Brocard Conjecture Project — Reproducibility Package

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, [email protected]