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 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 (harmonic phase coherence between A(log p;N) and A(2·log p;N)) is substantially extended and, in one important respect, corrected. The coherence law generalizes to harmonics k=3,4,5 with two independent null models agreeing closely, and its specificity to real zeta zeros at log(p) frequencies is now confirmed directly against four surrogates — Poisson, GUE (exact tridiagonal random-matrix construction), gap-shuffled, and random-frequency controls — with the real result significant (p=0.001) at every prime count tested while all four surrogates remain statistically indistinguishable from their own nulls (p=0.11–0.91). An additive phase law across prime pairs is significant but plateaus around R=0.62–0.69; three independent methods (regression screening, holdout-SVD, and a nonlinear/periodic search) all found no structure in this plateau, a clean negative result rather than an unsolved puzzle. An initial screening had suggested the fingerprint's amplitude tracks log(p)/√p; a considerably more rigorous test (multi-family AIC/BIC comparison, genuine extrapolation to primes outside the training range, and an order-respecting block bootstrap) does not support this — no functional family is stable across P_max, extrapolation R² is negative or negligible throughout, and a freely-fit exponent's sign is not even consistent across P_max — so both the amplitude and phase functional forms of the fingerprint are now reported as open rather than settled. Also new in this version: Section 4.5's theoretical discussion is substantially strengthened by a direct, comprehensive test of the von Mangoldt weighting the paper's heuristic already relies on. Classifying every integer n from 2 to 1000 by Λ(n) — prime, prime power (Λ(p^k)=log p, same as primes), or composite with two or more distinct prime factors (Λ(n)=0 by construction) — and computing A(log n;N) for all of them shows a sharp categorical split: primes and prime powers both show essentially perfect phase alignment (Rayleigh R=1.0000 for both) with amplitude normalized by log(p)/√n converging to the same |α(N)|≈180,100–180,150 established independently in Section 3.0, while composites show amplitude roughly 3000-fold smaller and no alignment toward π (a 1000-permutation test found no resample approaching the observed prime/composite gap, p=0.0000). This gives a mechanistic, not merely empirical, explanation for why Section 3.9's additive-law residual is unstructured noise: Λ(pq)=0 for distinct primes, so there is no diagonal term for that frequency to align to in the first place. A follow-up bias-corrected analysis confirms composites do retain non-trivial residual clustering (R≈0.55, but toward phase 0 rather than π) that is not explained by their smallest prime factor once small-sample bias is properly controlled for (weighted permutation test, p=0.95) — left as an explicitly open, secondary question. Taken together with an extensive, honestly-reported negative program — testing whether the amplitude law's residual R(p,N), the fingerprint's phase, or the additive law's residual Delta(p,q) are explained by any of roughly thirty natural candidates (smooth functions of p, discrete/modular residues, pair combinations, splines, Fourier bases, and three variants of an off-diagonal spectral-leakage model), all reported negative with appropriate controls — this version's contribution is less a new empirical law than what we describe in the text as a negative map of the space of possible explanations: a considerably narrower set of constraints any future mechanism must simultaneously satisfy, even without yet supplying that mechanism. 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.