揭示大模型验证链在相关性下的可靠性极限，提出可测量的理论框架。
Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings
- 用潜变量建模验证器相关性，推导出对数似然随门数呈凹函数关系。
- 在相关条件下失败率仅多项式下降，且存在无法突破的可靠性天花板。
- 实测表明独立假设会严重低估错误率，该理论可精准预测真实性能。
串行验证门是大模型系统中提升可靠性的核心机制：仅当k个验证器均通过时才返回答案。在条件独立假设下，近期的几率定律（arXiv:2606.15712）表明后验对数似然随k线性增长，失败率呈指数衰减，但指出‘部分相关验证链的紧致理论仍待解决’。本文给出最小化理论：将生成器自身错误上的每实例误接受率建模为潜变量α∼G（de Finetti），则级联后验为ℓ_k = ℓ_0 - ln m_k，其中m_k为G的第k阶矩。结果：(i) 对任意非退化G，ℓ_k在k上严格凹，几率定律为其在第一门处的切线与上界；(ii) 当G为Beta(a,b)分布时，失败率多项式衰减，1−r_k ≍ k^{-b}，相关参数ρ_v = 1/(a+b+1)；(iii) α=1处的盲点原子质量为1−π，限制证据提取上限为−ln(1−π)纳特，导致可靠性无法达1；(iv) 若真接受率也变化（β∼H），则出现三类行为——始终有益、平台化或反而有害，由G、H的上尾指数决定，存在闭式交叉点k†。机制本质为幸存者偏差：通过验证的错误是高α者。理论可测量：每个样本的R次判别可识别G的前R阶矩，两次判别即能估计ρ_v；通过贝塔-二项似然与NPMLE可恢复可靠性曲线及病态上限。合成测试显示，独立假设在k=5时低估失败率20倍，在k=10时高达3000倍；基于R=8的关联拟合准确跟踪保留深度。实践关键在于去相关——改变模型族、模态或证据来源，而非堆叠更多门。
原文摘要 · Abstract (English)
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.
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