{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09245",
  "author_total": 2,
  "authors": [
    {
      "name": "Pranav Singh"
    },
    {
      "name": "Prashant Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/evfky8nhks1a9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09245",
  "fulltext_url": null,
  "id": "evfky8nhks1a9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09245",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7212143567",
  "pdf_url": "https://arxiv.org/pdf/2609.09245",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/evfky8nhks1a9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:22Z",
      "native_id": "2609.09245",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09245"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "What Fixed-Rollout pass@k Evaluations Can Identify",
  "title_text": "What Fixed-Rollout pass@k Evaluations Can Identify",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:22Z",
  "url": "https://research.draconitic.com/paper/evfky8nhks1a9",
  "venue": "arXiv",
  "versions": []
}