Phase Moments on the Critical Strip

Abstract

A window of the critical strip can be read by integrating along its boundary, and this note works out what such a reading returns. One integration returns the winding vector and nothing else; under the reflection ι(s) = 1 − s̄ the value splits into an offset channel and a counting channel, and since the weight s − ½ − it₀ is odd for every real t₀ the centre can be moved without touching the offset. Beyond that, homology reaches exactly the linear moments of the phase error S(t) — Turing's bound is its zeroth — while a path-ordered reading at length three, through Chen's iterated integrals, returns a quantity quadratic in the zeros that no weight produces. Read formally as a wave, S(t) is one sine per prime power, of frequency k log p and amplitude 1/(πk·p^(k/2)); its spectrum, computed from 2500 zeros of ζ, shows a line at log p of height 1/(π√p), and its variance matches Selberg's asymptotic plus Goldston's second-order term to within 2%. An off-line pair is caught by the zeroth moment, which is what Turing's method already checks; the second moment adds a screen for close on-line pairs, ranking them at AUC 0.87 against a null that Berry's formula predicts. A function-field control closes the note — there the same second moment is equivalent to the Riemann Hypothesis for a numerator not known to satisfy it, by Weil's bootstrap read through Parseval. For ζ the equivalence is lost not because Σ 1/p diverges — off the line the corresponding mean square converges and still decides nothing — but because an average cannot see an exceptional set of density zero. Nothing here is a proof: Selberg's second moment is unconditional, so no measurement of it can decide the hypothesis. What the note offers is an instrument built from the functional equation alone, pushed to the point where the one missing inequality can be named.