Arithmetic satellites of the Bragg comb of the Riemann zero lattice: residue classes, characters, and a blind test of the Bogomolny–Keating mirror law

Companion notes found that the per-gap response of the Riemann zeros to the prime waves of the explicit formula is partly cancelled by the waves beyond any finite truncation, through a kernel concentrated on the Bragg comb of the lattice of gap midpoints. We study the satellites of that comb. Their positions form a height-independent grid Δω = ω − log(t/2π) = log(a/b). A stationary-phase analysis predicts that a satellite reads the residue class of the contributing prime power modulo a: the naive prediction of rotating class phases fails, while the refined law — class shares cos(2πrb/a)/μ(a) — passes a blind test at +log 10 (two other satellites follow its signs but exceed its magnitudes). On the zeros of real Dirichlet L-functions, satellites involving a prime that divides the conductor vanish, and the cancellation keeps the sign of the zeta case for both parities. The Euler product of the Bogomolny–Keating pair correlation gives the satellite amplitudes in closed form. It implies a mirror law — on the negative side the envelope is exactly 1/x at Δω = −log x — which a pre-registered blind test in a previously unexamined band confirms at 1.023 ± 0.023 of the prediction, excluding a line-density rival by more than 12σ. Coefficients with a positive exponent are transferred with a factor below one that relaxes towards one with height (exponent 1.36 ± 0.21 over three windows; a constant is excluded at 6.4σ); since the ratios-conjecture pair correlation has no such term, this is a property of our mixed zero–prime observable. Nothing here bears on a proof of the Riemann Hypothesis. Code, data, pre-registration files and audit reports: https://github.com/azizran/riemann-zero-statistics (release v1.1). Not peer reviewed. Prepared with extensive AI assistance (Claude, Anthropic); the author takes responsibility for all content.