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

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)=∑ne−iγnτA(\tau)=\sum_n e^{-i\gamma_n\tau}A(τ)=n∑e−iγnτ contain computationally extractable information about prime numbers when evaluated directly from finite sets of zeta zeros? Rather than proposing a new theory, the paper documents a series of computational observations subjected to extensive negative controls and robustness tests. Main results The repository reports evidence for three connected observations. 1. Spectral geometry At frequencies τ=log⁡p,\tau=\log p,τ=logp, the complex amplitudes form an extremely narrow one-dimensional geometric structure whose transverse width decreases approximately as N−1,N^{-1},N−1, where NNN is the number of zeros included. This behavior was reproduced across four independent reference heights spanning nearly five orders of magnitude. 2. Generality The observed structure survives numerous robustness checks, including independent spectral windows, varying numbers of zeros, different numbers of test primes, bootstrap analysis, structured poisoning, sensor subsampling, rank-generalization experiments, matched Poisson controls, genuine unfolded GUE spectra, phase scrambling, permutation testing. Several initially promising analyses that failed under closer inspection were deliberately removed or reported as negative results. 3. Prime information Several independent experiments suggest that the phase geometry contains measurable information correlated with the prime distribution. These include blind rank prediction, supervised prime discovery, continuous blind spectral scans, wide-range blind spectral scans. Across the reported experiments, real zeta zeros consistently outperform both Poisson and GUE controls. Repository contents The repository includes full manuscript, Python source code, statistical analysis scripts, robustness tests, blind discovery experiments, rank prediction experiments, structured poisoning tests, bootstrap analyses, wide spectral scan pipeline, generated result tables. Philosophy This work is intentionally presented as an empirical computational investigation rather than a proof. No theoretical mechanism is claimed. Instead, the emphasis is placed on reproducibility, independent controls, explicit negative results, robustness against alternative explanations, transparent reporting of failed ideas and corrected analyses.