CoRD: Conjecture-and-Refutation Decoding for Evidence-Gated Overrides in Collaborative Reasoning

Token-level collaboration enables an LLM to assist an SLM at selected reasoning states, but existing methods often rely on Teacher preference to rank local candidates. Such preference does not establish whether replacing the SLM's current action improves its own subsequent reasoning. We propose CoRD(Conjecture-and-Refutation Decoding), which reformulates local collaboration as intervention verification. At each triggered state, CoRD forms an incumbent–challenger pair, rolls out short candidate-specific Student futures, and asks an LLM verifier to determine whether the incumbent exposes a reasoning error and the challenger avoids it. An override is made only under this asymmetric evidence, followed by a short verified suffix to stabilize the correction. Across six mathematical reasoning benchmarks and two Teacher scales, CoRD outperforms Teacher takeover and S2T in 11 of 12 settings; with a 32B Teacher it raises Math Avg. from 24.81% to 40.50% over Student-only decoding while keeping generation under Student control. Frozen-state counterfactual analysis further shows that accepted interventions achieve 27.27 percentage points of downstream utility, compared with only 1.20 percentage points for rejected interventions under a 16-token verified suffix. Suffix commitment also eliminates the observed token-only damage. These results support moving token-level collaboration from candidate preference toward evidence-gated correction.