Spectral Rigidity and Zero Exclusion via Residue Projection: A Finite-Rank Framework for Dirichlet L-Functions

We develop a spectral–geometric framework for excluding zeros of Dirichlet L-functions based on the structural rigidity of arithmetic residue operators. The central mechanism is a proof of incompatibility between zero-induced forcing—which amplifies high-prime contributions—and a spectral ceiling imposed by mixing on the residue dictionary after the removal of a finite coherent subspace. The paper establishes two primary results: Numerical Certification: For the modulus set Q0={3,5,7,11,13}, we certify the coercivity of the projected Gram operator by direct computation for packet scales X∈[100,5000], yielding unconditional zero exclusion in this regime. Analytic Stability (RC∞): We prove that this coercivity persists asymptotically. After projection onto the mean-zero residue subspace, the Gram operator converges to a strictly positive definite limit (c∞=1/12), establishing the Rank Condition at Infinity. The framework operates through spectral rigidity rather than traditional dispersion or density estimates. By isolating the remaining arithmetic input into a "Moment Pumping" hypothesis, we show that zero-free regions are a structural necessity of the residue-operator geometry.