The central claim of this paper is a precision gap, not a new analytic theorem. The qualitative picture — that primes act as resonant frequencies of the non-trivial zeros via the explicit formula, with amplitude scale log (p) /√p and negative-real phase orientation — follows from the classical diagonal approximation to the Guinand-Weil explicit formula, and we do not claim otherwise (Section 4. 5. 1 states this explicitly). What this heuristic does not predict is the precision with which finite collections of zeros realize that picture: across several near-independent observables — a 0. 005% negative-real alignment at N=2, 000, 000, a residual convergence rate (N^−0. 85) slower to vanish than the N^−1 a standard truncation bound implies, and a harmonic-coherence law so sharply localized at τ=log (p) that every tested perturbation, however small, collapses it — the measured accuracy consistently exceeds what the shared underlying heuristic, or any simple extension of it we tested, accounts for. We call this gap between predicted leading-order behavior and observed quantitative precision the paper's central finding, and its explanation the principal open problem this work poses. The supporting contribution is the first comprehensive experimental characterization of the phase structure of A (τ) = Σ⏒≤ₓ exp (−iγτ) at prime logarithms needed to establish that gap rigorously: quantitative convergence measurements precise enough to expose it, statistical validation against random-matrix (GUE) and Poisson controls establishing that the effect is specific to the actual zeros and not a generic consequence of matched spacing statistics, and a systematic, controlled falsification of roughly thirty alternative explanations for the open residuals this characterization surfaces. 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). 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 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. Section 3. 9 (harmonic phase coherence between A (log p;N) and A (2·log p;N) ) 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 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). A direct test of localization sharpness — recomputing the same coherence statistic at ten perturbed frequencies near log (p) — found every perturbation, however small (including a fixed shift of just 0. 05), collapses coherence by 47–97%, confirming the effect is narrow and specific rather than a broad resonance. An additive phase law across prime pairs is significant but plateaus around R=0. 62–0. 69; three independent methods found no structure in this plateau, a clean negative result. 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, and an order-respecting block bootstrap) does not support this — both the amplitude and phase functional forms of the fingerprint are reported as open rather than settled. Section 4. 5's theoretical discussion is 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 (pᵏ) =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 (p=0. 0000). This gives a mechanistic 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. Taken together with an extensive, honestly-reported negative program — testing whether R (p, N), the fingerprint's phase, or the additive law's residual are explained by any of roughly thirty natural candidates (smooth functions of p, discrete/modular residues, pair combinations, splines, Fourier bases, off-diagonal spectral-leakage models), all reported negative with appropriate controls — this work's contribution is less a new empirical law than a negative map of the space of possible explanations: a considerably narrower set of constraints any future mechanism must simultaneously satisfy, and a precisely located gap between what current heuristics predict and what finite-N realizations of the zeros actually deliver. 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.
Serhii Kanivets (Fri,) studied this question.