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 3.8 extends the exponent test underlying the core amplitude law from the previously-tested range (p<1200) 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, with the fitted normalization constant agreeing with the independently-fit value from the complex-law analysis above to within 0.03%. Above p~25,000 the point estimate becomes non-monotonic (dipping to 0.486 before rising to 0.526) rather than drifting smoothly. Three independent diagnostics — a joint parameter-landscape analysis (with Hessian and principal-component characterization), a data-independent basis-collinearity check, and an exact, discretization-free profile of the residual as a function of the exponent alone with the normalization constant optimized analytically at every point — together show that for p>10,000 the exponent 1/2 costs at most 0.1–0.5% in fit quality relative to the true optimum, on an objective surface flat enough that a window of exponent values 0.08–0.18 wide fits nearly as well as the reported point estimate. We read this as a parameter-identifiability limit (the exponent and the normalization constant become poorly separated once the residual approaches the noise floor), not as evidence for a genuine higher-order correction to the law, and report it accordingly as an open question rather than an eighth confirmed result. This directly answers a question the previous version of this paper left explicitly open. The dedication has also been extended with a short passage on why this particular subject was chosen. 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.