Agreement Before Diversity: Verification-First Complementarity for Heterogeneous Language-Model Coordination

Heterogeneous language-model ensembles expand the space of candidate responses, yet they lack a principled criterion for when a newly generated answer should supersede an already supported one. We decouple candidate headroom from replacement authority, rendering the latter as an explicit, auditable object. Our proposed method, Agreement-Before-Diversity (ABD), is a frozen, label-free decision rule: an anchor answer is retained if two additional trusted samples corroborate it under a fixed equivalence relation; otherwise, it is replaced by a heterogeneous synthesis. For this gating mechanism, we prove two exact identities. The first shows that the accuracy gap relative to unconditional synthesis is determined jointly by the agreement coverage and the anchor's advantage on the protected subset. The second shows that the gap relative to never synthesizing reflects a contrast between authorized recovery and authorized destruction. Neither identity assumes independence or calibrated confidence, and the expected inference cost is approximately eight minus five times the coverage in number of calls. Under blind, exact-ID evaluation, ABD achieves 59.43% on the complete LiveCodeBench-v6 (vs. 52.57% for Single9 and 52.00% for HAC; n = 175) and 75.00% on an untouched GPQA-Diamond split (both controls at 72.78%; n = 180). Furthermore, these identities localize every aggregate difference to an enumerable protected stratum: no discordant items occur among the 3 protected cases on LiveCodeBench, where coverage bounds the gate's contribution to 1.71 points a priori; 13 versus 8 discordant cases among 132 on GPQA-Diamond; and 12 versus 0 among 71 under a frozen anchor perturbation. Diversity supplies potential; verification structure supplies authority.