A Spectral Stability Framework for Finite Gram Matrices Derived from the Weil Explicit Formula

We present a Fast Diagonalization framework that transforms explicit-formula analysis into aspectral stability criterion for finite Gram systems. Logarithmic coordinates reveal phase coherenceon the arithmetic side, imposing symmetric constraints on the spectral side through windowedlocalization.The framework establishes a hierarchy of verifiable conditions: numerical diagnostics, densityonestability, and uniform spectral stability. A modular Rigidity Interface ensures robustness underrefinement. Numerical implementation shows that increasing height T amplifies rather than obscurespotential deviations, enabling global spectral monitoring instead of local verification. Only time will tell if the decision taken by arXiv to not accept the article was the right one.