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When the marginal cost of generating scientific prose, code, and artifact-shaped analyses approaches zero, the scarce input that anchors scientific credibility becomes verification: time, compute, experimental runs, audits, replications, and methodological discipline deployed to discriminate truth from error. This paper introduces the \emph{verification budget} as a first-class institutional variable and proposes a resource law for modern science: holding fixed a community's verification capacity, increases in claim generation mechanically dilute verification per claim, reducing expected reliability unless institutions explicitly reprice credibility by making verification budgets legible, comparable, and rewarded. We define claims, evidence, and verification budgets; model credibility as a function of verification allocation using Bayesian information acquisition and microeconomic models with information asymmetry and costly signaling; and analyze three institutional equilibria---generation-driven, verification-driven, and regulatory-driven-that emerge as verification becomes the binding constraint. We derive two paradoxes of the near-zero generation regime: paper inflation, where publication volume can grow faster than the stock of verified knowledge, and evidence dilution, where average evidential strength per claim declines as claim supply explodes. To operationalize the framework, we propose a mandatory, machine-readable Verification Statement that inventories key claims, maps each to verification modalities, and declares budgets in standardized units, supported by stake-based calibration and random audits. The proposal aims to reduce credibility-asymmetry between authors and evaluators, mitigate adverse selection in scientific claims, and create scalable governance compatible with AI-accelerated research production.