Recent gains in language‑model reasoning stem from reinforcement learning on reasoning traces, and these gains are concentrated in domains that have a cheap, sound verifier. We argue that the field’s binding constraint is the verification gap: the lack of a scalable, incorruptible reward for reasoning outside formal domains. This paper makes four contributions. (1) Theory: In a joint‑Gaussian model of best‑of‑N selection, the correlation $\rho$ between verifier output and gold answers is the exact exchange rate between test‑time compute and capability; an unsound verifier incurs a polynomial penalty $N^{1//\rho^2}$. A margin‑free copula formulation predicts the realized soundness of real LLM judges with a median error of about 4%. (2) Demonstration: In program‑synthesis testbeds with executable ground truth (including a pre‑registered scaled replication), unsound verifiers lose Soundness‑under‑Pressure as optimization scale grows ( $\rho$ drops from 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, shrinking the hacking gap from ~0.27 to ~0; soundness scales log‑linearly with settled labels, and on‑policy settlement is roughly 10× 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.
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