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 K_max(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(ρ_surrogate≥ρ_real)=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 (previous version) 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 reports a second, independently-motivated empirical law, at an earlier stage of verification than the amplitude law above. Rather than the magnitude of A at a single frequency, it concerns the relationship between two frequencies for the same prime: the phase of A(2·log p; N) tracks twice the phase of A(log p; N), with a coherence statistic rising from 0.17 at N=100 to 0.9999 at N=2,000,000 (reproduced to machine precision against its own historical values), against a flat gap-shuffle control throughout. This coherence is shown to be a stationary property of the spectrum rather than an artifact of low-lying zeros, measured independently across 39 windows of 100,000 zeros spanning a 27-fold range in height (mean 0.966, essentially flat trend). A rank-1 structure test for a prime-specific residual component beyond simple clustering — run with both a full-sample and a genuinely out-of-sample (train/holdout) design, across a tenfold range of prime count (46 to 430 primes) — is significant throughout, with the out-of-sample separation from its null growing monotonically with prime count (6.4 to 17.6 null standard deviations). We report this second law explicitly as less mature than the amplitude law: it rests on two positive, controlled results rather than the six-plus-artifact-hunting standard the rest of the paper was held to, and several designed follow-up tests (higher harmonics, an additive phase law across prime pairs, and modeling the fingerprint's functional form) remain in progress and are listed as such. 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.