Low-Rank Zero-Mode Cancellation in the Riemann–von Mangoldt Explicit Formula via Operator-Coupled Kernels

This deposit contains a standalone preprint proposing a concrete finite-rank operator coupling acting on smoothing kernels in the Riemann–von Mangoldt explicit formula. The coupling is constructed to null the weighted contributions of a prescribed finite set of nontrivial zeros (“zero-modes”) by solving a finite linear system Mc=bMc=bMc=b. The note proves existence/uniqueness under an invertibility condition, gives stability bounds controlled by ∥M−1∥\|M^{-1}\|∥M−1∥ (conditioning), and defines a non-ambiguous projected residual EaE_aEa via orthogonal projection in a weighted Hilbert space on a finite window [X,2X][X,2X][X,2X]. Included are reproducible numerical diagnostics (envelope fits and ratio plots) illustrating finite-window oscillation reduction under low-rank cancellation, plus a minimal script scaffold for experiments.