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

Abstract

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)=∑ne−iγnτ A(\tau)=\sum_n e^{-i\gamma_n\tau} A(τ)=∑ne−iγnτ over the non-trivial zeros of the Riemann zeta function, evaluated at τ=log⁡p \tau=\log p τ=logp. The work is independent of the companion study “The Precision Gap” (DOI: 10.5281/zenodo.21538616) and investigates a different question: can A(τ) A(\tau) 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. Main results Spectral geometry. At frequencies τ=log⁡p \tau=\log p τ=logp, 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) \arg A(\log p) argA(logp) is locked to within a median deviation of order 1.5×10−3 1.5\times10^{-3} 1.5×10−3 radians across primes. The transverse width of the locus decreases as a clean N−1 N^{-1} N−1 power law (coordinate-independent measure: exponents −0.99 -0.99 −0.99 to −0.97 -0.97 −0.97, R2>0.998 R^2>0.998 R2>0.998 on every height). The structure is absent in matched Poisson, gap-shuffled, and genuine unfolded GUE surrogates. Reproduced across four independent reference heights spanning nearly five orders of magnitude in T T T. 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 21 21–42% 42\% 42% versus chance 50% 50\% 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) A(\tau) 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, strengthened, then further qualified. An earlier version reported that the response at prime logarithms obeys an asymptotically exact linear coherent-accumulation law α→1.00 \alpha\to 1.00 α→1.00, based on a narrow set of 99 primes (p≤523 p\le523 p≤523). A subsequent investigation widening the range to p≤105 p\le10^5 p≤105 (9590 primes) and applying four independent methodologies found that α \alpha α is not universal: it appeared as a systematically declining function of log⁡p \log p logp, with opposite sign to matched non-prime controls, reproduced on all four heights. A further, more careful re-examination — extending N N N to 1.5×106 1.5\times10^6 1.5×106 and tracking the local slope of α \alpha α versus log⁡p \log p logp as a function of N N N itself — shows that the prime-size dependence is a finite-N N N transient. The slope starts strongly negative at small N N N and decays monotonically to zero (∣b∣<2×10−4 |b|<2\times10^{-4} ∣b∣<2×10−4, non-significant) at the largest N N N tested; simultaneously the median α \alpha α over primes rises to exactly 1.0000 1.0000 1.0000 on every height, while matched non-prime-power frequencies converge to α≈0 \alpha\approx0 α≈0. The class separation therefore sharpens, rather than weakens, with N N N. An unbounded coherence statistic ∣A(τ)∣/N |A(\tau)|/\sqrt{N} ∣A(τ)∣/N separates primes from non-prime-powers by a factor of 16.8 16.8 16.8–20× 20\times 20× on every height; prime powers pk p^k pk (k≥2 k\ge2 k≥2) sit cleanly between the two, following the support of the von Mangoldt function. The complementary phase-width exponent β(p) \beta(p) β(p) shows the same declining pattern in the original range, but has not yet been re-measured at large N N N; that gap is marked explicitly. A nested model avoiding a non-identifiability problem further qualifies the residual p p p-dependence. We now regard the origin of the (now-transient) pattern as open among four candidate explanations — a genuine zero-process property, an estimator artifact, a finite-N N N transient, or already-known explicit-formula geometry — each independently testable. We regard this as a stronger position than the single claim it replaces, even though it is a more modest one. As with every previous correction, it was found and reported through continued application of this repository’s own verification discipline. Repository contents Full manuscript; Python source code; statistical analysis and robustness-test scripts; supervised rank-shift and blind-scan experiments; structured-perturbation tests; bootstrap analyses; wide-range spectral-scan pipeline; per-prime scaling-exponent pipeline (including the large-N N N transient analysis); generated result tables. Scope and epistemic status This work is presented as an empirical computational investigation, not a proof. No theoretical mechanism is claimed for why the observed structure exists. The emphasis throughout is on reproducibility, independent negative controls, explicit reporting of negative results and retractions, robustness against alternative explanations, and transparent documentation of failed ideas and corrected analyses — including where an error was found in our own code and is reported alongside the fix.