Brocard Conjecture Project — Reproducibility Package

Version 2 (2026-10-01): This version includes an AI-use disclosurein the manuscript, added in response to community standards fortransparent reporting of AI assistance. This deposit contains the code, data, figures, and manuscript for the paper"An Empirically Guided Cascade Pipeline for the Strengthened-Legendre Routeto Brocard's Conjecture" by Michael Jee Shiu (September 2026). The paper presents a computational pipeline for testing the strengthenedLegendre route to Brocard's conjecture. The pipeline combines apre-computable primorial survivor screen, a composite anomaly score, and acalibrated statistical analysis. Applied to k <= 2,762,695 square gaps andn <= 10^6 Brocard intervals, the pipeline establishes: - Brocard's conjecture holds empirically over n <= 10^6, with b_min(10^6) = 5 at n = 2 (margin 1 above the conjectured floor of 4).- The strengthened Legendre hypothesis holds empirically over k <= 2,762,695 with no violations.- The composite anomaly score recovers 98% of the low-density tail at K = 500. The pipeline is not a proof of Brocard's conjecture; it is an efficientsurvey tool and an empirical characterization. Contents:- Paper/ -- LaTeX source and compiled PDF- scripts/ -- Python scripts for all pipeline stages- results/ -- CSV output files- figures/ -- 8 figures as PNG- logs/ -- execution logs Requirements: Python 3.14, Miniconda, primecountpy, sympy, numpy, pillow.Runtime: approximately 24 hours, dominated by exact prime counting. Contact: Michael Jee Shiu, mjshiu@stanfordalumni.org