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What Fixed-Rollout pass@k Evaluations Can Identify

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Updated 6 h ago · first seen 11 Sept 2026

paper_01M294GKY6VPGCD6C7RJX1VNM4

Published: 11 Sept 2026; T1 · 6 h ago
arXiv: 2609.09245; T1 · 6 h ago
Category: stat.ML; T1 · 6 h ago

Abstract

Repeated-sampling evaluations increasingly extrapolate pass@k far beyond the number n of samples collected per problem. We show that, in the pooled/random-task conditional-Binomial model, fixed-n success counts identify only the n free moments of the latent per-task success distribution. Consequently, direct pass@k is identified for k <= n, but generic extrapolated pass@k, tail exponents, and tail constants are not identified for k > n, even with arbitrarily many exchangeable tasks at the same rollout budget. This is stronger than the observation that the usual estimator is undefined beyond n: it characterizes the information missing from the fixed-depth count-law experiment. We give exact count-law-preserving constructions with incompatible extrapolations, state the exceptional unique-extension case, and compute sharp population identified intervals through Hausdorff principal representations. On the public 10,000-rollout-per-problem release of Brown et al., counterfactual n = 16 evaluations leave failure at k = 1000 ambiguous by factors from 1.5 to over 2,600 across four MATH/GSM8K/CodeContests configurations. The calibration shows that intermediate-scale failure share alone does not determine width. Our result does not reject parametric inference-time scaling laws; it supplies the nonparametric baseline against which their assumptions can be evaluated. We give an exact, conservative one-coordinate finite-task confidence certificate and a reporting standard separating direct estimates, identified sets, and model-conditioned forecasts.

Authors 2

Pranav Singh, Prashant Singh

Specification

Official page: arxiv.org/abs/2609.09245
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Arxiv announce type: cross
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
arXiv id: 2609.09245
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Categories: stat.ML, cs.AI, cs.LG
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
PDF: arxiv.org/pdf/2609.09245
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Primary category: stat.ML
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh

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T19

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6 h ago

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9 claims · 9 properties

Official pageofficial_url1

Claim history for Official page
Value	Valid from → to	Status	Source	Confidence	Extractor
https://arxiv.org/abs/2609.09245	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Abstractabstract1

Claim history for Abstract
Value	Valid from → to	Status	Source	Confidence	Extractor
Repeated-sampling evaluations increasingly extrapolate pass@k far beyond the number n of samples collected per problem. We show that, in the pooled/random-task conditional-Binomial model, fixed-n success counts identify only the n free moments of the latent per-task success distribution. Consequently, direct pass@k is identified for k <= n, but generic extrapolated pass@k, tail exponents, and tail constants are not identified for k > n, even with arbitrarily many exchangeable tasks at the same rollout budget. This is stronger than the observation that the usual estimator is undefined beyond n: it characterizes the information missing from the fixed-depth count-law experiment. We give exact count-law-preserving constructions with incompatible extrapolations, state the exceptional unique-extension case, and compute sharp population identified intervals through Hausdorff principal representations. On the public 10,000-rollout-per-problem release of Brown et al., counterfactual n = 16 evaluations leave failure at k = 1000 ambiguous by factors from 1.5 to over 2,600 across four MATH/GSM8K/CodeContests configurations. The calibration shows that intermediate-scale failure share alone does not determine width. Our result does not reject parametric inference-time scaling laws; it supplies the nonparametric baseline against which their assumptions can be evaluated. We give an exact, conservative one-coordinate finite-task confidence certificate and a reporting standard separating direct estimates, identified sets, and model-conditioned forecasts.	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Arxiv announce typearxiv_announce_type1

Claim history for Arxiv announce type
Value	Valid from → to	Status	Source	Confidence	Extractor
cross	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

arXiv idarxiv_id1

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Value	Valid from → to	Status	Source	Confidence	Extractor
2609.09245	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Authorsauthors1

Claim history for Authors
Value	Valid from → to	Status	Source	Confidence	Extractor
Pranav Singh, Prashant Singh	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Categoriescategories1

Claim history for Categories
Value	Valid from → to	Status	Source	Confidence	Extractor
stat.ML, cs.AI, cs.LG	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

PDFpdf_url1

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https://arxiv.org/pdf/2609.09245	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Primary categoryprimary_category1

Claim history for Primary category
Value	Valid from → to	Status	Source	Confidence	Extractor
stat.ML	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Publishedpublished_at1

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Value	Valid from → to	Status	Source	Confidence	Extractor
11 Sept 2026	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.AI	feed	T1· Official	6 h ago	1

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