"An Empirical Amplitude Law for the Spectral Form Factor of 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. " Version 4 updates the paper's title to reflect the complex-analytic law as its foundational result, with minor textual clarifications throughout; the core empirical claims and numerical results are unchanged from v3. 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.