A Reduction Framework for the Collatz Conjecture: Dual-Phase Structure, Criticality, and Quantitative Targets for Per-Orbit Mixing

# A Reduction Framework for the Collatz Conjecture: Dual-Phase Structure, Criticality, and Quantitative Targets for Per-Orbit Mixing **ETDS Submission – Final Independent Version** **Date: ** 2026-07-14 **Author: ** Zhongming Xiong **Affiliation: ** Wuhan University of Technology, Wuhan, China --- ## Overview This repository contains the final submission version of the manuscript *A Reduction Framework for the Collatz Conjecture: Dual-Phase Structure, Criticality, and Quantitative Targets for Per-Orbit Mixing*, submitted to *Ergodic Theory and Dynamical Systems*. The paper presents a reduction framework that converts the Collatz conjecture from an integer-arithmetic problem to a question in dynamical-systems statistics. The framework comprises **three unconditional structural results**, **three unconditional probabilistic results**, and **one conditional reduction**. **Key features of this version: **- Fully self-contained: no cross-citations to unpublished works or companion papers- All spectral results in Appendix D are proved in-line- Only published reference to related work: the Zenodo Complete Record (DOI: 10. 5281/zenodo. 21350247) --- ## File Contents | File | Description ||------|-------------|| `ETDS SUBMISSION (REVISION 3 — FINAL, INDEPENDENT). tex` | Main LaTeX source of the manuscript (Revision 3 — Final, Independent). || `Supplementary Materials. tex` | Supplementary Materials: complete experimental data, methodology, Python code, and reproducibility information for numerical observations. || `README. md` | This file. | --- ## Paper Summary ### Unconditional Structural Results 1. **Criticality correction** (Proposition 3. 1): The expected one-step contraction factor is E = 1 (critical), not e^E = 9/16, resolving a Jensen inequality conflation. 2. **Dual-phase structure** (Lemma 5. 1): For n₀ = 2E + 1 with E 3 (i. e. , n₀ 1 4), Phase I admits the closed form nₖ = 3ᵏ 2^E-2k + 1 with sₖ = vₖ = 1 for 0 k (E-3) /2, yielding r₄₅₅ = 2. **Phase I is proved only for n₀ 1 4; extension to general n₀ is Gap D. ** 3. **Empirical -mixing** (Definition 6. 1): A rigorous definition for deterministic sequences via single-symbol block-frequency measures, enabling the McLeish SLLN without a probability space or stationarity. ### Unconditional Probabilistic Results (Appendix D) 4. **Spectral radius bound** (Proposition D. 1): \|M_\|₁ = 1/4, hence (M_) 1/4 1. 5;2. The bounded sixth moment condition holds;3. The empirical marginal converges to Geom (1/2) (Gap B) ; then the orbit converges to 1 and non-trivial cycles are excluded. ### Numerical Observations Six experiments spanning n₀ from 2^2640 to 2^13000 provide quantitative targets: - r₄₅₅ = 2. 000 (bit divergence) - d_*/E = 0. 338 (scale law) - _ consistently above 1. 5 by at least 35\%- Phase II bits: mean absolute deviation 0. 020 from 0. 5- Carry chain: EL₃₄ₓ 29, ₃₄ₓ 372- F 2ⁿ has no non-trivial cycles for n 11 ### Open Gaps Six open gaps are identified: Gap A (map mixing), Gap B (measure-to-orbit transfer), Gap C (Collatz mixing), Gap D (general n₀), Gap E (a. e. to pointwise), Gap F (coset vs. Haar uniformity). --- ## Relationship to Companion Archival Record A companion archival record ** (DOI: 10. 5281/zenodo. 21350247) ** contains the complete research history, including: - All phases of the analysis (V₂ potential → martingale structure) ;- 14 excluded attack routes;- Biprime classification framework (single-prime vs. bi-prime maps) ;- FSA numerical experiments;- Cross-domain analogy with UES/CCRF;- Proven Results Register. **This ETDS paper is independent and self-contained**; it references the archival record only as a source for the complete weak Gibbs–Markov verification (Remark D. 1) and as a published record of related work. --- ## Compilation Instructions This manuscript requires a standard LaTeX distribution with the following packages: - `amsmath`, `amsthm`, `amssymb`, `amsfonts`- `booktabs`, `geometry`, `enumitem`- `xcolor`, `framed`, `hyperref`, `cleveref`- `appendix`, `microtype` To compile the main manuscript: ```bashpdflatex "ETDS SUBMISSION (REVISION 3 — FINAL, INDEPENDENT). tex"