{
  "abstract": "Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if $k$ verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in $k$, so failure decays exponentially, and states that \"a tight theory of partially correlated verifier cascades remains open.\" This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable $α\\sim G$ (de Finetti), the exact cascade posterior is $\\ell_k = \\ell_0 - \\ln m_k$, with $m_k$ the $k$-th moment of $G$. Then: (i) $\\ell_k$ is concave in $k$ for every non-degenerate $G$ -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta$(a,b)$ latents, failure decays polynomially, $1-r_k \\asymp k^{-b}$, with correlation parameter $ρ_v = 1/(a+b+1)$; (iii) a blind-spot atom of mass $1-π$ at $α=1$ caps the evidence extractable from any number of gates at $-\\ln(1-π)$ nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary ($β\\sim H$) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of $G$ and $H$, with closed-form crossover $k^\\dagger$. The mechanism is survivorship: errors surviving gates are the high-$α$ ones. The theory is measurable: $R$ repeated verdicts per instance identify the first $R$ moments of $G$, so two verdicts identify $ρ_v$; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at $k=5$ and ~3000x at $k=10$; the correlated fit at $R=8$ tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.13918",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiangang Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1bjaste1vq4dc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.13918",
  "fulltext_url": null,
  "id": "1bjaste1vq4dc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.13918",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
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      "arxiv",
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    [
      "openalex",
      "1702|Adversarial Robustness in Machine Learning"
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  "openalex": "W7168617473",
  "pdf_url": "https://arxiv.org/pdf/2607.13918",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1bjaste1vq4dc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2607.13918",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.13918"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings",
  "title_text": "Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/1bjaste1vq4dc",
  "venue": "arXiv",
  "versions": []
}