Layered Recursive Compression as a Structural Proof of the Collatz Conjecture (Version 92)

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.

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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.

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[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

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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.