All Research

Abstract

We develop a quantitative covariance formulation of the Riemann hypothesis through arithmetic Weil positivity. In the original odd weighted realization, RH is equivalent to liminf as M tends to infinity of sqrt(M)(gamma_M - 1)_+ = 0, where gamma_M is the complete covariance norm at ridge 1/M after any sequence of finite genuine-null restrictions. We prove that the complete covariance norm differs from one by O_{b,epsilon}(M^(-1/2+epsilon)) for every epsilon > 0 at each prescribed ridge M^(-b), b > 0 fixed, uniformly over all finite genuine-null restrictions. The estimate includes every negative-source frequency and all mixed terms. Its proof combines a complete arithmetic source majorant, theta-tail bounds, the Vinogradov-Korobov prime-number error and a uniform actual-inverse construction. Genuine derivative nulls then provide complete source coordinates and signed Schur certificates. A complete complementary cap bounded by a fixed q < 1, together with source-normalized physical lift cost o(sqrt(M)), suffices for RH. We prove uniform small-set inverse bounds throughout [0, exp(A sqrt(log(M)))] for every fixed A > 0, unconditionally. Factorial physical lift budgets and actual inverse-weighted exterior and null-spectrum bounds quantify the construction. These results retain all mixed terms and give complete signed criteria for vanishing original-norm allowances and the Riemann hypothesis. A second cofinal RH criterion uses explicit M^(o(1))-dimensional Schur matrices retaining the exact infinite receiving inverse, with a proved complete receiving floor and only O(log(M)) positive Green responses. The response reduction preserves the original-norm vanishing-allowance scale. A finite-sine core gives joint Schur error O(sqrt(log(M))/M^2) in physical core coordinates.