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 **Key features of this version:**- Fully self-contained: no cross-citations to unpublished works or companion papers- All spectral results in Appendix D are proved in-line- Only published reference to related work: the Zenodo Complete Record (DOI: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[ρ] = 1 (critical), not eE[λ] = 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 · 2E-2k + 1 with sₖ = vₖ = 1 for 0 ≤ k ≤ (E-3)/2, yielding reff = 2. **Phase I is proved only for n₀ ≡ 1 4; extension to general n₀ is Gap D.** 3. **Empirical α-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_μ\|₁ = 1/4, hence ρ(M_μ) ≤ 1/4 < 1. 5. **Stopping-time exponential moment** (Proposition D.2): E[eλ' σ(N)] < ∞ for λ' < ln 2, for logarithmic-density a.e. N — an unconditional result relying on Tao's published Proposition (rach), holding only for logarithmic-density a.e. N, not pointwise. 6. **Geometric ergodicity** (Proposition D.3): The Collatz–Syracuse system exhibits geometric ergodicity in the subsampled drift sense of Connor–Fort. ### Conditional Reduction (Theorem 7.1) If **all three** of the following hold:1. \(sᵢ, vᵢ)\ satisfies empirical α-mixing with β > 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 2²⁶⁴⁰ to 2¹³⁰⁰⁰ provide quantitative targets:- reff = 2.000 (bit divergence)- d_*/E = 0.338 (scale law)- β_∞ consistently above $1.5$ by at least $35%$- Phase II bits: mean absolute deviation ≈ 0.020 from $0.5$- Carry chain: E[L]det ≈ 29, τdet ≈ 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"