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

Update 09 Feb 2026 (v94. 8. 426) This is a presentation + reproducibility refinement of the previous Zenodo version. The main certified claim remains the same (strict, branch-robust artifact for Weak Covering (13, 30) and the corresponding termination pipeline), but the exposition and auditability were strengthened. Highlights * Proof sketches eliminated: remaining “Proof sketch” fragments were replaced by full proofs. * Notation unified: parameterized occurrences are standardized (e. g. , mH). * Soundness write-up tightened: the checker-soundness argument was made fully consistent with the dominance/monotonicity lemmas used in the termination pipeline. * Artifact bundle cleaned + pinned: the verification bundle was cleaned (no _MACOSX noise) and pinned with SHA256 checksums (certificate/logs/verifier and bundle checksum). * Reproducibility remains one-command: independent users can reproduce the acceptance result by following README. md. Files (typical bundle) - PDF manuscript (this record) - collatzᵥ94₈₄26. zip (certificate + strict verifier + README + checksums) - sha256. txt (checksums; provided as a separate attachment if not included in the ZIP) ## Independent verification reports welcome If you reproduce the verification result on your own machine, please consider posting a short report (in a comment, email, or linked issue/discussion). The most useful minimal report includes: * OS and CPU architecture (e. g. , Ubuntu 22. 04 x86₆4, macOS arm64, Windows 11) * Python version (python3 --version) * The exact command you ran * The line showing the final verdict (expected: "ACCEPT") * (Optional) wall-clock runtime Such independent reproductions strengthen confidence in the artifact-certified result. 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 verifyᵥ2. py (branch|minimal). The strict checker verifyᵥ2branchₛtrict. py accepts no --mode or --sha flags. Typesetting hygiene. Long monospaced strings use … to avoid overfull boxes; code snippets use a single code … code environment; bibliography wrapped with … 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 (verifyᵥ2branchₛtrict. 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 tₖ >= 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 2ᵐ set as 2^\, m. Artifacts curated (filenames unchanged): cert. txt, sha256. txt, verify. py (stdlib-only), optional certbranchsafe. 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.