A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion

Abstract

This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--P\'olya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law $A_k(a) = \log(1/a) + \log(k-2) + B_0 + O(a)$, with $B_0 = \log q - 2\log 2$, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient $B(\nu)$ and shows $B_0$ depends only on the conductor $q$, independent of the Archimedean parameter. (R2b) The degree $d$ of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, $S(a) \sim (2a)^3/6$. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line $\sigma^*(a)$ descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue $\lambda_1(a)$ is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function $W(a,0;z)$ is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.