Decodable but Not Faithful: Coupling Natural-Language Rationales to Programmatic Verifiers

Language models can generate plausible rationales for their predictions, but these explanations may not faithfully represent the model's internal reasoning. We propose verifier-coupled reasoning, a framework that inserts inline claims into reasoning traces and trains an auxiliary consistency head to predict programmatic verifier outputs from rationale-span hidden states. The central finding is a gap between decodability and faithfulness: consistency training reliably makes verifier information decodable from rationale representations, but decodability does not guarantee faithful generation. In LeanCheck (formal theorem proving), rationale-only and proof-only pooling achieve perfect directional separation under counterfactual conflict. In KataGo (Go engine), commentary spans encode 10-way win-rate buckets at 81\% accuracy. Yet in a code setting, the model achieves 98.6% coupling while its generated explanations remain unfaithful: fluent prose with correct structured claims, but describing unrelated algorithms; a controlled pretrained-vs-from-scratch comparison shows the gap is not capacity-driven. Synthetic activation patching confirms causal influence (73-89\% vs. 31\% baseline), FEVER reveals that evidence-only pooling isolates genuine evidence sensitivity at the cost of raw accuracy, and per-claim analysis shows that consistency loss disproportionately benefits fine-grained claims over binary ones. These results establish that consistency losses are effective diagnostics and representation-shaping tools, but not sufficient conditions for faithful reasoning.