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Life sciences · Preprint
arXiv · September 9, 2026
Raises a question worth testing. It does not answer one.
This preprint proposes a theoretical framework (joint-Gaussian model of verifier-gold correlation) and empirical approach ('proof-carrying cognition' with reality-settled rewards) to reduce reward hacking in language-model reasoning. Experiments show that unsound verifiers degrade under optimization pressure in program synthesis, while reality-anchored settlement and stronger judges are more robust, with potential ~10-fold label efficiency gain. However, the work remains unrefereed, is confined to formal domains with executable ground truth, lacks clear applicability to open-ended reasoning, and does not report absolute sample sizes or generalization evidence.
Mixed methods: theoretical joint-Gaussian model, controlled program-synthesis experiments with executable ground truth, and uncontrolled LLM judge robustness assessments. Program-synthesis tasks with executable ground truth; language models with weak and stronger LLM judges; systems trained with GRPO. No human subjects or clinical population.. Intervention: Reality-anchored settlement of reasoning rewards; refit reward model on 10% settlement stream; proper scoring rules; self-built world model trained on held-out reality.. Compared with: Frozen verifier; unsound verifier; weak LLM judge; random labeling; selection alone without reward model refit..
Unsound verifier soundness degrades from 0.94 to 0.32 at N=4096 under optimization; sound verifier improves monotonically. Reality-anchored settlement reduces hacking gap from ~0.27 to ~0 under adversarial pressure. Soundness scales log-linearly with settled labels; on-policy settlement ~10x more label-efficient than random labeling.
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This preprint presents theoretical modeling, controlled demonstrations in formal domains (program synthesis), and a proposed paradigm without peer review or evidence of real-world clinical or deployed reasoning system validation.
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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.
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