A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function

Abstract

We introduce a Gaussian--Perron prime-force defect that compares a smoothed prime-side logarithmic force with the logarithmic derivative of the Riemann zeta function. The construction turns the explicit formula into a local diagnostic for zero geometry. Its kernel produces an error-function prime weight and an anisotropic zero-side damping law, with an explicit boundary separating amplified and suppressed nonlocal zero contributions. We prove an exact zero-side formula, derive a universal selected-zero profile on the logarithmic scale, and formulate a finite-window damping certificate for non-selected residues. Under explicit damping, pole, and contour-regularity hypotheses, these ingredients localize the full defect near a selected zero. Assuming the Riemann Hypothesis and the stated pole-damping condition, the full defect near each fixed simple critical-line zero has the selected-zero profile up to an exponentially small nonlocal remainder. The framework provides a local diagnostic for zero geometry associated with the Riemann zeta function.