Prime–Boundary Synchrony for the Riemann Zeta Function: Certified Instruments for a Weil-Positivity Program

Abstract

This Zenodo record archives the canonical materials for Aaron Woffinden's 2026 preprint Prime–Boundary Synchrony for the Riemann Zeta Function: Certified Instruments for a Weil-Positivity Program, affiliated with the Institute of Unitive Cross-Domain Morphology. The record contains the manuscript PDF, a LaTeX source archive, a supplementary code archive, and this README. The code reproduces every number in the manuscript's verification sections: the certification of both Synchrony identities with singularity-aware quadrature split at the zeta-zero ordinates, the six-identity auxiliary battery, the Pick-orientation samples, and the finite-prime truncation ladder. The paper supplies the instrument-certification layer for a research program targeting the positivity of Weil's quadratic functional, a statement equivalent to the Riemann Hypothesis. It fixes exact normalizations for, proves, and machine-certifies an order-zero Prime–Boundary Synchrony equation — a Poisson average of log|(s−1)ζ(s)| over the critical line equals an explicit prime-power sum plus a nonnegative defect vanishing identically if and only if RH holds — and its order-one derivative form, in which the arithmetic side sharpens to the raw von Mangoldt spectrum. Certified residuals reach 10⁻⁵ at order zero and 10⁻⁷ at order one. The record also documents a numerical-methodology finding of independent interest: fixed-grid quadrature fabricates a spurious, cutoff-stable defect of order 10⁻², because the defect channel lives inside the logarithmic dips at the zeros — singularity-aware sampling is mandatory. An unconditional half-line positivity corollary and a certification protocol for the Weil functional are included. The paper proves no case of the Riemann Hypothesis, claims no progress on Weil positivity, and asserts no new conjecture; the identities assemble classical facts, and their contribution is certified exactness of the stated normalizations.