Spectral Riccati--Gamma Concavity, Symmetric Zero Cancellation, and Conditional Criteria for the Riemann Hypothesis

Abstract

We examine a Riccati--Gamma approach to the logarithmic derivative of the completed Riemann zeta function. The first part proves, in full local detail, that a naive two-sided vertical concavity criterion for $\Xi'/\Xi$ cannot be a proof of the Riemann Hypothesis, because every zero produces opposite vertical curvatures on the two horizontal sides of the pole of the logarithmic derivative. The second part replaces this obstruction by a rigorously formulated finite spectral averaging framework. We prove cancellation at the critical line, positivity of the off-critical paired contribution on the left of the critical line under a concrete low-frequency kernel condition, a conditional zero-density consequence, and a precise conditional theorem showing which additional localisation hypotheses would imply the Riemann Hypothesis. The results are therefore not presented as an unconditional proof of RH. They give a partial resolution of the Riccati--Gamma question: one natural route is ruled out unconditionally, a second symmetric mechanism is proved at the finite spectral level, and the remaining step is isolated as explicit analytic hypotheses. Reproducible Python routines and numerical figures accompany the analytic discussion.