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. This repository contains the complete computational pipeline, source code, and manuscript for an exploratory study of phase-coherent structures in exponential sums over the non-trivial zeros of the Riemann zeta function. The work is independent of the companion study "The Precision Gap" (DOI: 10.5281/zenodo.21538616) and investigates a different question: can the exponential sum A(τ) = Σₙ e^(−iγₙτ) contain computationally extractable information about prime numbers when evaluated directly from finite sets of zeta zeros? The classical explicit formula already establishes the theoretical link between zeros and primes; this repository does not dispute or extend that link theoretically. It reports a direct test of the most natural naive explanation — the elementary diagonal term of the explicit formula, used as a model on its own — and finds that it reproduces neither the magnitude nor the shape of what is observed here. Rather than proposing a new theory, the repository documents a series of computational observations subjected to extensive negative controls and robustness tests. Main results 1. Spectral geometry. At frequencies τ = log p, the complex amplitudes form an extremely narrow, nearly one-dimensional geometric structure whose transverse width decreases approximately as N⁻¹, where N is the number of zeros included. Reproduced across four independent reference heights spanning nearly five orders of magnitude. 2. Generality. The structure survives numerous robustness checks: independent spectral windows; varying numbers of zeros; varying numbers of test primes; bootstrap analysis; structured perturbation experiments; rank-shift generalization across heights and window sizes, with p-values in some configurations below double-precision floating-point resolution; matched Poisson controls; genuine, correctly-unfolded 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 rather than kept. 3. Prime information. Supervised classification of primes from |A(log n)| alone, without normalizing by any known amplitude law, together with a fully blind, continuous scan of A(τ) with no prior arithmetic information supplied, both show local structure that lands on logarithms of primes significantly more often than for matched Poisson or GUE controls. Real zeta zeros consistently outperform both null models across every combination of height, zero count, and integer range tested. 4. Empirical scaling law of coherent accumulation. Across four independent height windows and over more than three orders of magnitude in the number of included zeros (up to N=10⁶), the spectral response at prime logarithms obeys |A(log p; N)| ∝ N — essentially exact linear coherent accumulation (α → 1.00) — while composite and random control frequencies remain statistically consistent with no growth at all (α ≈ 0). An independently measured, complementary quantity — the angular spread of the locked phase — shrinks as N⁻¹ (β → 1.00) over the same range, exactly mirroring the amplitude result. This law is established experimentally, not derived analytically, and is offered as a phenomenon that future theoretical descriptions should explain. Repository contents Full manuscript; Python source code; statistical analysis and robustness-test scripts; supervised-classification and blind-scan experiments; rank-shift prediction experiments; structured-perturbation tests; bootstrap analyses; wide-range spectral-scan pipeline; 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, 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.