Decomposing Wrong-Consensus Agreement in LLM Self-Consistency

Abstract

Agreement among repeated samples of a language model is routinely read as evidence about answer reliability, yet wrong answers can agree just as strongly as right ones. This paper asks what information wrong-consensus agreement actually contains, and answers with a quantitative decomposition. A pluralistic agreement index Gamma, normalized by the reference scale d=(1-p)/(C-1), is split into a mechanical component (agreement delivered by a per-case answer preference alone) and a preference-unexplained residual. The mechanical reference is leak-free: each case's preference and accuracy are estimated from its other runs only. On public GPT-4.1 per-run data, coverage phi (the mechanical/empirical ratio) shows a benchmark-associated direction: 0.81-0.93 on multiple-choice GPQA-Diamond against 0.59-0.78 on open-domain AIME, where a residual of 1.54-2.80 Gamma units survives, more than absorbed by a calibrated run-level preference-heterogeneity reference. A controlled replication under one fixed protocol (four runs per question, K=32 votes) on five open-weights checkpoints (Qwen3.5-9B/122B, Qwen3.8-27B, Gemma4-26B/31B) finds near-complete mechanical coverage in all ten cells (phi approximately 1, with a small overshoot consistent with a quantified finite-donor plug-in bias), robust to a two-run design; the largest cell (qwen3.5-122b, p=0.222) sits inside the GPT-4.1 AIME accuracy range and still saturates (phi=1.041). A cross-system contrast at comparable aggregate accuracy contrasts near-complete mechanical agreement in the open-weights models against a larger preference-unexplained residual in the frontier family. This contrast is confounded with sampling protocol by design. Agreement is graded evidence, not certification. No new voting method is proposed; code and evidence are committed.