Machine-Checkable Weak Covering for the Collatz (3x+1) Problem

Update 23 Oct 2025 (v94.8-4)

Highlights

• Paper ⇄ Artifact fully synchronized. Strict examples no longer show --sha; checksum is performed separately via sha256.txt. README, Sec. 8, and the 6-file ZIP now match.

• “Unconditional” clarified. A one-paragraph definition was added in Sec. 8: unconditional means the results follow solely from acceptance of the released machine-checkable artifact (no extra assumptions), together with the proved implications (acceptance ⇒ Weak Covering and Weak Covering ⇒ termination).

• References and labels corrected. All “see Theorem 6.12” instances fixed to Corollary 6.12; “see Corollary 8.1” unified; cross-refs audited.

• Interface clarification. --mode applies only to verify_v2.py (branch|minimal). The strict checker verify_v2_branch_strict.py accepts no --mode or --sha flags.

• Typesetting hygiene. Long monospaced strings use {\ttfamily\nolinkurl{…}} to avoid overfull boxes; code snippets use a single \begin{code} … \end{code} environment; bibliography wrapped with \begingroup\sloppy … \endgroup to curb underfull boxes; all link colors set to black.

• Sec. 14.5 refreshed. File names and CSV mentions updated to match the shipped bundle (verify_v2_branch_strict.py, etc.).

• Reproducibility simplified. Two steps only: checksum, then one-command strict acceptance. Logs behavior documented: empty when stdout isn’t redirected; contains [ACCEPT] when piped/tee’d.

•  

Update 25/Sep/2025 (v94.8-3)

• Parameters locked: Weak Covering(13,30) as the main target; added a small "Parameter note (27 vs 30)" box in the Artifact section; standardized the example command.
• Short-window thresholds consolidated: Theta2 = 5, Theta3 = 6 with end-window anchor t_k >= 2; macros aligned across statements and figures.
• Checker ↔ Theory alignment table placed immediately after Theorem (checker soundness) and before the certified corollary.
• Typesetting hygiene: T1 + lmodern + microtype, xurl, pdfstring safe commands; all overfull/underfull resolved; first occurrence of 2^m set as 2^{\,m}.
• Artifacts curated (filenames unchanged): cert.txt, sha256.txt, verify.py (stdlib-only), optional cert_branchsafe.txt; README includes one-command verification.
• Data/Code Availability updated to point to this version DOI; keywords adjusted for automated reasoning / formal verification.

One-command verification:
python3 verify.py --cert cert.txt --sha sha256.txt --m 13 --H 30

Update 29/Aug/2025(v94.8-2)

• Unified notation: accelerated map as “C”, compression map as “D”.

• Aligned short-window thresholds and documented the minimal vs. sufficient settings.

• Added the “terminal valuation inheritance” lemma to carry bounds to actual trajectories.

• Clarified the proof narrative (from sufficient window to monotone envelope to termination).

• Standardized references (label-based) and cleaned up formatting.

• Curated artifacts: CSV certificate, verifier script, and published SHA256 for integrity.

Update 19/Aug/2025(v94.7.20)

• Title: Short-Window Anchors and a Finite-Cover Certificate for Collatz (Unconditional)

• Authors: Yoshihito Kawanishi

• Version: V94-7-20

• Description:
  “Using short-window thresholds (Θ₂=5, Θ₃=6), a finite-cover certificate at M=8192 (H=27), and a distance potential φ(r)=0.60·d(r), this package provides the paper and machine-checkable artifacts certifying unconditional anchor hitting and subsequent termination. Includes CSV/LP/PNG, verification scripts, and SHA256 checksums.”

Update August 2025(v93.2):

This update strengthens the structural proof with:

• Clear formulation of strong induction for all odd integers (Theorem 7)

• Improved theorem structure and logical flow

• Enhanced visual diagrams (e.g., Z = 27 compression tree)

• Expanded glossary and refined definitions

• Sharper ZFC-based loop exclusion

This version replaces V92.3 with a clearer and more rigorous presentation.

---

Update July 2025 (v92.3):
 This version presents the proof in a clearer and more reader-friendly structure, aligned with the expectations of top mathematical journals.

---

[Update July 2025] The updated version (v92.1.2) is titled:  
**"A Structural Proof of the Collatz Conjecture via Recursive Compression"**  
This reflects the refined formulation and educational emphasis in the final version.- Added full section numbering for journal submission
- Improved Glossary (Appendix E) with alphabetized terms
- Enhanced structural clarity and notation consistency

---

This document presents a complete structural proof of the Collatz Conjecture based on a recursive compression framework. By applying a unified arithmetic transformation compress(Z) = (Z + 1) / 2 and a hierarchical convergence structure, the paper shows that all positive integers, both even and odd, must converge to 1 in a finite number of steps. Visual appendices include flowcharts and hierarchical induction diagrams to support the logical structure of the proof.