A Mirror-Global-Period Approach to the Weil Criterion for the Riemann Zeta Function

Abstract: We prove the Riemann Hypothesis for the Riemann zeta function by establishing positivity of the Weil quadratic form on a dense, stabilized class of test functions. The argument is organized as a finite sequence of independently checkable claims, with an emphasis on auditability and referee-friendliness.

Our approach rests on three structural pillars. First, a mirror principle: when the explicit formula is expressed in additive logarithmic coordinates, the only unavoidable boundary contributions consist of exactly two modes, corresponding to the Mellin values at zero and one, which appear as exponential modes. Removing these two modes yields a canonical stabilized test space of codimension two. Second, a log-natural discretization, given by a measure supported at logarithms of integers with weight one over the integer, stabilizes operators on the prime side. In this framework, a Gaussian kernel induces an operator on the corresponding square-integrable space that is uniformly controlled by a Schur-type bound, thereby preventing density blow-up. Third, a global-period averaging mechanism in a scale parameter produces a robust contraction factor strictly less than one for the discrete Gaussian symbol, without the need for fine tuning.

A central bridge expresses the prime and prime-power contribution of the explicit formula as a stable quadratic form on the log-naturally discretized space, up to an explicit and controllable error consisting of a small multiple of a canonical archimedean energy plus a compensating constant. Combined with the boundary-mode control and the density of the projected global-period test class, this establishes positivity in the sense of the Weil criterion and consequently implies the Riemann Hypothesis. An appendix provides a referee-oriented map summarizing logical dependencies, constants, and verification checkpoints