verification\. Figure[1](https://arxiv.org/html/2610.33379#S1.F1)illustrates the difference between Packed\-SS and existing schedules\.

The two key observations behind this design are as follows\. First, a diffusion drafter predicts tokens independently given its conditioning prefix, so a rejection at one position does not invalidate later draft tokens\. Second, draft freshness affects speed rather than correctness: standard acceptance and residual correction remain exact when they use the proposal distribution that generated each draft token\. Thus, an older draft may be accepted less often, but does not bias the output\.

These observations remove the synchronization between drafting and verification\. With an appropriate attention mask, they can be packed into one forward pass\. Each iteration thus requires only one model call, while the number of evaluated positions remains linear inWW\.

Our contributions are summarized as follows:

1. 1\.We introducePacked\-SS, which simultaneously verifies the previous draft and constructs the next draft in a single model call, with input size linear in the speculative width, while preserving the target model’s distribution\.
2. 2\.We theoretically prove that, assuming that correct draft predictions remain correct as verified context grows, Packed\-SS requires no more model calls than Linear\-SS to generate any prescribed output length\.
3. 3\.We evaluate Packed\-SS on Nemotron 3B/8B\([Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)and SetDLM 1\.7B\([Arriola and Kuleshov, 2026](https://arxiv.org/html/2610.33379#bib.bib10)\)\. Across the primary greedy evaluations, Packed\-SS achieves higher throughput than Linear\-SS and Quadratic\-SS\. Under temperature\-1 sampling, it consistently outperforms Linear\-SS while remaining competitive with Quadratic\-SS\.

First model callSecond model callLinearVerify old draftaccept some new tokensDraft using new tokensQuadraticVerify old draftDraft if 0 tokens acceptedDraft if 1 token accepted⋮\\vdotsDraft ifWWtokens acceptedPackedVerify old draftDraft using old prefixToken evals\.2​W\+12W\+1\(W\+1\)2\(W\+1\)^\{2\}≤2​W\\leq 2W

Figure 1:Where drafting happens in one call\.Outlined boxes are model calls\. Linear\-SS must finish verifying before it can draft, so an iteration costs two calls\. Quadratic\-SS fits in one call by drafting a continuation for every place verification might stop, so its input grows quadratically with the speculative widthWW\. Packed\-SS also uses one call, building a single draft from the previously verified prefix alone\. The right column gives total draft and verifier prediction counts per iteration, including predictions that may later be discarded\.

## 2Background

### 2\.1Notation

We write𝒱\\mathcal\{V\}for the vocabulary andθ\\thetafor model parameters\. For a sequencexxwe writex\(<i\)=\(x\(1\),…,x\(i−1\)\)x^\{\(<i\)\}=\(x^\{\(1\)\},\\ldots,x^\{\(i\-1\)\}\), and\(x,y\)\(x,y\)denotes the sequencexxfollowed by itemyy\. We use\[u\]\+=max⁡\{u,0\}\[u\]\_\{\+\}=\\max\\\{u,0\\\}to denote the positive part ofuu\. A complete list of variables can be found in[AppendixA](https://arxiv.org/html/2610.33379#A1)\.

### 2\.2Preliminaries

#### Speculative decoding\.

Given prefixPP, the drafter proposesWWdraft tokensD=\(d\(1\),…,d\(W\)\)D=\(d^\{\(1\)\},\\ldots,d^\{\(W\)\}\), drawing eachd\(i\)d^\{\(i\)\}from a drafter distributionq\(i\)q^\{\(i\)\}, whereiicounts positions afterPP\. In one model call, the target modelpθp\_\{\\theta\}usesPPandDDto return target distributionsp\(i\)=pθ\(⋅∣P,d\(<i\)\)p^\{\(i\)\}=p\_\{\\theta\}\(\\cdot\\mid P,d^\{\(<i\)\}\)for everyi≤W\+1i\\leq W\+1\. The draft tokens are then verified from left to right, where each draft tokend\(i\)d^\{\(i\)\}is accepted with probabilitymin⁡\{1,p\(i\)​\(d\(i\)\)/q\(i\)​\(d\(i\)\)\}\\min\\\{1,\\,p^\{\(i\)\}\(d^\{\(i\)\}\)/q^\{\(i\)\}\(d^\{\(i\)\}\)\\\}\. At the first rejection, the decoder replaces that draft token with a correction token from the residual distribution∝\[p\(i\)−q\(i\)\]\+\\propto\[p^\{\(i\)\}\-q^\{\(i\)\}\]\_\{\+\}\. If all tokens in a draft are accepted, the target model yields one extra bonus token drawn fromp\(W\+1\)=pθ\(⋅∣P,D\)p^\{\(W\+1\)\}=p\_\{\\theta\}\(\\cdot\\mid P,D\)\. The rejection\-sampling correction ensures that the generated text follows the target distribution exactly, so a low\-quality drafter costs speed but never quality\.

In practice, we use the target model to sample at a certain temperature, with temperature11being the model’s original distribution and lower temperature yielding a sharper distribution\. At temperature00, the output distribution mass is on itsarg⁡max\\arg\\maxtoken\. This is also calledgreedy decoding\.

#### Masked diffusion language models\.

A masked diffusion language model \(MDLM\) generates text by repeatedly filling missing token positions, represented by a special mask tokenMask\([Sahoo et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib14);[Shi et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib49)\)\. Following the denoising framework of discrete diffusion\([Hoogeboom et al\., 2021](https://arxiv.org/html/2610.33379#bib.bib54);[Austin et al\., 2021a](https://arxiv.org/html/2610.33379#bib.bib13);[Campbell et al\., 2022](https://arxiv.org/html/2610.33379#bib.bib47);[Lou et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib48)\), the model is trained to predict original tokens from sequences in which randomly selected tokens have been replaced byMask\. During inference, letCCdenote the visible tokens and their positions, andUUthe remaining masked positions\. One model call predicts, for each positioni∈Ui\\in U, a distributionqθ\(i\)\(⋅∣C\)q\_\{\\theta\}^\{\(i\)\}\(\\cdot\\mid C\)that approximates the conditional marginalp\(i\)\(⋅∣C\)p^\{\(i\)\}\(\\cdot\\mid C\)of the data distributionpp\. Sampling each position independently from its prediction approximates the joint distributionp⁡\(zU∣C\)p\(z\_\{U\}\\mid C\)by the product of marginalsqθ​\(zU∣C\)=∏i∈Uqθ\(i\)​\(zi∣C\)q\_\{\\theta\}\(z\_\{U\}\\mid C\)=\\prod\_\{i\\in U\}q\_\{\\theta\}^\{\(i\)\}\(z\_\{i\}\\mid C\)\. This approximation incurs error whenever the tokens inUUare not conditionally independent givenCC, incurring intrinsic trade\-off between sampling efficiency and accuracy\([Li and Cai, 2025](https://arxiv.org/html/2610.33379#bib.bib2);[Chen et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib9);[Zhao and Cai, 2026](https://arxiv.org/html/2610.33379#bib.bib8)\)\. To keep this error small, the decoder fills only a subset of the positions in each call, often those with the most confident predictions\([Ben\-Hamu et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib1);[Nie et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib51);[Cai and Li, 2026](https://arxiv.org/html/2610.33379#bib.bib3)\), and then repeats the model call with the updated context to predict the remaining positions\. Block diffusion applies this procedure within blocks generated from left to right, completing each block before starting the next\([Arriola et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib11)\)\. A block size of one recovers autoregressive generation, while a single block covering the full sequence recovers full\-sequence masked diffusion\.

#### Shared\-weight drafting and verification\.

A model with shared\-weight AR and diffusion modes can perform both roles in speculative decoding: its masked mode serves as the*drafter*qθq\_\{\\theta\}, proposing multiple tokens after the verified prefix in parallel, while its causal mode serves as the*verifier*pθp\_\{\\theta\}\. When these modes differ only in their attention masks, Packed\-SS can combine them in one forward pass using a custom attention mask\([Miao et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib36);[Spector and Ré, 2023](https://arxiv.org/html/2610.33379#bib.bib43);[Fu et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib15)\)\. Nemotron\-Labs\-Diffusion supports both modes through joint training on autoregressive and diffusion objectives\([Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)\. SetDLM also supports this construction through its any\-order formulation: single\-position conditionals in left\-to\-right order provide AR verification, while parallel predictions of unresolved positions provide diffusion drafting\([Arriola and Kuleshov, 2026](https://arxiv.org/html/2610.33379#bib.bib10)\)\. See[AppendixD](https://arxiv.org/html/2610.33379#A4)for model\-specific input layouts and attention\-mask constructions\.

### 2\.3Related work

#### Sequential draft–verify decoding\.

Most speculative decoding methods improve drafting within a sequential draft–verify schedule\([Leviathan et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib12);[Chen et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib35)\)\. Self\-speculative methods reuse the target network: Draft & Verify\([Zhang et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib40)\)skips selected layers without retraining, whereas LayerSkip\([Elhoushi et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib39)\)trains intermediate layers for early\-exit drafting\. EAGLE\([Li et al\., 2024b](https://arxiv.org/html/2610.33379#bib.bib38);[Li et al\., 2025b](https://arxiv.org/html/2610.33379#bib.bib4)\)variants instead train auxiliary drafters using target features\. Parallel drafting reduces sequential work, from blockwise parallel decoding\([Stern et al\., 2018](https://arxiv.org/html/2610.33379#bib.bib42)\)and SpecDec\([Xia et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib53)\)to diffusion\-based drafters such as SpecDiff\([Christopher et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib26)\)and DiffuSpec\([Li et al\., 2025a](https://arxiv.org/html/2610.33379#bib.bib27)\)\. DEER\([Cheng et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib7)\)aligns prefix\-conditioned drafts with an AR teacher, while DFlash\([Chen et al\., 2026a](https://arxiv.org/html/2610.33379#bib.bib20)\)uses a shallow block\-diffusion drafter conditioned on target features\. DSpark\([Cheng et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib41)\)adds confidence\-based verification\-length scheduling, while DFlow\([Zhang et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib21)\)reuses verifier features from rejected suffixes to improve subsequent drafts\.

#### Other speculative decoding schedules\.

SpecInfer\([Miao et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib36)\)and Medusa\([Cai et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib37)\)use tree attention to verify multiple candidate continuations, while EAGLE\-2\([Li et al\., 2024a](https://arxiv.org/html/2610.33379#bib.bib5)\)adaptively constructs draft trees using confidence estimates\. Tree\-structured verification covers more candidate continuations, but does not by itself remove the dependency between drafting and verification\. Jacobi decoding\([Santilli et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib6)\)takes a different approach, updating multiple tentative positions in parallel through fixed\-point iteration\. Lookahead Decoding\([Fu et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib15)\)turns these trajectories into cachednn\-gram candidates and combines a lookahead branch with a verification branch in one forward pass\. For diffusion decoding, SSD\([Gao et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib45)\)and FreeDave\([Wu and Zhang, 2025](https://arxiv.org/html/2610.33379#bib.bib46)\)batch multiple speculative decoding states, with FreeDave also reusing accepted\-state predictions for subsequent drafting\. Within shared\-weight models, the quadratic schedules in[Samragh et al\. \(2025\)](https://arxiv.org/html/2610.33379#bib.bib33), TiDAR\([Liu et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib32)\), and Nemotron\([Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)prepare continuations at every possible acceptance boundary, combining drafting and verification at the cost ofO⁡\(W2\)O\(W^\{2\}\)evaluated positions for speculative widthWW\. Apple’s linear decoding\([Samragh et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib33)\)and Introspective DLMs\([Yu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib44)\)discard continuations conditioned on tentative tokens after rejection, which frequently fails to simultaneously yield a draft\. Speculative Jacobi Decoding\([Teng et al\., 2024](https://arxiv.org/html/2610.33379#bib.bib56)\)reuses predictions after rejection, but its proposals also condition on tentative tokens\.

## 3Our Method

In this section, we first present the high\-level description of Packed\-SS\. We then describe the procedure under greedy decoding in[Section3\.1](https://arxiv.org/html/2610.33379#S3.SS1), where the main idea is clearest\.[Section3\.2](https://arxiv.org/html/2610.33379#S3.SS2)describes the general procedure\.

At a high level, iterationttof Packed\-SS reads the verified prefixPt−1P\_\{t\-1\}and the current draftDt−1D\_\{t\-1\}, and performs verification and drafting simultaneously and independently in one model call\. The verifier acceptsAtA\_\{t\}tokens from the standard speculative decoding process \(including one correction or bonus token\)\. TheseAtA\_\{t\}tokens are extended toPt−1P\_\{t\-1\}to form the next prefixPtP\_\{t\}\. For speculative widthWW, the drafter reads onlyPt−1P\_\{t\-1\}and generates a new proposalZt=\(zt\(i\)\)i=1WZ\_\{t\}=\(z\_\{t\}^\{\(i\)\}\)\_\{i=1\}^\{W\}for the nextWWpositions\. The firstAtA\_\{t\}of those positions have just been filled by the verifier, so their proposal tokens are discarded, and the rest become the draftDtD\_\{t\}that iterationt\+1t\+1verifies\. See[Figure2](https://arxiv.org/html/2610.33379#S3.F2)for an illustration of one iteration under greedy decoding\.

### 3\.1Packed self\-speculative procedure under greedy decoding

We now state the procedure under greedy decoding, and summarize it in[Algorithm1](https://arxiv.org/html/2610.33379#alg1)\.

One model callVerified prefixPt−1P\_\{t\-1\}\+1\+1\+2\+2\+3\+3\+4\+4IterationttreadsPt−1P\_\{t\-1\}dt−1\(1\)d\_\{t\-1\}^\{\(1\)\}dt−1\(2\)d\_\{t\-1\}^\{\(2\)\}dt−1\(3\)d\_\{t\-1\}^\{\(3\)\}DraftDt−1D\_\{t\-1\}AR==≠\\neqVerifierpθp\_\{\\theta\}vt\(1\)v\_\{t\}^\{\(1\)\}vt\(2\)v\_\{t\}^\{\(2\)\}vt\(3\)v\_\{t\}^\{\(3\)\}vt\(4\)v\_\{t\}^\{\(4\)\}AR predictionsComputed, not reachedAdvanceAt=ht\+1=2A\_\{t\}=h\_\{t\}\{\+\}1=2Drafterqθq\_\{\\theta\}zt\(1\)z\_\{t\}^\{\(1\)\}zt\(2\)z\_\{t\}^\{\(2\)\}zt\(3\)z\_\{t\}^\{\(3\)\}zt\(4\)z\_\{t\}^\{\(4\)\}ProposalZtZ\_\{t\}DiffusionIterationt\+1t\{\+\}1readsPt−1P\_\{t\-1\}dt−1\(1\)d\_\{t\-1\}^\{\(1\)\}vt\(2\)v\_\{t\}^\{\(2\)\}zt\(3\)z\_\{t\}^\{\(3\)\}zt\(4\)z\_\{t\}^\{\(4\)\}DraftDtD\_\{t\}Verified prefixPtP\_\{t\}DraftDtD\_\{t\}DiscardedKept

Figure 2:One iteration of greedy Packed\-SS\.Columns are token positions\. The old draft tokendt−1\(i\)d\_\{t\-1\}^\{\(i\)\}, the verifier predictionvt\(i\)v\_\{t\}^\{\(i\)\}, and the new proposal tokenzt\(i\)z\_\{t\}^\{\(i\)\}are in the same position\. A single forward pass takes the verified prefixPt−1P\_\{t\-1\}and draftDt−1D\_\{t\-1\}as input and produces the two middle rows: the AR verifier’s next\-token predictionsvt\(i\)v\_\{t\}^\{\(i\)\}, conditioned onPt−1P\_\{t\-1\}anddt−1\(1\),…,dt−1\(i−1\)d\_\{t\-1\}^\{\(1\)\},\\dots,d\_\{t\-1\}^\{\(i\-1\)\}, and the diffusion drafter’s proposalZt=\(zt\(i\)\)i=1WZ\_\{t\}=\(z\_\{t\}^\{\(i\)\}\)\_\{i=1\}^\{W\}, conditioned onPt−1P\_\{t\-1\}alone\. Here, one matching draft token \(ht=1h\_\{t\}=1\) and the correction tokenvt\(2\)v\_\{t\}^\{\(2\)\}yieldAt=2A\_\{t\}=2output tokens\. Proposals at these two positions are discarded, leavingDt=\(zt\(3\),zt\(4\)\)D\_\{t\}=\(z\_\{t\}^\{\(3\)\},z\_\{t\}^\{\(4\)\}\)as the next draft\.#### Initialization\.

Given a promptPP, start with the prefixP0=PP\_\{0\}=Pand draftD0=∅D\_\{0\}=\\varnothing\.

#### Iterationt≥1t\\geq 1\.

Given the prefixPt−1P\_\{t\-1\}and draftDt−1=\(dt−1\(1\),dt−1\(2\),…\)D\_\{t\-1\}=\(d\_\{t\-1\}^\{\(1\)\},d\_\{t\-1\}^\{\(2\)\},\\ldots\)followingPt−1P\_\{t\-1\}, iterationttevaluates the verifier and drafter in one forward pass\.

For verification, the AR mode predicts the token at each position of the previous draft plus one position after it, conditioned on the prefixPt−1P\_\{t\-1\}and the draft tokens before that position:

vt\(i\)=arg⁡maxx∈𝒱​pθ​\(x∣Pt−1,dt−1\(<i\)\),1≤i≤\|Dt−1\|\+1\.v\_\{t\}^\{\(i\)\}=\\arg\\max\_\{x\\in\\mathcal\{V\}\}p\_\{\\theta\}\(x\\mid P\_\{t\-1\},d\_\{t\-1\}^\{\(<i\)\}\),\\qquad 1\\leq i\\leq\|D\_\{t\-1\}\|\+1\.\(1\)Lethth\_\{t\}be the number of leading draft tokens that match the verifier’s predictions:

ht=max⁡\{0≤h≤\|Dt−1\|:dt−1\(i\)=vt\(i\)​for every​i≤h\}\.h\_\{t\}=\\max\\\{0\\leq h\\leq\|D\_\{t\-1\}\|:d\_\{t\-1\}^\{\(i\)\}=v\_\{t\}^\{\(i\)\}\\text\{ for every \}i\\leq h\\\}\.\(2\)Verification yields the accepted draft prefix followed byvt\(ht\+1\)v\_\{t\}^\{\(h\_\{t\}\+1\)\}, which corrects the first mismatch or supplies a bonus after full acceptance\. Thus, a total ofAt=ht\+1A\_\{t\}=h\_\{t\}\+1tokens are accepted\.

In the same call, the diffusion mode drafts theWWpositions that follow the previous prefixPt−1P\_\{t\-1\}:

Zt=\(zt\(1\),…,zt\(W\)\),zt\(i\)=arg⁡maxx∈𝒱​qθ\(i\)​\(x∣Pt−1\)\.Z\_\{t\}=\(z\_\{t\}^\{\(1\)\},\\ldots,z\_\{t\}^\{\(W\)\}\),\\qquad z\_\{t\}^\{\(i\)\}=\\arg\\max\_\{x\\in\\mathcal\{V\}\}q\_\{\\theta\}^\{\(i\)\}\(x\\mid P\_\{t\-1\}\)\.\(3\)It is important to note that the draft tokensZtZ\_\{t\}are generated conditioned on the previous prefixPt−1P\_\{t\-1\}, rather than in Linear\-SS the new prefixPtP\_\{t\}\. This distinction allows drafting and verification to run in parallel while Linear\-SS needs to be evaluated sequentially\. On nonterminal iterations, the suffix ofZtZ\_\{t\}after its firstAtA\_\{t\}tokens becomes the new draft:

Dt=\(zt\(At\+1\),…,zt\(W\)\)\.D\_\{t\}=\(z\_\{t\}^\{\(A\_\{t\}\+1\)\},\\ldots,z\_\{t\}^\{\(W\)\}\)\.\(4\)

#### Termination\.

Stop at the first accepted end\-of\-sequence \(EOS\) token or afterLLnew tokens\.

Algorithm 1Packed\-SS with LTR greedy decoding0:prompt

PP, speculative width

WW, output length limit

LL, verifier

pθp\_\{\\theta\}, drafter

qθq\_\{\\theta\}
1:

P0←PP\_\{0\}\\leftarrow P,

D0←∅D\_\{0\}\\leftarrow\\varnothing,

t←1t\\leftarrow 1
2:while

\|Pt−1\|−\|P\|<L\|P\_\{t\-1\}\|\-\|P\|<Land \(

t=1t=1or the last token of

Pt−1P\_\{t\-1\}is not

EOS\\mathrm\{EOS\}\)do

2:In one model call:

3:

vt\(i\)←arg⁡maxx​pθ​\(x∣Pt−1,dt−1\(<i\)\)v\_\{t\}^\{\(i\)\}\\leftarrow\\arg\\max\_\{x\}p\_\{\\theta\}\(x\\mid P\_\{t\-1\},d\_\{t\-1\}^\{\(<i\)\}\)for

i=1,…,\|Dt−1\|\+1i=1,\\ldots,\|D\_\{t\-1\}\|\+1⊳\\trianglerightVerifier predictions

4:

zt\(i\)←arg⁡maxx​qθ\(i\)​\(x∣Pt−1\)z\_\{t\}^\{\(i\)\}\\leftarrow\\arg\\max\_\{x\}q\_\{\\theta\}^\{\(i\)\}\(x\\mid P\_\{t\-1\}\)for

i=1,…,Wi=1,\\ldots,W⊳\\trianglerightDrafter proposals

4:end

5:

ht←max⁡\{0≤h≤\|Dt−1\|:dt−1\(i\)=vt\(i\)​for all​i≤h\}h\_\{t\}\\leftarrow\\max\\\{0\\leq h\\leq\|D\_\{t\-1\}\|:d\_\{t\-1\}^\{\(i\)\}=v\_\{t\}^\{\(i\)\}\\text\{ for all \}i\\leq h\\\}⊳\\trianglerightCount matches

6:

Ct←\(dt−1\(1\),…,dt−1\(ht\),vt\(ht\+1\)\)C\_\{t\}\\leftarrow\(d\_\{t\-1\}^\{\(1\)\},\\ldots,d\_\{t\-1\}^\{\(h\_\{t\}\)\},v\_\{t\}^\{\(h\_\{t\}\+1\)\}\),

At←ht\+1A\_\{t\}\\leftarrow h\_\{t\}\+1⊳\\trianglerightAdd verifier token

7:

a←min⁡\(\{At,L\+\|P\|−\|Pt−1\|\}∪\{1≤i≤At:Ct\(i\)=EOS\}\)a\\leftarrow\\min\\bigl\(\\\{A\_\{t\},L\+\|P\|\-\|P\_\{t\-1\}\|\\\}\\cup\\\{1\\leq i\\leq A\_\{t\}:C\_\{t\}^\{\(i\)\}=\\mathrm\{EOS\}\\\}\\bigr\)⊳\\trianglerightEnding condition

8:

Ct←\(Ct\(1\),…,Ct\(a\)\)C\_\{t\}\\leftarrow\(C\_\{t\}^\{\(1\)\},\\ldots,C\_\{t\}^\{\(a\)\}\)⊳\\trianglerightPrune by stopping index

9:

Pt←\(Pt−1,Ct\)P\_\{t\}\\leftarrow\(P\_\{t\-1\},C\_\{t\}\)⊳\\trianglerightPrepare next prefix, append accepted tokens

10:

Dt←\(zt\(a\+1\),…,zt\(W\)\)D\_\{t\}\\leftarrow\(z\_\{t\}^\{\(a\+1\)\},\\ldots,z\_\{t\}^\{\(W\)\}\)⊳\\trianglerightPrepare next draft,zt\(i\)z\_\{t\}^\{\(i\)\}is generated in line 4

11:

t←t\+1t\\leftarrow t\+1
12:endwhile

13:Output:

Pt−1P\_\{t\-1\}

#### Comparison with Linear\-SS and Quadratic\-SS\.

Linear\-SS alternates between drafting\-only and verification\-only calls, whereas Packed\-SS performs both in each call\. By retaining the unresolved suffix ofZtZ\_\{t\}as the next draft, Packed\-SS avoids a separate drafting call\. This overlap comes at the cost of a shorter draft conditioned on less verified context: on nonterminal iterations of[Algorithm1](https://arxiv.org/html/2610.33379#alg1), Packed\-SS retains\|Dt\|=W−At≤W−1\|D\_\{t\}\|=W\-A\_\{t\}\\leq W\-1tokens by equation[4](https://arxiv.org/html/2610.33379#S3.E4), whereas Linear\-SS draftsWWtokens conditioned onAtA\_\{t\}more verified tokens\. In addition, both Packed\-SS and Quadratic\-SS draft and verify simultaneously in one model call\. However, Packed\-SS evaluatesO⁡\(W\)O\(W\)positions per iteration rather thanO⁡\(W2\)O\(W^\{2\}\)\. The trade\-off is conditioning context: Quadratic\-SS’s selected proposal excludes only the final verifier token fromPtP\_\{t\}, whereas Packed\-SS conditions onPt−1P\_\{t\-1\}, excluding allAtA\_\{t\}newly appended tokens\. Conceptually, Packed\-SS can be viewed as Quadratic\-SS with only the branch for zero accepted draft tokens\.

#### Theoretical guarantees\.

Under greedy decoding and the monotonicity assumption below, Packed\-SS requires no more forward passes than Linear\-SS to generate a prescribed number of tokens from the same prompt\.

Fix a prompt and speculative widthWW, and letx1,x2,…x\_\{1\},x\_\{2\},\\ldotsdenote the target model’s greedy continuation\. Letx^k​\(c\)\\widehat\{x\}\_\{k\}\(c\)denote the deterministic drafter’s prediction at positionkk, conditioned on the prompt andx1,…,xcx\_\{1\},\\ldots,x\_\{c\}\.

###### Assumption 3\.1\(Monotonicity of draft correctness\)\.

For each output positionkk, there exists a thresholdck⋆≤kc\_\{k\}^\{\\star\}\\leq ksuch that𝟙\{x^k\(c\)=xk\}=𝟙\{c≥ck⋆\}\\mathds\{1\}\\\{\\widehat\{x\}\_\{k\}\(c\)=x\_\{k\}\\\}=\\mathds\{1\}\\\{c\\geq c\_\{k\}^\{\\star\}\\\}for everyc<kc<kevaluated by either schedule\.

Here,ck⋆=kc\_\{k\}^\{\\star\}=kindicates that positionkkis never correctly predicted before it is verified\. Thus, additional verified context cannot turn a correct prediction into an incorrect one\. The thresholds need not increase withkk: a later position may become correctly predictable with less context than an earlier one\.

###### Theorem 3\.2\(Pathwise dominance\)\.

Suppose Packed\-SS and Linear\-SS use the same target model, drafter, and speculative widthW≥1W\\geq 1, and start from a common verified prefix\. Under[Assumption3\.1](https://arxiv.org/html/2610.33379#S3.Thmtheorem1), Packed\-SS’s verified prefix after2​t2tforward passes is at least as long as Linear\-SS’s afterttiterations, for everyt≥0t\\geq 0\.

LetNFES​\(L\)\\mathrm\{NFE\}\_\{S\}\(L\)denote the number of forward passes scheduleSSrequires to generateLLtokens afterPP\. Since each Linear\-SS iteration uses two forward passes, for everyL≥0L\\geq 0, one has

NFEPacked\-SS​\(L\)≤NFELinear\-SS​\(L\)\.\\mathrm\{NFE\}\_\{\\text\{Packed\-SS\{\}\}\}\(L\)\\leq\\mathrm\{NFE\}\_\{\\text\{Linear\-SS\{\}\}\}\(L\)\.\(5\)

Since both schedules follow the same greedy continuation, the bound also holds when stopping at the first emitted EOS token\. See[SectionB\.1](https://arxiv.org/html/2610.33379#A2.SS1)for the proof\.

### 3\.2General packed self\-speculative decoding

We now present Packed\-SS for general sampling\. Compared with the greedy procedure in[Algorithm1](https://arxiv.org/html/2610.33379#alg1)from[Section3\.1](https://arxiv.org/html/2610.33379#S3.SS1), the general sampling requires two minor modifications\.

Consider iterationtt\. First, we store beyond each draft tokendt\(i\)d\_\{t\}^\{\(i\)\}the distribution from which it was sampled,Qt\(i\)=qt\(a\+i\)=qθ\(a\+i\)\(⋅∣Pt−1\)Q\_\{t\}^\{\(i\)\}=q\_\{t\}^\{\(a\+i\)\}=q\_\{\\theta\}^\{\(a\+i\)\}\(\\cdot\\mid P\_\{t\-1\}\)\. At temperature00,Qt−1\(i\)Q\_\{t\-1\}^\{\(i\)\}collapses to a point mass ondt−1\(i\)d\_\{t\-1\}^\{\(i\)\}\. So storing the draft token alone is sufficient as in[Algorithm1](https://arxiv.org/html/2610.33379#alg1)\.

Second, when verifying each draft tokendt−1\(i\)d\_\{t\-1\}^\{\(i\)\}, we replace the deterministic match testdt−1\(i\)=vt\(i\)d\_\{t\-1\}^\{\(i\)\}=v\_\{t\}^\{\(i\)\}in[Equation2](https://arxiv.org/html/2610.33379#S3.E2)with the standard rejection\-sampling acceptance rule\. Withpt\(i\)=pθ\(⋅∣Pt−1,dt−1\(<i\)\)p\_\{t\}^\{\(i\)\}=p\_\{\\theta\}\(\\cdot\\mid P\_\{t\-1\},d\_\{t\-1\}^\{\(<i\)\}\)the target conditional at that position andQt−1\(i\)Q\_\{t\-1\}^\{\(i\)\}its stored drafter distribution, we accept the draft tokendt−1\(i\)d\_\{t\-1\}^\{\(i\)\}with probabilitymin⁡\{1,pt\(i\)​\(dt−1\(i\)\)/Qt−1\(i\)​\(dt−1\(i\)\)\}\\min\\\{1,p\_\{t\}^\{\(i\)\}\(d\_\{t\-1\}^\{\(i\)\}\)/Q\_\{t\-1\}^\{\(i\)\}\(d\_\{t\-1\}^\{\(i\)\}\)\\\}\. Crucially, the denominator is the stored drafter distributionQt−1\(i\)Q\_\{t\-1\}^\{\(i\)\}from whichdt−1\(i\)d\_\{t\-1\}^\{\(i\)\}was originally sampled, even though it may have been computed from an older prefix\. At the first rejection, we draw a correction from the normalized residual distribution∝\[pt\(i\)−Qt−1\(i\)\]\+\\propto\[p\_\{t\}^\{\(i\)\}\-Q\_\{t\-1\}^\{\(i\)\}\]\_\{\+\}\. If the entire draft is accepted, we draw a bonus token frompt\(\|Dt−1\|\+1\)p\_\{t\}^\{\(\|D\_\{t\-1\}\|\+1\)\}\.

We summarize the resulting procedure in[Algorithm2](https://arxiv.org/html/2610.33379#alg2)in[SectionB\.2](https://arxiv.org/html/2610.33379#A2.SS2)\. The following theorem, proved in[AppendixB](https://arxiv.org/html/2610.33379#A2), states that Packed\-SS preserves the distribution of the AR mode\. Since greedy decoding is recovered at temperature00, it also implies the correctness of[Algorithm1](https://arxiv.org/html/2610.33379#alg1)\.

###### Theorem 3\.3\(Distributional correctness\)\.

The continuation generated by Packed\-SS after the promptPPhas the same distribution as sampling from the AR modelpθp\_\{\\theta\}, stopped at the first EOS or at the output length limitLL\.

Finally, storing the drafter distributionsQt−1\(i\)Q\_\{t\-1\}^\{\(i\)\}need not require a probability for every vocabulary token at each draft position\. With top\-kk\([Fan et al\., 2018](https://arxiv.org/html/2610.33379#bib.bib22)\)or top\-pptruncation\([Holtzman et al\., 2020](https://arxiv.org/html/2610.33379#bib.bib23)\), we need to store only the nonzero entries ofQt−1\(i\)Q\_\{t\-1\}^\{\(i\)\}\. For instance, Qwen3 recommends top\-k=20k=20for a vocabulary of151,936151\{,\}936tokens\([Qwen Team, 2025](https://arxiv.org/html/2610.33379#bib.bib24)\), and the reference implementation of Llama 3 uses top\-p=0\.9p=0\.9by default\([Grattafiori and others, 2024](https://arxiv.org/html/2610.33379#bib.bib25)\)\.

## 4Experiments

#### Models and datasets\.

We evaluate Packed\-SS on Nemotron\-Labs\-Diffusion\([Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)and SetDLM\([Arriola and Kuleshov, 2026](https://arxiv.org/html/2610.33379#bib.bib10)\)\. Nemotron 3B and 8B are evaluated on five datasets: GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2610.33379#bib.bib28)\)and MATH\-500\([Hendrycks et al\., 2021](https://arxiv.org/html/2610.33379#bib.bib31)\)for mathematical reasoning, HumanEval\([Chen et al\., 2021](https://arxiv.org/html/2610.33379#bib.bib29)\)and MBPP\([Austin et al\., 2021b](https://arxiv.org/html/2610.33379#bib.bib30)\)for code generation, and WritingBench\([Wu et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib55)\)for generation speed on writing prompts\. For SetDLM, we evaluate two separately fine\-tuned checkpoints on GSM8K released by\([Arriola and Kuleshov, 2026](https://arxiv.org/html/2610.33379#bib.bib10)\)\.

#### Experimental setup\.

All experiments use batch size one on a single A100 80GB GPU, with Nemotron in BF16 and SetDLM in FP32\. Generations are capped at 512 tokens for code and writing and 1,024 tokens for mathematics, to avoid repetitive responses dominating the comparison\. Because output lengths can differ across schedules, our primary efficiency metric is aggregate token throughput, which decomposes as

Throughput=total generated tokenstotal decoder wall time=Tokens/NFETime/NFE\.\\text\{Throughput\}=\\frac\{\\text\{total generated tokens\}\}\{\\text\{total decoder wall time\}\}=\\frac\{\\text\{Tokens/NFE\}\}\{\\text\{Time/NFE\}\}\.Tokens per NFE \(Tokens/NFE\) and decoder time per NFE \(Time/NFE\) are recorded to separate token yield from cost per NFE\. Detailed setup and results are provided in[AppendicesE](https://arxiv.org/html/2610.33379#A5)and[F](https://arxiv.org/html/2610.33379#A6)\.

### 4\.1Greedy decoding

#### Nemotron\.

Linear\-SS and Packed\-SS use diffusion block sizeB=32B=32, while Quadratic\-SS usesB=8B=8\. Each queried block includes the final token of the verified prefix, so the effective speculative width isW=B−1W=B\-1\. This gives Packed\-SS and Quadratic\-SS comparable maximum input sizes of 64 and 72 token rows, respectively\. These block sizes also give the highest throughput among the settings tested in the small\-scale experiment in[SectionF\.1](https://arxiv.org/html/2610.33379#A6.SS1)\.

0\.80\.81\.01\.01\.11\.1MBPPHumanEvalGSM8KMATH\-500WritingBenchMBPPHumanEvalGSM8KMATH\-500WritingBenchNemotron 3BNemotron 8B1\.01\.01\.21\.21\.41\.41\.01\.01\.21\.21\.41\.4−4\-40044N/AN/A0\.50\.51\.01\.01\.41\.4GSM8K,W=8W=8GSM8K,W=16W=16SetDLM1\.01\.01\.51\.52\.02\.01\.01\.02\.02\.03\.03\.0−10\-10001010Throughput→\\rightarrowTokens/NFE→\\rightarrowTime/NFE←\\leftarrowΔ\\DeltaAccuracy \(%\)→\\rightarrowPacked\-SSQuadratic\-SSNative SetDLMLinear\-SSFigure 3:Greedy decoding efficiency and accuracy\.Efficiency metrics are ratios to Linear\-SS; accuracy differences are in percentage points\. Error bars show 95% paired bootstrap confidence intervals\. Native is the released non\-speculative SetDLM diffusion decoder, shown for reference\. WritingBench measures speed only\.Results are summarized in[Figure3](https://arxiv.org/html/2610.33379#S4.F3), with numerical results in[Table6](https://arxiv.org/html/2610.33379#A6.T6)\. Packed\-SS achieves the highest measured throughput in all ten Nemotron settings\. Compared with Linear\-SS, throughput improves by3\.53\.5–9\.3%9\.3\\%, with all ten 95% confidence intervals for the throughput ratio above one\. The improvement comes from a13\.413\.4–20\.3%20\.3\\%increase in tokens per NFE, which more than offsets the8\.98\.9–10\.1%10\.1\\%increase in decoder time per NFE\. Compared with Quadratic\-SS, throughput improves by3\.43\.4–32\.3%32\.3\\%at the evaluated block sizes\. Accuracy differences between schedules are observed, but they do not provide a direct comparison of decoding quality because they arise from finite\-precision effects\.

#### SetDLM\.

The checkpoints with training horizonsSmax=16S\_\{\\max\}=16andSmax=32S\_\{\\max\}=32are evaluated at speculative widthsW=8W=8andW=16W=16, respectively\. All three speculative schedules use the same width within each checkpoint\.

Packed\-SS improves throughput over Linear\-SS by6\.0%6\.0\\%atW=8W=8and1\.5%1\.5\\%atW=16W=16, as in[Figures3](https://arxiv.org/html/2610.33379#S4.F3)and[8](https://arxiv.org/html/2610.33379#A6.T8)\. The additional token yield outweighs the higher cost per NFE: tokens per NFE increase by8\.58\.5–8\.9%8\.9\\%, while decoder time per NFE increases by2\.72\.7–7\.0%7\.0\\%\. Packed\-SS also has higher throughput than Quadratic\-SS in both settings\. The improvements extend to confidence\-based verification order; details are provided in[SectionF\.5](https://arxiv.org/html/2610.33379#A6.SS5)\.

We also test the native SetDLM decoding schedule, which is diffusion decoding instead of speculative decoding\. AtW=8W=8andW=16W=16, Packed\-SS achieves higher throughput than Native, with accuracy higher by8\.728\.72and7\.057\.05percentage points, respectively\.

### 4\.2Decoding at temperature 1

We further evaluate the sampling procedure of[Section3\.2](https://arxiv.org/html/2610.33379#S3.SS2)at temperature 1, where draft tokens are verified probabilistically\. We use the same five datasets and both Nemotron scales as in greedy decoding, with block sizeB=32B=32for Linear\-SS and Packed\-SS andB=8B=8for Quadratic\-SS\.

#### Scope and baselines\.

Linear\-SS with speculative sampling is the standard stochastic setting\([Leviathan et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib12);[Chen et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib35)\)\. However, prior empirical evaluations of Quadratic\-SS are reported under greedy decoding\([Liu et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib32);[Samragh et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib33);[Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)\. We therefore use Linear\-SS as the primary baseline at temperature 1\. For completeness, we extend Quadratic\-SS to rejection sampling by applying the same probabilistic verification to its selected branch as a secondary comparison\. More details are in[SectionE\.4](https://arxiv.org/html/2610.33379#A5.SS4)\.

0\.80\.81\.01\.01\.11\.1MBPPHumanEvalGSM8KMATH\-500WritingBenchMBPPHumanEvalGSM8KMATH\-500WritingBenchNemotron 3BNemotron 8B1\.01\.01\.21\.21\.41\.41\.01\.01\.21\.21\.41\.4−4\-40044N/AN/AThroughput→\\rightarrowTokens/NFE→\\rightarrowTime/NFE←\\leftarrowΔ\\DeltaAccuracy \(%\)→\\rightarrowPacked\-SSQuadratic\-SSLinear\-SSFigure 4:Efficiency and accuracy at temperature 1\.Efficiency metrics are ratios to Linear\-SS, accuracy differences are in percentage points\. Error bars show 95% paired bootstrap confidence intervals\. WritingBench measures speed only\.
#### Results\.

Results are presented in[Figure4](https://arxiv.org/html/2610.33379#S4.F4), with numerical results in[Table7](https://arxiv.org/html/2610.33379#A6.T7)\. Compared with Linear\-SS, Packed\-SS achieves higher throughput in all ten settings, improving throughput by2\.02\.0–6\.3%6\.3\\%, with all ten 95% confidence intervals for the throughput ratio above one\. The improvement again comes from increased token yield: Packed\-SS generates14\.814\.8–21\.5%21\.5\\%more tokens per NFE, which outweighs the11\.911\.9–15\.4%15\.4\\%increase in decoder time per NFE\.

Compared with Quadratic\-SS, Packed\-SS shows comparable performance in terms of throughput\. Its throughput point estimate is higher in four settings and lower in six\. The paired 95% confidence intervals lie above one in three settings, below one in two, and include one in the remaining five\.

## 5Discussion and Limitations

We presented Packed\-SS, which overlaps drafting and verification in one forward pass per iteration while evaluating onlyO⁡\(W\)O\(W\)positions\. Under greedy decoding and a monotonicity assumption, we showed that Packed\-SS requires no more forward passes than Linear\-SS to generate a prescribed number of tokens from the same prompt at the same width\. Experiments across multiple tasks and model families demonstrate that Packed\-SS can improve throughput over existing self\-speculative decoding schedules\.

Several limitations and directions for future work remain\. Our single\-call implementation requires shared\-weight models supporting parallel drafting and autoregressive verification\. Combining this schedule with drafters based on intermediate\-layer representations remains unexplored\. The current efficiency analysis relies on an idealized assumption; extending it to weaker conditions and stochastic sampling would broaden its applicability\. Finally, preparing only a subset of the continuation branches used by Quadratic\-SS could provide intermediate designs between Packed\-SS and Quadratic\-SS, potentially improving upon the current algorithm\.

## Acknowledgements

C\. Cai is supported by the NSF CAREER Award CCF\-2541600 and the NSF grant DMS\-2515333\.

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## Appendix ANotation

[Table1](https://arxiv.org/html/2610.33379#A1.T1)collects the symbols used in[Section3](https://arxiv.org/html/2610.33379#S3)\. In left\-to\-right \(LTR\) decoding,dt\(i\)d\_\{t\}^\{\(i\)\}is indexed afterPtP\_\{t\}, whilezt\(i\)z\_\{t\}^\{\(i\)\}andvt\(i\)v\_\{t\}^\{\(i\)\}are indexed afterPt−1P\_\{t\-1\}\. In the any\-order procedure, position indices are fixed output positions after the prompt\.

Table 1:Symbols\.Sizes and indices, decoder state, and distributions\.
## Appendix BProofs

### B\.1Pathwise dominance

We prove[Theorem3\.2](https://arxiv.org/html/2610.33379#S3.Thmtheorem2)using two lemmas\. The first describes the progress of fresh drafting and verification; the second compares this progress with two forward passes of Packed\-SS\. The comparisons below concern verified output lengths, excluding the prompt, before applying EOS or length truncation; stopping is handled at the end of the theorem’s proof\.

###### Lemma B\.1\(Monotonicity of verified output length\)\.

Count verified output tokens after the prompt bycc, and define

F⁡\(c\)=min⁡\{c\+W\+1,min⁡\{k\>c:ck⋆\>c\}\},F\(c\)=\\min\\bigl\\\{c\+W\+1,\\ \\min\\\{k\>c:c\_\{k\}^\{\\star\}\>c\\\}\\bigr\\\},\(6\)with the minimum of an empty set read as\+∞\+\\infty\. ThenFFis nondecreasing\. Under[Assumption3\.1](https://arxiv.org/html/2610.33379#S3.Thmtheorem1),F⁡\(c\)F\(c\)is the verified output length after freshly draftingWWtokens from a context ofccverified output tokens and verifying that draft, for any such context evaluated by either schedule\. Thus,F⁡\(c\)−cF\(c\)\-ccounts the newly accepted tokens, including the correction or bonus token\.

###### Proof\.

We first identify the number of tokens accepted by fresh drafting and verification\. Fix a context ofccverified output tokens\. If the draft contains an incorrect proposal, letkkbe its first incorrect position\. By[Assumption3\.1](https://arxiv.org/html/2610.33379#S3.Thmtheorem1), this is the first position inc\+1,…,c\+Wc\+1,\\ldots,c\+Wwithck⋆\>cc\_\{k\}^\{\\star\}\>c\. Verification accepts the earlier proposals and adds the corrected token atkk, giving a verified output length ofkk\. If allWWproposals are correct, verification accepts them and adds a bonus token atc\+W\+1c\+W\+1\. These two cases give the expression forF⁡\(c\)F\(c\)in[Equation6](https://arxiv.org/html/2610.33379#A2.E6)\.

We next prove monotonicity\. Takec′≥cc^\{\\prime\}\\geq c\. Ifc′≥F⁡\(c\)c^\{\\prime\}\\geq F\(c\), then

F⁡\(c′\)≥c′\+1\>F⁡\(c\)\.F\(c^\{\\prime\}\)\\geq c^\{\\prime\}\+1\>F\(c\)\.It remains to considerc′<F⁡\(c\)c^\{\\prime\}<F\(c\)\. For every positionkksatisfyingc′<k<F⁡\(c\)c^\{\\prime\}<k<F\(c\), the definition ofF⁡\(c\)F\(c\)gives

ck⋆≤c≤c′\.c\_\{k\}^\{\\star\}\\leq c\\leq c^\{\\prime\}\.Thus, the first position withck⋆\>c′c\_\{k\}^\{\\star\}\>c^\{\\prime\}cannot occur beforeF⁡\(c\)F\(c\)\. The other term in the minimum definingF⁡\(c′\)F\(c^\{\\prime\}\)also satisfies

c′\+W\+1≥c\+W\+1≥F⁡\(c\)\.c^\{\\prime\}\+W\+1\\geq c\+W\+1\\geq F\(c\)\.Both terms are therefore at leastF⁡\(c\)F\(c\), provingF⁡\(c′\)≥F⁡\(c\)F\(c^\{\\prime\}\)\\geq F\(c\)\. ∎

###### Lemma B\.2\(Verified output length after two forward passes\)\.

Under[Assumption3\.1](https://arxiv.org/html/2610.33379#S3.Thmtheorem1), if a Packed\-SS iteration begins withccverified output tokens and the next iteration ends withc′′c^\{\\prime\\prime\}, thenc′′≥F⁡\(c\)c^\{\\prime\\prime\}\\geq F\(c\)\.

###### Proof\.

Letc′c^\{\\prime\}be the verified output length after the first forward pass\. During this pass, Packed\-SS verifies its stored draft and produces new proposals for positionsc\+1,…,c\+Wc\+1,\\ldots,c\+W, using the context of lengthcc\. Every pass accepts at least one token, soc′\>cc^\{\\prime\}\>c\.

Ifc′≥F⁡\(c\)c^\{\\prime\}\\geq F\(c\), the desired inequality already holds, since the next pass can only increase the verified output length\. Otherwise, consider the positions

c′\+1,…,F⁡\(c\)−1\.c^\{\\prime\}\+1,\\ldots,F\(c\)\-1\.The new proposals at these positions are correct by the definition ofF⁡\(c\)F\(c\)\. They also belong to the newly drafted block, becauseF⁡\(c\)−1≤c\+WF\(c\)\-1\\leq c\+W\. Since these positions lie beyond the updated prefix, their proposals are retained for the next pass\.

The next pass therefore accepts all of these retained proposals\. Its correction or bonus token is added at a position at leastF⁡\(c\)F\(c\), so its final verified output length satisfiesc′′≥F⁡\(c\)c^\{\\prime\\prime\}\\geq F\(c\)\. This argument also covers an empty range of retained positions, since the next pass still accepts at least one token\. ∎

###### Proof of[Theorem3\.2](https://arxiv.org/html/2610.33379#S3.Thmtheorem2)\.

We first compare the two schedules without EOS or length truncation\. Letℓt\\ell\_\{t\}be the number of verified output tokens afterttiterations of Linear\-SS, and letctc\_\{t\}be the corresponding number afterttforward passes of Packed\-SS\. Both counts exclude the prompt, andc0=ℓ0=0c\_\{0\}=\\ell\_\{0\}=0\. A Linear\-SS iteration drafts from its current verified prefix and then verifies that draft, so

ℓt\+1=F⁡\(ℓt\)\.\\ell\_\{t\+1\}=F\(\\ell\_\{t\}\)\.
We show by induction thatc2​t≥ℓtc\_\{2t\}\\geq\\ell\_\{t\}for everyt≥0t\\geq 0\. The claim holds att=0t=0\. Suppose it holds for somett\. Then

c2​t\+2≥F⁡\(c2​t\)≥F⁡\(ℓt\)=ℓt\+1\.c\_\{2t\+2\}\\geq F\(c\_\{2t\}\)\\geq F\(\\ell\_\{t\}\)=\\ell\_\{t\+1\}\.The first inequality follows from[LemmaB\.2](https://arxiv.org/html/2610.33379#A2.Thmtheorem2); the second follows from the induction hypothesis and the monotonicity in[LemmaB\.1](https://arxiv.org/html/2610.33379#A2.Thmtheorem1)\. This proves the induction step\. The argument includes initialization: the first Packed\-SS pass starts with an empty draft and generates one target token, so[LemmaB\.2](https://arxiv.org/html/2610.33379#A2.Thmtheorem2)applies fromc0=0c\_\{0\}=0\.

Now fixL\>0L\>0, and letTTbe the first Linear\-SS iteration for whichℓT≥L\\ell\_\{T\}\\geq L\. Since each Linear\-SS iteration uses two forward passes, its total cost is2​T2T\. The inequalityc2​T≥ℓT≥Lc\_\{2T\}\\geq\\ell\_\{T\}\\geq Lshows that Packed\-SS reaches the same output length within those2​T2Tpasses\. Therefore,

NFEPacked\-SS​\(L\)≤2​T=NFELinear\-SS​\(L\)\.\\mathrm\{NFE\}\_\{\\text\{Packed\-SS\{\}\}\}\(L\)\\leq 2T=\\mathrm\{NFE\}\_\{\\text\{Linear\-SS\{\}\}\}\(L\)\.ForL=0L=0, neither schedule makes a call\.

Finally, both schedules produce the same greedy continuation\. If that continuation contains an EOS token, its first occurrence is therefore at the same position for both schedules\. Apply the same comparison to that position or the length limitLL, whichever is reached first\. Truncating the final block changes the returned output but does not remove the forward pass that produced it\. The same bound thus holds with EOS and length stopping\. ∎

### B\.2Exact sampling

[Algorithm2](https://arxiv.org/html/2610.33379#alg2)gives the full LTR sampling procedure corresponding to[Section3\.2](https://arxiv.org/html/2610.33379#S3.SS2)\. Compared with[Algorithm1](https://arxiv.org/html/2610.33379#alg1), the decoder stores the sampling distributionQt−1\(i\)Q\_\{t\-1\}^\{\(i\)\}with each draft token and replaces greedy matching with rejection sampling\.

Algorithm 2Packed\-SS with LTR rejection sampling0:prompt

PP, speculative width

WW, output length limit

LL, verifier

pθp\_\{\\theta\}, drafter

qθq\_\{\\theta\}
1:

P0←PP\_\{0\}\\leftarrow P,

D0,Q0←∅D\_\{0\},Q\_\{0\}\\leftarrow\\varnothing,

t←1t\\leftarrow 1
2:while

\|Pt−1\|−\|P\|<L\|P\_\{t\-1\}\|\-\|P\|<Land \(

t=1t=1or the last token of

Pt−1P\_\{t\-1\}is not

EOS\\mathrm\{EOS\}\)do

2:In one model call:

3:

pt\(i\)←pθ\(⋅∣Pt−1,dt−1\(<i\)\)p\_\{t\}^\{\(i\)\}\\leftarrow p\_\{\\theta\}\(\\cdot\\mid P\_\{t\-1\},d\_\{t\-1\}^\{\(<i\)\}\)for

i=1,…,\|Dt−1\|\+1i=1,\\ldots,\|D\_\{t\-1\}\|\+1⊳\\trianglerightVerifier distributions

4:

qt\(j\)←qθ\(j\)\(⋅∣Pt−1\)q\_\{t\}^\{\(j\)\}\\leftarrow q\_\{\\theta\}^\{\(j\)\}\(\\cdot\\mid P\_\{t\-1\}\)for

j=1,…,Wj=1,\\ldots,W⊳\\trianglerightDrafter proposals

4:end

5:

ui∼Unif⁡\(0,1\)u\_\{i\}\\sim\\mathrm\{Unif\}\(0,1\),

αi←min⁡\{1,pt\(i\)​\(dt−1\(i\)\)/Qt−1\(i\)​\(dt−1\(i\)\)\}\\alpha\_\{i\}\\leftarrow\\min\\\{1,\\,p\_\{t\}^\{\(i\)\}\(d\_\{t\-1\}^\{\(i\)\}\)/Q\_\{t\-1\}^\{\(i\)\}\(d\_\{t\-1\}^\{\(i\)\}\)\\\}for

i=1,…,\|Dt−1\|i=1,\\ldots,\|D\_\{t\-1\}\|⊳\\trianglerightAcceptance tests

6:

ht←max⁡\{0≤h≤\|Dt−1\|:ui<αi​for all​i≤h\}h\_\{t\}\\leftarrow\\max\\\{0\\leq h\\leq\|D\_\{t\-1\}\|:u\_\{i\}<\\alpha\_\{i\}\\text\{ for all \}i\\leq h\\\}⊳\\trianglerightCount accepted tokens

7:

c∼Rc\\sim R,

R∝\{\[pt\(ht\+1\)−Qt−1\(ht\+1\)\]\+,ht<\|Dt−1\|,pt\(\|Dt−1\|\+1\),ht=\|Dt−1\|\.R\\propto\\begin\{cases\}\[p\_\{t\}^\{\(h\_\{t\}\+1\)\}\-Q\_\{t\-1\}^\{\(h\_\{t\}\+1\)\}\]\_\{\+\},&h\_\{t\}<\|D\_\{t\-1\}\|,\\\\ p\_\{t\}^\{\(\|D\_\{t\-1\}\|\+1\)\},&h\_\{t\}=\|D\_\{t\-1\}\|\.\\end\{cases\}⊳\\trianglerightCorrection or bonus

8:

Ct←\(dt−1\(1\),…,dt−1\(ht\),c\)C\_\{t\}\\leftarrow\(d\_\{t\-1\}^\{\(1\)\},\\ldots,d\_\{t\-1\}^\{\(h\_\{t\}\)\},c\),

At←ht\+1A\_\{t\}\\leftarrow h\_\{t\}\+1⊳\\trianglerightAdd verifier token

9:

a←min⁡\(\{At,L\+\|P\|−\|Pt−1\|\}∪\{1≤i≤At:Ct\(i\)=EOS\}\)a\\leftarrow\\min\\bigl\(\\\{A\_\{t\},L\+\|P\|\-\|P\_\{t\-1\}\|\\\}\\cup\\\{1\\leq i\\leq A\_\{t\}:C\_\{t\}^\{\(i\)\}=\\mathrm\{EOS\}\\\}\\bigr\)⊳\\trianglerightEnding condition

10:

Ct←\(Ct\(1\),…,Ct\(a\)\)C\_\{t\}\\leftarrow\(C\_\{t\}^\{\(1\)\},\\ldots,C\_\{t\}^\{\(a\)\}\)⊳\\trianglerightPrune by stopping index

11:

Pt←\(Pt−1,Ct\)P\_\{t\}\\leftarrow\(P\_\{t\-1\},C\_\{t\}\)⊳\\trianglerightAppend to prefix

12:

Qt\(j\)←qt\(a\+j\)Q\_\{t\}^\{\(j\)\}\\leftarrow q\_\{t\}^\{\(a\+j\)\},

dt\(j\)∼Qt\(j\)d\_\{t\}^\{\(j\)\}\\sim Q\_\{t\}^\{\(j\)\}for

j=1,…,max⁡\(0,W−a\)j=1,\\ldots,\\max\(0,W\-a\)⊳\\trianglerightSample retained draft

13:

Dt←\(dt\(1\),…,dt\(max⁡\(0,W−a\)\)\)D\_\{t\}\\leftarrow\(d\_\{t\}^\{\(1\)\},\\ldots,d\_\{t\}^\{\(\\max\(0,W\-a\)\)\}\)⊳\\trianglerightPrepare next draft

14:

t←t\+1t\\leftarrow t\+1
15:endwhile

16:Output:

Pt−1P\_\{t\-1\}

We now prove[Theorem3\.3](https://arxiv.org/html/2610.33379#S3.Thmtheorem3)\. First, we show that the acceptance and correction steps produce the correct target distribution for one token\. Then, we show that each draft token still follows its stored sampling distribution when it is checked\.

#### Conditioning information\.

Fix the prompt\. Let𝒢t\\mathcal\{G\}\_\{t\}collect the information used for conditioning immediately before sampling the draftDtD\_\{t\}\. It includes the verified prefixPt−1P\_\{t\-1\}at the start of calltt, the draftDt−1D\_\{t\-1\}checked in that call, and the distributionsQt−1Q\_\{t\-1\}used to sample that draft\. It also includes the tokensCtC\_\{t\}accepted after applying the stopping rule and the new drafter distributionsqt\(1\),…,qt\(W\)q\_\{t\}^\{\(1\)\},\\ldots,q\_\{t\}^\{\(W\)\}, computed fromPt−1P\_\{t\-1\}\. Define

𝒢t:=\(Pt−1,Dt−1,Qt−1,Ct,\(qt\(j\)\)j=1W\)\.\\mathcal\{G\}\_\{t\}:=\\left\(P\_\{t\-1\},D\_\{t\-1\},Q\_\{t\-1\},C\_\{t\},\(q\_\{t\}^\{\(j\)\}\)\_\{j=1\}^\{W\}\\right\)\.Conditional on𝒢t\\mathcal\{G\}\_\{t\}, the advancea=\|Ct\|a=\|C\_\{t\}\|, the updated prefixPtP\_\{t\}, and the stored distributionsQt\(i\)Q\_\{t\}^\{\(i\)\}for the draft tokens are fixed\. All draft sampling, acceptance tests, and correction draws use independent random numbers\.

Writeut\+1,ju\_\{t\+1,j\}for the uniform random number used to test draft tokendt\(j\)d\_\{t\}^\{\(j\)\}in callt\+1t\+1\. For1≤i≤\|Dt\|1\\leq i\\leq\|D\_\{t\}\|, add the earlier draft tokens and the random numbers used to test them:

𝒢t,i:=\(𝒢t,\(dt\(j\),ut\+1,j\)1≤j<i\)\.\\mathcal\{G\}\_\{t,i\}:=\\left\(\\mathcal\{G\}\_\{t\},\(d\_\{t\}^\{\(j\)\},u\_\{t\+1,j\}\)\_\{1\\leq j<i\}\\right\)\.Conditional on𝒢t,i\\mathcal\{G\}\_\{t,i\}, the preceding acceptance decisions are fixed, so whether verification reaches positioniiis also fixed\.

###### Lemma B\.3\(One\-token correction using the original sampling distribution\)\.

Consider theii\-th draft token reached by left\-to\-right verification in callt\+1t\+1\. LetH=\(Pt,dt\(<i\)\)H=\(P\_\{t\},d\_\{t\}^\{\(<i\)\}\)be the verified prefix, including the draft tokens already accepted in this call\. LetP⁡\(x\)=p⁡\(x∣H\)P\(x\)=p\(x\\mid H\)be the target distribution for the next token\. LetD=dt\(i\)D=d\_\{t\}^\{\(i\)\}be the draft token andQ=Qt\(i\)Q=Q\_\{t\}^\{\(i\)\}the distribution used to sample it\. AssumePr⁡\(D=x∣𝒢t,i\)=Q⁡\(x\)\\Pr\(D=x\\mid\\mathcal\{G\}\_\{t,i\}\)=Q\(x\)for every tokenxx\. AcceptDDwith probabilitymin⁡\{1,P⁡\(D\)/Q⁡\(D\)\}\\min\\\{1,P\(D\)/Q\(D\)\\\}, and otherwise emit an independent draw from the normalized positive part ofP−QP\-Q\. Then, given𝒢t,i\\mathcal\{G\}\_\{t,i\}, the emitted token has distributionPP\.

###### Proof\.

Condition on𝒢t,i\\mathcal\{G\}\_\{t,i\}, so thatPPandQQare fixed\. LetXXdenote the emitted token andAAthe event that the draft token is accepted\. We compute the probability of emitting a given tokenxxthrough each of the two possible branches\.

The acceptance branch contributes

Pr⁡\(X=x,A∣𝒢t,i\)=min⁡\{P⁡\(x\),Q⁡\(x\)\}\.\\Pr\(X=x,A\\mid\\mathcal\{G\}\_\{t,i\}\)=\\min\\\{P\(x\),Q\(x\)\\\}\.ForQ⁡\(x\)\>0Q\(x\)\>0, this follows by multiplying the proposal probabilityQ⁡\(x\)Q\(x\)by the acceptance probabilitymin⁡\{1,P⁡\(x\)/Q⁡\(x\)\}\\min\\\{1,P\(x\)/Q\(x\)\\\}\. ForQ⁡\(x\)=0Q\(x\)=0, both sides are zero\. It follows that the total acceptance probability is

β:=Pr⁡\(A∣𝒢t,i\)=∑xmin⁡\{P⁡\(x\),Q⁡\(x\)\}\.\\beta:=\\Pr\(A\\mid\\mathcal\{G\}\_\{t,i\}\)=\\sum\_\{x\}\\min\\\{P\(x\),Q\(x\)\\\}\.
The rejection probability is

1−β=∑x\(P⁡\(x\)−min⁡\{P⁡\(x\),Q⁡\(x\)\}\)=∑x\[P⁡\(x\)−Q⁡\(x\)\]\+\.1\-\\beta=\\sum\_\{x\}\\bigl\(P\(x\)\-\\min\\\{P\(x\),Q\(x\)\\\}\\bigr\)=\\sum\_\{x\}\[P\(x\)\-Q\(x\)\]\_\{\+\}\.Whenβ<1\\beta<1, this is exactly the normalizing constant of the correction distribution\. The rejection branch therefore contributes

Pr⁡\(X=x,Ac∣𝒢t,i\)=\(1−β\)​\[P⁡\(x\)−Q⁡\(x\)\]\+1−β=\[P⁡\(x\)−Q⁡\(x\)\]\+\.\\Pr\(X=x,A^\{\\mathrm\{c\}\}\\mid\\mathcal\{G\}\_\{t,i\}\)=\(1\-\\beta\)\\frac\{\[P\(x\)\-Q\(x\)\]\_\{\+\}\}\{1\-\\beta\}=\[P\(x\)\-Q\(x\)\]\_\{\+\}\.Adding the two contributions yields

Pr⁡\(X=x∣𝒢t,i\)=min⁡\{P⁡\(x\),Q⁡\(x\)\}\+\[P⁡\(x\)−Q⁡\(x\)\]\+=P⁡\(x\)\.\\Pr\(X=x\\mid\\mathcal\{G\}\_\{t,i\}\)=\\min\\\{P\(x\),Q\(x\)\\\}\+\[P\(x\)\-Q\(x\)\]\_\{\+\}=P\(x\)\.
Ifβ=1\\beta=1, thenP=QP=Qand rejection has probability zero, so the same conclusion holds without a correction draw\. Also,Q⁡\(D\)\>0Q\(D\)\>0with probability one becauseDDis sampled fromQQ, so the acceptance test never divides by zero\. Neither distribution needs to assign positive probability to every token\. ∎

This is the standard speculative\-sampling rule\([Leviathan et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib12);[Chen et al\., 2023](https://arxiv.org/html/2610.33379#bib.bib35)\)\. The next lemma shows that each draft token follows its stored sampling distribution when checked, as required by[LemmaB\.3](https://arxiv.org/html/2610.33379#A2.Thmtheorem3)\.

###### Lemma B\.4\(Distribution of a draft token when checked\)\.

For each draft position1≤i≤\|Dt\|1\\leq i\\leq\|D\_\{t\}\|reached by verification in callt\+1t\+1, and every tokenxx,

Pr⁡\(dt\(i\)=x∣𝒢t,i\)=Qt\(i\)​\(x\)\.\\Pr\(d\_\{t\}^\{\(i\)\}=x\\mid\\mathcal\{G\}\_\{t,i\}\)=Q\_\{t\}^\{\(i\)\}\(x\)\.\(7\)

###### Proof\.

Condition on𝒢t\\mathcal\{G\}\_\{t\}\. The advanceaa, the positions to retain, and the stored distributionsQt\(j\)=qt\(a\+j\)Q\_\{t\}^\{\(j\)\}=q\_\{t\}^\{\(a\+j\)\}are fixed\. Algorithm[2](https://arxiv.org/html/2610.33379#alg2)samples the tokens inDtD\_\{t\}independently from these distributions\. Thus,dt\(i\)d\_\{t\}^\{\(i\)\}has distributionQt\(i\)Q\_\{t\}^\{\(i\)\}and is independent of the earlier draft tokens given𝒢t\\mathcal\{G\}\_\{t\}\.

By definition,𝒢t,i\\mathcal\{G\}\_\{t,i\}adds only the earlier draft tokens and the random numbers used to test them to𝒢t\\mathcal\{G\}\_\{t\}\. These random numbers are independent of\(𝒢t,Dt\)\(\\mathcal\{G\}\_\{t\},D\_\{t\}\)\. Conditioning on this additional information therefore leaves the distribution ofdt\(i\)d\_\{t\}^\{\(i\)\}unchanged, proving[Equation7](https://arxiv.org/html/2610.33379#A2.E7)\. Whether verification reaches positioniiis determined by𝒢t,i\\mathcal\{G\}\_\{t,i\}, so the same identity holds at every position that is checked\. ∎

###### Proof of[Theorem3\.3](https://arxiv.org/html/2610.33379#S3.Thmtheorem3)\.

Fix the promptPPand consider[Algorithm2](https://arxiv.org/html/2610.33379#alg2)without EOS or length stopping\. LetXkX\_\{k\}be itskk\-th output token after the prompt\. We prove by induction onmmthat

Pr\(X1:m=x1:m∣P\)=∏k=1mpθ\(xk∣P,x<k\)\.\\Pr\(X\_\{1:m\}=x\_\{1:m\}\\mid P\)=\\prod\_\{k=1\}^\{m\}p\_\{\\theta\}\(x\_\{k\}\\mid P,x\_\{<k\}\)\.SinceD0=∅D\_\{0\}=\\varnothing, the first call drawsX1X\_\{1\}directly frompθ\(⋅∣P\)p\_\{\\theta\}\(\\cdot\\mid P\), proving the claim form=1m=1\.

Suppose the identity holds form=k−1m=k\-1, wherek≥2k\\geq 2\. Lett\+1t\+1be the call that producesXkX\_\{k\}\. Its position within that call isi=k−\(\|Pt\|−\|P\|\)i=k\-\(\|P\_\{t\}\|\-\|P\|\)\. Thus,1≤i≤\|Dt\|\+11\\leq i\\leq\|D\_\{t\}\|\+1, withi=\|Dt\|\+1i=\|D\_\{t\}\|\+1denoting the bonus position\. Letℋk\\mathcal\{H\}\_\{k\}collecttt,ii,𝒢t\\mathcal\{G\}\_\{t\}, the earlier draft tokens, and the random numbers used to test them:

ℋk:=\(t,i,𝒢t,\(dt\(j\),ut\+1,j\)1≤j<i\)\.\\mathcal\{H\}\_\{k\}:=\\left\(t,i,\\mathcal\{G\}\_\{t\},\(d\_\{t\}^\{\(j\)\},u\_\{t\+1,j\}\)\_\{1\\leq j<i\}\\right\)\.The firsti−1i\-1draft tokens have been accepted, so

\(P,X<k\)=\(Pt,dt\(<i\)\),pt\+1\(i\)=pθ\(⋅∣P,X<k\)\.\(P,X\_\{<k\}\)=\(P\_\{t\},d\_\{t\}^\{\(<i\)\}\),\\qquad p\_\{t\+1\}^\{\(i\)\}=p\_\{\\theta\}\(\\cdot\\mid P,X\_\{<k\}\)\.
First supposei≤\|Dt\|i\\leq\|D\_\{t\}\|, so the algorithm checks a draft token\. For fixedttandii,𝒢t,i\\mathcal\{G\}\_\{t,i\}determines whether the preceding draft tokens are accepted and whether\|Pt\|−\|P\|\+i=k\|P\_\{t\}\|\-\|P\|\+i=k\. Hence[LemmaB\.4](https://arxiv.org/html/2610.33379#A2.Thmtheorem4)givesPr⁡\(dt\(i\)=x∣ℋk\)=Qt\(i\)​\(x\)\\Pr\(d\_\{t\}^\{\(i\)\}=x\\mid\\mathcal\{H\}\_\{k\}\)=Q\_\{t\}^\{\(i\)\}\(x\)\. Applying[LemmaB\.3](https://arxiv.org/html/2610.33379#A2.Thmtheorem3)with targetpt\+1\(i\)p\_\{t\+1\}^\{\(i\)\}and proposalQt\(i\)Q\_\{t\}^\{\(i\)\}yields

Pr⁡\(Xk=x∣ℋk\)=pθ​\(x∣P,X<k\)\.\\Pr\(X\_\{k\}=x\\mid\\mathcal\{H\}\_\{k\}\)=p\_\{\\theta\}\(x\\mid P,X\_\{<k\}\)\.Ifi=\|Dt\|\+1i=\|D\_\{t\}\|\+1, the algorithm draws a bonus token directly frompt\+1\(i\)p\_\{t\+1\}^\{\(i\)\}, so the same equality holds\.

Since\(P,X<k\)\(P,X\_\{<k\}\)is determined byℋk\\mathcal\{H\}\_\{k\}, the law of total expectation gives

Pr⁡\(Xk=x∣P,X<k\)\\displaystyle\\Pr\(X\_\{k\}=x\\mid P,X\_\{<k\}\)=𝔼\[Pr\(Xk=x∣ℋk\)∣P,X<k\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\Pr\(X\_\{k\}=x\\mid\\mathcal\{H\}\_\{k\}\)\\mid P,X\_\{<k\}\\right\]=pθ​\(x∣P,X<k\)\.\\displaystyle=p\_\{\\theta\}\(x\\mid P,X\_\{<k\}\)\.Multiplying byPr⁡\(X<k=x<k∣P\)\\Pr\(X\_\{<k\}=x\_\{<k\}\\mid P\)and applying the induction hypothesis proves the identity form=km=k\.

Now restore stopping\. Let

τ:=min⁡\(\{1≤k≤L:Xk=EOS\}∪\{L\}\)\.\\tau:=\\min\\left\(\\\{1\\leq k\\leq L:X\_\{k\}=\\mathrm\{EOS\}\\\}\\cup\\\{L\\\}\\right\)\.Using the same random numbers, the stopped algorithm produces the same tokens through positionτ\\tauand returns\(P,X1:τ\)\(P,X\_\{1:\\tau\}\)\. Takingm=Lm=Lin the identity above shows thatX1:LX\_\{1:L\}has the AR distribution\. Applying the same stopping rule therefore gives the same distribution of returned continuations\. ∎

#### Greedy decoding\.

For greedy decoding, let each model assign probability one to the argmax token chosen by its fixed tie\-breaking rule, and zero to all other tokens\. Then[Theorem3\.3](https://arxiv.org/html/2610.33379#S3.Thmtheorem3)gives the greedy correctness of[Algorithm1](https://arxiv.org/html/2610.33379#alg1), including EOS and length stopping\.

## Appendix CAny\-Order Extension

A later token can be easier to predict than an earlier one\. In LTR verification, the first mismatch ends acceptance for that call, so an uncertain early token can prevent later, easier tokens from being accepted\. Checking tokens that are more likely to be accepted first may therefore let a call accept more draft tokens\.

An any\-order AR verifier enables this change: it conditions on tokens earlier in the chosen verification order, even when they occur later in the output sequence\. For a permutationσ\\sigma, writexσ\(<r\)=\(xσ⁡\(1\),…,xσ⁡\(r−1\)\)x\_\{\\sigma\(<r\)\}=\(x\_\{\\sigma\(1\)\},\\ldots,x\_\{\\sigma\(r\-1\)\}\)for the tokens earlier in that order\. The model defines

pθσ\(x1:L∣P\)=∏r=1Lpθ\(xσ⁡\(r\)∣P,xσ\(<r\);σ\)\.p\_\{\\theta\}^\{\\sigma\}\(x\_\{1:L\}\\mid P\)=\\prod\_\{r=1\}^\{L\}p\_\{\\theta\}\(x\_\{\\sigma\(r\)\}\\mid P,x\_\{\\sigma\(<r\)\};\\sigma\)\.The order specifies the conditionals, so different orders can produce different greedy outputs\.

Self\-speculation with separate draft and verification passes already appears in*Any\-Subset Speculative Decoding*\(ASSD\)\([Guo and Ermon, 2026](https://arxiv.org/html/2610.33379#bib.bib16)\)\. ASSD uses the same model to draft tokens in parallel and verify them in a second pass, matching the two\-call schedule we call Linear\-SS\. Prism also describes self\-speculative decoding and, separately, choosing the next position by confidence\([Li, 2025](https://arxiv.org/html/2610.33379#bib.bib17), Chapter 5\)\. Here we extend the packed schedule to such verification orders\.

#### Procedure\.

In LTR decoding, accepted tokens form a prefix, and all unresolved positions lie after it\. With any\-order verification, a later position can be accepted while an earlier one remains unresolved\. We therefore keep each accepted or draft token associated with its output position, numbered from 1 after the prompt\. At the start of iterationtt,Pt−1P\_\{t\-1\}contains the prompt and accepted position–token pairs in acceptance order, andDt−1D\_\{t\-1\}maps positions to available draft tokens\. Position superscripts now refer to these fixed output positions rather than offsets after a prefix\.

Each iteration starts its active window at the leftmost unresolved position and includes up toWWconsecutive output positions\. Positions whose tokens have already been accepted are skipped, leaving the setUtU\_\{t\}ofut=\|Ut\|u\_\{t\}=\|U\_\{t\}\|unresolved positions to process\. Some positions inUtU\_\{t\}may have no draft token yet; in particular,D0=∅D\_\{0\}=\\varnothingin the first iteration\.

The order policy then chooses a permutationπt\\pi\_\{t\}ofUtU\_\{t\}using information available before the model call, and this order remains fixed throughout the call\. Here,πt​\(r\)\\pi\_\{t\}\(r\)is the output position checked at rankrr; changing the order does not move a draft token to a different output position\. For example, confidence ordering checks positions in decreasing order of scores computed in an earlier iteration \(see[SectionE\.6](https://arxiv.org/html/2610.33379#A5.SS6.SSS0.Px1)for the scores used in our experiments\)\.

Verification followsπt\\pi\_\{t\}, using draft tokens at earlier ranks as context\. Writedom⁡\(Dt−1\)\\mathrm\{dom\}\(D\_\{t\-1\}\)for the positions with available draft tokens, and letκt\\kappa\_\{t\}be the number of consecutive entries ofπt\\pi\_\{t\}in this set before the first missing draft token\. A*same\-position*verifier predicts a token at each queried output position\. Writingpθ,jp\_\{\\theta,j\}for its distribution at positionjjunder the chosen order, its prediction at rankrris

vt\(πt​\(r\)\)=arg⁡maxx​pθ,πt​\(r\)​\(x∣Pt−1,\(πt​\(s\),dt−1\(πt​\(s\)\)\)s<r\),v\_\{t\}^\{\(\\pi\_\{t\}\(r\)\)\}=\\arg\\max\_\{x\}p\_\{\\theta,\\pi\_\{t\}\(r\)\}\\bigl\(x\\mid P\_\{t\-1\},\(\\pi\_\{t\}\(s\),d\_\{t\-1\}^\{\(\\pi\_\{t\}\(s\)\)\}\)\_\{s<r\}\\bigr\),\(8\)forr=1,…,min⁡\(κt\+1,ut\)r=1,\\ldots,\\min\(\\kappa\_\{t\}\+1,u\_\{t\}\)\. Accept matching draft tokens in this order until the first mismatch or missing draft token, then accept the verifier’s prediction at that position\. If allutu\_\{t\}positions have draft tokens and all match, accept those tokens without an extra token outside the window\. Thus, ifhth\_\{t\}draft tokens match consecutively, the number accepted before applying the stopping rule isAt=min⁡\(ht\+1,ut\)A\_\{t\}=\\min\(h\_\{t\}\+1,u\_\{t\}\)\.

In the same call as verification, Packed\-SS drafts every position inUtU\_\{t\}fromPt−1P\_\{t\-1\}\. After acceptance, predictions at positions that remain unresolved become the next draft; predictions at newly accepted positions are discarded\. For Quadratic\-SS, the proposal branches followπt\\pi\_\{t\}instead of textual order, and the branch selected after verification supplies the next draft at the remaining unresolved positions\.

[Algorithm3](https://arxiv.org/html/2610.33379#alg3)gives the formal procedure for Packed\-SS with any\-order greedy decoding\. LetEEbe the last output position to generate\. It starts at the length limitLL; accepting EOS at positionjjchanges it tomin⁡\(E,j\)\\min\(E,j\)\. The algorithm fills all remaining unresolved positions up toEEbefore returning\. For an example with confidence\-ordered verification on SetDLM, see[SectionF\.5](https://arxiv.org/html/2610.33379#A6.SS5)\.

In[Algorithm3](https://arxiv.org/html/2610.33379#alg3),dom⁡\(Pt\)\\mathrm\{dom\}\(P\_\{t\}\)denotes the output positions with accepted tokens inPtP\_\{t\}, excluding the prompt, andPt​\(j\)P\_\{t\}\(j\)is the token stored at positionjj\. The restriction⋅\|≤E\\left\.\\cdot\\right\|\_\{\\leq E\}removes output pairs beyondEE, preserving the prompt and the order of the remaining pairs\. The completed sequence is returned with the prompt included and its continuation restored to textual order\.

Algorithm 3Packed\-SS with any\-order greedy decoding0:prompt

PP, speculative width

W≥1W\\geq 1, output length limit

L\>0L\>0, verifier

pθp\_\{\\theta\}, drafter

qθq\_\{\\theta\}, order policy

1:

P0←PP\_\{0\}\\leftarrow P,

D0←∅D\_\{0\}\\leftarrow\\varnothing,

E←LE\\leftarrow L,

t←1t\\leftarrow 1
2:while

\{1,…,E\}⊈dom⁡\(Pt−1\)\\\{1,\\ldots,E\\\}\\nsubseteq\\mathrm\{dom\}\(P\_\{t\-1\}\)do

3:

ℓ←min⁡\(\{1,…,E\}∖dom⁡\(Pt−1\)\)\\ell\\leftarrow\\min\\bigl\(\\\{1,\\ldots,E\\\}\\setminus\\mathrm\{dom\}\(P\_\{t\-1\}\)\\bigr\)⊳\\trianglerightLeftmost unresolved position

4:

Ut←\{ℓ,…,min⁡\(E,ℓ\+W−1\)\}∖dom⁡\(Pt−1\)U\_\{t\}\\leftarrow\\\{\\ell,\\ldots,\\min\(E,\\ell\+W\-1\)\\\}\\setminus\\mathrm\{dom\}\(P\_\{t\-1\}\),

ut←\|Ut\|u\_\{t\}\\leftarrow\|U\_\{t\}\|⊳\\trianglerightActive positions

5:Choose

πt\\pi\_\{t\}over

UtU\_\{t\}using the order policy and information available before this call

6:

κt←max⁡\{0≤k≤ut:πt​\(r\)∈dom⁡\(Dt−1\)​for all​r≤k\}\\kappa\_\{t\}\\leftarrow\\max\\\{0\\leq k\\leq u\_\{t\}:\\pi\_\{t\}\(r\)\\in\\mathrm\{dom\}\(D\_\{t\-1\}\)\\text\{ for all \}r\\leq k\\\}⊳\\trianglerightAvailable draft prefix

6:In one model call:

7:Compute

vt\(πt​\(r\)\)v\_\{t\}^\{\(\\pi\_\{t\}\(r\)\)\}by[Equation8](https://arxiv.org/html/2610.33379#A3.E8)for

r=1,…,min⁡\(κt\+1,ut\)r=1,\\ldots,\\min\(\\kappa\_\{t\}\+1,u\_\{t\}\)⊳\\trianglerightVerifier predictions

8:

zt\(j\)←arg⁡maxx​qθ\(j\)​\(x∣Pt−1\)z\_\{t\}^\{\(j\)\}\\leftarrow\\arg\\max\_\{x\}q\_\{\\theta\}^\{\(j\)\}\(x\\mid P\_\{t\-1\}\)for

j∈Utj\\in U\_\{t\}⊳\\trianglerightDrafter proposals

8:end

9:

ht←max⁡\{0≤h≤κt:dt−1\(πt​\(r\)\)=vt\(πt​\(r\)\)​for all​r≤h\}h\_\{t\}\\leftarrow\\max\\\{0\\leq h\\leq\\kappa\_\{t\}:d\_\{t\-1\}^\{\(\\pi\_\{t\}\(r\)\)\}=v\_\{t\}^\{\(\\pi\_\{t\}\(r\)\)\}\\text\{ for all \}r\\leq h\\\}⊳\\trianglerightCount matches

10:

At←min⁡\(ht\+1,ut\)A\_\{t\}\\leftarrow\\min\(h\_\{t\}\+1,u\_\{t\}\),

Ct←\(\(,,,\)\)r=1AtC\_\{t\}\\leftarrow\\bigl\(\(\\pi\_\{t\}\(r\),v\_\{t\}^\{\(\\pi\_\{t\}\(r\)\)\}\)\\bigr\)\_\{r=1\}^\{A\_\{t\}\}⊳\\trianglerightCollect accepted tokens

11:

E←min⁡\(\{E\}∪\{j:\(j,EOS\)∈Ct\}\)E\\leftarrow\\min\\bigl\(\\\{E\\\}\\cup\\\{j:\(j,\\mathrm\{EOS\}\)\\in C\_\{t\}\\\}\\bigr\)⊳\\trianglerightEnding condition

12:

Ct←Ct\|≤EC\_\{t\}\\leftarrow\\left\.C\_\{t\}\\right\|\_\{\\leq E\},

Pt←Pt−1\|≤EP\_\{t\}\\leftarrow\\left\.P\_\{t\-1\}\\right\|\_\{\\leq E\}⊳\\trianglerightDiscard positions afterEE

13:

Pt←\(Pt,Ct\)P\_\{t\}\\leftarrow\(P\_\{t\},C\_\{t\}\)⊳\\trianglerightAppend accepted tokens

14:

Jt←\{j∈Ut∖dom⁡\(Pt\):j≤E\}J\_\{t\}\\leftarrow\\\{j\\in U\_\{t\}\\setminus\\mathrm\{dom\}\(P\_\{t\}\):j\\leq E\\\}⊳\\trianglerightRetained draft positions

15:

Dt←\{j↦zt\(j\):j∈Jt\}D\_\{t\}\\leftarrow\\\{j\\mapsto z\_\{t\}^\{\(j\)\}:j\\in J\_\{t\}\\\}⊳\\trianglerightPrepare next draft

16:

t←t\+1t\\leftarrow t\+1
17:endwhile

18:Output:

\(P,Pt−1​\(1\),…,Pt−1​\(E\)\)\(P,P\_\{t\-1\}\(1\),\\ldots,P\_\{t\-1\}\(E\)\)⊳\\trianglerightTextual order

###### Proposition C\.1\(Greedy correctness under the chosen verification orders\)\.

Fix a run of[Algorithm3](https://arxiv.org/html/2610.33379#alg3)\. Consider a*serial reference*that starts from the same prompt and, in iterationtt, generatesAtA\_\{t\}tokens greedily at positionsπt​\(1\),…,πt​\(At\)\\pi\_\{t\}\(1\),\\ldots,\\pi\_\{t\}\(A\_\{t\}\), conditioning each prediction on its ownPt−1P\_\{t\-1\}and the tokens generated earlier in that group\. The reference uses[Algorithm3](https://arxiv.org/html/2610.33379#alg3)’s verifier, tie\-breaking rule, and rules for updating the accepted tokens and handling EOS\. Then the two procedures return the same output sequence\.

###### Proof of[PropositionC\.1](https://arxiv.org/html/2610.33379#A3.Thmtheorem1)\.

We show by induction that the two procedures have the samePtP\_\{t\}andEEafter every iteration\. The claim holds initially, since both start from the promptPPwithE=LE=L\.

Suppose the claim holds before iterationtt\. Letyry\_\{r\}be the serial reference’s token at positionπt​\(r\)\\pi\_\{t\}\(r\)in this iteration\. We prove, in verification order, that

yr=vt\(πt​\(r\)\),r=1,…,At\.y\_\{r\}=v\_\{t\}^\{\(\\pi\_\{t\}\(r\)\)\},\\qquad r=1,\\ldots,A\_\{t\}\.At rank 1, both predictions condition only onPt−1P\_\{t\-1\}, so they agree\. Now taker\>1r\>1and suppose agreement holds at all earlier ranks\. Sincer≤At=min⁡\(ht\+1,ut\)r\\leq A\_\{t\}=\\min\(h\_\{t\}\+1,u\_\{t\}\), everys<rs<rsatisfiess≤hts\\leq h\_\{t\}\. These earlier draft tokens match their verifier predictions by the definition ofhth\_\{t\}\. Thus,

dt−1\(πt​\(s\)\)=vt\(πt​\(s\)\)=ys,s<r\.d\_\{t\-1\}^\{\(\\pi\_\{t\}\(s\)\)\}=v\_\{t\}^\{\(\\pi\_\{t\}\(s\)\)\}=y\_\{s\},\\qquad s<r\.Substituting these equalities into[Equation8](https://arxiv.org/html/2610.33379#A3.E8)gives

yr\\displaystyle y\_\{r\}=arg⁡maxx​pθ,πt​\(r\)​\(x∣Pt−1,\(πt​\(s\),ys\)s<r\)\\displaystyle=\\arg\\max\_\{x\}p\_\{\\theta,\\pi\_\{t\}\(r\)\}\\bigl\(x\\mid P\_\{t\-1\},\(\\pi\_\{t\}\(s\),y\_\{s\}\)\_\{s<r\}\\bigr\)=vt\(πt​\(r\)\)\.\\displaystyle=v\_\{t\}^\{\(\\pi\_\{t\}\(r\)\)\}\.This proves agreement at every rank up toAtA\_\{t\}, including the final verifier token whenht<uth\_\{t\}<u\_\{t\}\.

The two procedures therefore generate the sameAtA\_\{t\}position–token pairs before applying the stopping rule\. They find EOS at the same positions, updateEEidentically, and remove the same pairs beyondEE\. The resulting values ofPtP\_\{t\}agree, completing the induction step\.

Each iteration accepts at least one token at a previously unresolved position at or before the updatedEE, whileEEnever increases\. Hence the number of unresolved positions up toEEstrictly decreases, and both procedures terminate\. They return the same tokens in the same output\-position order\. ∎

## Appendix DAttention Mask Layouts

Packed\-SS combines verification of the current draft and generation of the next proposal in one model call\. This is achieved by a custom attention mask\. Below we show the packed inputs, masks, and cache updates for Nemotron and SetDLM\.

### D\.1Nemotron: sequence packing and padding

Letnt−1=\|Pt−1\|n\_\{t\-1\}=\|P\_\{t\-1\}\|\. Each call after initialization processes the last token of the verified prefix,Pt−1\(nt−1\)P\_\{t\-1\}^\{\(n\_\{t\-1\}\)\}, followed byB−1B\-1draft slots; earlier prefix tokens are read from the cache\. The first\-call is slightly different and discussed in[SectionE\.2](https://arxiv.org/html/2610.33379#A5.SS2)\.

#### Physical layout\.

LetD¯t−1\\bar\{D\}\_\{t\-1\}beDt−1D\_\{t\-1\}followed by enough masks to occupyB−1B\-1slots\. The Packed\-SS input is

\[Pt−1\(nt−1\),D¯t−1∣Pt−1\(nt−1\),MaskB−1\],\[P\_\{t\-1\}^\{\(n\_\{t\-1\}\)\},\\bar\{D\}\_\{t\-1\}\\mid P\_\{t\-1\}^\{\(n\_\{t\-1\}\)\},\\textsc\{Mask\}^\{B\-1\}\],concatenated along the sequence axis\. Earlier tokens are stored in KV cache and are not part of the input\. Its shape is\[1,2​B\]\[1,2B\]\.

LetTBT\_\{B\}be theB×BB\\times Blower\-triangular Boolean matrix including its diagonal, andJBJ\_\{B\}theB×BB\\times Ball\-ones matrix\. The following matrix specifies which rows in the current input may attend to each other:

\(TB00JB\),\\begin\{pmatrix\}T\_\{B\}&0\\\\ 0&J\_\{B\}\\end\{pmatrix\},and every row additionally sees the cached prefixPt−1\(<nt−1\)P\_\{t\-1\}^\{\(<n\_\{t\-1\}\)\}\. The causal region yields the shifted next\-token verifier predictions\([Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)\. The proposal region contains another copy of the last prefix token, whose output is unused, andB−1B\-1mask queries, whose outputs are the entries ofZtZ\_\{t\}\. Padded verifier rows are evaluated but cannot extend acceptance, since verification stops at the first absent draft token, represented physically by a mask\.

#### Cache update\.

After each call, the cache contains the verified prefix except its last token, which is processed in the next call\. Keys and values from the proposal region and unaccepted verifier inputs are discarded\.

### D\.2SetDLM: position\-aligned queries and ordered context

#### Duplicate masks\.

SetDLM predicts a token at the position of its input mask\. Its training construction represents each position twice: once with the token and once with a mask, both using the same position ID\([Arriola and Kuleshov, 2026](https://arxiv.org/html/2610.33379#bib.bib10), Section 5\.3 and Appendix K\)\. The mask can attend to tokens earlier in the generation order but cannot see its own target token or later tokens, allowing all conditional predictions to be computed in one pass\. We use the same construction for verification, placing draft tokens in the token entries and comparing them with predictions from the corresponding masks\. Packed\-SS adds a second mask at each position to generate the next draft from the prompt and accepted tokens\.

#### Packed input and attention\.

LetUtU\_\{t\}contain the unresolved positions in the active window, and writeut=\|Ut\|u\_\{t\}=\|U\_\{t\}\|\. LetNtN\_\{t\}contain the correction token, if any, that has not yet been processed as an input\. LetOtO\_\{t\}contain the previous draftDt−1D\_\{t\-1\}, with masks for missing entries;MtM\_\{t\}contain masks for verifier predictions; andRtR\_\{t\}contain masks for new draft predictions\. The input is

\[Nt∣Ot​∣Mt∣​Rt\],\|Nt\|=p∈\{0,1\},\|Ot\|=\|Mt\|=\|Rt\|=ut\.\[N\_\{t\}\\mid O\_\{t\}\\mid M\_\{t\}\\mid R\_\{t\}\],\\qquad\|N\_\{t\}\|=p\\in\\\{0,1\\\},\\quad\|O\_\{t\}\|=\|M\_\{t\}\|=\|R\_\{t\}\|=u\_\{t\}\.All regions attend to the cached keys and values of the prompt and accepted tokens\. The regionsOt,Mt,RtO\_\{t\},M\_\{t\},R\_\{t\}also attend toNtN\_\{t\}, if present, and the row inNtN\_\{t\}attends to itself\.

WriteOr,Mr,RrO\_\{r\},M\_\{r\},R\_\{r\}for the rows ofOt,Mt,RtO\_\{t\},M\_\{t\},R\_\{t\}at verification rankrr\. Let𝒦r\\mathcal\{K\}\_\{r\}contain the earlier rankss<rs<rin the orderπt\\pi\_\{t\}that have a stored draft token\. UsingX←YX\\leftarrow Yto mean that rowXXmay attend to the rows inYY, the remaining attention connections are

Or←\{Os:s∈𝒦r\}∪\{Or\},Mr←\{Os:s∈𝒦r\}∪\{Mr\},Rr←\{Rr\}\.O\_\{r\}\\leftarrow\\\{O\_\{s\}:s\\in\\mathcal\{K\}\_\{r\}\\\}\\cup\\\{O\_\{r\}\\\},\\qquad M\_\{r\}\\leftarrow\\\{O\_\{s\}:s\\in\\mathcal\{K\}\_\{r\}\\\}\\cup\\\{M\_\{r\}\\\},\\qquad R\_\{r\}\\leftarrow\\\{R\_\{r\}\\\}\.The three rows at rankrrshare the textual position IDπt​\(r\)\\pi\_\{t\}\(r\)\. The physical call evaluates allutu\_\{t\}verifier queries, while the logical algorithm needs predictions only through the first missing draft token or disagreement\.

#### Cache update\.

Only keys and values fromNtN\_\{t\}and accepted draft tokens inOtO\_\{t\}are added to the cache\. A mask’s keys and values do not represent its predicted token, so a correction is processed in the next call throughNt\+1N\_\{t\+1\}\.

### D\.3Input\-size comparison

[Table2](https://arxiv.org/html/2610.33379#A4.T2)compares the input sizes of these layouts with those of Linear\-SS and Quadratic\-SS\.

Table 2:Execution cost per iteration with cached prompt context\.Evaluated uncached rows count token representations processed by the transformer, including duplicates and evaluated padding, summed across calls\. Nemotron uses block sizeB=W\+1B=W\+1\. In SetDLM,ut≤Wu\_\{t\}\\leq Wpositions remain unresolved in the active window, andp=\|Nt\|∈\{0,1\}p=\|N\_\{t\}\|\\in\\\{0,1\\\}counts the correction token processed in this call; maximum row counts takeut=Wu\_\{t\}=Wandp=1p=1\. Prompt rows are processed once and depend on prompt length\.Evaluated uncached rows per iterationSetDLM maximumMethodNFE/iterationNemotronW=8W=8W=16W=16SetDLM generalLinear\-SS22​B2B2549p\+3​utp\+3u\_\{t\}Quadratic\-SS1B⁡\(B\+1\)B\(B\+1\)81289p\+2​ut\+ut2p\+2u\_\{t\}\+u\_\{t\}^\{2\}Packed\-SS \(Ours\)12​B2B2549p\+3​utp\+3u\_\{t\}For SetDLM, Linear\-SS processesp\+utp\+u\_\{t\}rows for drafting and2​ut2u\_\{t\}for verification, cachingNtN\_\{t\}between calls\. Packed\-SS processes the same total in one call\. Quadratic\-SS usesp\+2​utp\+2u\_\{t\}rows for verification andutu\_\{t\}proposal regions ofutu\_\{t\}rows each\.

## Appendix EExperimental Details

### E\.1Hardware, models, and tasks

All runs use batch size one on a single A100 80GB GPU\. Nemotron is evaluated in bfloat16 \(BF16\), and SetDLM in float32 \(FP32\)\. Nemotron uses left\-to\-right AR verification in every main setting\. Both scales run the same four tasks—HumanEval, MBPP, MATH\-500, and GSM8K\([Chen et al\., 2021](https://arxiv.org/html/2610.33379#bib.bib29);[Austin et al\., 2021b](https://arxiv.org/html/2610.33379#bib.bib30);[Hendrycks et al\., 2021](https://arxiv.org/html/2610.33379#bib.bib31);[Cobbe et al\., 2021](https://arxiv.org/html/2610.33379#bib.bib28)\)—under greedy decoding and under rejection sampling\. Each cell draws on a fixed prompt set: 164 HumanEval problems, 378 MBPP problems, 500 MATH\-500 problems, and all 1,319 GSM8K test questions, and these counts are the number of examples behind every mean and interval reported for that cell\. Both scales additionally run all 555 English WritingBench prompts\([Wu et al\., 2025](https://arxiv.org/html/2610.33379#bib.bib55)\)at both temperatures\. The SetDLM experiments allocate 1,024 output positions and use active windows of size 8 and 16; because its two windows come from separately adapted checkpoints, comparing them is not a same\-weights window\-size ablation\. At both temperatures, Linear\-SS and Packed\-SS use Nemotron block sizeB=32B=32, while Quadratic\-SS usesB=8B=8, with speculative widthW=B−1W=B\-1\.

#### Training horizon versus inference window\.

The SetDLM training construction uses a desired prediction budgetS∈\{4,8,16\}S\\in\\\{4,8,16\\\}and permits schedules reaching2​S2Sfuture positions\([Arriola and Kuleshov, 2026](https://arxiv.org/html/2610.33379#bib.bib10)\)\. Checkpoints are named by their maximum horizon\. Hence smax32 is theS=16S=16, maximum\-horizon\-32 model, and its matched active evaluation window isW=16W=16rather than 32\.

### E\.2Initialization and stopping

#### Initial calls\.

In our Nemotron implementation, the drafting part of a regular call’s input contains the last accepted token followed byWWmasks\. During initialization, the drafting part containsW\+1W\+1masks, giving one extra draft position\. For SetDLM, Packed\-SS builds the prompt cache in one counted call before entering the empty\-draft loop of[Algorithm3](https://arxiv.org/html/2610.33379#alg3)\.

At temperature 1, our implementation first processes and caches the prompt, then samples the first output token from its last prediction\. A second call proposes draft tokens conditioned on the prompt and this first output token\. Both calls count toward NFE and supply a full draft before verification, whereas[Algorithm2](https://arxiv.org/html/2610.33379#alg2)starts from an empty draft\.

#### Output caps\.

Generations are capped at 512 tokens for code and writing and 1,024 tokens for mathematics\.

#### SetDLM stop sequences\.

An end marker in the SetDLM evaluation can consist of several tokens\. We recognize it only after all of those tokens have been accepted\. We setEE, the last output position, to the marker’s final position and fill any earlier unresolved positions before returning\. If several markers are recognized, we use the one that ends earliest in the output\.

### E\.3Prompts and scoring

Nemotron uses zero\-shot prompts: function completion for HumanEval, visible original test cases for MBPP, and requests for boxed final answers on the mathematics tasks\. MBPP uses the 378\-problem EvalPlus subset and its original base tests\. MATH\-500 uses boxed\-answer exact match, and SetDLM GSM8K scoring compares normalized final answers\. The code and mathematics runs use the same prompts and graders at both temperatures\.

For temperature\-1 WritingBench runs, each complete query uses the checkpoint’s user\-message chat template with thinking disabled; prompts are not truncated\. Scored\-task tables report accuracy or pass@1 at temperature 0 and mean per\-response correctness at temperature 1\. WritingBench measures speed only, without judge\-based scoring\.

### E\.4Sampling and repetitions

At temperature 1, rejection sampling can produce different generation trajectories for the same prompt, varying output length and decoding cost\. To reduce this source of variance on the code tasks, each method draws 20 responses per HumanEval prompt and 7 per MBPP prompt\. MATH\-500, GSM8K, and WritingBench use one response per prompt and method\. Sampling uses float32 probability normalization and the full vocabulary, without top\-kkor top\-ppfiltering\. For Quadratic\-SS, the verification result selects the branch matching the accepted draft prefix before its next draft tokens are sampled\.

### E\.5Timing and confidence intervals

Decoder wall time is CUDA\-synchronized and includes prefill and all decoding work, excluding model loading, prompt preparation, tokenization, output\-text decoding, and scoring\. Each model forward pass counts as one NFE, including prompt processing\. Generated\-token counts include stop and special tokens and any generated reasoning text\.

We compute 95% percentile confidence intervals from 20,000 bootstrap resamples within each checkpoint, task, decoding mode, and verification order\. Each resample draws prompt indices with replacement, uses the same indices for all methods, and recomputes the reported statistics\. We bootstrap over prompts, using all repeated responses for each sampled prompt\. The intervals describe variation across prompts rather than timing noise for a fixed generation, and are not adjusted for multiple comparisons\.

### E\.6SetDLM decoding settings

#### Confidence scores\.

For confidence ordering, Packed\-SS and Quadratic\-SS score each unresolved position by the probability assigned to its draft token when that token was proposed; Quadratic\-SS uses the selected proposal branch\. Linear\-SS uses the probability of the verifier’s most likely token at each position in the previous iteration\. Positions are ranked by decreasing score, with ties broken by textual position\. Positions without a draft token or a finite score come last in textual order, so the first iteration uses LTR order\. Each order is fixed before the next model call; for Linear\-SS, this is the drafting call\.

#### Native baseline\.

Native SetDLM uses the released sliding\-window decoder with argmax token selection and confidence threshold0\.90\.9\. It uses the same GSM8K prompts, number of output positions, window sizes, and stopping convention as the speculative schedules\.

## Appendix FAdditional Experiments and Tables

This appendix collects additional experiments, absolute measurements, and numerical values supporting the main figures\.

### F\.1Nemotron block size and execution cost

To select the block size for each schedule, we run a small probe on Nemotron 8B using the first ten GSM8K test questions \([Table3](https://arxiv.org/html/2610.33379#A6.T3)\)\. We choose the size with the highest token throughput \(tokens/s\) among those tested\. All runs use a total output limit of 512 new tokens\.

Table 3:Nemotron 8B GSM8K block\-size probe\.Each row uses the same ten questions\. Tokens/s is calculated from the displayed means\. Wall is mean total decoder time per response, including prompt processing; total NFE and output\-token counts are also per\-response means\. Accuracy is reported for comparison\. Rows/call is the maximum number of input rows per call after initialization\.
### F\.2Proposal\-only LoRA accounting

We evaluate the proposal\-only low\-rank adapter trained by the Nemotron authors\([Fu et al\., 2026](https://arxiv.org/html/2610.33379#bib.bib19)\)on Nemotron 8B and all GSM8K test questions, changing diffusion drafting while preserving the AR verifier\. The base and LoRA measurements in this study use a total output cap of 512 new tokens, including thinking and answer tokens\. This cap differs from the 1,024\-token GSM8K setting in[Section4](https://arxiv.org/html/2610.33379#S4), so absolute timings are not directly comparable with[Table6](https://arxiv.org/html/2610.33379#A6.T6)\. Packed\-SS reduces NFE by 12\.0% relative to Linear\-SS but takes 1\.749 rather than 1\.698 seconds per example, a 3\.0% slowdown \([Table4](https://arxiv.org/html/2610.33379#A6.T4)\)\.

Table 4:Nemotron 8B GSM8K with proposal\-only LoRA \(n=1,319n=1\{,\}319\)\.Means; model execution time and non\-model decoder time sum to inclusive decoder wall time\. Emitted length and throughput accompany latency, and accuracy is reported for comparison\.In calls with the adapter enabled, the implementation computes an update for every input row but applies it only to proposal rows\. Packed therefore computes 64 updates but applies only 32, whereas Linear places its 32 proposal rows in a separate call and disables the adapter for its 32\-row verifier\. Quadratic applies updates to 64 proposal rows but computes them for all 72 input rows\.

Table 5:Nemotron 8B GSM8K model execution time per iteration\.Linear’s iteration contains both calls\. “Selected” counts rows receiving adapter updates; “evaluated” counts rows for which those updates are computed\.With LoRA, mean model execution time is nearly equal for Packed\-SS and Linear\-SS \(1\.570 versus 1\.561 seconds\)\. After the first call, Packed\-SS averages 47\.44 iterations versus Linear\-SS’s 27\.02, but its model time per iteration is 32\.37 versus 56\.77 milliseconds, so these differences almost cancel\. Non\-model decoder time adds 0\.179 seconds for Packed\-SS against 0\.137 for Linear\-SS, and the Packed\-SS\-minus\-Linear\-SS wall\-time difference is0\.05130\.0513seconds with paired 95% interval\[0\.0243,0\.0788\]\[0\.0243,0\.0788\]\. Token throughput is also lower: Packed\-SS emits 274\.2 tokens on average against Linear\-SS’s 272\.2, and its throughput ratio is0\.9780\.978with paired interval\[0\.969,0\.987\]\[0\.969,0\.987\]\. Shared, densely fine\-tuned weights for drafting and verification would eliminate these extra LoRA computations and could reduce the latency penalty\. The Nemotron authors have not released such a checkpoint\.

### F\.3Model execution and non\-model decoder time

To separate model evaluation from decoder overhead, we normalize each task’s component times by its Linear\-SS total and then average tasks equally within each model scale, so that long\-output tasks do not dominate the decomposition\. The decomposition covers GSM8K and MATH\-500 only\.

LinearQuadraticPackedLinearQuadraticPacked000\.50\.5118B3BTime / Linear totalTemperature 0: GSM8K \+ MATH\-500, cap 1,024Model execution timeNon\-model decoder timeFigure 5:Normalized Nemotron timing decomposition on GSM8K and MATH\-500\.Greedy decoding with a 1,024\-token cap\. Each task’s component times are divided by that task’s Linear\-SS total, then the two tasks are averaged with equal weight within each model scale\. The dashed line marks Linear\-SS’s total\. Non\-model decoder time covers draft token construction, selection, and state updates\.Packed\-SS cuts model execution time to 0\.860 and 0\.854 of Linear\-SS’s total at 3B and 8B while raising non\-model decoder time by about 40%, for totals of 0\.964 and 0\.951\. Quadratic\-SS cuts model execution time comparably, to 0\.898 and 0\.876, but its non\-model decoder time is 2\.9 and 2\.7 times Linear\-SS’s, so its totals exceed one at 1\.106 and 1\.062\.

### F\.4Absolute measurements

Table 6:Nemotron absolute measurements at temperature 0\.Tokens, wall time, and NFE are means per response; rates use aggregate totals\. Bold marks the highest tokens/s within each model and task\.Table 7:Nemotron absolute measurements at temperature 1\.Tokens, wall time, and NFE are means per response; rates use aggregate totals\. Bold marks the highest tokens/s within each model and task\.Table 8:SetDLM absolute measurements on GSM8K\.LTR verification and native decoding on all 1,319 test questions\. Tokens, wall time, and NFE are means per response; rates use aggregate totals\. Bold marks the highest tokens/s within each window\.
### F\.5Confidence\-ordered verification on SetDLM

Table 9:SetDLM measurements on GSM8K\.All 1,319 test questions\. Top: LTR verification and native decoding; bottom: confidence\-ordered verification\. Tokens, inclusive wall time, and total NFE are means per response; rates use aggregate totals\. Bold marks the highest tokens/s within each window and order\.The training horizons areSmax=16S\_\{\\max\}=16and3232forW=8W=8and1616, respectively\. ForW=8W=8and1616, respectively, maximum recurrent rows/call are\(16,32\)\(16,32\)for Linear\-SS,\(81,289\)\(81,289\)for Quadratic\-SS,\(25,49\)\(25,49\)for Packed\-SS, and\(16,28\)\(16,28\)for Native\. Native commits at confidence threshold0\.90\.9and has no verification order \([SectionE\.6](https://arxiv.org/html/2610.33379#A5.SS6.SSS0.Px2)\)\.

Table 10:Paired SetDLM efficiency comparisons\.Wall\-time differences are Packed\-SS minus baseline; throughput ratios are Packed\-SS/baseline\. Brackets give paired 95% confidence intervals\.Under confidence order, Packed\-SS remains fastest at both windows \([Table9](https://arxiv.org/html/2610.33379#A6.T9)\), reducing Linear\-SS’s wall time by 9\.5% atW=8W=8and 9\.7% atW=16W=16and its total NFE by 12\.4% and 16\.5%\. Its throughput gains over Linear\-SS are 10\.2% and 12\.8%, and atW=16W=16it reaches2\.20×2\.20\\timesQuadratic\-SS’s throughput; all eight SetDLM paired intervals favour Packed\-SS \([Table10](https://arxiv.org/html/2610.33379#A6.T10)\)\. Two effects should be kept apart, because only one of them is a cost\. The three LTR schedules have identical per\-example correctness labels within each checkpoint, so Packed\-SS’s LTR speedup incurs no observed accuracy cost\. Changing the order can affect accuracy\. Confidence ordering lowers Packed\-SS’s latency by a further 4\.1% atW=8W=8and 13\.1% atW=16W=16relative to LTR, but accuracy falls from 68\.39% to 65\.35% and from 66\.57% to 59\.06%; the second drop is large enough to give up the advantage over SetDLM’s own threshold decoder, which scores 59\.51% there\. Adaptive position choices alter the greedy trajectory even in exact arithmetic, and single\-step confidence is in any case a weak criterion for choosing which tokens to accept\([Chen et al\., 2026b](https://arxiv.org/html/2610.33379#bib.bib18)\)\. We therefore report the ordering as a speed–accuracy trade\-off of the decoding policy rather than a property of the schedule: Packed\-SS gives the lowest latency in each regime\.

### F\.6Accuracy and output trajectories

We report paired accuracy differences to assess variation in task scores At temperature 1, the schedules draw different random samples, so their continuations and measured accuracies can differ\. Under greedy decoding, small floating\-point differences can also change token choices and subsequent continuations\.[Table11](https://arxiv.org/html/2610.33379#A6.T11)reports the Nemotron comparisons in percentage points, with paired 95% confidence intervals\.

For SetDLM, the LTR schedules have identical correctness labels despite different outputs on 5 of 1,319 smax16 questions and 13 of 1,319 smax32 questions\. Their accuracy gains over Native are8\.728\.72points atW=8W=8\(95% CI\[6\.37,11\.07\]\[6\.37,11\.07\]\) and7\.057\.05points atW=16W=16\(95% CI\[4\.78,9\.33\]\[4\.78,9\.33\]\)\.

Table 11:Paired Nemotron accuracy differences\.
### F\.7Paired efficiency comparisons

[Tables12](https://arxiv.org/html/2610.33379#A6.T12)and[13](https://arxiv.org/html/2610.33379#A6.T13)report Packed\-SS minus baseline wall time and Packed\-SS/baseline throughput, with 95% paired intervals\. Negative differences and ratios above one favor Packed\-SS\.

Table 12:Paired Nemotron efficiency comparisons at temperature 0\.Table 13:Paired Nemotron efficiency comparisons at temperature 1\.All ten paired throughput intervals against Linear\-SS lie above one \([Table13](https://arxiv.org/html/2610.33379#A6.T13)\), supporting a throughput advantage in the measured settings\. Against the Quadratic\-SS extension, three intervals lie above one, two lie below it, and five include it; the last group is inconclusive rather than evidence of equivalence\.

#### Reference schedules\.

At temperature 0, Quadratic\-SS has lower throughput than Linear\-SS on all eight code and mathematics settings but higher throughput on WritingBench; at temperature 1, it has higher throughput on seven of the eight code and mathematics settings\. Shorter accepted prefixes require more iterations, so eliminating a second call can outweigh the cost of its fixed 72\-row pass \([SectionF\.3](https://arxiv.org/html/2610.33379#A6.SS3)\)\. On SetDLM, Linear\-SS and Packed\-SS are faster than Native at both windows; Quadratic\-SS is faster only atW=8W=8\([Table8](https://arxiv.org/html/2610.33379#A6.T8)\)\.