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

Formal verification guarantees proof validity but not formalization faithfulness. For natural-language logical reasoning, where models construct axiom systems from scratch without library constraints, this gap between valid proofs and faithful translations is especially acute. We investigate whether frontier models exploit this gap when generating Lean 4 proofs, a behavior we term formalization gaming. We evaluate GPT-5 and DeepSeek-R1 on 303 first-order logic problems (203 from FOLIO, 100 from Multi-LogiEval), comparing unified generation against a two-stage pipeline that separates formalization from proving. Despite compilation rates of 87-99%, we find no evidence of systematic gaming in unified generation: models prefer reporting failure over forcing proofs, even under prompting designed to encourage it. However, unfaithfulness that evades our detection signals may still occur. The two-stage pipeline reveals two distinct modes of unfaithfulness: GPT-5 fabricates axioms during proof generation, a reactive fallback detectable via cross-stage comparison, while DeepSeek-R1 mistranslates premises during formalization, producing internally consistent outputs that evade detection entirely. These findings show that high compilation rates or accuracies should not be equated with faithful reasoning. Code and data are available at https://github.com/koreankiwi99/formalization-gaming.

Chain-of-thought (CoT) reasoning is useful for monitoring language models only when the reasoning trace faithfully reflects the computation that produces the final answer. However, models can rely on prompt-to-answer shortcuts that bypass the CoT, making the visible reasoning trace misleading even when it appears plausible. We study CoT faithfulness through a structural information-flow perspective: faithful reasoning should route answer-relevant information through the mediated path from prompt to CoT to answer, rather than through a direct prompt-to-answer shortcut. This perspective yields a task-agnostic framework based on three complementary properties, sufficiency, completeness, and necessity, which we instantiate with entropy-based, masked-KL, and gradient-based diagnostics. We show that these metrics recover externally judged faithfulness differences in hinted reasoning, and identify a low-entropy failure mode of KL-based diagnostics where gradient-based measures remain more stable. Building on this analysis, we introduce update-time interventions for verifier-based on-policy RL, including attention masking, backward-only gradient masking, CoT gradients, and adversarial perturbations of prompt representations. Across hinted arithmetic, reward-hackable code repair, and DAPO-Math models trained without hints but evaluated under wrong-hint injection, our interventions shift behavioral and structural indicators toward stronger CoT mediation. In particular, they make shortcut and reward-hacking behavior more transparent in the CoT and improve task-agnostic faithfulness metrics, while in some settings also reducing wrong-hint susceptibility. Our results suggest that controlling information flow during training is a practical route toward more faithful and monitorable CoT reasoning. Code is available at https://github.com/safety-research/faithful-cot.

Frontier gains in language-model reasoning come from reinforcement learning on reasoning traces and are concentrated in domains with a cheap, sound verifier. We argue the field's binding constraint is the verification gap: no scalable, incorruptible reward for reasoning outside formal domains. We make four contributions. (1) Theory: in a joint-Gaussian model of best-of-N selection, verifier-gold correlation rho is the exact exchange rate between test-time compute and capability, and an unsound verifier pays a polynomial penalty N^(1/rho^2); a margin-free copula form predicts realized soundness of real LLM judges to 4% median error. (2) Demonstration: in program-synthesis testbeds with executable ground truth, including a pre-registered scaled replication, unsound verifiers lose Soundness-under-Pressure as optimization grows (0.94 to 0.32 at N=4096) while a sound verifier improves monotonically; reality-anchored settlement beats a frozen verifier under i.i.d. and adversarial pressure, driving the hacking gap from ~0.27 to ~0; soundness scales log-linearly with settled labels, with on-policy settlement ~10x more label-efficient than random labeling. With real LLM judges and unit-test execution as gold, a weak judge loses soundness under best-of-N (p<0.001), a stronger judge is more robust, and selection alone manufactures +0.53 hacking gaps from honest samples. Under real GRPO training, a frozen reward model traces the full overoptimization curve (executed reward collapses 90%) while the same model refit on a 10% settlement stream preserves 6x the executed reward. (3) Paradigm: proof-carrying cognition, where reasoning steps are typed probabilistic claims priced by a self-built world model trained only on held-out reality and settled by proper scoring rules. (4) Benchmark: we specify Soundness-under-Pressure as the headline metric for a reality-settled reasoning benchmark.