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." New in this version: Section 3.6 reports δ(p), a newly surfaced function on the primes governing the local rate of convergence to the complex law's leading-order prediction. δ(p) ranges from 0.284 to 1.213 across 196 tested primes (p<1200), decreases systematically with p (Spearman ρ=−0.60, p<10⁻⁶), is stable under substantial changes to the checkpoint grid, persists (weaker) under an unnormalized residual, and shows strong spatial smoothness (neighboring primes' δ(p) values cluster far outside chance, permutation percentile 0%). We report this as a new object surfaced by the data, explicitly not yet meeting the bar of an established empirical law: block-independent and block-bootstrap controls remain to be performed. Earlier findings (GUE/Poisson surrogate comparison, shuffled-spacing control, C(N)=T(N)²/(4π²N) resolution) are retained unchanged. 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. Given the real, non-hypothetical risk of losing the underlying data, code, and results to renewed strikes, this deposit is published in its current, evolving state rather than held back for ideal conditions — with the intention of updating it further as work continues. 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.