A Complex-Analytic Law for Riemann Zeta Zeros at Prime Frequencies

For the spectral form factor (SFF) of the first N non-trivial zeros of ζ (s), K (τ) = |Σₙ exp (−iτγₙ) |²/N, the amplitude at frequencies τ=k·log (p) corresponding to prime powers follows the empirical formula Kₘax (k·log p) = C· (log p) ²/pᵏ, motivated by the weight Λ (p) =log (p) of the von Mangoldt function in the explicit formula. We verify this relation at N=2, 000, 000 on 92 primes (5<p<499), obtaining C=16220. 37 with R²=0. 9999997 on correctly-localized measurements, robust to the choice of localization threshold (θ∈0. 80, 0. 99). An independent, unconstrained power-law fit recovers α=−1. 0003 and β=2. 0007 without imposing these exponents, confirmed monotonic and unique via a multi-start, cross-N (100k–1M) check. The paper's central, logically prior result is a complex-analytic law: the raw complex sum A (τ) =Σₙ exp (−iτγₙ), evaluated at τ=log (p), aligns with the negative real axis to within 0. 005% at N=2, 000, 000: A (log p;N) = α (N) ·log (p) /√p + R (p, N), verified across six independent N (50k–2M) with the residual converging as N^ (−0. 85). K = |A|²/N follows as an algebraic consequence rather than an independently-fitted law; two independent regressions (on A and on K) agree on the derived quantity |α (N) | to within 0. 02–0. 14% at every tested N. Further findings: resolution of an apparent N² scaling discrepancy in C (N) =T (N) ²/ (4π²N) (0. 01–0. 04% across five N) ; a shuffled-spacing control showing the law depends on phase structure, not merely density; and a matched-N, matched-density comparison against true GUE and Poisson surrogates showing the law essentially absent in both (Monte Carlo P (ρₛurrogate≥ρᵣeal) =0. 0000 over 200 surrogates). The precision of the negative-real alignment exceeds what the paper's existing heuristic (diagonal approximation to the explicit formula) predicts on its own — closing this gap is now the sharpest open question raised by this work. C and α remain measured empirical invariants, not yet derived from first principles; we term the relationship an empirical formula rather than a "law. " Section 3. 8 extends the exponent test underlying the core amplitude law from p<1, 200 out to p<65, 000, testing the exponent freely rather than fixed, across ten disjoint prime ranges. The exponent 1/2 is confirmed exactly for p<3, 000; above p~25, 000 the point estimate becomes non-monotonic rather than drifting smoothly, and three independent diagnostics (joint parameter-landscape analysis, a data-independent basis-collinearity check, and an exact discretization-free residual profile) together show this is best read as a parameter-identifiability limit, not a genuine higher-order correction — reported as an open question, not an eighth confirmed result. New in this version: Section 3. 9 substantially extends the second, independently-motivated empirical law introduced previously — harmonic phase coherence between A (log p; N) and A (2·log p; N), still reported at an earlier stage of verification than the amplitude law above. The coherence statistic (0. 17 at N=100 to 0. 9999 at N=2, 000, 000, verified to machine precision against its own historical values) is confirmed stationary across the tested spectrum (39 windows spanning a 27-fold range in height) and generalizes to harmonics k=3, 4, and 5 with two independent null models (gap-shuffle and phase randomization) agreeing closely throughout. A rank-1 structure test for a prime-specific residual (a "fingerprint" F (p) ) is significant across a tenfold range of prime count with a genuinely out-of-sample design, and — the key new result this version — its specificity was tested directly against four surrogate models (Poisson, GUE via an exact tridiagonal random-matrix construction, gap-shuffled, and real zeros probed at random rather than logarithm-of-prime frequencies): the real result is significant at every prime count tested (p=0. 001) while all four surrogates are statistically indistinguishable from their own nulls throughout (p=0. 11–0. 91), extending the amplitude law's own GUE/Poisson specificity standard to this second law. A distinct additive phase law across pairs of different primes is also significant but plateaus around R=0. 62–0. 69 rather than approaching 1; three independent methods (regression screening, holdout-SVD stability, and a nonlinear/periodic search) all found no structure in this plateau, pointing to a statistically additive law with a genuine ceiling below 1, not a hidden correction — a clear contrast with the fingerprint's own reproducibility, independently confirmed by a second, split-half method (amplitude Pearson r=0. 83, p=0. 001; phase circular correlation below its own null, p=0. 993). The fingerprint's amplitude is well explained by the same log (p) /√p scale as the main amplitude law; what determines its phase remains open and is, in our assessment, this paper's most promising untested direction going forward. This work is conducted independently, without institutional resources, in Kyiv, Ukraine, under Russia's ongoing war of aggression. Computation and writing continue through repeated air raids and periods without power or internet access. The risk of losing the underlying data, code, and results to renewed strikes is not hypothetical, and this deposit is published in its current, evolving state for that reason, rather than held back for ideal conditions that may not come. Dedicated to the memory of Mykhailo Novikov and Mykhailo Palamarchuk, who served alongside the author in the 25th Separate Airborne Sicheslav Brigade and did not return.