Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings
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∼ G (de Finetti), the exact cascade posterior is ℓk = ℓ0 − ln mk, with mk the k-th moment of G. Then: (i) ℓk 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−rk asymp k−b, with correlation parameter ρv = 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∼ 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†. 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 ρv; 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.
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BibTeX
@misc{han2026partially,
title = {{Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings}},
author = {Jiangang Han},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.13918},
eprint = {2607.13918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13918},
}
RIS
TY - UNPB
TI - Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings
AU - Jiangang Han
PY - 2026
DA - 2026/07/15
PB - arXiv
DO - 10.48550/arxiv.2607.13918
UR - https://arxiv.org/abs/2607.13918
AB - 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∼ G (de Finetti), the exact cascade posterior is ℓ_k = ℓ_0 − ln m_k, with m_k the k-th moment of G. Then: (i) ℓ_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∼ 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^†. 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.
ER -