Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings

TL;DR — A theoretical note modeling serial verifier cascades in LLM harnesses (where an answer is returned only if all k verifier gates accept), extending the "Odds Law" to the realistic case where verifier errors are correlated. It matters because assuming independent verifiers drastically overstates reliability, misinforming how agentic verification pipelines are designed.

• Treating the per-instance false-accept rate as a latent variable α∼G (de Finetti), the exact cascade posterior log-odds is ℓ_k = ℓ₀ − ln m_k (m_k = k-th moment of G); this is concave in k, so the linear Odds Law is only a tangent/upper bound, not the true decay.
• Correlation changes decay regime: Beta(a,b) latents give polynomial (not exponential) failure decay, 1−r_k ∼ k^(−b), with correlation parameter ρ_v = 1/(a+b+1); a "blind-spot" mass at α=1 caps total extractable evidence at −ln(1−π) nats, so reliability saturates below 1.
• Letting the true-accept rate also vary (β∼H) yields a trichotomy — added gates eventually help, plateau, or actively harm — determined by upper-tail exponents of G and H, with a closed-form crossover depth k†.
• The theory is measurable (R repeated verdicts identify the first R moments; 2 verdicts give ρ_v via beta-binomial/NPMLE); synthetic tests show independence-based extrapolation underestimates failure ~20× at k=5 and ~3000× at k=10, and the key practical lever is decorrelation (changing model family/modality/evidence source), not stacking more gates.