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Abstract

We give a finite-verifiable criterion for the Riemann hypothesis through arithmetic Weil positivity on its original odd weighted domain. RH is equivalent to the existence of strictly positive finite certificates along an unbounded sequence of explicit growing supports. Each certificate retains every active prime power, the polar constraint, all mixed terms, and the entire receiving complement. Its entries use finite function integrals and a finite constraint-Gram inverse. Under RH, every admitted cutoff has a strict certificate from a computable trial family. A complete physical Green-tail estimate, a positive dyadic theta-ridge floor, and a stable finite-sine core establish this conditional completeness with explicit residual and rounding tolerances. A quantitative receiving-return enclosure gives an explicit sufficient finite trial-size bound. The core has subpolynomial dimension; response and potential representations use logarithmically many cells and features up to a log-log factor. We also prove a complete arithmetic covariance estimate of order M^(-1/2+epsilon) at every fixed polynomial ridge, uniformly over finite genuine-null restrictions. Exact signed identities, physical lift estimates, and complete complementary bounds connect that quantitative formulation to the same Weil positivity endpoint. All realization-specific supporting proofs are included; an indexed technical supplement provides alternative exact-response methods and reproducible numerical applications.