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

Abstract

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 $+24.2$pp pivotal gain, whereas a role-reversed configuration gives $-11.2$pp; an exploratory code stress test (14 pivotal rows across 9 tasks) gives $-24.5$pp. 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$\to$44\% in the primary configuration), while error correlations provide a descriptive error-association diagnostic. Controlled degradation and a $k\in \{3,5,7\}$ 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 $-0.9$pp 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.