VOUCH for Mathematical Reasoning: Counterfactual Contracts Cure the Self-Verification Pathology

Multi-agent verification of chain-of-thought math typically asks one large language model to commit to a counterfactual prediction (“if I change a number in the problem, my answer becomes y′”) and rewards the agent when its post-intervention output matches the commitment. We prove that this single-side self-verification is unsound: a self-coherent reasoner satisfies its commitment by construction, independent of correctness, so the protocol cannot distinguish a careful solver from a confidently wrong one. We document the pathology on GSM8K, the algebra subset of MATH, and SVAMP under a controlled adversarial-prompt regime, where task accuracy falls by up to 28 percentage points while the protocol’s self-verification rate rises by up to 11.6 points. Our fix is a dual-side counterfactual contract: the proposer also commits to a second agent’s post-intervention prediction, a cross-side claim that self-coherence cannot fulfill. Across all six math cells the dual-side construction reduces the hurt:caught ratio by a mean of 46% (per-cell 27% to 68%), with 95% bootstrap CIs strictly above zero on every cell. A same-family LLM-as-judge baseline (Llama-3.1-8B) and self-consistency at k = 5 both inherit the same self-coherent pathology, confirming that the fix is structural rather than a function of sampling or arbitration.