Published October 2, 2025
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Exclusion of Non-Trivial Cycles and Emergence of Contractive Windows in Collatz Dynamics
Description
Together these mechanisms establish bounded drift and rule out non-trivial cycles,
leaving only the trivial cycle: 1 → 2 → 4 → 1.
This repository includes:
• Exclusion of Non-Trivial Cycles and Emergence of Contractive Windows in Collatz Dynamics.pdf —
full LaTeX-rendered paper (with appendices, references, and Zenodo-linked reproducibility).
• rho_star_scan.py — script to reproduce the odd-density ceiling ρ* ≈ 0.627 numerical scans.
• trajectory_sim.py — Collatz trajectory simulator and Δ_k computation.
• crt_penalty.py — computation of uniform CRT penalty constant (κ = 1/48).
• README.md — usage instructions and reproduction guide.
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Exclusion of Non-Trivial Cycles and Emergence of Contractive Windows in Collatz Dynamics.pdf
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Additional details
References
- Applegate, D., Lagarias, J. C., & Sloane, N. J. A. (1993). On the distribution of stopping times in the 3x+1 problem. Mathematics of Computation, 61(203), 437–457.
- Garner, L. (1981). On the Collatz 3n+1 algorithm. Proceedings of the American Mathematical Society, 82(1), 19–22.
- Korec, I. (1994). A density estimate related to the 3x+1 problem. Acta Arithmetica, 66(2), 147–160.
- Lagarias, J. C. (Ed.). (2010). The Ultimate Challenge: The 3x+1 Problem. American Mathematical Society.
- Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. arXiv preprint arXiv:1909.03562.
- Tao, T. (2022). Almost all orbits of the Collatz map attain almost bounded values. Forum of Mathematics, Pi, 10:e8.
- Wirsching, G. J. (1998). The Dynamical System Generated by the 3n+1 Function. Springer.