Phase Moments on the Critical Strip

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 returns a quantity quadratic in the zeros that no weight produces. Read 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), four primes carry half its power, and its second moment separates tight zero configurations at AUC 0.868 where Turing's own statistic is at chance. A function-field control closes the note — there the same second moment is equivalent to the Riemann Hypothesis, by Weil's bootstrap read through Parseval, and what removes the equivalence for ζ is not the absence of an Euler product but the divergence of Σ 1/p. 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.