Spectral Geometry, Generality, and Prime Information in the Riemann Zero Spectrum/First public research preprint

Dedicated to the memory of Mykhailo Novikov and Mykhailo Palamarchuk — young scientists whose lives were cut short by Russia's war of aggression against Ukraine — and to all soldiers who gave their lives defending Ukraine. Full biographical dedication in the manuscript's Acknowledgments section. This work is carried out and published in the first person plural — in their memory. Every computation reported here was carried out in Kyiv during the ongoing aggression against Ukraine; that circumstance is part of why this repository is published now, in a form still being actively refined, rather than held back for a more polished draft. A note the author hesitated over including, and includes now for methodological rather than personal reasons: the author sustained a severe concussion and a physical injury in the course of this war. This is stated here not as an appeal for leniency, but as a direct explanation for why the volume of verification in this repository is not excess caution to be trimmed for concision. Under these circumstances, catching one's own errors reliably on a single pass is not something the author can take for granted. Repeated, independently re-run checks; deliberate negative controls; retraction of analyses that do not survive scrutiny — these are not stylistic thoroughness here. They are elementary personal methodological hygiene, applied because the alternative is publishing mistakes the author would not otherwise catch. The primary motive for this work remains the dedication above; this note explains the discipline behind it, not the reason for it. This repository contains the complete computational pipeline, source code, and manuscript for an exploratory study of phase-coherent structures in the exponential sum A(τ) = Σₙ e^(−iγₙτ) over the non-trivial zeros of the Riemann zeta function, evaluated at τ = log p. The work is independent of the companion study "The Precision Gap" (DOI: 10.5281/zenodo.21538616) and investigates a different question: can A(τ) contain computationally extractable information about prime numbers when evaluated directly from finite sets of zeta zeros, without invoking the explicit formula, the von Mangoldt function, or any other arithmetic input? The classical explicit formula already establishes the theoretical link between zeros and primes; this repository does not dispute or extend that link theoretically. A direct test of the most natural naïve explanation — the elementary diagonal term of the explicit formula, used as a model on its own — reproduces neither the magnitude nor the shape of what is observed. No new theory is proposed. The repository documents a series of computational observations subjected to extensive negative controls, robustness tests, and explicit retractions. All results below are reproduced on four independent reference heights, from T ≈ 3.7×10⁸ to T ≈ 9.4×10⁹ (a factor of about 25 in T). Main results Spectral geometry. At frequencies τ = log p, the complex amplitudes form an extremely narrow, nearly one-dimensional geometric locus aligned with the negative real axis. The individual phase arg A(log p) is locked to within a median deviation of order 1.5×10⁻³ radians across primes. The transverse width of the locus decreases as a clean N⁻¹ power law (coordinate-independent measure: exponents −0.99 to −0.97, R² > 0.998 on every height). The structure is absent in matched Poisson, gap-shuffled, and genuine unfolded GUE surrogates. An objection worth stating explicitly is addressed in the manuscript: writing γₙ = T₀ + δₙ makes every value carry a common prefactor e^(−iT₀ log p) whose direction depends on T₀ and p alone, so the locus might be a kinematic artifact of where the window origin was placed. It is not, for three independent reasons — the prefactor predicts directions scattered essentially uniformly across the tested primes (their circular concentration is ≈ 0.24), whereas the observed cloud has Re A < 0 for every prime on every height; the prefactor is a single constant angle added to every partial sum, so it cannot change their dispersion (the phase-concentration statistic is identical to nine decimal places with and without it); and the surrogate ensembles share the same T₀, window and frequencies yet do not produce the locus. Sliding the window by up to 70000 zeros rotates arg A(log p) by less than 0.007 radians, where the kinematic account predicts rotations of tens of thousands of radians. Spectral rigidity, measured directly. At frequencies far from the support of Λ (at least five peak widths from any prime power), the coherence statistic |A(τ)|/√N reads 0.062–0.091, against the value 0.8326 expected for independent random phases. This is a suppression by a factor of 9.2–11.1 (mean 10.1, spread only 1.9 across heights) — a direct measurement of GUE-like spectral rigidity in the zero sequence. It is reported here as a standalone result: it depends on no arithmetic claim and on none of the contested control constructions, and it also supplies the correct empirical null against which every arithmetic contrast below is measured. Using the Poisson expectation instead would understate those contrasts by an order of magnitude. Generality. The structure survives numerous robustness checks: independent spectral windows; varying numbers of zeros and test primes; bootstrap analysis; structured perturbation experiments; rank-shift generalization across heights and window sizes (true-zero percentile ranks 21–42% versus chance 50%; phase-scramble collapses to chance); matched Poisson controls; genuine GUE spectra; phase scrambling; permutation testing. Several initially promising analyses that failed under closer inspection — including one un-normalized secondary metric and one implementation error in a phase-scramble control — were deliberately retracted and reported as negative results. Both Poisson and genuine GUE collapse to chance-level performance on the rank-shift task: local level-repulsion statistics alone do not explain the global, prime-indexed structure. Prime information. Supervised rank-shift prediction of a held-out zero's position, and a fully blind continuous scan of A(τ) with no arithmetic information supplied, both recover structure that lands on logarithms of primes significantly more often than for matched Poisson or GUE controls. Real zeros consistently outperform both null models across every combination of height, zero count, and integer range tested. Empirical scaling of coherent accumulation — corrected twice, and the second correction reverses the first. An earlier version reported that the response at prime logarithms obeys an asymptotically exact linear coherent-accumulation law α → 1.00, based on a narrow set of 99 primes (p ≤ 523). A subsequent investigation widening the range to p ≤ 10⁵ (9590 primes) and applying four independent methodologies found that α appeared instead to be a systematically declining function of log p, with opposite sign to matched non-prime controls, reproduced on all four heights. We argued at the time that widening the prime range steepened the slope, which is not what a finite-size artifact would do. That argument licensed a conclusion only about the prime-range axis. The number of zeros N is a separate axis, and it had been tested only to N = 10⁵. Extending N to 1.5×10⁶ and tracking the local slope of α against log p as a function of N itself shows that the two axes behave in opposite ways: the prime-size dependence is a finite-N transient. The slope starts strongly negative at small N (b between −0.11 and −0.16 at N = 2000) and decays monotonically to zero, reaching |b| < 2×10⁻⁴ at the largest N with permutation p-values between 0.23 and 0.97 — not significant on any height. We therefore retract the claim that α(p) obeys a prime-size-dependent scaling law. The effect is real in the regime where it was measured and reproduces across heights; neither prevents it from vanishing asymptotically, which is what it does. What the same measurement establishes instead. As N grows, prime frequencies converge to α = 1.0000 exactly on every height — fully coherent accumulation, |A_N| ∝ N — while matched non-prime-power frequencies converge to α ≈ 0. The class separation is therefore sharper at large N than in the regime where the transient law was measured. Three independent measurements agree: comparing each prime against a control at essentially the same τ (an integer with two distinct prime factors, matched in log n) gives a median difference in α of +0.95 to +1.05 with permutation p = 0.0000 on all four heights, and the same answer under a second, differently constituted control pool; the unbounded coherence statistic |A(τ)|/√N separates primes from non-prime-powers by a factor of 16.8–20.0 on every height; and prime powers p^k with k ≥ 2 sit cleanly between the two, at 6.9–8.9 times the empirical null, so the ordering follows the support of the von Mangoldt function rather than primality in the narrower sense. The scale of the control sets works against these numbers rather than for them: the non-prime-power controls sit at smaller log n, where the classical weight Λ(n)/√n is larger, and still read at the null level. A limit on how this can be worded. Comparing individual per-prime exponents between two spectral heights, the values do not reproduce: Pearson r ≈ 0.14 for α and 0.17 for β, and the overlap of the top-decile primes by either exponent is at chance level. Only aggregate, class-level statements replicate across heights, not the values attached to particular primes. Whatever these exponents describe, it is therefore not a stable numerical fingerprint of the individual integer. Remaining gaps, marked explicitly. The complementary phase-width exponent β(p) shows the same declining pattern in the original N ≤ 10⁵ range but has not yet been re-measured at large N; since it tracks the same underlying dependence as α, the most likely outcome is that its slope vanishes in