Paper Repository

Spectral Structure of Prime Gap Correlations — Reproduction Package

This repository accompanies the manuscript Spectral Structure of Prime Gap Correlations: The Log-Periodic Tower, Variance Asymptotics, and Structural Limits (P. R. Rubino). It contains the full manuscript together with all code, data, pre-registered tests, and working notes required to reproduce every empirical and numerical claim in the paper.

The work studies the statistical structure of consecutive prime gaps g_n = p_{n+1}-p_n across 18 logarithmic windows totalling 1.41×10111.41\times10^{11} 1.41×1011 gaps. It reports: (i) a parameter-free transfer operator on the multiplicative residues (\mathbb{Z}/Q\mathbb{Z})^* realizing the Hardy–Littlewood singular-series content as an explicit spectrum; (ii) an empirically validated closed-form asymptotic for the gap variance (χ2/dof=0.40\chi^2/\mathrm{dof}=0.40 χ2/dof=0.40); (iii) a spectral mechanism for the log-periodic tower in the lag-$k$ autocovariance, proved as a theorem in the function field Fq[T]\mathbb{F}_q[T] Fq[T]; and (iv) a $3.90\sigma$ falsification of geometric self-similarity in the Bogomolny–Keating hierarchy, plus a quantified generative no-go.

Repository contents

• manuscript/ — the paper (PDF and LaTeX source) and figures.
• code/ — Python scripts reproducing all numerical results: variance and moment computations, autocovariance/tower analysis, the function-field finite-$q$ carrier derivation (ff_finite_q_carrier.py), CMI hierarchy, calibration, and audit checks.
• data/ — computed statistics (variance per window, higher moments, CMI tables, autocovariance at high lag) in CSV/JSON.
• prereg/ — pre-registered test protocols (ladder-spacing test; zeros-signature test) fixed in advance of measurement.
• notes/ — working notes, correction logs, and result branches.

Reproducibility. Each substantive claim in the paper carries an explicit rigor tag (proved / derived / empirical / conditional / open / no-go). Numerical claims are reproducible from the scripts in code/ against the data in data/; see README.md for the run order and dependencies (Python with numpy, scipy, mpmath).

Disclosure. Large language models were used as computational and editorial assistants (code, stress-testing, literature search, drafting); all mathematical statements, proofs, and numerical results were checked and are the author's responsibility.

Author. Paolo Raffaello Rubino — info@paolorubino.it — Milan, Italy.