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

arXiv: https://arxiv.org/abs/2607.13918 · HTML: https://mlsanity.com/paper/2607.13918/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 22 references that are in the map.
- First posted on arXiv 15 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Large Language Models Cannot Self-Correct Reasoning Yet](https://mlsanity.com/paper/2310.01798/) (2023)
- [When Can LLMs Actually Correct Their Own Mistakes? A Critical Survey of Self-Correction of LLMs](https://mlsanity.com/paper/2406.01297/) (2024)
- [Provable Scaling Laws for the Test-Time Compute of Large Language Models](https://mlsanity.com/paper/2411.19477/) (2024)
- [Nine Judges, Two Effective Votes: Correlated Errors Undermine LLM Evaluation Panels](https://mlsanity.com/paper/2605.29800/) (2026)
- [Self-Correction Bench: Uncovering and Addressing the Self-Correction Blind Spot in Large Language Models](https://mlsanity.com/paper/2507.02778/) (2025)
- [Variation in Verification: Understanding Verification Dynamics in Large Language Models](https://mlsanity.com/paper/2509.17995/) (2025)
- [When Does Combining Language Models Help? A Co-Failure Ceiling on Routing, Voting, and Mixture-of-Agents Across 67 Frontier Models](https://mlsanity.com/paper/2606.27288/) (2026)
- [Demystifying LLM-as-a-Judge: Analytically Tractable Model for Inference-Time Scaling](https://mlsanity.com/paper/2512.19905/) (2025)
- [Phase Transition for Budgeted Multi-Agent Synergy](https://mlsanity.com/paper/2601.17311/) (2026)
- [Jailbreak Scaling Laws for Large Language Models: Polynomial-Exponential Crossover](https://mlsanity.com/paper/2603.11331/) (2026)
- [Two Calls, Two Moments, and the Vote-Accuracy Curve of Repeated LLM Inference](https://mlsanity.com/paper/2605.03379/) (2026)
- [Odds Law: The Decomposition Algebra On How Intelligence Organizes Itself to Solve Difficult Problems Reliably](https://mlsanity.com/paper/2606.15712/) (2026)
- [When Can Voting Help, Hurt, or Change Course? Exact Structure of Binary Test-Time Aggregation](https://mlsanity.com/paper/2605.05592/) (2026)

## Built on by (most cited first, 2 of 2)

- [Engineering Reliable Commit Gates for Agentic AI: Cost-Aware Verification Portfolios under Common-Mode Data Failures](https://mlsanity.com/paper/2609.10969/) (2026)
- [Requirement-Bound Verified Commissioning: A Frozen Four-Billion-Parameter Local Model as a Candidate Generator under an External Acceptance Layer with Verification and Release Authority](https://mlsanity.com/paper/2609.30219/) (2026)

## Abstract (arXiv)

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^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 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.