The Error Structure of a One-Shot Riemann Zero Predictor

The one number to take from this file. Computing the Selberg prime sum directly at several cut-offs — content that sits above the Nyquist limit and that no periodogram of this data can see — removes 44.299% of held-out residual variance on 130,012 zeros, and 43.631% on a second block of 130,012 zeros that no fit in this deposit has ever touched. 0.668 points apart. Every regressor is evaluated at the prediction, never at the true zero. Every band that buys anything survives a circular-shift sweep and a matched-frequency surrogate. With the prime tones added it is 53.307%, and a predictor goes from 3.48% to 2.26% of a mean gap. 44.299%/43.631% is the pre-registered, five-cut-off grid (§3.6a lists all three pre-specified dictionaries). It is not the largest number in this file: extending the same grid one band further, to , takes cumulative held-out variance removed to 54.695%/53.937% (§3.6) — a later, exploratory extension, reported honestly as such rather than folded into the headline. Not everything else in this file is smaller than 44.3%; most of it is smaller, set aside, or a confirmation of something already known. The band, §3.3b, carries the sign the convention predicts once the test's rows are aligned: the band's sign matches the missing-tail direction it is checked against (§3.3b), and the exact tail past , evaluated at the true zero, accounts for close to half (48.7%/49.1%) of the corrected predictor's own held-out residual variance — the current reading of this figure and a reported anti-correlation no longer hold below. Both compared the residual against quantities whose position is fixed relative to the true zero itself, so each partly compared the true zero's own local position against a copy of itself. Evaluated instead at points that do not move with the true zero, the genuinely smooth tail's relation to the residual falls to a small, correctly-signed , and the useful bands' relation to that same content collapses to (both blocks, down from an apparent ). The bands are best read as resolving the local zero configuration around the prediction, not as recovering missing prime content (§3.3b). The test described as "not yet run" above has now been run: one damped Newton step on , taken from the already-corrected prediction with damping , removes % of the corrected predictor's held-out residual variance on its own — one free parameter, against 54.695%/53.937% from the six-coefficient band regression at the same cut-off (§3.6). This record holds one paper on its own and is refreshed every couple of months. The complete deposit, 10.5281/zenodo.21788037, holds every paper with its code and data and is updated first; for the newest version of this paper, go there.