Spectral Rigidity of the Euler Product and the Resolvent Inequality in the Weil Quadratic Functional

We investigate both analytically and experimentally the spectral rigidity of the extended Weil quadratic functional Q_delta across families of L-functions in GL(1) and GL(2). We prove that Dirichlet–Hecke multiplicativity algebraically annihilates the compound remainder operator (R = 0), thereby confining the operator spectrum to a Hilbert space Pi_0 governed by the uniform continuous resolvent inequality ||A^(-1/2) P A^(-1/2)||L^2 <= 1 - epsilon. In contrast, for linear combinations lacking an Euler product (such as Davenport–Heilbronn type functions and non-automorphic modular mixtures), the remainder operator R induces constructive non-harmonic interferences that overwhelm the archimedean envelope, opening negative Minkowski cones in Pontryagin spaces Pi_kappa with kappa > 0. We validate this analytical framework via intensive out-of-sample computational benchmarks: Achieved a 92%–97% detection rate of off-critical-line zeros in non-Euler functions (DH41, DH29, and DH5 in high frequencies up to t = 1000). Confirmed an identically empty map (zero negative eigenvalue detections) for all evaluated automorphic L-functions endowed with an Euler product (chi_3, chi_4, chi_5, chi_7, chi_8, and elliptic curve 11a1). Demonstrated a massive two-order-of-magnitude spectral divergence in GL(2) between pure Hecke eigenforms and artificial non-automorphic linear combinations. Complete source execution scripts in Python and tabulated numerical datasets are included.