{
  "abstract": "Reinforcement learning with verifiable rewards (RLVR) makes a language model solve more problems on the first try and fewer when given many tries: it raises pass@1 while the base model overtakes it at large sampling budget k. Is this a failure of RL to add reasoning, or a transient that more training removes? We settle the question with a theorem. The reversal is a price, and we compute its exact rate. Modeling RLVR as an exponential tilt of a base policy over strategies shared across prompts, we prove that under exclusive specialization every update that raises pass@1 strictly lowers the success of some prompt; a strictly positive price on prompts (Stiemke duality) is the exact condition, and being merely the best strategy on a prompt is provably not enough. When the loss reaches the hardest prompts, it forces a pass@k crossover, and the number of crossings is at most the sign variation of the shift in per-prompt success, a totally positive bound that is tight and that recovers the known single-crossover result as its order-preserving case. The crossover is not a bug to patch: a constrained update traces the exact pass@1 versus pass@k frontier, and removing the crossover entirely costs the entire pass@1 gain. The same tilt identity yields entropy collapse and the entropy-performance law with an identifiable exponent, unifying two literatures under one mechanism. Reinforcement cannot widen the reasoning frontier; it can only move along it, at a price we make measurable by a diagnostic that reads the bound, the order-preserving test, and the price from any evaluation log. There is no free lunch, only a price.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39295",
  "author_total": 1,
  "authors": [
    {
      "name": "Yinuo Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4b6hskfgt1k9h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39295",
  "fulltext_url": null,
  "id": "4b6hskfgt1k9h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39295",
  "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",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4b6hskfgt1k9h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "There Is No Free Lunch in RLVR: Every pass@1 Gain Has a pass@k Price",
  "title_text": "There Is No Free Lunch in RLVR: Every pass@1 Gain Has a pass@k Price",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:00Z",
  "url": "https://research.draconitic.com/paper/4b6hskfgt1k9h",
  "venue": "arXiv",
  "versions": []
}