# When Verifiers Vote Backwards under Verdict Substitution: Signed Pivotal Value in Correlated Self-Consistency

arXiv: https://arxiv.org/abs/2609.26144 · HTML: https://mlsanity.com/paper/2609.26144/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 0 of its 12 references that are in the map.
- First posted on arXiv 8 Aug 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Abstract (arXiv)

Replacing one ballot can change a majority decision only on queries decided by a single vote; this structural fact requires no independence assumption. We study the sign of that change using a labeled, verdict-style intervention: one correctness signal replaces one correctness-indicator ballot in k=7 self-consistency panels. This diagnostic intervention is not identical to deployed answer-identity plurality. A primary MATH-500 experiment (n=570) gives a different-model verifier a +24.2pp pivotal gain, whereas a role-reversed configuration gives -11.2pp; an exploratory code stress test (14 pivotal rows across 9 tasks) gives -24.5pp. An exact signed-gain decomposition accounts for all observed signs through the verifier's state-specific accuracy and the composition of the two one-vote tally states, rather than global accuracy or model provenance. Same-source signals lose accuracy on the pivotal stratum (65→44% in the primary configuration), while error correlations provide a descriptive error-association diagnostic. Controlled degradation and a k∈3,5,7 subset sensitivity analysis probe the stability of the observed pattern around this accounting. Under the evaluated ties-incorrect answer-identity plurality analysis, the structural zero and strong-verifier benefit persist, but the role-reversed harm attenuates to -0.9pp and is not significant. The results therefore establish harmful verdict substitution, not universally harmful deployed plurality, and motivate a testable but unverified hypothesis for negative process-reward-model weights.