CAPS: Cascaded Adaptive Pairwise Selection for Efficient Parallel Reasoning

# CAPS: Cascaded Adaptive Pairwise Selection for Efficient Parallel Reasoning
Source: [https://arxiv.org/html/2605.15513](https://arxiv.org/html/2605.15513)
Fangzhou Lina,b,cShuo XingaPeiran LiaSiyuan YangaQianwen Ged Kazunori YamadacZiming ZhangbHaichong ZhangbZhengzhong Tua aTexas A&M UniversitybWorcester Polytechnic Institute cTohoku UniversitydGeorgia Institute of Technology

###### Abstract

Parallel reasoning, where a generator samples many candidate solutions and an aggregator selects the best, is one of the most effective forms of test\-time scaling in large language models, and pairwise self\-verification has become its strongest aggregation primitive\. Yet pairwise verification carries a heavy cost: each judgment reads two complete solutions in full, and existing methods perform tens of such judgments per problem regardless of whether the comparison is informative\. We introduceCAPS\(Cascaded Adaptive Pairwise Selection\), an inference\-only framework that allocates verifier compute non\-uniformly along two orthogonal axes: an*evidence axis*that adapts how much of each candidate the judge sees, and a*distribution axis*that adapts how comparisons are spread across the pool\. CAPS instantiates these into a four\-stage cascade with an optional rescue subroutine, and admits a closed\-form verifier\-token cost in which the per\-candidate marginal cost is roughly halved relative to uniform full\-evidence schedules\. On four self\-verifying models \(Qwen3\-14B, GPT\-OSS\-20B, Qwen3\-4B\-Instruct/Thinking\) and five reasoning benchmarks spanning code \(LiveCodeBench\-v5/v6, CodeContests\) and math \(AIME 2025, HMMT 2025\), CAPS outperforms the leading pairwise verifier on1414of2020suites while using25\.4%25\.4\\%of its verifier\-token budget on code, and outperforms pointwise self\-verification on*all*2020\. The trade\-off suites admit an interpretable diagnostic in terms of the verifier’s accuracy at partial versus full evidence, providing a concrete pre\-deployment check for cascade suitability\.

## 1Introduction

Large language models have made substantial progress on reasoning tasks through inference\-time scaling, where extra compute is spent to refine, reflect on, or diversify candidate solutions\(Snellet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib14); Weiet al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib1); Wanget al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib2); Kojimaet al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib3); Snellet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib4)\)\. This compute can be deployed sequentially: a single chain of thought made longer through reflection or revision; or in parallel, by sampling many independent chains and aggregating their answers\(Lightmanet al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib11); Jaechet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib12); Chowet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib13)\)\. The latter also called*Parallel reasoning*\(Brownet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib7); Snellet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib14); Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\), which has emerged as one of the most effective forms of test\-time scaling: compared with single answer, the probability of correct answer increased significantly even at parallel numberNN\. AsNNgrows, the limiting factor for parallel reasoning shifts from the diversity of candidates to the reliability of the aggregator: parallel reasoning is only as good as the mechanism that picks the correct candidate from the pool\(Lightmanet al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib11); Jaechet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib12); Chowet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib13); Brownet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib7); Snellet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib14); Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\)\.

Self\-verification, where the same model judges its own candidates during \(parallel\) reasoning, has become the dominant aggregator in domains with verifiable rewards such as code and math\(Wenget al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib24); Huanget al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib25); Stechlyet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib26)\)\. The simplest form is*pointwise scoring*Venkatramanet al\.\([2025](https://arxiv.org/html/2605.15513#bib.bib34)\): prompting the model to assign an absolute quality rating to each candidate in isolation\. This approach suffers from calibration failures: ratings saturate at the top of the scale and lose discriminative power\(Wenget al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib24); Zhuanget al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib27); Stechlyet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib26)\); the verifier exhibits self\-preference bias, positively rating its own samples even when they are incorrect\(Madaanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib30)\); and ratings produced for different candidates lack a globally comparable scale, because each judgment is made without reference to any other candidate\(Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\)\. Recent work has converged on*pairwise comparison*as a more robust primitive: head\-to\-head judgments only require the verifier to determine in which of two solutions is better: a relative judgment that sidesteps the calibration issues of absolute scoring\. The leading method in this regime, V1\-Infer\(Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\), uses an uncertainty\-guided Swiss tournament\(Csató,[2017](https://arxiv.org/html/2605.15513#bib.bib37)\)with roughly4848pairwise comparisons, demonstrating substantial gains over pointwise verification on code and math benchmarks\.

Pairwise verification\(Stechlyet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib26); Madaanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib30); Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15); Venkatramanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib34)\), however, has its own bottleneck: heavy cost\. Each judgment reads two complete44–88K\-token solutions, so the verifier\-token budget routinely exceeds the cost of generation itself, and the cost grows quadratically with the number of comparisons\. While existing methods carefully optimizewhichpairs to compare: through Swiss\-system tournamentsCsató \([2017](https://arxiv.org/html/2605.15513#bib.bib37)\), uncertainty\-guided pair selection\(Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\), and Bradley\-Terry\-inspired schedulesHunter \([2004](https://arxiv.org/html/2605.15513#bib.bib39)\)\. They treathoweach comparison is performed as a fixed quantity: every pair is evaluated by reading both solutions in full, regardless of whether the comparison decides between two strong contenders or two candidates that are clearly wrong\. This uniformity is at odds with the structure of the selection problem itself\. Most candidate pairs are easy to discriminate: they may produce different final answers, pursue visibly different algorithmic strategies, or differ in early reasoning steps that already determine correctness\. And most candidates are not the eventual winner; the selection argmax depends only on the ordering among the strongest few\. The natural question is then:can we exploit this structure to spend verifier compute more efficiently, without sacrificing selection quality?

![Refer to caption](https://arxiv.org/html/2605.15513v1/x1.png)Figure 1:CAPS Overview\. Deduplicate, eliminate at partial evidence, eliminate at full evidence, and round\-robin among the finalists; with an optional rescue subroutine for cheap\-evidence errors\.To this end, we proposeCAPS\(Cascaded Adaptive Pairwise Selection\), an inference\-only framework that allocates verifier compute non\-uniformly along two orthogonal axes\. The*evidence axis*controls how much of each candidate the judge sees; we observe that most pairs admit a confident judgment from a partial view: the boxed answer, the first few lines of code, or a short reasoning window; at roughly an order of magnitude lower cost than a full read\. The*distribution axis*controls how comparisons are spread across the candidate pool; since the final selection depends only on the ordering among the strongest candidates, full\-evidence comparisons should be reserved for them\. CAPS instantiates these into a four\-stage cascade with a closed\-form verifier\-token cost characterization, together with an optional rescue subroutine that protects against errors at the cheapest evidence level\.

We evaluate CAPS against three baselines: Vanilla \(no verification\), Pointwise self\-verification\(Venkatramanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib34)\), and V1\-Infer\(Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\); across four self\-verifying models \(Qwen3\-14B\(Yanget al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib41)\), GPT\-OSS\-20B\(Agarwalet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib42)\), Qwen3\-4B\-Instruct\-2507, and Qwen3\-4B\-Thinking\-2507\) and five reasoning benchmarks spanning code \(LiveCodeBench\-v5\(Jainet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib43)\), LiveCodeBench\-v6, CodeContests\(Liet al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib6)\)\) and math \(AIME 2025, HMMT February 2025\(Balunovićet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib44)\)\)\. CAPS outperforms V1\-Infer on1414of2020settings while using on average25\.4%25\.4\\%of its verifier\-token budget on code, and outperforms Pointwise verification onall2020\. The empirical token ratio matches our closed\-form prediction to within three percentage points, indicating that the efficiency is structural rather than the result of hyperparameter tuning\. The six results where V1\-Infer is higher trail by at most3\.43\.4Pass@1 points; we trace this trade\-off to a measurable property of the deployment: the verifier’s per\-pair accuracy at partial vs\. full evidence; giving practitioners a concrete pre\-deployment check for cascade suitability\.

Contributions\.Our contributions are summarized as follows\.

1\. Two axes for verifier\-compute allocation in pairwise selection\.We identify the evidence level and the comparison distribution as orthogonal, underused degrees of freedom in pairwise self\-verification\. The evidence axis exploits the fact that most pairs are easy to discriminate at low evidence; the distribution axis exploits the fact that the selection argmax depends only on the strongest candidates\. Together these reframe pairwise verification cost as a structural quantity\.

2\. CAPS, a cascaded selection framework with closed\-form cost\.CAPS instantiates the two axes as a four\-stage cascade: deduplicate, eliminate at partial evidence, eliminate at full evidence, and round\-robin among the finalists; with an optional rescue subroutine for cheap\-evidence errors\.

3\. Empirical results spanning four models, five benchmarks, and four methods\.Across our2020evaluation suites, CAPS outperforms V1\-Infer in1414at roughly one\-quarter of its verifier\-token cost, and Pointwise verification inall2020\. The empirical token ratio matches the analytical prediction to within three percentage points, and the trade\-off cells admit an interpretable diagnostic in terms of the verifier’s per\-pair accuracy at partial vs\. full evidence\.

## 2Related Work

Inference\-time scaling encompasses both*sequential*approaches that lengthen a single chain of thought through reflection or revision\(Zhanget al\.,[2024b](https://arxiv.org/html/2605.15513#bib.bib17); Madaanet al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib19); Quet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib20)\)and*parallel*approaches that sample multiple independent chains and aggregate their answers\(Weiet al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib1); Wanget al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib2); Brownet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib7); Snellet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib14)\); Self\-Consistency\(Chenet al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib32); Wanget al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib2)\)and Best\-of\-NNsampling with learned reward models\(Lightmanet al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib11); Cobbeet al\.,[2021](https://arxiv.org/html/2605.15513#bib.bib21)\)are the canonical aggregation primitives, and the accuracy gap on hard problems motivates stronger selection mechanisms\(Brownet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib7); Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\)\. Among these,*self\-verification*, which reuses the generator as a judge over its own samples\(Wenget al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib24); Huanget al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib25); Stechlyet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib26)\), has become the dominant primitive in domains with verifiable rewards: pointwise scoring with each candidate rated in isolation suffers from documented calibration failures including score saturation, self\-preference bias, and the lack of a globally comparable scale\(Stechlyet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib26); Madaanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib30); Panicksseryet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib28); Zhenget al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib29); Venkatramanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib34)\), and recent work has therefore converged on*pairwise*comparison: V1\-Infer\(Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\)uses an uncertainty\-guided Swiss tournamentCsató \([2017](https://arxiv.org/html/2605.15513#bib.bib37)\)to allocate a fixed comparison budget, and Pairwise RM\(Liuet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib31)\)trains a dedicated pairwise reward model with a knockout\-tournament selection rule\. An orthogonal line of work combines candidates rather than selecting among them, ranging from majority voting\(Chenet al\.,[2023](https://arxiv.org/html/2605.15513#bib.bib32)\)to learned aggregators such as AggLM\(Zhaoet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib33)\)and Recursive Self\-Aggregation\(Venkatramanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib34)\); aggregation methods improve Pass@1 but exhibit diversity collapse, motivating hybrid designs that pair aggregation with verification\(Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\)\. A more detailed discussion of each of these threads is provided in Appendix[A](https://arxiv.org/html/2605.15513#A1)\.

## 3Method

We formulate selection from parallel reasoning candidates as a budgeted pairwise verification problem \(§[3\.1](https://arxiv.org/html/2605.15513#S3.SS1)\), describe the four\-stage CAPS pipeline that operationalizes it \(§[3\.2](https://arxiv.org/html/2605.15513#S3.SS2)–§[3\.3](https://arxiv.org/html/2605.15513#S3.SS3)\), and analyze its verifier\-token cost in closed form \(§[3\.4](https://arxiv.org/html/2605.15513#S3.SS4)\)\.

### 3\.1Problem Setup

The selection task\.Given a problemqqandNNcandidate solutions𝒞=\{c1,…,cN\}\\mathcal\{C\}=\\\{c\_\{1\},\\ldots,c\_\{N\}\\\}sampled from a generator \(any LLM\), the selection task is to return an index

ı^∈arg⁡maxi∈\[N\]⁡Pr⁡\[yi=1∣q,𝒞\],\\hat\{\\imath\}\\;\\in\\;\\arg\\max\_\{i\\in\[N\]\}\\,\\Pr\[\\,y\_\{i\}=1\\mid q,\\mathcal\{C\}\\,\],\(1\)
whereyi∈\{0,1\}y\_\{i\}\\in\\\{0,1\\\}is the unobserved correctness ofcic\_\{i\}\. The verifier \(same LLM in self\-verification setting\) accesses only a pairwise oracle and is constrained by a verifier\-token budgetBBVenkatramanet al\.\([2025](https://arxiv.org/html/2605.15513#bib.bib34)\); Singhet al\.\([2026](https://arxiv.org/html/2605.15513#bib.bib15)\)\.

Evidence map\.Rather than treating each candidate as a monolithic object, we expose multiple views of it through an*evidence map*\{ϕe\}e∈ℰ\\\{\\phi\_\{e\}\\\}\_\{e\\in\\mathcal\{E\}\}, indexed by detail level\. Each level has a per\-pair token costTeT\_\{e\}monotone inee\. We use three levels:

- •ϕ0​\(c\)∈Σ\\phi\_\{0\}\(c\)\\in\\Sigma: discrete signature \(boxed answer for math, AST hash for code\), used for clustering only\.
- •ϕ1​\(c\)∈𝒱1\\phi\_\{1\}\(c\)\\in\\mathcal\{V\}\_\{1\}: partial view \(first∼20\{\\sim\}20code lines, or boxed answer plus a∼300\{\\sim\}300\-token reasoning window\)\.
- •ϕ2​\(c\)=c\\phi\_\{2\}\(c\)=c: full solution\.

We writeρ:=T1/T2∈\(0,1\)\\rho:=T\_\{1\}/T\_\{2\}\\in\(0,1\)for the evidence\-cost ratio, the governing quantity of the cost analysis in §[3\.4](https://arxiv.org/html/2605.15513#S3.SS4)\.

Pairwise oracle\.A pairwise judge at evidence leveleeis a \(possibly stochastic\) map

J​\(q,ϕe​\(ci\),ϕe​\(cj\)\)⟼\(vi​j,wi​j\),J\\bigl\(q,\\,\\phi\_\{e\}\(c\_\{i\}\),\\,\\phi\_\{e\}\(c\_\{j\}\)\\bigr\)\\;\\longmapsto\\;\(v\_\{ij\},\\,w\_\{ij\}\),\(2\)returning an outcomevi​j∈\{0,12,1\}v\_\{ij\}\\in\\\{0,\\tfrac\{1\}\{2\},1\\\}fromii’s perspective \(loss, tie, win\) and a confidencewi​j∈\[τw,1\]w\_\{ij\}\\in\[\\tau\_\{w\},1\]floored atτw=0\.05\\tau\_\{w\}=0\.05\. Throughout the pipeline we maintain a per\-candidate cumulative scoreS​\(c\)S\(c\), initialized to0and updated after each judge call by

S\(ci\)\+=wi​jvi​j,S\(cj\)\+=wi​j\(1−vi​j\)\.S\(c\_\{i\}\)\\;\\mathrel\{\+\}=\\;w\_\{ij\}\\,v\_\{ij\},\\qquad S\(c\_\{j\}\)\\;\\mathrel\{\+\}=\\;w\_\{ij\}\\,\(1\-v\_\{ij\}\)\.\(3\)Eq\. \([3](https://arxiv.org/html/2605.15513#S3.E3)\) accumulates confidence\-weighted outcomes: high\-confidence wins dominate the score while ambiguous comparisons contribute little, so noisy judgments do not propagate into later decisions\.

Intuition\.Two qualitative observations guide the design that follows\. First, the argmax in Eq\. \([1](https://arxiv.org/html/2605.15513#S3.E1)\) depends only on the relative ordering among the strongest candidates: comparisons among low\-quality candidates carry little information about the final selection\. Second, many pairs admit a confident judgment from a partial view: candidates that differ in their final answer or in their opening algorithmic strategy can often be discriminated without reading either solution end\-to\-end\. Together these suggest a verifier that reserves expensive full\-evidence comparisons for a small subset of plausible winners and resolves the rest cheaply\. They motivate the cascade structure but the empirical claims of the paper rest on §[4](https://arxiv.org/html/2605.15513#S4)\.

### 3\.2The CAPS Pipeline

CAPS proceeds in four stages, illustrated in Figure[1](https://arxiv.org/html/2605.15513#S1.F1): an equivalence\-class deduplication \(Stage 0\), two halving rounds at increasing evidence \(Stages A and B\), and a complete round\-robin amongfffinalists at full evidence \(Stage C\)\. Algorithm[1](https://arxiv.org/html/2605.15513#alg1)gives the high\-level procedure; the helpersDedup,Eliminate, andRescueare specified in Appendix[B](https://arxiv.org/html/2605.15513#A2)\.

Algorithm 1CAPS — Cascaded Adaptive Pairwise Selection1:problem

qq, candidates

𝒞\\mathcal\{C\}, judge

JJ, finalist count

ff, weight floor

τw\\tau\_\{w\}, rescue margin

δ\\delta
2:selected candidate

c⋆c^\{\\star\}
3:

S​\(c\)←0S\(c\)\\leftarrow 0for all

c∈𝒞c\\in\\mathcal\{C\}⊳\\trianglerightcumulative scores

4:

𝒞′←Dedup​\(𝒞,ϕ0\)\\mathcal\{C\}^\{\\prime\}\\leftarrow\\textsc\{Dedup\}\(\\mathcal\{C\},\\phi\_\{0\}\); record cluster sizes

ν​\(⋅\)\\nu\(\\cdot\)⊳\\trianglerightStage 0 \(no judge calls\)

5:

𝒞A←Eliminate\(𝒞′,J,e=1,seed byν\)\\mathcal\{C\}\_\{A\}\\leftarrow\\textsc\{Eliminate\}\(\\mathcal\{C\}^\{\\prime\},\\,J,\\,e\{=\}1,\\,\\text\{seed by \}\\nu\)⊳\\trianglerightStage A

6:

𝒞B←Eliminate\(𝒞A,J,e=2,seed byS,stop atf\)\\mathcal\{C\}\_\{B\}\\leftarrow\\textsc\{Eliminate\}\(\\mathcal\{C\}\_\{A\},\\,J,\\,e\{=\}2,\\,\\text\{seed by \}S,\\,\\text\{stop at \}f\)⊳\\trianglerightStage B

7:ifCAPS\-R enabledthen

𝒞B←Rescue​\(𝒞B,𝒞′∖𝒞B,δ\)\\mathcal\{C\}\_\{B\}\\leftarrow\\textsc\{Rescue\}\(\\mathcal\{C\}\_\{B\},\\,\\mathcal\{C\}^\{\\prime\}\\setminus\\mathcal\{C\}\_\{B\},\\,\\delta\)⊳\\trianglerightEq\. \([5](https://arxiv.org/html/2605.15513#S3.E5)\)

8:endif

9:foreach unordered pair

\{ci,cj\}⊂𝒞B\\\{c\_\{i\},c\_\{j\}\\\}\\subset\\mathcal\{C\}\_\{B\}do⊳\\trianglerightStage C round\-robin

10:

\(vi​j,wi​j\)←J​\(q,ϕ2​\(ci\),ϕ2​\(cj\)\)\(v\_\{ij\},w\_\{ij\}\)\\leftarrow J\(q,\\,\\phi\_\{2\}\(c\_\{i\}\),\\,\\phi\_\{2\}\(c\_\{j\}\)\)
11:endfor

12:return

c⋆←arg⁡maxc∈𝒞B⁡sC​\(c\)c^\{\\star\}\\leftarrow\\arg\\max\_\{c\\in\\mathcal\{C\}\_\{B\}\}s\_\{C\}\(c\)via Eq\. \([4](https://arxiv.org/html/2605.15513#S3.E4)\)

Stage 0: Equivalence\-class deduplication\.We partition𝒞\\mathcal\{C\}via the equivalence relationci∼cj⇔ϕ0​\(ci\)=ϕ0​\(cj\)c\_\{i\}\\sim c\_\{j\}\\iff\\phi\_\{0\}\(c\_\{i\}\)=\\phi\_\{0\}\(c\_\{j\}\)and keep one representative per cluster, producing𝒞′\\mathcal\{C\}^\{\\prime\}of sizeN′≤NN^\{\\prime\}\\leq N\. Cluster sizesν​\(c\)\\nu\(c\)are recorded as metadata for tie\-breaking but*do not contribute toS​\(c\)S\(c\)*, preserving one\-solution\-one\-vote semantics: a popular\-but\-wrong cluster majority should not outvote a correct singleton\. Whenϕ0\\phi\_\{0\}is informative \(math benchmarks with closed\-form answers\),N′≪NN^\{\\prime\}\\ll N; on free\-form code,ϕ0\\phi\_\{0\}rarely distinguishes candidates and Stage 0 reduces to a no\-operation\.

Stage A: Halving at E1\.We pair theN′N^\{\\prime\}survivors using*slaughter pairing*Sziklaiet al\.\([2022](https://arxiv.org/html/2605.15513#bib.bib40)\): theii\-th and\(N′\+1−i\)\(N^\{\\prime\}\{\+\}1\{\-\}i\)\-th seeds, sorted byν\\nudescending; query the judge at evidence level E1, updateSSvia Eq\. \([3](https://arxiv.org/html/2605.15513#S3.E3)\), and retain the higher\-scoring member of each pair\. It maximizes the seed gap within each pair, making the cheap E1 judgment more reliable: large quality differentials are easier to discriminate from a partial view than small ones\. Stage A makes⌊N′/2⌋\\lfloor N^\{\\prime\}/2\\rfloorE1 calls, and its principal residual risk is mis\-elimination at E1; this is the failure mode that the rescue mechanism \(§[3\.3](https://arxiv.org/html/2605.15513#S3.SS3)\) addresses\.

Stage B: Reduction toffat E2\.Stage B halves the pool repeatedly at full evidence until the finalist countffis reached, with pairing seeded by the current cumulative scoreSSrather thanν\\nu\. For\|𝒞A\|≤2​f\|\\mathcal\{C\}\_\{A\}\|\\leq 2f\(the regimeN≤4​fN\\leq 4f, which covers our defaultN=16,f=4N=16,f=4\), Stage B is a single round of\|𝒞A\|/2=N′/4\|\\mathcal\{C\}\_\{A\}\|/2=N^\{\\prime\}/4E2 calls; for larger pools it iterates until\|𝒞cur\|=f\|\\mathcal\{C\}\_\{\\text\{cur\}\}\|=f\. Each round updatesSS, so Stage B’sSSvalues reflect the integrated signal from both E1 and E2 rounds\.

Stage C: Round\-robin at E2\.Thefffinalists are the candidates among which the argmax most plausibly lies\. Stage C concentrates the budget here: every unordered pair in𝒞B\\mathcal\{C\}\_\{B\}is judged once at E2, totaling\(f2\)\\binom\{f\}\{2\}comparisons\. Each finalist’s Stage\-C score is the confidence\-normalized win rate

sC​\(c\)=∑c′∈𝒞B∖\{c\}w​\(c,c′\)​v​\(c,c′\)∑c′∈𝒞B∖\{c\}w​\(c,c′\),s\_\{C\}\(c\)\\;=\\;\\frac\{\\sum\_\{c^\{\\prime\}\\in\\mathcal\{C\}\_\{B\}\\setminus\\\{c\\\}\}w\(c,c^\{\\prime\}\)\\,v\(c,c^\{\\prime\}\)\}\{\\sum\_\{c^\{\\prime\}\\in\\mathcal\{C\}\_\{B\}\\setminus\\\{c\\\}\}w\(c,c^\{\\prime\}\)\},\(4\)computed solely from Stage\-C calls; the pre\-Stage\-CSSenters only as a tiebreaker \(followed byν\\nu, then random\)\. The denominator normalizes by aggregate confidence sosCs\_\{C\}measures the proportion of confidence\-weighted wins rather than their unnormalized total\. We use round\-robin\(Shreedhar and Varghese,[1996](https://arxiv.org/html/2605.15513#bib.bib36); Fürnkranz,[2002](https://arxiv.org/html/2605.15513#bib.bib35)\)rather than a Swiss\(Csató,[2017](https://arxiv.org/html/2605.15513#bib.bib37)\)or bracket\(Tonon,[2004](https://arxiv.org/html/2605.15513#bib.bib38)\)because, on a small finalist pool \(f=4f=4\),\(f2\)\\binom\{f\}\{2\}is already affordable and round\-robin gives a path\-independent ranking signal: every finalist is compared against every other finalist at full evidence\. We returnc⋆=arg⁡maxc∈𝒞B⁡sC​\(c\)c^\{\\star\}=\\arg\\max\_\{c\\in\\mathcal\{C\}\_\{B\}\}s\_\{C\}\(c\)\.

### 3\.3CAPS\-R: Rescue of Mis\-Eliminated Candidates

Stage A operates at E1 and is the most error\-prone step\. CAPS\-R is an optional subroutine that admits one previously eliminated candidate into the finalist round when its elimination evidence appears weak\. Letc\+:=arg⁡maxc∈𝒞′∖𝒞B⁡S​\(c\)c^\{\+\}:=\\arg\\max\_\{c\\in\\mathcal\{C\}^\{\\prime\}\\setminus\\mathcal\{C\}\_\{B\}\}S\(c\)be the strongest eliminated candidate andcmin:=arg⁡minc∈𝒞B⁡S​\(c\)c\_\{\\min\}:=\\arg\\min\_\{c\\in\\mathcal\{C\}\_\{B\}\}S\(c\)the weakest finalist\. CAPS\-R triggers if either

\|S​\(c\+\)−S​\(cmin\)\|≤δ⏟margin: lost narrowlyorν​\(c\+\)=1​and​\|S​\(c\+\)−S​\(cmin\)\|≤2​δ⏟rarity: correct singleton,\\underbrace\{\|S\(c^\{\+\}\)\-S\(c\_\{\\min\}\)\|\\,\\leq\\,\\delta\}\_\{\\text\{margin: lost narrowly\}\}\\quad\\text\{or\}\\quad\\underbrace\{\\nu\(c^\{\+\}\)=1\\;\\text\{and\}\\;\|S\(c^\{\+\}\)\-S\(c\_\{\\min\}\)\|\\,\\leq\\,2\\delta\}\_\{\\text\{rarity: correct singleton\}\},\(5\)in which casec\+c^\{\+\}joins𝒞B\\mathcal\{C\}\_\{B\}and the round\-robin expands from\(f2\)\\binom\{f\}\{2\}to\(f\+12\)\\binom\{f\+1\}\{2\}E2 calls \(overheadf⋅T2f\\cdot T\_\{2\}\)\. We useδ=0\.15\\delta=0\.15\. The two conditions cover distinct failure modes:*margin*handles statistical noise on close pairs, while*rarity*handles correct singletons whose stylistic distinctiveness can mislead a low\-evidence judge; the relaxed threshold2​δ2\\deltafor the rarity condition reflects the higher prior of misranking under stylistic mismatch\.

### 3\.4Verifier\-Token Cost Analysis

LetTCAPS​\(N′,f\)T\_\{\\text\{CAPS\}\}\(N^\{\\prime\},f\)denote the total verifier\-token cost of the deterministic pipeline\. By construction of the four stages,

TCAPS​\(N′,f\)=⌊N′/2⌋​T1⏟Stage A\+\(∑r=1rB⌊Nr/2⌋\)​T2⏟Stage B\+\(f2\)​T2⏟Stage C,T\_\{\\text\{CAPS\}\}\(N^\{\\prime\},f\)\\;=\\;\\underbrace\{\\lfloor N^\{\\prime\}/2\\rfloor\\,T\_\{1\}\}\_\{\\text\{Stage A\}\}\\;\+\\;\\underbrace\{\\Bigl\(\\sum\_\{r=1\}^\{r\_\{B\}\}\\lfloor N\_\{r\}/2\\rfloor\\Bigr\)\\,T\_\{2\}\}\_\{\\text\{Stage B\}\}\\;\+\\;\\underbrace\{\\binom\{f\}\{2\}\\,T\_\{2\}\}\_\{\\text\{Stage C\}\},\(6\)whererB:=⌈log2⁡\(⌈N′/2⌉/f\)⌉r\_\{B\}:=\\lceil\\log\_\{2\}\(\\lceil N^\{\\prime\}/2\\rceil/f\)\\rceilis the number of Stage\-B halving rounds andNrN\_\{r\}the round\-rrentrant count\. Closing the Stage\-B geometric sum by telescoping \(Appendix[C\.2](https://arxiv.org/html/2605.15513#A3.SS2)\):

TCAPS​\(N′,f\)=N′2​\(T1\+T2\)−f​T2\+\(f2\)​T2\+O​\(log⁡N′\)\.T\_\{\\text\{CAPS\}\}\(N^\{\\prime\},f\)\\;=\\;\\frac\{N^\{\\prime\}\}\{2\}\(T\_\{1\}\+T\_\{2\}\)\\,\-\\,f\\,T\_\{2\}\\,\+\\,\\binom\{f\}\{2\}\\,T\_\{2\}\\,\+\\,O\(\\log N^\{\\prime\}\)\.\(7\)The dominant term is linear inN′N^\{\\prime\}with slope12​\(T1\+T2\)\\tfrac\{1\}\{2\}\(T\_\{1\}\+T\_\{2\}\):*the marginal cost of an additional candidate is half a Stage\-A E1 call plus half a Stage\-B E2 call*, rather than the fullT2T\_\{2\}that a uniform full\-evidence schedule would incur\. This replacement ofT2T\_\{2\}by12​\(T1\+T2\)\\tfrac\{1\}\{2\}\(T\_\{1\}\+T\_\{2\}\)in the per\-candidate marginal cost is the structural source of CAPS’s efficiency\.

For the standard configurationN=N′=16,f=4N=N^\{\\prime\}=16,f=4\(the worst case for CAPS, where Stage 0 yields no savings\), Eq\. \([6](https://arxiv.org/html/2605.15513#S3.E6)\) givesrB=1r\_\{B\}=1,N1=8N\_\{1\}=8, and

TCAPS​\(16,4\)=8​T1\+4​T2\+6​T2=8​T1\+10​T2=\(10\+8​ρ\)​T2\.T\_\{\\text\{CAPS\}\}\(16,4\)\\;=\\;8\\,T\_\{1\}\+4\\,T\_\{2\}\+6\\,T\_\{2\}\\;=\\;8\\,T\_\{1\}\+10\\,T\_\{2\}\\;=\\;\(10\+8\\rho\)\\,T\_\{2\}\.\(8\)With empirically measuredρ∈\[0\.10,0\.15\]\\rho\\in\[0\.10,0\.15\]\(Appendix[B\.2\.1](https://arxiv.org/html/2605.15513#A2.SS2.SSS1)\), this evaluates to roughly11​T211\\,T\_\{2\}: the entire pipeline costs approximately1111full\-evidence comparison\-equivalents per problem\. The cost is determined byNN,ff, andρ\\rho, all of which are properties of the deployment rather than free parameters\.

## 4Experiments

We evaluate CAPS against three baselines on five reasoning benchmarks across four self\-verifying models\. Our goal is to measure \(i\) selection quality of the cascaded pipeline relative to existing pairwise verifiers, \(ii\) the verifier\-token cost reduction afforded by progressive evidence, and \(iii\) the contributions of individual CAPS components\.

### 4\.1Experimental Settings

Models and Benchmarks\.We evaluate four models that span two families and two scale regimes:Qwen3\-14B\(Yanget al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib41)\),GPT\-OSS\-20B\(Agarwalet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib42)\),Qwen3\-4B\-Instruct\-2507\(Yanget al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib41)\), andQwen3\-4B\-Thinking\-2507\(Yanget al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib41)\)\. For code generation we useLiveCodeBench\-v5\(Jainet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib43)\)\(279 problems, 24\.08–25\.02\),LiveCodeBench\-v6\(Jainet al\.,[2024](https://arxiv.org/html/2605.15513#bib.bib43)\)\(131 problems, 25\.02–25\.05\), andCodeContests\(Liet al\.,[2022](https://arxiv.org/html/2605.15513#bib.bib6)\)\(165 problems\)\. For math reasoning we useAIME 2025\(30 problems\) andHMMT February 2025\(30 problems\)\(Balunovićet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib44)\)\. We sampleN=16N=16candidates per problem with temperature0\.60\.6for code and1\.01\.0for math \(top\-p=0\.95p=0\.95throughout\); the same model serves as both generator and pairwise judge\. Generation prompts, full hyperparameters, and dataset sources are reproduced in Appendix[E](https://arxiv.org/html/2605.15513#A5)\.

Table 1:Main results: Pass@1 \(%\)\.Selection accuracy across four methods, four models, and five benchmarks atN=16N=16, totaling2020\(model, benchmark\) suites\.Bold: best method per \(model, benchmark\) row\. AIME 2025 and HMMT 2025 each contain3030problems, so a single problem flip corresponds to3\.33\.3percentage points\.Methods Compared\.We compare four selection strategies on the same candidate pools:

- •Vanilla: first\-sampled candidate, no verification \(the Pass@1 baseline of unaided sampling\)\.
- •Pointwise: each candidate receives an absolute11–1010rating from the same model in isolation; the highest\-rated candidate is selected, ties broken at random\(Venkatramanet al\.,[2025](https://arxiv.org/html/2605.15513#bib.bib34)\)\.
- •V1\-Infer\(Singhet al\.,[2026](https://arxiv.org/html/2605.15513#bib.bib15)\)at the3×3\\timesbudget multiplier, the strongest pairwise verification configuration in the public literature: an uncertainty\-guided Swiss tournament with∼48\\sim 48pairwise comparisons atN=16N=16, all at full evidence\.
- •CAPS\(ours\) withf=4f=4finalists, rescue marginδ=0\.15\\delta=0\.15, weight floorτw=0\.05\\tau\_\{w\}=0\.05\.

All four methods receive the same input candidate pool\. CAPS prompts return categorical Winner/Tie verdicts with HIGH/LOW confidence \(Appendix[E\.1](https://arxiv.org/html/2605.15513#A5.SS1)\); for compatibility with V1\-Infer’s confidence\-weighted aggregation, the verdicts are mapped to11–1010ratings via Appendix Table[14](https://arxiv.org/html/2605.15513#A5.T14)\. Pointwise and V1\-Infer use the prompt templates ofSinghet al\.\([2026](https://arxiv.org/html/2605.15513#bib.bib15)\), reproduced in Appendix[E\.1](https://arxiv.org/html/2605.15513#A5.SS1)for completeness\. Random pair scheduling at matched comparison count, additional ablations, and budget\-matched comparisons are deferred to Appendix[D](https://arxiv.org/html/2605.15513#A4)\.

Metrics\.We reportPass@1\(selection accuracy under self\-verification\) andverifier\-token cost: the total input tokens consumed by judge calls, summed across the benchmark\. We do not include generation tokens since all four methods operate on a shared candidate pool\. Evaluation criteria with all\-tests\-pass for code, SymPy symbolic equivalence for math, are detailed in Appendix[E\.5](https://arxiv.org/html/2605.15513#A5.SS5)\.

### 4\.2Main Results

Table[1](https://arxiv.org/html/2605.15513#S4.T1)reports Pass@1Chenet al\.\([2021](https://arxiv.org/html/2605.15513#bib.bib46)\)across all selection methods on the five benchmarks for the full2020suites\. For a fair comparison, we repeated all experiments three times with the same random seeds and report the average results\.

CAPS is the strongest method on average across the suites\.The average Pass@1 across all2020measured suites is48\.2%48\.2\\%for CAPS,46\.5%46\.5\\%for V1\-Infer,40\.6%40\.6\\%for Pointwise, and37\.7%37\.7\\%for Vanilla\. The mean improvement of CAPS over V1\-Infer is\+1\.7\+1\.7points, over Pointwise is\+7\.6\+7\.6points, and over Vanilla is\+10\.5\+10\.5points\. The two methods that operate on pairwise comparisons \(V1\-Infer and CAPS\) both outperform Pointwise by a wide margin; CAPS further improves over V1\-Infer by allocating evidence and comparisons non\-uniformly\.

Largest gains appear on harder benchmarks\.On code benchmarks, the gains over V1\-Infer concentrate where the candidate pool is most diverse and the Pass@1 baseline lowest: Qwen3\-4B\-Instruct on CodeContests improves from7\.9%7\.9\\%\(V1\) to14\.5%14\.5\\%\(CAPS\), an84%84\\%relative gain; Qwen3\-4B\-Instruct on LCB\-v5 gains\+6\.4\+6\.4points, and Qwen3\-4B\-Thinking on LCB\-v5 gains\+6\.4\+6\.4\. On math, CAPS dominates V1\-Infer on44of88suites: Qwen3\-14B AIME \(\+3\.3\+3\.3\), Qwen3\-4B\-Thinking AIME and HMMT \(\+3\.3\+3\.3each\), and GPT\-OSS\-20B AIME \(\+3\.3\+3\.3\)\.

Trade\-off suites are narrow\.Six suites favor V1\-Infer over CAPS, all within1\.51\.5–3\.43\.4Pass@1 points\. The four math regressions \(Qwen3\-14B HMMT, Qwen3\-4B\-Instruct AIME and HMMT, GPT\-OSS\-20B HMMT\) each correspond to a single problem flip on the3030\-problem benchmarks, and CAPS remains above Pointwise and Vanilla in all four cases\. The two GPT\-OSS\-20B code regressions \(LCB\-v6, CodeContests\) lose by1\.81\.8and1\.51\.5points respectively, while still using only2828–34%34\\%of V1\-Infer’s verifier\-token budget \(Left of Figure[2](https://arxiv.org/html/2605.15513#S4.F2)\)\.

Table 2:Per\-pair verifier accuracy \(%\) under V1\-Infer \(full evidence\) vs\. CAPS \(mixed E1/E2 evidence\)on representative \(model, benchmark\) suites\.Δ\\Deltais the accuracy change under partial\-evidence judgments\. The sign ofΔ\\Deltapredicts whether CAPS dominates V1\-Infer in Pass@1 \(Table[1](https://arxiv.org/html/2605.15513#S4.T1)\):Δ≥0\\Delta\\geq 0implies CAPS wins;Δ≲−5\\Delta\\lesssim\-5implies regression\.When does CAPS dominate? A verifier\-accuracy diagnostic\.The trade\-off suites admit a clean explanation in terms of a measurable quantity: the verifier’s per\-pair accuracy at partial vs\. full evidence\. Table[2](https://arxiv.org/html/2605.15513#S4.T2)reports per\-pair verifier accuracy \(the fraction of judge calls whose chosen winner is the ground\-truth\-correct candidate, computed over pairs distinguishable by ground truth\) on representative suites\. The pattern is consistent: when CAPS’s mixed E1/E2 accuracy is comparable to or higher than V1\-Infer’s all\-E2 accuracy \(Δ≥0\\Delta\\geq 0\), CAPS dominates V1\-Infer in Pass@1; when partial\-evidence accuracy degrades sharply:Δ≲−5\\Delta\\lesssim\-5percentage point \(pp\), CAPS regresses\. The Qwen3\-4B\-Instruct math suites lose∼9\\sim 9pp of verifier accuracy under partial evidence because the Instruct model relies on the full reasoning chain to discriminate between math candidates, while the Qwen3\-4B\-Thinking model maintains accuracy on math because its boxed answer plus terminal reasoning window already encodes the algorithmic decision\. GPT\-OSS\-20B exhibits a∼4\.5\\sim 4\.5pp accuracy drop on code\. The full table across all evaluated suites is provided in Appendix[D\.3](https://arxiv.org/html/2605.15513#A4.SS3)\. This diagnostic suggests a deployment\-time check for cascade suitability: practitioners can compare the verifier’s accuracy underϕ1\\phi\_\{1\}vs\.ϕ2\\phi\_\{2\}on a small held\-out set and predict whether CAPS will dominate V1\-Infer for their model\.

Verifier\-Token CostLeft of Figure[2](https://arxiv.org/html/2605.15513#S4.F2)shows the verifier\-token cost of CAPS relative to V1\-Infer\. On code benchmarks, CAPS uses on average25\.4%25\.4\\%of V1\-Infer’s verifier\-token budget across1212suites\. On math benchmarks, CAPS averages50\.4%50\.4\\%across88suites \(a2\.0×2\.0\\timesreduction\); the higher ratio reflects the smaller absoluteT2T\_\{2\}on math problems against which Stage A’sϕ1\\phi\_\{1\}overhead is amortized\. The empirical code\-benchmark mean closely matches the closed\-form prediction of §[3\.4](https://arxiv.org/html/2605.15513#S3.SS4): withρ=T1/T2≈0\.12\\rho=T\_\{1\}/T\_\{2\}\\approx 0\.12, Eq\. \([8](https://arxiv.org/html/2605.15513#S3.E8)\) predictsT%≈22​–​23%T\\%\\approx 22\\text\{\-\-\}23\\%for theN=16,f=4N=16,f=4configuration\.

![Refer to caption](https://arxiv.org/html/2605.15513v1/x2.png)Figure 2:On the left: Pass@1 \(%\), selection accuracy across five benchmarks with pointwise, V1\-infer and CAPS; On the right: and Verifier\-token cost between V1\-infer and CAPS\.The empiricalT%T\\%matches the closed\-form prediction\.The meanT%T\\%over the twelve code\-benchmark suites is25\.4%25\.4\\%, within33percentage points of the analytical value\(10\+8​ρ\)/48≈22​–​23%\(10\+8\\rho\)/48\\approx 22\\text\{\-\-\}23\\%from Eq\. \([8](https://arxiv.org/html/2605.15513#S3.E8)\)\. On math benchmarks,T%T\\%is higher \(4242–70%70\\%\) because the pipeline performs the same number of judge calls but each call’sϕ2\\phi\_\{2\}view is shorter on average: the\(f2\)=6\\binom\{f\}\{2\}=6Stage C comparisons are amortized over a smaller per\-callT2T\_\{2\}, while Stage A’sϕ1\\phi\_\{1\}calls retain a similar size, raising the empiricalρ\\rho\. The cost reduction in raw token terms \(1\.41\.4–2\.4×2\.4\\times\) remains substantial\.

### 4\.3Analysis and Ablations

Performance gains across difficulty levels\.Following the analysis protocol ofSinghet al\.\([2026](https://arxiv.org/html/2605.15513#bib.bib15)\), we stratify LCB\-v6 problems by their official difficulty label and report Pass@1 for each method on Qwen3\-14B in Table[3](https://arxiv.org/html/2605.15513#S4.T3)\. The gap between Vanilla and the verification\-based methods widens with difficulty: on easy problems all methods achieve7070–83%83\\%, but on hard problems Vanilla is at20\.3%20\.3\\%while CAPS reaches23\.8%23\.8\\%, with V1\-Infer in between\. CAPS matches V1\-Infer on easy problems and edges ahead on medium and hard problems despite using a fraction of the verifier\-token budget\.

Table 3:Difficulty stratificationon LCB\-v6, Qwen3\-14B,N=16N=16\. Pass@1 \(%\) by official benchmark difficulty label\. The gap over Vanilla widens with difficulty, where verification matters most\.Random pair scheduling\.To isolate the contribution of*informative*pair selection \(whether through V1\-Infer’s uncertainty\-guided Swiss tournament or CAPS’s cascaded slaughter pairing\) from mere candidate filtering, we evaluate a Random baseline that performs4848pairwise comparisons at full evidence over uniformly random pairs and aggregates them with the same confidence\-weighted scheme as V1\-Infer\. Table[4](https://arxiv.org/html/2605.15513#S4.T4)reports Random alongside Vanilla and the verification methods on three \(model, benchmark\) suites\. Random selection at full evidence achieves Pass@1 close to \(and sometimes below\) Vanilla itself: with no informative pair\-selection signal, the comparisons it performs are noise that occasionally flips the selection toward a weaker candidate\. The substantial gap between Random and V1\-Infer/CAPS confirms that the gains in Table[1](https://arxiv.org/html/2605.15513#S4.T1)arise from the structure of pair selection, not from the verification interface alone\.

Table 4:Random pair schedulingat matched comparison count to V1\-Infer \(∼48\\sim 48pairs atN=16N=16\), full evidenceϕ2\\phi\_\{2\}\. Random is close to or below Vanilla, indicating that informative pair selection \(not just running pairwise judge calls\) is what produces the gains in Table[1](https://arxiv.org/html/2605.15513#S4.T1)\. Full table in Appendix[D\.2](https://arxiv.org/html/2605.15513#A4.SS2)\.Scaling in candidate countNN\.The closed\-form cost analysis of §[3\.4](https://arxiv.org/html/2605.15513#S3.SS4)predicts that the cost ratioT%T\\%approaches the asymptotic limit\(1\+ρ\)/\(2​k\)≈0\.19\(1\+\\rho\)/\(2k\)\\approx 0\.19asNNgrows \(Eq\. \([7](https://arxiv.org/html/2605.15513#S3.E7)\)\), since the fixed Stage C round\-robin cost is amortized over a larger candidate pool\. Table[5](https://arxiv.org/html/2605.15513#S4.T5)reports Pass@1 andT%T\\%forN∈\{8,16,32\}N\\in\\\{8,16,32\\\}on Qwen3\-14B / LCB\-v6\. The empiricalT%T\\%tracks the closed\-form prediction closely atN=16N=16andN=32N=32\(within33pp\); the larger gap atN=8N=8reflects per\-call overhead \(system prompt, problem statement\) that the closed\-form abstracts away and is amortized poorly when so few comparisons are made\. The Pass@1 advantage over V1\-Infer grows monotonically withNN\(\+0\.0\+0\.0,\+0\.8\+0\.8,\+2\.3\+2\.3\), indicating that CAPS’s structural efficiency advantage compounds as the candidate pool grows\.

Table 5:Scaling inNNon Qwen3\-14B / LCB\-v6 withf=4f=4\. EmpiricalT%T\\%tracks the closed\-form prediction within33pp atN≥16N\\geq 16; the larger gap atN=8N=8reflects per\-call overhead amortized poorly at small candidate counts\. CAPS’s Pass@1 advantage over V1\-Infer grows withNN\.Component ablation\.Table[6](https://arxiv.org/html/2605.15513#S4.T6)disables each component of CAPS one at a time on Qwen3\-14B\. Removing the rescue subroutine \(CAPS\-R\) costs−2\.3\-2\.3Pass@1 on LCB\-v6 and−0\.6\-0\.6on CodeContests; the trigger fires on∼10\\sim 10–15%15\\%of instances\. Disabling Stage 0 deduplication has minimal effect on code benchmarks \(whereϕ0\\phi\_\{0\}rarely distinguishes candidates\) but a larger effect on math \(deferred to Appendix[D](https://arxiv.org/html/2605.15513#A4)\)\. Replacing the evidence cascade with all\-E2 evaluation removes the cost advantage entirely without proportional quality gain\. Replacing slaughter pairing with random pairing in Stage A reduces Stage\-A elimination accuracy and propagates errors into the finalist set\.

Table 6:Component ablation on Qwen3\-14B\.Δ\\Delta: change in Pass@1 relative to full CAPS\. Each row reports Pass@1 when an certain component is disabled\.The E1 cascade row clarifies the role of partial evidence: running all stages at full evidence gives a slightly higher Pass@1 \(\+0\.6\+0\.6\) but at∼4×\\sim 4\\timesthe verifier\-token cost, eliminating the efficiency advantage that motivates CAPS in the first place\. The cascade is therefore best understood as a cost mechanism: it preserves selection quality \(within11point on these benchmarks\) at a fraction of the budget\. The largest single Pass@1 contribution comes from slaughter pairing \(−2\.9\-2\.9on LCB\-v6,−2\.1\-2\.1on CodeContests\), confirming that the seeding logic in Stage A is materially helpful at the cheapest evidence level\. The full ablation grid across all four models is reported in Appendix[D\.6](https://arxiv.org/html/2605.15513#A4.SS6), and per\-stage prompt and evidence specifications are given in Appendices[E\.1](https://arxiv.org/html/2605.15513#A5.SS1)–[E\.2](https://arxiv.org/html/2605.15513#A5.SS2)\.

Qualitative analysis: why CAPS reaches V1\-Infer at lower cost\.We examine LCB\-v6 problems where CAPS and V1\-Infer select different solutions \(Qwen3\-14B; full examples in Appendix[D\.7](https://arxiv.org/html/2605.15513#A4.SS7)\)\. A recurring pattern: V1\-Infer’s full Swiss tournament spreads its∼48\\sim 48comparisons across all1616candidates, with most comparisons spent disambiguating between candidates that the model would never select as the final answer\. CAPS uses cheap E1 calls to filter such candidates and concentrates its66E2 finalist comparisons on the candidates that genuinely contend for the top\. The resulting selection is identical to V1\-Infer’s on the majority of problems, with occasional gains on problems where the rescue mechanism recovers a strong candidate from a noisy E1 round\.

## 5Conclusion

In this work, we introduceCAPS, an inference\-only framework for pairwise self\-verification that allocates verifier compute non\-uniformly along two orthogonal axes: an*evidence axis*that adapts how much of each candidate the judge sees, and a*distribution axis*that concentrates full\-evidence comparisons on the strongest candidates; through a four\-stage cascade with an optional rescue subroutine\. We derive a closed\-form characterization of CAPS’s verifier\-token cost showing that the per\-candidate marginal cost is roughly halved relative to uniform full\-evidence schedules, and our experiments confirm that the empirical efficiency is structural rather than a hyperparameter effect\. Across four self\-verifying models and five reasoning benchmarks, CAPS outperforms V1\-Infer at3×3\\timesbudget on1414of2020settings while using a fraction of its verifier\-token cost\.

Limitations\.CAPS does not achieve state\-of\-the\-art selection on every suites of our evaluation: V1\-Infer outperforms CAPS on66of2020settings, primarily on math benchmarks where the verifier relies on the full reasoning chain to discriminate between candidates and on GPT\-OSS\-20B code where partial\-view accuracy degrades modestly\.

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## Appendix AExtended Related Work

This appendix expands on the related\-work overview from Section[2](https://arxiv.org/html/2605.15513#S2)of the main text, providing additional context for parallel reasoning, test\-time scaling, self\-verification, and self\-aggregation\.

##### Parallel reasoning\.

Parallel reasoning in LLMs generates multiple independent solution attempts to a problem and selects or combines among them, in contrast to sequential approaches that produce a single chain of thought, possibly with refinement\[[47](https://arxiv.org/html/2605.15513#bib.bib1),[44](https://arxiv.org/html/2605.15513#bib.bib2),[17](https://arxiv.org/html/2605.15513#bib.bib3),[37](https://arxiv.org/html/2605.15513#bib.bib4),[15](https://arxiv.org/html/2605.15513#bib.bib58),[24](https://arxiv.org/html/2605.15513#bib.bib52),[16](https://arxiv.org/html/2605.15513#bib.bib56),[30](https://arxiv.org/html/2605.15513#bib.bib57)\]\. The basic premise dates to early ensemble decoding\[[22](https://arxiv.org/html/2605.15513#bib.bib6),[5](https://arxiv.org/html/2605.15513#bib.bib32),[45](https://arxiv.org/html/2605.15513#bib.bib5),[50](https://arxiv.org/html/2605.15513#bib.bib54),[54](https://arxiv.org/html/2605.15513#bib.bib53)\]and has been refined through methods that exploit the diversity of independent samples to cover a larger fraction of the problem’s solution space than any single chain can\[[3](https://arxiv.org/html/2605.15513#bib.bib7),[46](https://arxiv.org/html/2605.15513#bib.bib8),[55](https://arxiv.org/html/2605.15513#bib.bib55)\]\. Self\-Consistency\[[5](https://arxiv.org/html/2605.15513#bib.bib32),[43](https://arxiv.org/html/2605.15513#bib.bib9),[40](https://arxiv.org/html/2605.15513#bib.bib10)\]demonstrated that majority voting over independently sampled chains substantially improves arithmetic reasoning\. Best\-of\-NNsampling with learned reward models\[[23](https://arxiv.org/html/2605.15513#bib.bib11),[13](https://arxiv.org/html/2605.15513#bib.bib12),[6](https://arxiv.org/html/2605.15513#bib.bib13),[21](https://arxiv.org/html/2605.15513#bib.bib47)\]extends this to settings without an objective ground truth answer\. More recent work has shown that the gap between Pass@1 and Pass@NN, the probability that the correct answer exists somewhere in the candidate pool, remains large for hard problems even at moderateNN\[[3](https://arxiv.org/html/2605.15513#bib.bib7),[36](https://arxiv.org/html/2605.15513#bib.bib14),[35](https://arxiv.org/html/2605.15513#bib.bib15)\], motivating the development of stronger selection mechanisms that close this gap\.

##### Test\-time scaling\.

A complementary line of work treats inference\-time compute as a continuous axis along which model performance can be improved without changing model weights\[[51](https://arxiv.org/html/2605.15513#bib.bib16),[37](https://arxiv.org/html/2605.15513#bib.bib4),[25](https://arxiv.org/html/2605.15513#bib.bib50),[19](https://arxiv.org/html/2605.15513#bib.bib51)\]\. Two families have emerged:*sequential scaling*, in which a model performs longer chains of reasoning, reflection, or revision\[[53](https://arxiv.org/html/2605.15513#bib.bib17),[52](https://arxiv.org/html/2605.15513#bib.bib18),[27](https://arxiv.org/html/2605.15513#bib.bib19),[32](https://arxiv.org/html/2605.15513#bib.bib20),[9](https://arxiv.org/html/2605.15513#bib.bib49)\], and*parallel scaling*, in which multiple independent attempts are aggregated\[[7](https://arxiv.org/html/2605.15513#bib.bib21),[29](https://arxiv.org/html/2605.15513#bib.bib22),[33](https://arxiv.org/html/2605.15513#bib.bib23),[20](https://arxiv.org/html/2605.15513#bib.bib48)\]\. The two are largely orthogonal and can be combined\[[36](https://arxiv.org/html/2605.15513#bib.bib14),[3](https://arxiv.org/html/2605.15513#bib.bib7)\]\. Empirical scaling laws relate test\-time compute to performance, and Snell et al\.\[[36](https://arxiv.org/html/2605.15513#bib.bib14)\]showed that, for a fixed compute budget, the optimal allocation between train\-time and test\-time compute is task\-dependent and often favors the latter for hard problems\. Within parallel scaling, the principal compute budget consists of two components: the cost of generation \(samplingNNcandidates\) and the cost of selection \(verifying them\)\. The relative weight of the two has shifted as generation has become cheaper through speculative decoding, batch inference, and small\-model speedups, with verification cost now frequently dominating the test\-time budget for verifiable\-reward settings such as code and math\.

##### Self\-verification\.

Self\-verification methods use the same model that generated the candidates to assess their correctness, avoiding the need for a separately trained reward model or external verifier\[[48](https://arxiv.org/html/2605.15513#bib.bib24),[11](https://arxiv.org/html/2605.15513#bib.bib25),[38](https://arxiv.org/html/2605.15513#bib.bib26)\]\. Early work used pointwise scoring, prompting the model to assign an absolute quality score to each candidate\[[48](https://arxiv.org/html/2605.15513#bib.bib24),[58](https://arxiv.org/html/2605.15513#bib.bib27)\]\. Recent studies have systematic failure modes of pointwise self\-verification: scores lack a globally comparable scale\[[38](https://arxiv.org/html/2605.15513#bib.bib26),[28](https://arxiv.org/html/2605.15513#bib.bib30),[42](https://arxiv.org/html/2605.15513#bib.bib34)\], models exhibit a self\-preference bias toward their own samples even when those samples are incorrect\[[31](https://arxiv.org/html/2605.15513#bib.bib28),[57](https://arxiv.org/html/2605.15513#bib.bib29)\], and verifiers saturate at the top of the scale, assigning near\-maximum scores to most candidates and losing discriminative power\. The current state of the art is V1\-Infer\[[35](https://arxiv.org/html/2605.15513#bib.bib15)\], which replaces pointwise scoring with*pairwise*comparison: candidates are compared head\-to\-head, and an uncertainty\-guided Swiss\-system tournament dynamically allocates the comparison budget to the most ambiguous pairs\. V1\-Infer reports up to\+10%\+10\\%Pass@1 over pointwise verification on code and math benchmarks, and the same paper introduces V1\-PairRL, an RL framework that co\-trains a single model as both generator and pairwise verifier so that the verifier remains in\-distribution as the generator evolves\. Pairwise RM\[[26](https://arxiv.org/html/2605.15513#bib.bib31)\]similarly trains a dedicated pairwise reward model and applies a knockout tournament with answer\-based grouping for Best\-of\-NNselection \.

##### Self\-aggregation\.

Self\-aggregation methods take a different approach to using a candidate pool: rather than selecting one of the existing candidates, they prompt the model to combine or refine the pool into a new solution\[[28](https://arxiv.org/html/2605.15513#bib.bib30),[42](https://arxiv.org/html/2605.15513#bib.bib34)\]\. Majority voting\[[5](https://arxiv.org/html/2605.15513#bib.bib32)\]is the simplest instance, applicable when the answer admits an exact\-match comparison\. Beyond majority, recent work has explored learned aggregation: AggLM\[[56](https://arxiv.org/html/2605.15513#bib.bib33)\]trains an aggregator via reinforcement learning to synthesize a final answer from a pool, and Recursive Self\-Aggregation \(RSA\)\[[42](https://arxiv.org/html/2605.15513#bib.bib34)\]iteratively consolidates groups ofkkcandidates into single solutions over multiple rounds, producing an evolutionary refinement loop\. RSA and similar methods improve Pass@1 in many settings but exhibit*diversity collapse*: Pass@NNmonotonically decreases with aggregation steps as correct outlier solutions are absorbed into majority\-aligned refinements\[[42](https://arxiv.org/html/2605.15513#bib.bib34),[35](https://arxiv.org/html/2605.15513#bib.bib15)\]\. This has motivated hybrid designs that combine aggregation with verification, using the latter as a fitness signal that preserves Pass@NNacross aggregation rounds\. The two paradigms, selection\-based and aggregation\-based, are largely orthogonal and can be combined when verification is reliable\.

## Appendix BMethod: Helper Procedures and Implementation Details

This appendix specifies the helper procedures invoked in Algorithm[1](https://arxiv.org/html/2605.15513#alg1)of the main text—Dedup,Slaughter,Eliminate, andRescue—together with the implementation details and design rationale deferred from §[3](https://arxiv.org/html/2605.15513#S3)\. Section[B\.1](https://arxiv.org/html/2605.15513#A2.SS1)gives the full pseudocode; Section[E](https://arxiv.org/html/2605.15513#A5)covers token\-cost decomposition, representative selection, and other implementation choices; Section[B\.3](https://arxiv.org/html/2605.15513#A2.SS3)presents the design rationale for the evidence cascade\.

### B\.1Helper Procedures

##### Shared state\.

We maintain a state𝒯=\(\{S​\(c\)\}c∈𝒞′,\{ν​\(c\)\}c∈𝒞′\)\\mathcal\{T\}=\(\\\{S\(c\)\\\}\_\{c\\in\\mathcal\{C\}^\{\\prime\}\},\\,\\\{\\nu\(c\)\\\}\_\{c\\in\\mathcal\{C\}^\{\\prime\}\}\)whereS​\(c\)S\(c\)is the cumulative confidence\-weighted score of candidatecc\(initialized to0in Algorithm[1](https://arxiv.org/html/2605.15513#alg1)of the main text\) andν​\(c\)\\nu\(c\)is the cluster size from Stage 0\. Score updates use Eq\. \([3](https://arxiv.org/html/2605.15513#S3.E3)\); we abbreviate this asUpdateScore​\(𝒯,i,j,vi​j,wi​j\)\\textsc\{UpdateScore\}\(\\mathcal\{T\},i,j,v\_\{ij\},w\_\{ij\}\)in the procedures below\.

#### B\.1\.1Dedup: Stage 0

Deduppartitions the candidate pool by their E0 signatures and returns one representative per cluster\. Cluster sizes are recorded as metadata inν​\(⋅\)\\nu\(\\cdot\)but, by design, do not contribute to the comparison scoreS​\(⋅\)S\(\\cdot\)\.

Algorithm 2Dedup\(𝒞,ϕ0\)\(\\mathcal\{C\},\\phi\_\{0\}\)— Equivalence\-class deduplication1:candidate set

𝒞=\{c1,…,cN\}\\mathcal\{C\}=\\\{c\_\{1\},\\ldots,c\_\{N\}\\\}, signature map

ϕ0\\phi\_\{0\}
2:deduplicated pool

𝒞′\\mathcal\{C\}^\{\\prime\}, cluster sizes

ν​\(⋅\)\\nu\(\\cdot\)
3:Compute

σi←ϕ0​\(ci\)\\sigma\_\{i\}\\leftarrow\\phi\_\{0\}\(c\_\{i\}\)for

i=1,…,Ni=1,\\ldots,N
4:Partition

𝒞\\mathcal\{C\}into clusters

\{K1,…,KM\}\\\{K\_\{1\},\\ldots,K\_\{M\}\\\}where

Km=\{ci:σi=σm\}K\_\{m\}=\\\{c\_\{i\}:\\sigma\_\{i\}=\\sigma\_\{m\}\\\}
5:

𝒞′←∅\\mathcal\{C\}^\{\\prime\}\\leftarrow\\emptyset
6:for

m=1,…,Mm=1,\\ldots,Mdo

7:

rm←arg⁡maxc∈Km⁡\|c\|r\_\{m\}\\leftarrow\\arg\\max\_\{c\\in K\_\{m\}\}\|c\|⊳\\trianglerightrepresentative: longest solution by token count

8:

ν​\(rm\)←\|Km\|\\nu\(r\_\{m\}\)\\leftarrow\|K\_\{m\}\|
9:

𝒞′←𝒞′∪\{rm\}\\mathcal\{C\}^\{\\prime\}\\leftarrow\\mathcal\{C\}^\{\\prime\}\\cup\\\{r\_\{m\}\\\}
10:endfor

11:return

𝒞′,ν​\(⋅\)\\mathcal\{C\}^\{\\prime\},\\,\\nu\(\\cdot\)

The longest\-solution heuristic for representative selection prefers more complete reasoning chains and avoids selecting truncated outputs as cluster representatives\. Section[B\.2\.2](https://arxiv.org/html/2605.15513#A2.SS2.SSS2)discusses the sensitivity of CAPS to this choice; in our experiments, alternative tiebreakers \(first\-sampled, shortest, random\) yield within\-noise differences in Pass@1\.

#### B\.1\.2Slaughter: Pairing rule

Slaughterproduces disjoint pairs by matching the strongest seeds against the weakest\. Given an ordered list of candidates and a seeding key, the pairing is fully determined\.

Algorithm 3Slaughter\(𝒳,key\)\(\\mathcal\{X\},\\mathrm\{key\}\)— Slaughter pairing1:candidate set

𝒳\\mathcal\{X\}, seeding key

key:𝒳→ℝ\\mathrm\{key\}:\\mathcal\{X\}\\to\\mathbb\{R\}
2:disjoint pair set

𝒫\\mathcal\{P\}, optional bye candidate

bb
3:

π←\\pi\\leftarrowsort

𝒳\\mathcal\{X\}in descending order of

key​\(⋅\)\\mathrm\{key\}\(\\cdot\), breaking ties uniformly at random

4:

n←\|𝒳\|n\\leftarrow\|\\mathcal\{X\}\|;

𝒫←∅\\mathcal\{P\}\\leftarrow\\emptyset
5:for

i=1,…,⌊n/2⌋i=1,\\ldots,\\lfloor n/2\\rfloordo

6:

𝒫←𝒫∪\{\(πi,πn\+1−i\)\}\\mathcal\{P\}\\leftarrow\\mathcal\{P\}\\cup\\\{\(\\pi\_\{i\},\\pi\_\{n\+1\-i\}\)\\\}
7:endfor

8:if

nnis oddthen

9:

b←π⌈n/2⌉b\\leftarrow\\pi\_\{\\lceil n/2\\rceil\}⊳\\trianglerightbye candidate skips this round

10:else

11:

b←⊥b\\leftarrow\\bot
12:endif

13:return

𝒫,b\\mathcal\{P\},\\,b

Slaughter pairing maximizes seed dispersion within each pair: the highest\-seeded candidate is matched against the lowest\-seeded, the second\-highest against the second\-lowest, and so on\. Large quality differentials are easier to discriminate from a partial view than small ones, so pairs with maximal seed gap admit the most reliable cheap\-evidence judgments\. This is what makes Stage A’s E1 calls effective\.

#### B\.1\.3Eliminate: Generic elimination round

Eliminateis the workhorse helper used by both Stage A \(single round,e=1e=1, seeded byν\\nu\) and Stage B \(repeated rounds,e=2e=2, seeded bySS, stopping atff\)\. The unified procedure factors out the common structure: pair, judge, update scores, retain winners, repeat if needed\.

Algorithm 4Eliminate\(𝒳,J,e,key,𝒯,\[stop​\_​at=f\]\)\(\\mathcal\{X\},J,e,\\mathrm\{key\},\\mathcal\{T\},\[\\mathrm\{stop\\\_at\}=f\]\)— Halving\-elimination round\(s\)1:candidate pool

𝒳\\mathcal\{X\}, judge

JJ, evidence level

e∈\{1,2\}e\\in\\\{1,2\\\}, seeding key

key\\mathrm\{key\}, state

𝒯\\mathcal\{T\}, optional stop count

ff
2:surviving candidate pool

𝒳out\\mathcal\{X\}\_\{\\text\{out\}\}
3:

𝒳cur←𝒳\\mathcal\{X\}\_\{\\text\{cur\}\}\\leftarrow\\mathcal\{X\}
4:repeat

5:

𝒫,b←Slaughter​\(𝒳cur,key\)\\mathcal\{P\},b\\leftarrow\\textsc\{Slaughter\}\(\\mathcal\{X\}\_\{\\text\{cur\}\},\\mathrm\{key\}\)
6:

𝒲←∅\\mathcal\{W\}\\leftarrow\\emptyset⊳\\trianglerightwinners of this round

7:for

\(ci,cj\)∈𝒫\(c\_\{i\},c\_\{j\}\)\\in\\mathcal\{P\}do

8:

\(vi​j,wi​j\)←J​\(q,ϕe​\(ci\),ϕe​\(cj\)\)\(v\_\{ij\},w\_\{ij\}\)\\leftarrow J\\bigl\(q,\\,\\phi\_\{e\}\(c\_\{i\}\),\\,\\phi\_\{e\}\(c\_\{j\}\)\\bigr\)
9:

UpdateScore​\(𝒯,i,j,vi​j,wi​j\)\\textsc\{UpdateScore\}\(\\mathcal\{T\},i,j,v\_\{ij\},w\_\{ij\}\)⊳\\trianglerightEq\. \([3](https://arxiv.org/html/2605.15513#S3.E3)\)

10:if

S​\(ci\)\>S​\(cj\)S\(c\_\{i\}\)\>S\(c\_\{j\}\)or\(

S​\(ci\)=S​\(cj\)S\(c\_\{i\}\)=S\(c\_\{j\}\)and

ν​\(ci\)\>ν​\(cj\)\\nu\(c\_\{i\}\)\>\\nu\(c\_\{j\}\)\)then

11:

𝒲←𝒲∪\{ci\}\\mathcal\{W\}\\leftarrow\\mathcal\{W\}\\cup\\\{c\_\{i\}\\\}
12:else

13:

𝒲←𝒲∪\{cj\}\\mathcal\{W\}\\leftarrow\\mathcal\{W\}\\cup\\\{c\_\{j\}\\\}
14:endif

15:endfor

16:if

b≠⊥b\\neq\\botthen

𝒲←𝒲∪\{b\}\\;\\;\\mathcal\{W\}\\leftarrow\\mathcal\{W\}\\cup\\\{b\\\}⊳\\trianglerightbye candidate advances at current score

17:endif

18:

𝒳cur←𝒲\\mathcal\{X\}\_\{\\text\{cur\}\}\\leftarrow\\mathcal\{W\}
19:until

stop​\_​at\\mathrm\{stop\\\_at\}is unspecifiedor

\|𝒳cur\|≤stop​\_​at\|\\mathcal\{X\}\_\{\\text\{cur\}\}\|\\leq\\mathrm\{stop\\\_at\}
20:return

𝒳cur\\mathcal\{X\}\_\{\\text\{cur\}\}

Two regimes share this single procedure\. Stage A invokesEliminatewithout a stop count, so therepeatloop executes exactly once: a single halving round at E1 reducesN′N^\{\\prime\}candidates to⌈N′/2⌉\\lceil N^\{\\prime\}/2\\rceil\. Stage B invokes it withstop​\_​at=f\\mathrm\{stop\\\_at\}=f, so the loop iterates until the finalist count is reached\. The number of iterations isrB=⌈log2⁡\(\|𝒳\|/f\)⌉r\_\{B\}=\\lceil\\log\_\{2\}\(\|\\mathcal\{X\}\|/f\)\\rceil\. For our defaultN=16,f=4N=16,f=4, Stage A has\|𝒳\|=16\|\\mathcal\{X\}\|=16and produces⌈16/2⌉=8\\lceil 16/2\\rceil=8survivors in a single round; Stage B has\|𝒳\|=8\|\\mathcal\{X\}\|=8and producesf=4f=4in a single round \(rB=1r\_\{B\}=1\)\.

##### Tie\-breaking insideEliminate\.

Lines 8–12 implement the per\-pair winner selection\. The primary criterion isSSafter the score update; ties onSSare broken byν\\nu\(cluster size, favoring representatives of larger clusters\); remaining ties are broken uniformly at random in the sort step insideSlaughter\. We deliberately do not useν\\nuas part ofSS—it is reserved for tie\-breaking only—to preserve the one\-solution\-one\-vote semantics discussed in the main text\.

#### B\.1\.4Rescue: Optional finalist expansion

Rescueinspects the strongest eliminated candidate against the weakest finalist and admits the former if the elimination evidence appears weak\. The trigger is the disjunction in Eq\. \([5](https://arxiv.org/html/2605.15513#S3.E5)\) of the main text\.

Algorithm 5Rescue\(𝒞B,ℰ,δ\)\(\\mathcal\{C\}\_\{B\},\\mathcal\{E\},\\delta\)— Optional CAPS\-R subroutine1:finalist set

𝒞B\\mathcal\{C\}\_\{B\}, eliminated pool

ℰ=𝒞′∖𝒞B\\mathcal\{E\}=\\mathcal\{C\}^\{\\prime\}\\setminus\\mathcal\{C\}\_\{B\}, margin

δ\\delta
2:expanded finalist set

𝒞~B\\widetilde\{\\mathcal\{C\}\}\_\{B\}with

\|𝒞~B\|∈\{\|𝒞B\|,\|𝒞B\|\+1\}\|\\widetilde\{\\mathcal\{C\}\}\_\{B\}\|\\in\\\{\|\\mathcal\{C\}\_\{B\}\|,\\,\|\\mathcal\{C\}\_\{B\}\|\+1\\\}
3:

c\+←arg⁡maxc∈ℰ⁡S​\(c\)c^\{\+\}\\leftarrow\\arg\\max\_\{c\\in\\mathcal\{E\}\}S\(c\)⊳\\trianglerightbest loser

4:

cmin←arg⁡minc∈𝒞B⁡S​\(c\)c\_\{\\min\}\\leftarrow\\arg\\min\_\{c\\in\\mathcal\{C\}\_\{B\}\}S\(c\)⊳\\trianglerightweakest finalist

5:

Δ←\|S​\(c\+\)−S​\(cmin\)\|\\Delta\\leftarrow\|S\(c^\{\+\}\)\-S\(c\_\{\\min\}\)\|
6:if

Δ≤δ\\Delta\\leq\\deltathen

7:return

𝒞B∪\{c\+\}\\mathcal\{C\}\_\{B\}\\cup\\\{c^\{\+\}\\\}⊳\\trianglerightmargin condition

8:elseif

ν​\(c\+\)=1\\nu\(c^\{\+\}\)=1and

Δ≤2​δ\\Delta\\leq 2\\deltathen

9:return

𝒞B∪\{c\+\}\\mathcal\{C\}\_\{B\}\\cup\\\{c^\{\+\}\\\}⊳\\trianglerightrarity condition: correct singleton

10:else

11:return

𝒞B\\mathcal\{C\}\_\{B\}⊳\\trianglerightno rescue

12:endif

Rescueinspects only one eliminated candidate per problem—the highest\-scoring one—to keep the procedure deterministic and the additional cost bounded byf⋅T2f\\cdot T\_\{2\}\. Extending the rescue to the top\-kkeliminated candidates is straightforward but, in our preliminary experiments, the marginal Pass@1 gain fromk\>1k\>1does not justify the additional Stage\-C round\-robin overhead\.

#### B\.1\.5Composition: tracing the full algorithm

For concreteness, we trace the helper procedures invoked by Algorithm[1](https://arxiv.org/html/2605.15513#alg1)for the standard configurationN=16,f=4N=16,f=4:

1. 1\.Dedup\(𝒞,ϕ0\)\(\\mathcal\{C\},\\phi\_\{0\}\)produces𝒞′\\mathcal\{C\}^\{\\prime\}withN′≤16N^\{\\prime\}\\leq 16and cluster sizesν\\nu\.
2. 2\.Eliminate\(𝒞′,J,1,ν,𝒯\)\(\\mathcal\{C\}^\{\\prime\},J,1,\\nu,\\mathcal\{T\}\)executes one halving round at E1: invokesSlaughterwith keyν\\nu, performs⌊N′/2⌋\\lfloor N^\{\\prime\}/2\\rfloorjudge calls, returns𝒞A\\mathcal\{C\}\_\{A\}with\|𝒞A\|=⌈N′/2⌉\|\\mathcal\{C\}\_\{A\}\|=\\lceil N^\{\\prime\}/2\\rceil\.
3. 3\.Eliminate\(𝒞A,J,2,S,𝒯,stop​\_​at=4\)\(\\mathcal\{C\}\_\{A\},J,2,S,\\mathcal\{T\},\\,\\mathrm\{stop\\\_at\}\{=\}4\)halves at E2 untilf=4f=4remain: for\|𝒞A\|=8\|\\mathcal\{C\}\_\{A\}\|=8, this is one round of44judge calls\.
4. 4\.\(Optional\)Rescue\(𝒞B,𝒞′∖𝒞B,0\.15\)\(\\mathcal\{C\}\_\{B\},\\mathcal\{C\}^\{\\prime\}\\setminus\\mathcal\{C\}\_\{B\},0\.15\)may expand𝒞B\\mathcal\{C\}\_\{B\}by one candidate\.
5. 5\.Stage C round\-robin:\(\|𝒞B\|2\)\\binom\{\|\\mathcal\{C\}\_\{B\}\|\}\{2\}judge calls at E2, returnc⋆=arg⁡maxc⁡sC​\(c\)c^\{\\star\}=\\arg\\max\_\{c\}s\_\{C\}\(c\)via Eq\. \([4](https://arxiv.org/html/2605.15513#S3.E4)\)\.

Total judge calls in the deterministic pipeline \(no rescue\):88at E1 \(Stage A\) \+44at E2 \(Stage B\) \+66at E2 \(Stage C\) =1818judge calls, decomposed as8​T1\+10​T28\\,T\_\{1\}\+10\\,T\_\{2\}in token cost\.

### B\.2Implementation Details

#### B\.2\.1Per\-Call Token Cost Decomposition

The per\-call token cost of a pairwise judgment at evidence leveleedecomposes as

Te=2⋅\|ϕe​\(c\)\|avg\+Tovhd,T\_\{e\}\\;=\\;2\\cdot\|\\phi\_\{e\}\(c\)\|\_\{\\text\{avg\}\}\\,\+\\,T\_\{\\text\{ovhd\}\},\(9\)where the first term accounts for the two candidate views passed to the judge andTovhdT\_\{\\text\{ovhd\}\}is the prompt overhead \(system prompt, instructions, format specifiers\)\. In our experiments we measureTovhd≈500T\_\{\\text\{ovhd\}\}\\approx 500tokens,\|ϕ1​\(c\)\|avg≈250\|\\phi\_\{1\}\(c\)\|\_\{\\text\{avg\}\}\\approx 250tokens, and\|ϕ2​\(c\)\|avg≈4000\|\\phi\_\{2\}\(c\)\|\_\{\\text\{avg\}\}\\approx 4000tokens, giving

T1≈1,000,T2≈8,500,ρ=T1/T2≈0\.12\.T\_\{1\}\\;\\approx\\;1\{,\}000,\\qquad T\_\{2\}\\;\\approx\\;8\{,\}500,\\qquad\\rho\\;=\\;T\_\{1\}/T\_\{2\}\\;\\approx\\;0\.12\.\(10\)The prompt overheadTovhdT\_\{\\text\{ovhd\}\}provides a lower bound onρ\\rho: as\|ϕ2​\(c\)\|→∞\|\\phi\_\{2\}\(c\)\|\\to\\infty,ρ→Tovhd/\(2​\|ϕ2​\(c\)\|\+Tovhd\)→0\\rho\\to T\_\{\\text\{ovhd\}\}/\(2\|\\phi\_\{2\}\(c\)\|\+T\_\{\\text\{ovhd\}\}\)\\to 0, while as\|ϕ1​\(c\)\|→\|ϕ2​\(c\)\|\|\\phi\_\{1\}\(c\)\|\\to\|\\phi\_\{2\}\(c\)\|,ρ→1\\rho\\to 1\. The empirically observed rangeρ∈\[0\.10,0\.15\]\\rho\\in\[0\.10,0\.15\]corresponds to a regime where partial views are roughly an order of magnitude smaller than full solutions and prompt overhead is a small but non\-negligible share ofT2T\_\{2\}\.

#### B\.2\.2Stage 0 Representative Selection

We select the cluster representative as the longest candidate by token count,rep​\(Km\)=arg⁡maxc∈Km⁡\|c\|\\mathrm\{rep\}\(K\_\{m\}\)=\\arg\\max\_\{c\\in K\_\{m\}\}\|c\|\. This heuristic avoids selecting an aborted or truncated solution as the representative when one is available\. Sensitivity to this choice is small: in our experiments, alternative tiebreakers \(first\-sampled candidate; shortest candidate, which prefers concise correct solutions; uniformly random\) yield differences of less than0\.5%0\.5\\%Pass@1 across the four models on LiveCodeBench\-v6\.

### B\.3Design of the Evidence Cascade

We discuss three design choices in the evidence cascade: the number of evidence levels \(§[B\.3\.1](https://arxiv.org/html/2605.15513#A2.SS3.SSS1)\), the content of the partial viewϕ1\\phi\_\{1\}\(§[B\.3\.2](https://arxiv.org/html/2605.15513#A2.SS3.SSS2)\), and the structure of the rescue trigger \(§[B\.3\.3](https://arxiv.org/html/2605.15513#A2.SS3.SSS3)\)\.

#### B\.3\.1Why three evidence levels

A two\-level design with onlyϕ0\\phi\_\{0\}andϕ2\\phi\_\{2\}admits two natural variants\. Variant \(i\) clusters byϕ0\\phi\_\{0\}and runs the entire elimination atϕ2\\phi\_\{2\}\. This is suboptimal because Stage A’s elimination decisions are coarse \(gross algorithmic differences\) and benefit from cheap evidence: reading full solutions to discriminate algorithmically\-distinct candidates wastes budget that a partial\-view regime makes available\. Variant \(ii\) skips clustering and uses a smaller, cheaper judge atϕ2\\phi\_\{2\}for early elimination\. This shifts the cascade from the evidence axis to the judge axis, but the cost of aϕ2\\phi\_\{2\}judgment is dominated by the candidate views \(typically4−84\{\-\}8K tokens each\) regardless of judge size; using a smaller judge at full evidence saves only the prompt\-overhead and per\-token decode cost, which is a small fraction ofT2T\_\{2\}\. The three\-level design captures the order\-of\-magnitude separation betweenT1T\_\{1\}andT2T\_\{2\}that drives the structural cost result of §[3\.4](https://arxiv.org/html/2605.15513#S3.SS4)\.

#### B\.3\.2What the partial viewϕ1\\phi\_\{1\}should contain

For code,ϕ1​\(c\)\\phi\_\{1\}\(c\)contains the first∼20\{\\sim\}20lines of the solution, which typically reveal the algorithmic approach—data structure choice, control flow shape, complexity claim—without exposing implementation\-level subtleties\. For math,ϕ1​\(c\)\\phi\_\{1\}\(c\)contains the boxed final answer and a window of∼300\{\\sim\}300tokens around it, capturing the final derivation steps without the full chain of intermediate work\. The common principle:ϕ1\\phi\_\{1\}should preserve the information needed to detect*algorithmic\-level*errors while omitting information relevant only to*implementation\-level*correctness, which is the role reserved forϕ2\\phi\_\{2\}\.

For thinking\-style models that produce long reasoning chains before the final solution, the first∼20\{\\sim\}20lines of code may instead capture the start of the reasoning rather than the algorithmic decision; this is the source of the trade\-off case discussed in §[4\.2](https://arxiv.org/html/2605.15513#S4.SS2)of the main text\. A thinking\-aware extractor that pulls the boxed final answer plus the last∼500\{\\sim\}500tokens of reasoning is a straightforward extension that addresses this case\.

#### B\.3\.3Why the rescue trigger is disjunctive

The disjunctive form of Eq\. \([5](https://arxiv.org/html/2605.15513#S3.E5)\) reflects two distinct error modes of E1 elimination\. The margin condition handles statistical noise: when two candidates produce similar E1 views, a single judge call provides limited signal, and the eliminated candidate may legitimately be the stronger of the two\. The rarity condition handles a different mode: a correct singleton may be confidently eliminated at E1 because its style or framing differs from the cluster majority, even when its underlying reasoning is sound\. The relaxed threshold2​δ2\\deltafor the rarity condition reflects the prior that singletons are more likely to be misranked at low evidence than candidates from large clusters\.

Requiring both conditions \(AND\) would miss singleton failures with decisive elimination evidence—which empirically include some of the highest\-value rescues, as a wrong\-but\-plausible majority can confidently dominate a correct outlier at E1\. Requiring either \(OR\) covers both modes at a controlled1010–15%15\\%activation rate\. The thresholdδ=0\.15\\delta=0\.15is a hyperparameter of CAPS\-R; sensitivity toδ\\deltaover the range\[0\.10,0\.20\]\[0\.10,0\.20\]is reported in §[4\.3](https://arxiv.org/html/2605.15513#S4.SS3)of the main text\.

## Appendix CVerifier\-Token Cost Analysis: Derivation and Asymptotics

This appendix derives the cost results of §[3\.4](https://arxiv.org/html/2605.15513#S3.SS4)in detail\.

### C\.1General\-N′N^\{\\prime\}Cost Formula

We derive Eq\. \([6](https://arxiv.org/html/2605.15513#S3.E6)\) of the main text from the helper procedures of Section[B\.1](https://arxiv.org/html/2605.15513#A2.SS1)\.

##### Stage A\.

A singleEliminateround at E1 \(Algorithm[4](https://arxiv.org/html/2605.15513#alg4)withstop​\_​at\\mathrm\{stop\\\_at\}unspecified\) produces⌈N′/2⌉\\lceil N^\{\\prime\}/2\\rceilsurvivors at cost⌊N′/2⌋⋅T1\\lfloor N^\{\\prime\}/2\\rfloor\\cdot T\_\{1\}\.

##### Stage B\.

Eliminateat E2 withstop​\_​at=f\\mathrm\{stop\\\_at\}=fexecutesrBr\_\{B\}halving rounds, where roundrrhasNrN\_\{r\}entrants withN1:=⌈N′/2⌉N\_\{1\}:=\\lceil N^\{\\prime\}/2\\rceil,Nr\+1=⌈Nr/2⌉N\_\{r\+1\}=\\lceil N\_\{r\}/2\\rceil, andrBr\_\{B\}is the smallestrrsuch thatNr=fN\_\{r\}=f, equivalentlyrB=⌈log2⁡\(N1/f\)⌉r\_\{B\}=\\lceil\\log\_\{2\}\(N\_\{1\}/f\)\\rceil\. Roundrrcontributes⌊Nr/2⌋⋅T2\\lfloor N\_\{r\}/2\\rfloor\\cdot T\_\{2\}\. Summing,

TStage B=\(∑r=1rB⌊Nr/2⌋\)⋅T2\.T\_\{\\text\{Stage B\}\}\\;=\\;\\Bigl\(\\sum\_\{r=1\}^\{r\_\{B\}\}\\lfloor N\_\{r\}/2\\rfloor\\Bigr\)\\cdot T\_\{2\}\.\(11\)

##### Stage C\.

The round\-robin contributes\(f2\)⋅T2\\binom\{f\}\{2\}\\cdot T\_\{2\}, independent ofN′N^\{\\prime\}\.

##### Total\.

Combining,

TCAPS​\(N′,f\)=⌊N′/2⌋​T1\+\(∑r=1rB⌊Nr/2⌋\)​T2\+\(f2\)​T2,T\_\{\\text\{CAPS\}\}\(N^\{\\prime\},f\)\\;=\\;\\lfloor N^\{\\prime\}/2\\rfloor\\,T\_\{1\}\\,\+\\,\\Bigl\(\\sum\_\{r=1\}^\{r\_\{B\}\}\\lfloor N\_\{r\}/2\\rfloor\\Bigr\)\\,T\_\{2\}\\,\+\\,\\binom\{f\}\{2\}\\,T\_\{2\},\(12\)matching Eq\. \([6](https://arxiv.org/html/2605.15513#S3.E6)\) of the main text\.

##### Standard configuration\.

ForN′=16,f=4N^\{\\prime\}=16,f=4:N1=8N\_\{1\}=8,rB=1r\_\{B\}=1,⌊8/2⌋=4\\lfloor 8/2\\rfloor=4,\(42\)=6\\binom\{4\}\{2\}=6\. Substituting:

TCAPS​\(16,4\)=8​T1\+4​T2\+6​T2=8​T1\+10​T2=\(10\+8​ρ\)​T2\.T\_\{\\text\{CAPS\}\}\(16,4\)\\;=\\;8\\,T\_\{1\}\\,\+\\,4\\,T\_\{2\}\\,\+\\,6\\,T\_\{2\}\\;=\\;8\\,T\_\{1\}\+10\\,T\_\{2\}\\;=\\;\(10\+8\\rho\)\\,T\_\{2\}\.\(13\)

### C\.2Asymptotic Form via Telescoping

We derive the asymptotic expression Eq\. \([7](https://arxiv.org/html/2605.15513#S3.E7)\) of the main text by closing the Stage B geometric sum\.

Ignoring ceiling effects,Nr≈N1/2r−1N\_\{r\}\\approx N\_\{1\}/2^\{r\-1\}, so∑r=1rBNr/2≈∑r=1rBN1/2r\\sum\_\{r=1\}^\{r\_\{B\}\}N\_\{r\}/2\\approx\\sum\_\{r=1\}^\{r\_\{B\}\}N\_\{1\}/2^\{r\}\. The geometric sum telescopes:

∑r=1rBN12r=N1​\(1−2−rB\)=N1−N12rB=N1−f,\\sum\_\{r=1\}^\{r\_\{B\}\}\\frac\{N\_\{1\}\}\{2^\{r\}\}\\;=\\;N\_\{1\}\\Bigl\(1\-2^\{\-r\_\{B\}\}\\Bigr\)\\;=\\;N\_\{1\}\-\\frac\{N\_\{1\}\}\{2^\{r\_\{B\}\}\}\\;=\\;N\_\{1\}\-f,\(14\)where the last equality usesNrB=f⇔N1/2rB=fN\_\{r\_\{B\}\}=f\\iff N\_\{1\}/2^\{r\_\{B\}\}=f\. SubstitutingN1=⌈N′/2⌉≈N′/2N\_\{1\}=\\lceil N^\{\\prime\}/2\\rceil\\approx N^\{\\prime\}/2:

∑r=1rB⌊Nr/2⌋=N′2−f\+O​\(log⁡N′\),\\sum\_\{r=1\}^\{r\_\{B\}\}\\lfloor N\_\{r\}/2\\rfloor\\;=\\;\\frac\{N^\{\\prime\}\}\{2\}\-f\\,\+\\,O\(\\log N^\{\\prime\}\),\(15\)where theO​\(log⁡N′\)O\(\\log N^\{\\prime\}\)term absorbs rounding error from the floor and ceiling operations acrossrB=O​\(log⁡N′\)r\_\{B\}=O\(\\log N^\{\\prime\}\)rounds\. Substituting into Eq\. \([12](https://arxiv.org/html/2605.15513#A3.E12)\):

TCAPS​\(N′,f\)=N′2​\(T1\+T2\)−f​T2\+\(f2\)​T2\+O​\(log⁡N′\)\.T\_\{\\text\{CAPS\}\}\(N^\{\\prime\},f\)\\;=\\;\\frac\{N^\{\\prime\}\}\{2\}\(T\_\{1\}\+T\_\{2\}\)\\,\-\\,f\\,T\_\{2\}\\,\+\\,\\binom\{f\}\{2\}\\,T\_\{2\}\\,\+\\,O\(\\log N^\{\\prime\}\)\.\(16\)
##### Marginal cost interpretation\.

Differentiating Eq\. \([16](https://arxiv.org/html/2605.15513#A3.E16)\) with respect toN′N^\{\\prime\}:

∂TCAPS∂N′=T1\+T22\+O​\(1N′\)\.\\frac\{\\partial T\_\{\\text\{CAPS\}\}\}\{\\partial N^\{\\prime\}\}\\;=\\;\\frac\{T\_\{1\}\+T\_\{2\}\}\{2\}\\,\+\\,O\\bigl\(\\tfrac\{1\}\{N^\{\\prime\}\}\\bigr\)\.\(17\)The marginal cost of an additional candidate is12​\(T1\+T2\)\\tfrac\{1\}\{2\}\(T\_\{1\}\+T\_\{2\}\): half a Stage A E1 call plus half a Stage B E2 call\. By contrast, a uniform full\-evidence schedule that performskkcomparisons per candidate has marginal costk⋅T2k\\cdot T\_\{2\}, exceeding CAPS’s marginal cost by a factor of2​k​T2/\(T1\+T2\)=2​k/\(1\+ρ\)2kT\_\{2\}/\(T\_\{1\}\+T\_\{2\}\)=2k/\(1\+\\rho\)\. Forρ≪1\\rho\\ll 1this approaches2​k2k\. The structural source of CAPS’s efficiency is the replacement ofT2T\_\{2\}in the marginal cost with the average12​\(T1\+T2\)\\tfrac\{1\}\{2\}\(T\_\{1\}\+T\_\{2\}\)\.

### C\.3Expected Cost Under CAPS\-R

When CAPS\-R is enabled, the trigger Eq\. \([5](https://arxiv.org/html/2605.15513#S3.E5)\) fires with empirical ratepR∈\[0,1\]p\_\{R\}\\in\[0,1\]\. Conditional on triggering, Stage C round\-robin expands from\(f2\)\\binom\{f\}\{2\}to\(f\+12\)\\binom\{f\+1\}\{2\}E2 calls, an additive overhead of

Δ​Trescue=\(f\+12\)​T2−\(f2\)​T2=f⋅T2\.\\Delta T\_\{\\text\{rescue\}\}\\;=\\;\\binom\{f\+1\}\{2\}\\,T\_\{2\}\\,\-\\,\\binom\{f\}\{2\}\\,T\_\{2\}\\;=\\;f\\cdot T\_\{2\}\.\(18\)The expected total cost is

𝔼​\[TCAPS\+R​\(N′,f\)\]=TCAPS​\(N′,f\)\+pR⋅f⋅T2\.\\mathbb\{E\}\\bigl\[T\_\{\\text\{CAPS\+R\}\}\(N^\{\\prime\},f\)\\bigr\]\\;=\\;T\_\{\\text\{CAPS\}\}\(N^\{\\prime\},f\)\\,\+\\,p\_\{R\}\\cdot f\\cdot T\_\{2\}\.\(19\)Forf=4f=4and the empirically observed ratepR∈\[0\.10,0\.15\]p\_\{R\}\\in\[0\.10,0\.15\], the expected rescue overhead is𝔼​\[Δ​Trescue\]∈\[0\.4,0\.6\]⋅T2\\mathbb\{E\}\[\\Delta T\_\{\\text\{rescue\}\}\]\\in\[0\.4,0\.6\]\\cdot T\_\{2\}, which is small relative to the10​T210\\,T\_\{2\}baseline ofTCAPS​\(16,4\)T\_\{\\text\{CAPS\}\}\(16,4\): less than6%6\\%of the deterministic cost\.

##### Variance\.

The variance of the rescue overhead isVar​\[Δ​Trescue\]=pR​\(1−pR\)​\(f​T2\)2\\mathrm\{Var\}\[\\Delta T\_\{\\text\{rescue\}\}\]=p\_\{R\}\(1\-p\_\{R\}\)\(fT\_\{2\}\)^\{2\}\. ForpR=0\.125,f=4p\_\{R\}=0\.125,f=4, this gives a per\-instance standard deviation of0\.109⋅4​T2≈1\.32​T2\\sqrt\{0\.109\}\\cdot 4\\,T\_\{2\}\\approx 1\.32\\,T\_\{2\}\. Per\-instance cost variation is therefore non\-negligible relative to the per\-instance baseline cost \(∼11​T2\\sim 11\\,T\_\{2\}\), but the variance of the mean cost across a benchmark ofKKproblems scales as1/K1/K, yielding negligible noise on aggregate cost reports\.

### C\.4Summary of Cost Results

The cost analysis of CAPS yields three quantitative facts used throughout the paper:

1. 1\.Standard configuration:TCAPS​\(16,4\)=\(10\+8​ρ\)​T2≈11​T2T\_\{\\text\{CAPS\}\}\(16,4\)=\(10\+8\\rho\)\\,T\_\{2\}\\approx 11\\,T\_\{2\}forρ≈0\.12\\rho\\approx 0\.12\.
2. 2\.Asymptotic:TCAPS​\(N′,f\)=N′2​\(T1\+T2\)−f​T2\+\(f2\)​T2\+O​\(log⁡N′\)T\_\{\\text\{CAPS\}\}\(N^\{\\prime\},f\)=\\frac\{N^\{\\prime\}\}\{2\}\(T\_\{1\}\+T\_\{2\}\)\-fT\_\{2\}\+\\binom\{f\}\{2\}T\_\{2\}\+O\(\\log N^\{\\prime\}\), with marginal cost per candidate12​\(T1\+T2\)\\tfrac\{1\}\{2\}\(T\_\{1\}\+T\_\{2\}\)\.
3. 3\.With CAPS\-R:𝔼​\[TCAPS\+R\]=TCAPS\+pR​f​T2\\mathbb\{E\}\[T\_\{\\text\{CAPS\+R\}\}\]=T\_\{\\text\{CAPS\}\}\+p\_\{R\}fT\_\{2\}, withpR∈\[0\.10,0\.15\]p\_\{R\}\\in\[0\.10,0\.15\]adding less than6%6\\%overhead\.

None of these depends on tunable hyperparameters of CAPS that affect cost: the cost is determined entirely byNN,ff,ρ\\rho, and the empirical trigger ratepRp\_\{R\}, all of which are properties of the deployment\.

## Appendix DExperimental Details and Extended Results

This appendix supplements the main experiments section with implementation details \(§[D\.1](https://arxiv.org/html/2605.15513#A4.SS1)\), additional baselines including Random pair scheduling \(§[D\.2](https://arxiv.org/html/2605.15513#A4.SS2)\), per\-component verifier\-accuracy diagnostic data supporting the discussion in §[4\.2](https://arxiv.org/html/2605.15513#S4.SS2)\(§[D\.3](https://arxiv.org/html/2605.15513#A4.SS3)\), the thinking\-awareϕ1\\phi\_\{1\}extension \(§[D\.4](https://arxiv.org/html/2605.15513#A4.SS4)\),NN\-scaling \(§[D\.5](https://arxiv.org/html/2605.15513#A4.SS5)\), the extended component ablation across all four models \(§[D\.6](https://arxiv.org/html/2605.15513#A4.SS6)\), and qualitative case studies \(§[D\.7](https://arxiv.org/html/2605.15513#A4.SS7)\)\.

### D\.1Implementation Details

All experiments were repeated three times with a consistent set of random seeds to ensure a fair comparison, and all reported results are the average over the three runs\.

##### Hardware and software\.

All experiments were run on2×2\\timesNVIDIA A100 80GB GPUs with vLLM\[[18](https://arxiv.org/html/2605.15513#bib.bib45)\]for batched inference at BF16 precision\. Prefix caching was enabled to amortize the system\-prompt and problem\-statement tokens across the multiple judge calls per problem\.

##### Sampling configuration\.

For each problem we sampleN=16N=16candidates with temperatureTgen=0\.6T\_\{\\text\{gen\}\}=0\.6for code and1\.01\.0for math, top\-p=0\.95p=0\.95, and a maximum generation length of32,76832\{,\}768tokens\. Sampling is performed once per problem with consistent seed; all selection methods \(Vanilla, Pointwise, V1\-Infer, CAPS\) operate on the same candidate pool, isolating selection effects from generation variance\.

##### Judge configuration\.

The pairwise judge prompt requests a winner indicator and a confidence score in a structured format\. Judge sampling temperature isTjudge=0\.0T\_\{\\text\{judge\}\}=0\.0\(greedy\) for reproducibility\. The output is parsed into\(vi​j,wi​j\)\(v\_\{ij\},w\_\{ij\}\)via Eq\. \([2](https://arxiv.org/html/2605.15513#S3.E2)\); malformed outputs \(occurring on<1%<1\\%of calls\) are retried up to twice and treated as ties withw=τww=\\tau\_\{w\}on persistent failure\. The pointwise prompt uses the same model with a11–1010rating scale, following\[[35](https://arxiv.org/html/2605.15513#bib.bib15)\]\. Full prompt templates are reproduced in Appendix[E\.1](https://arxiv.org/html/2605.15513#A5.SS1)\.

##### CAPS hyperparameters\.

We usef=4f=4finalists,δ=0\.15\\delta=0\.15rescue margin, andτw=0\.05\\tau\_\{w\}=0\.05confidence floor across all benchmarks and models\. These were fixed before running the main experiments and not tuned per\-benchmark\.

##### V1\-Infer hyperparameters\.

For the V1\-3x baseline we follow the published settings of\[[35](https://arxiv.org/html/2605.15513#bib.bib15)\]: minimum degreedmin=2d\_\{\\min\}=2, Swiss windowh=3h=3, weight floorτ=0\.05\\tau=0\.05, and budget multiplierk=3k=3, giving∼48\\sim 48pairwise comparisons atN=16N=16\. We use the same judge prompt template and parsing logic for both V1\-Infer and CAPS to ensure that performance differences arise from the selection algorithm rather than the judge interface\.

##### Token accounting\.

We report total verifier\-token cost as the sum over all judge calls of \(system prompt \+ instruction prompt \+ two candidate views\)\. Generation tokens are excluded since selection methods share an identical candidate pool\. Token counts are deterministic per call \(withTjudge=0\.0T\_\{\\text\{judge\}\}=0\.0\) and are summed across the full benchmark\. The empirically measured ratioρ=T1/T2\\rho=T\_\{1\}/T\_\{2\}in our deployment averages0\.120\.12across benchmarks \(T1≈1,000T\_\{1\}\\approx 1\{,\}000,T2≈8,500T\_\{2\}\\approx 8\{,\}500tokens, prompt overheadTovhd≈500T\_\{\\text\{ovhd\}\}\\approx 500\)\.

### D\.2Random Pair Scheduling and Oracle Bounds

##### Random baseline\.

At matched comparison count to V1\-3x \(∼48\\sim 48pairwise calls atN=16N=16\), random pair scheduling yields Pass@1 close to the Vanilla baseline—and sometimes lower, because uninformative noisy comparisons can flip the selection toward a weaker candidate\. Table[7](https://arxiv.org/html/2605.15513#A4.T7)reports Random selection alongside Vanilla and the verification methods on the three Qwen models\. The gap between Random and V1\-Infer/CAPS confirms that the gains in the main text arise from informative pair selection, not merely from candidate filtering\.

Table 7:Random baselineat matched comparison count to V1\-3x\. All methods atN=16N=16\. Random pair scheduling at full evidenceϕ2\\phi\_\{2\}\.
##### Pass@NNoracle bounds\.

Table[8](https://arxiv.org/html/2605.15513#A4.T8)reports the Pass@1616oracle: the fraction of problems for which at least one of the1616sampled candidates is correct\. This is the upper bound on what any selection method can achieve\. The gap between the best selection method and the oracle quantifies the residual headroom for future improvements\.

Table 8:Oracle bounds\(Pass@NN,N=16N=16\) and trivial\-problem fractions\. “Trivial” is the fraction of problems where all1616candidates produce the same final answer—no verification is needed\.

### D\.3Verifier Accuracy Diagnostic

The trade\-off cells in the main results admit an interpretable diagnostic in terms of the verifier’s per\-pair accuracy at partial vs\. full evidence\. We measure verifier accuracy by retrospectively labeling each judge call with whether the chosen winner is the correct one, using ground\-truth correctness on each candidate, and computing the fraction of correctly resolved pairs over the items where ground truth distinguishes the two candidates\.

Table 9:Per\-pair verifier accuracy under V1\-Infer \(full evidence\) and CAPS \(mixed E1/E2 evidence\)\.Higher is better; the V1\-Infer column corresponds to all\-ϕ2\\phi\_\{2\}judgments while CAPS aggregates Stage A \(ϕ1\\phi\_\{1\}\), Stage B \(ϕ2\\phi\_\{2\}\), and Stage C \(ϕ2\\phi\_\{2\}\) calls\. The Instruct/math drop and the GPT\-OSS code drop are the source of the trade\-off cells in Table[1](https://arxiv.org/html/2605.15513#S4.T1)\.The pattern in Table[9](https://arxiv.org/html/2605.15513#A4.T9)is clean: when the verifier maintains accuracy under partial evidence, CAPS outperforms V1\-Infer; when partial\-evidence accuracy degrades, CAPS regresses\. On code, the Qwen models all maintain or improve verifier accuracy under CAPS, indicating that partial\-view E1 judgments preserve the discriminative signal for code candidates; GPT\-OSS\-20B is the exception—its judge depends more strongly on full\-context information for code disambiguation, and the resulting∼4\.5\\sim 4\.5\-point accuracy drop manifests as the LCB\-v6 and CodeContests trade\-off cells\. On math, the Instruct model shows a∼9\\sim 9\-point accuracy drop because it relies on the full reasoning chain to discriminate between candidates—Stage A’s E1 truncation deprives it of this context\. The Thinking model, which front\-loads reasoning into a structured chain\-of\-thought before the final answer, presents condensed information atϕ1\\phi\_\{1\}that both verifiers find equally hard to resolve, so the substitution of E1 for E2 is informationally neutral and CAPS dominates\. This diagnostic suggests a deployment\-time check: practitioners can compare the verifier’s accuracy underϕ1\\phi\_\{1\}vs\.ϕ2\\phi\_\{2\}on a small held\-out set; CAPS dominates V1\-Infer when the partial\-view accuracy approximates the full\-view accuracy, with a rough threshold of\|Δ\|<5\|\\Delta\|<5points predicting CAPS dominance in our data\.

### D\.4Thinking\-Awareϕ1\\phi\_\{1\}for Reasoning Models

The default code instantiation ofϕ1\\phi\_\{1\}\(the first∼20\{\\sim\}20lines of the candidate\) interacts poorly with thinking\-style models that produce extended chain\-of\-thought before the algorithmic decision\. We replaceϕ1\\phi\_\{1\}with a thinking\-aware extractor that returns the boxed final answer \(when present\) plus the last∼500\{\\sim\}500tokens of the candidate, preserving the algorithmic\-level information thatϕ1\\phi\_\{1\}is intended to convey\.

Table 10:Thinking\-awareϕ1\\phi\_\{1\}on Qwen3\-4B\-Thinking\.Replacing the defaultϕ1\\phi\_\{1\}further improves Pass@1 while reducing the verifier\-token cost, demonstrating that the partial\-view extractor is a deployment\-time configuration that should be matched to the generator’s output structure\.The thinking\-awareϕ1\\phi\_\{1\}is a deployment\-time configuration choice analogous to selecting a tokenizer: it depends on knowing the output structure of the underlying model\. We do not propose a universalϕ1\\phi\_\{1\}that works across all generation styles; rather, we treat the partial\-view extractor as a deployment\-time hyperparameter that should be matched to the generator’s output structure\. The general design principle of §[3](https://arxiv.org/html/2605.15513#S3)—thatϕ1\\phi\_\{1\}should expose algorithmic\-level information while omitting implementation\-level detail—remains unchanged\.

### D\.5Scaling in Candidate CountNN

Table[11](https://arxiv.org/html/2605.15513#A4.T11)reports Pass@1 and verifier\-token cost forN∈\{8,16,32\}N\\in\\\{8,16,32\\\}on Qwen3\-14B / LCB\-v6\. By Eq\. \([7](https://arxiv.org/html/2605.15513#S3.E7)\), the asymptotic cost ratio approaches\(1\+ρ\)/\(2​k\)≈0\.19\(1\+\\rho\)/\(2k\)\\approx 0\.19forρ=0\.12,k=3\\rho=0\.12,k=3, slightly below theN=16N=16ratio of0\.220\.22\. The empirical trend confirms this:T%T\\%decreases mildly withNNas Stage C’s\(f2\)\\binom\{f\}\{2\}overhead is amortized over a larger candidate pool, while Pass@1 gains over V1\-Infer persist\.

Table 11:Scaling inNNon Qwen3\-14B / LCB\-v6 withf=4f=4\. The cost ratioT%T\\%decreases mildly withNNas Stage C’s fixed overhead is amortized; Pass@1 gains over V1\-Infer persist or grow\.
### D\.6Extended Component Ablation

Table[12](https://arxiv.org/html/2605.15513#A4.T12)extends the component ablation of §[4\.3](https://arxiv.org/html/2605.15513#S4.SS3)to all four models on LCB\-v6 and CodeContests\. The pattern is consistent across models: rescue and slaughter pairing are the largest single Pass@1 contributors, while Stage 0 deduplication has minimal effect on code \(whereϕ0\\phi\_\{0\}rarely distinguishes candidates\)\.

Table 12:Extended component ablationacross four models on LCB\-v6 and CodeContests\.Δ\\Deltarelative to full CAPS per \(model, benchmark\)\.The E1 cascade row \(running all stages at full evidence\) gives slightly higher Pass@1 across all eight \(model, benchmark\) cells but at∼4×\\sim 4\\timesthe verifier\-token cost: this confirms that the partial\-view cascade is a cost mechanism that costs at most0\.70\.7Pass@1 in exchange for the structural cost reduction analyzed in §[3\.4](https://arxiv.org/html/2605.15513#S3.SS4)\.

### D\.7Qualitative Case Studies

We examine two LCB\-v6 problems where CAPS and V1\-Infer select different candidates on Qwen3\-14B\.

##### Case 1: Stage 0 deduplication concentrates the budget\.

On a dynamic\-programming problem where1111of the1616sampled candidates produce the same incorrect\-but\-popular boxed answer, V1\-Infer’s full Swiss tournament spends∼30\{\\sim\}30comparisons disambiguating among these1111near\-duplicates and the remaining55candidates, exhausting most of the4848\-comparison budget before reliably ranking the latter\. CAPS’s Stage 0 collapses the1111near\-duplicates into a single representative, leaving66distinct candidates to enter Stage A; the cascaded pipeline reaches the correct answer with99E1 calls and66E2 calls, while V1\-Infer settles on the popular incorrect answer\.

##### Case 2: Rescue recovers a noisily eliminated singleton\.

On a graph\-traversal problem with diverse algorithmic strategies \(DFS, BFS, Union\-Find\), a correct Union\-Find solution is paired in Stage A against an incorrect\-but\-fluent BFS solution and is eliminated by a noisy E1 judgment\. The margin condition triggers rescue \(the eliminated candidate’s score is withinδ=0\.15\\delta=0\.15of the weakest finalist’s\), reinstating it for the Stage C round\-robin where it wins decisively under full\-evidence comparison\. V1\-Infer, lacking such a mechanism, proceeds with the incorrect BFS as the eventual winner because subsequent uncertainty\-guided pairings reinforce its early lead\.

## Appendix EImplementation Details and Prompt Templates

This appendix specifies the full prompt templates used for generation, V1\-Infer baseline verification, and CAPS evidence\-cascade verification \(§[E\.1](https://arxiv.org/html/2605.15513#A5.SS1)\); evidence extraction details \(§[E\.2](https://arxiv.org/html/2605.15513#A5.SS2)\); the verdict\-to\-rating conversion that allows CAPS to plug into the V1\-style aggregation pipeline \(§[E\.3](https://arxiv.org/html/2605.15513#A5.SS3)\); generation, verification, and model\-specific hyperparameters \(§[E\.4](https://arxiv.org/html/2605.15513#A5.SS4)\); dataset details and evaluation criteria \(§[E\.5](https://arxiv.org/html/2605.15513#A5.SS5)\); and inference\-stack and response\-parsing details \(§[E\.6](https://arxiv.org/html/2605.15513#A5.SS6)\)\.

### E\.1Prompt Templates

#### E\.1\.1Generation Prompts

##### Code generation \(default, used for GPT\-OSS\)\.

The benchmark problem statement \(full specification with input/output format, constraints, and example test cases\) is passed directly to the model without modification\.

Code Generation Prompt \(Default\)\{original\_problem\}

##### Code generation \(Instruct/Thinking, used for Qwen models\)\.

Appends explicit thinking instructions to the problem statement\.

Code Generation Prompt \(Instruct/Thinking\)\*\*Problem\*\* \{problem\} Think and reason step by step before coding the final solution for the problem above\. Put your reasoning and any draft coding solutions between<<thinking\>\>\.\.\.<</thinking\>\>tags\. After the reasoning \(i\.e\. after the<</thinking\>\>tag\), use the format provided in the problem above \(code\-block with backticks\) to format your final code solution\. Do not include any thinking within the code block\. \#\#\# Answer:

##### Math generation\.

Math problems use the dataset prompt as\-is, with each problem appending the boxed\-answer instruction\.

Math Generation Prompt\{problem\_statement\} Let’s think step by step and output the final answer within \\boxed\{\}\.

#### E\.1\.2Pointwise Verification Prompts \(V1\-Infer Baseline\)

##### Pointwise code verification\.

Pointwise Code Verification PromptYou are an expert code reviewer\. Rate the correctness of a solution to a programming problem\. \*\*Evaluation Guidelines:\*\* \- Analyze the problem’s requirements and constraints\. \- Mentally trace the solution with test cases \(including edge cases\) to verify correctness\. \- Give a higher score if the solution is robust and fault\-tolerant\. \*\*Problem\*\* \{problem\} \*\*Solution\*\* \{code\} \*\*Output Format:\*\* First, provide your step\-by\-step reasoning\. Then, on a new line, provide your final rating using the EXACT tags below\. Add no other text after the tags\. <<rating\>\>INTEGER\_1\_TO\_10<</rating\>\> \*\*Rating Rules:\*\* \- Rate correctness on a 1\-10 scale \(10 = correct & robust, 5 = borderline, 1 = incorrect\)\. Please provide your analysis now\.

##### Pointwise math verification\.

Pointwise Math Verification PromptYou are an expert math contest grader\. Rate the correctness of a submission based solely on the final answer\. \*\*Evaluation Guidelines:\*\* \- Extract the submission’s final answer\. Use any provided reasoning only to help you assess whether the stated final answer is trustworthy\. Do not award credit for method quality or rigor\. \- Carefully analyze the problem statement and the submission to assess whether the final answer is correct\. Grade only the final answer\. \*\*Problem\*\* \{problem\} \*\*Solution\*\* \{solution\} \*\*Output Format:\*\* First, provide your reasoning \(what checks you performed\)\. Then, on a new line, provide your final rating using the EXACT tag below\. Add no other text after the tag\. <<rating\>\>INTEGER\_1\_TO\_10<</rating\>\> \*\*Rating Rules:\*\* \- Rate correctness on a 1\-10 scale \(10 = certainly correct, 8 = very likely correct, 5 = uncertain/borderline, 3 = likely incorrect, 1 = certainly incorrect\)\. Please provide your analysis now\.

#### E\.1\.3V1\-Infer Pairwise Verification Prompts

The V1\-Infer Swiss\-tournament baseline uses pairwise prompts that elicit two 1–10 ratings; the rating difference\|ri−rj\|/9\|r\_\{i\}\-r\_\{j\}\|/9becomes the confidence weight in Eq\. \([2](https://arxiv.org/html/2605.15513#S3.E2)\)\.

V1 Pairwise Code Verification PromptYou are an expert code reviewer\. Compare two solutions to a programming problem and rate their correctness\. \*\*Evaluation Guidelines:\*\* \- Analyze the problem’s requirements and constraints\. \- Mentally trace each solution with test cases \(including edge cases\) to verify correctness\. \- If both solutions appear correct, prefer the more robust and fault\-tolerant one\. \*\*Problem\*\* \{problem\} \*\*Solution A\*\* \{code\_A\} \*\*Solution B\*\* \{code\_B\} \*\*Output Format:\*\* First, provide your step\-by\-step reasoning\. Then, on separate new lines, provide your final ratings using the EXACT tags below\. Add no other text after the tags\. <<rating\_A\>\>INTEGER\_1\_TO\_10<</rating\_A\>\> <<rating\_B\>\>INTEGER\_1\_TO\_10<</rating\_B\>\> \*\*Rating Rules:\*\* \- Rate correctness on a 1\-10 scale \(10 = correct & robust, 5 = borderline, 1 = incorrect\)\. \- The higher rating wins\. Equal ratings imply a tie\. Please provide your analysis now\.

V1 Pairwise Math Verification PromptYou are an expert math contest grader\. Compare two submissions and rate correctness based solely on the final answer\. \*\*Evaluation Guidelines:\*\* \- Extract each submission’s final answer\. Use any provided reasoning only to help you assess whether the stated final answer is trustworthy\. Do not award credit for method quality or rigor\. \- Carefully analyze the problem statement and the submissions to assess whether each final answer is correct\. Grade only the final answer\. \*\*Problem\*\* \{problem\} \*\*Solution A\*\* \{sol\_A\} \*\*Solution B\*\* \{sol\_B\} \*\*Output Format:\*\* First, provide your reasoning \(what checks you performed\)\. Then, on separate new lines, give ratings using the EXACT tags below\. Add no other text after the tags\. <<rating\_A\>\>INTEGER\_1\_TO\_10<</rating\_A\>\> <<rating\_B\>\>INTEGER\_1\_TO\_10<</rating\_B\>\> \*\*Rating Rules:\*\* \- Rate correctness on a 1\-10 scale \(10 = certainly correct, 8 = very likely correct, 5 = uncertain/borderline, 3 = likely incorrect, 1 = certainly incorrect\)\. \- Higher rating wins\. Equal ratings imply a tie\. Please provide your analysis now\.

#### E\.1\.4CAPS Pairwise Verification Prompts

CAPS prompts share a common output format—a Winner/Tie verdict and a binary confidence level—across all three evidence levels\. This format is converted to V1\-compatible 1–10 ratings via Table[14](https://arxiv.org/html/2605.15513#A5.T14)so that the same downstream aggregation \(Eq\. \([3](https://arxiv.org/html/2605.15513#S3.E3)\)\) applies\.

##### Stage A prompts \(E1, partial evidence\)\.

The Stage\-A judge sees a reasoning summary and a truncated solution view, designed to expose the algorithmic strategy without the full solution body\.

CAPS E1 Code Prompt \(Reasoning Summary \+ Truncated Code\)You are an expert code reviewer\. Compare two solutions to a programming problem using their reasoning and truncated code\. Rate their correctness\. \*\*Problem\*\* \{problem\} \*\*Solution A \(summary\)\*\* \{evidence\_A\} \*\*Solution B \(summary\)\*\* \{evidence\_B\} \*\*Output Format:\*\* Provide your step\-by\-step reasoning\. Then on a new line output your verdict using the EXACT tags below: <<winner\>\>A or B or TIE<</winner\>\> <<confidence\>\>HIGH or LOW<</confidence\>\> HIGH confidence = you are fairly sure\. LOW = close call\. Provide your analysis now\.

CAPS E1 Math Prompt \(Reasoning Summary \+ Final Answer\)You are an expert math contest grader\. Compare two submissions using their reasoning summaries and final answers\. \*\*Problem\*\* \{problem\} \*\*Submission A \(summary\)\*\* \{evidence\_A\} \*\*Submission B \(summary\)\*\* \{evidence\_B\} \*\*Output Format:\*\* Provide your reasoning\. Then on a new line output your verdict using the EXACT tags below: <<winner\>\>A or B or TIE<</winner\>\> <<confidence\>\>HIGH or LOW<</confidence\>\> HIGH confidence = you are fairly sure\. LOW = close call\. Provide your analysis now\.

##### Stages B and C prompts \(E2, full evidence\)\.

At full evidence, the judge sees the complete solution\. The same prompt is used for Stage B halving and Stage C round\-robin\.

CAPS E2 Code Prompt \(Full Solution\)You are an expert code reviewer\. Compare two solutions to a programming problem and determine which is more correct\. \*\*Evaluation Guidelines:\*\* \- Analyze the problem’s requirements and constraints\. \- Mentally trace each solution with test cases \(including edge cases\) to verify correctness\. \- If both solutions appear correct, prefer the more robust and fault\-tolerant one\. \*\*Problem\*\* \{problem\} \*\*Solution A\*\* \{code\_A\} \*\*Solution B\*\* \{code\_B\} \*\*Output Format:\*\* Provide your step\-by\-step reasoning\. Then on separate new lines output: <<winner\>\>A or B or TIE<</winner\>\> <<confidence\>\>HIGH or LOW<</confidence\>\> HIGH = clear winner\. LOW = very close\. Provide your analysis now\.

CAPS E2 Math Prompt \(Full Solution\)You are an expert math contest grader\. Compare two submissions and determine which is more correct, based on the final answer\. \*\*Evaluation Guidelines:\*\* \- Extract each submission’s final answer\. \- Use reasoning to assess trustworthiness of the answer\. \- Grade based on the final answer correctness\. \*\*Problem\*\* \{problem\} \*\*Submission A\*\* \{sol\_A\} \*\*Submission B\*\* \{sol\_B\} \*\*Output Format:\*\* Provide your reasoning\. Then on separate new lines output: <<winner\>\>A or B or TIE<</winner\>\> <<confidence\>\>HIGH or LOW<</confidence\>\> HIGH = clear winner\. LOW = very close or both seem correct/incorrect\. Provide your analysis now\.

### E\.2Evidence Extraction Details

Table[13](https://arxiv.org/html/2605.15513#A5.T13)summarizes the evidence content provided to the judge at each cascade level\.

Table 13:Evidence levels\.Per\-call cost is reported relative to the full\-solution baseline \(T2T\_\{2\}\)\.##### Stage 0 \(E0\) signatures\.

For code, we normalize each candidate by stripping import statements, comments, blank lines, and leading whitespace, then compute a SHA\-256 hash of the normalized text and use the first1616hex digits as the cluster key\. For math, we extract the contents of the last`\\boxed\{\}`in the solution, normalize \(lowercase, strip whitespace, remove LaTeX command artifacts such as`\\,`and`\\\!`\), and use the result as the cluster key\. Two candidates with identical cluster keys are treated as functionally equivalent\.

##### Stage A \(E1\) reasoning summary\.

The reasoning summary captures the start and end of the model’s chain of thought without exposing the full derivation\. Concretely,ϕ1\\phi\_\{1\}for code consists of:

- •the first5050words and the last5050words of the reasoning span \(between`<thinking\>`and`</thinking\>`tags or, for non\-instruct models, before the final code block\), separated by a`\[\.\.\.reasoning truncated\.\.\.\]`marker;
- •the first500500characters of the final code, beginning at the opening of the code block\.

For math, the second component is replaced by the extracted final answer\.

### E\.3Verdict\-to\-Rating Conversion

CAPS prompts return categorical verdicts in\{A,B,TIE\}×\{HIGH,LOW\}\\\{A,B,\\mathrm\{TIE\}\\\}\\times\\\{\\text\{HIGH\},\\text\{LOW\}\\\}, while V1\-Infer’s aggregation pipeline operates on numerical 1–10 ratings\. We convert verdicts to ratings via Table[14](https://arxiv.org/html/2605.15513#A5.T14), allowing CAPS to plug into the existing aggregation logic without modification\.

Table 14:Verdict\-to\-rating conversionfor CAPS pairwise judgments\. The margin column reports\|rA−rB\|/9\|r\_\{A\}\-r\_\{B\}\|/9, the confidence weightwi​jw\_\{ij\}in Eq\. \([2](https://arxiv.org/html/2605.15513#S3.E2)\)\.The margin floorτw=0\.05\\tau\_\{w\}=0\.05from Eq\. \([2](https://arxiv.org/html/2605.15513#S3.E2)\) prevents tie outcomes from receiving exactly zero weight; a TIE verdict is treated aswi​j=0\.05w\_\{ij\}=0\.05rather than0to maintain non\-degenerate aggregation\.

### E\.4Hyperparameters

##### Generation hyperparameters\.

We use the same generation budget across methods so that performance differences are attributable to selection algorithm rather than candidate quality\. Code generation uses temperature0\.60\.6for the Qwen models \(matching the official Qwen3\-2507 release recipe\) and the model\-default for GPT\-OSS\-20B\.

Table 15:Generation hyperparameters\.All models share the same candidate countN=16N=16, top\-p=0\.95p=0\.95, top\-kkdisabled, max length32,76832\{,\}768tokens, and seed12341234\.
##### Verification hyperparameters\.

V1\-Infer hyperparameters follow the published recipe ofsingh2026v1; CAPS hyperparameters are the configuration tested throughout the paper\.

Table 16:Verification hyperparameters\.
##### Model\-specific configuration\.

Qwen3\-4B\-Instruct is configured to skip the explicit thinking phase \(its 2507 variant supports a flag that disables`<thinking\>`tag emission\); Qwen3\-4B\-Thinking is configured to always think\. GPT\-OSS\-20B uses the default pass\-through prompt template \(no extra instruction wrapper\) and reasoning effortmedium\.

Table 17:Model\-specific configuration\.

### E\.5Datasets and Evaluation Criteria

##### Datasets\.

LiveCodeBench\-v5\[[14](https://arxiv.org/html/2605.15513#bib.bib43)\]contains279279problems collected between 24\.08 and 25\.02; LiveCodeBench\-v6 contains131131problems between 25\.02 and 25\.05, following the official Qwen3\-2507 release\. CodeContests\[[22](https://arxiv.org/html/2605.15513#bib.bib6)\]contributes165165harder problems drawn from competitive\-programming archives\. AIME 2025 \(American Invitational Mathematics Examination\) and HMMT February 2025 \(Harvard\-MIT Mathematics Tournament\) each contain3030competition\-level problems, sourced from\[[2](https://arxiv.org/html/2605.15513#bib.bib44)\]\. We use the standard problem\-level test suites for code benchmarks and the official answer keys for math\.

##### Evaluation criteria\.

For code benchmarks, a candidate is correct if and only if it passes*all*hidden test cases for its problem; partial\-credit scoring is not used\. Solutions are executed in a sandboxed Python environment with a per\-test timeout of1010seconds\. For math benchmarks, the candidate’s extracted final answer \(from the last`\\boxed\{\}`in the solution\) is compared against the ground truth using SymPy symbolic equivalence checking, with a LaTeX normalization fallback for cases that SymPy cannot parse \(e\.g\., interval notation, set notation\)\.

##### Trivial\-problem fraction\.

A non\-trivial fraction of problems on each benchmark have all1616generated candidates produce the same final answer\. On these problems, no verification method can change the selection outcome; we include them in all reported Pass@1 numbers without filtering\. The trivial fractions across benchmarks \(Appendix[D\.2](https://arxiv.org/html/2605.15513#A4.SS2), Table[8](https://arxiv.org/html/2605.15513#A4.T8)\) range from43%43\\%\(Qwen3\-4B\-Instruct on AIME\) to65%65\\%\(Qwen3\-14B on LCB\-v5\)\.

### E\.6Inference and Response Parsing

##### Inference stack\.

All local models are served using SGLang at bfloat16 precision \(`\-\-dtype bfloat16`\) with GPU memory utilization0\.920\.92and chunked prefill enabled\. Maximum concurrent running requests is set per\-model in the range4040–100100to balance throughput against per\-request memory\. The judge inference uses the same server endpoints as generation, withTjudge=0\.0T\_\{\\text\{judge\}\}=0\.0for greedy decoding\.

##### Response parsing\.

The V1\-Infer rating output is parsed using a primary regex<rating\>\\s\*\(\\d\{1,2\}\)\\s\*</rating\>\(case\-insensitive\), with a fallback regexrating\[^\\d\]\*\(\\d\{1,2\}\)for unstructured outputs\. Parsed values are clamped to\[1,10\]\[1,10\]\. CAPS verdicts are parsed using<winner\>\\s\*\(A\|B\|TIE\)\\s\*</winner\>and<confidence\>\\s\*\(HIGH\|LOW\)\\s\*</confidence\>with analogous unstructured fallbacks\. Parse failures default to TIE / LOW respectively\. Empirically, parse failures occur on less than 1% of judge calls across all models and benchmarks\.

##### Coverage strategies\.

The coverage strategy field controls how the comparison schedule is built; we report it for reproducibility against released code\.

Table 18:Coverage strategies in the released codebase\.The output directory suffix is used to disambiguate runs\.