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Changes in this version (v3.1). This version supersedes Version 3.0 deposited under DOI 10.5281/zenodo.23483273. It restores the full author metadata (Ali Bourizk) in the LaTeX/PDF files for the public Zenodo record, adds the complete numerical validation script (test_pillar1.py) and the BibTeX database (references.bib) to the upload package for full reproducibility, updates the CHANGELOG.txt file, and adds a comprehensive README.md file. The mathematical results and proofs (including the exact threshold at J0 = 0.405Δ) remain strictly unchanged and locked from v3.0.