Machine-Checkable Weak Covering for the Collatz (3x+1) Problem
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2025
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| _version_ | 1866902338079293440 |
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| author | Kawanishi, Yoshihito |
| author_facet | Kawanishi, Yoshihito |
| contents | <h2>Update 23 Oct 2025 (v94.8-4)</h2> <p><strong>Highlights</strong></p> <ul> <li> <p><strong>Paper ⇄ Artifact fully synchronized.</strong> Strict examples no longer show <code>--sha</code>; checksum is performed <strong>separately</strong> via <code>sha256.txt</code>. README, Sec. 8, and the 6-file ZIP now match.</p> </li> <li> <p><strong>“Unconditional” clarified.</strong> A one-paragraph definition was added in Sec. 8: <em>unconditional</em> 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).</p> </li> <li> <p><strong>References and labels corrected.</strong> All “see Theorem 6.12” instances fixed to <strong>Corollary 6.12</strong>; “see Corollary 8.1” unified; cross-refs audited.</p> </li> <li> <p><strong>Interface clarification.</strong> <code>--mode</code> applies <strong>only</strong> to <code>verify_v2.py</code> (<code>branch|minimal</code>). The strict checker <code>verify_v2_branch_strict.py</code> accepts <strong>no</strong> <code>--mode</code> or <code>--sha</code> flags.</p> </li> <li> <p><strong>Typesetting hygiene.</strong> Long monospaced strings use <code>{\ttfamily\nolinkurl{…}}</code> to avoid overfull boxes; code snippets use a single <code>\begin{code} … \end{code}</code> environment; bibliography wrapped with <code>\begingroup\sloppy … \endgroup</code> to curb underfull boxes; all link colors set to <strong>black</strong>.</p> </li> <li> <p><strong>Sec. 14.5 refreshed.</strong> File names and CSV mentions updated to match the shipped bundle (<code>verify_v2_branch_strict.py</code>, etc.).</p> </li> <li> <p><strong>Reproducibility simplified.</strong> Two steps only: checksum, then one-command strict acceptance. Logs behavior documented: empty when stdout isn’t redirected; contains <code>[ACCEPT]</code> when piped/tee’d.</p> </li> <li> </li> </ul> <p><strong>Update 25/Sep/2025 (v94.8-3)</strong></p> <ul> <li>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.</li> <li>Short-window thresholds consolidated: Theta2 = 5, Theta3 = 6 with end-window anchor t_k >= 2; macros aligned across statements and figures.</li> <li>Checker ↔ Theory alignment table placed immediately after Theorem (checker soundness) and before the certified corollary.</li> <li>Typesetting hygiene: T1 + lmodern + microtype, xurl, pdfstring safe commands; all overfull/underfull resolved; first occurrence of 2^m set as 2^{\,m}.</li> <li>Artifacts curated (filenames unchanged): cert.txt, sha256.txt, verify.py (stdlib-only), optional cert_branchsafe.txt; README includes one-command verification.</li> <li>Data/Code Availability updated to point to this version DOI; keywords adjusted for automated reasoning / formal verification.</li> </ul> <p>One-command verification:<br>python3 verify.py --cert cert.txt --sha sha256.txt --m 13 --H 30</p> <p><strong>Update 29/Aug/2025(v94.8-2)</strong></p> <ul> <li> <p>Unified notation: accelerated map as “C”, compression map as “D”.</p> </li> <li> <p>Aligned short-window thresholds and documented the minimal vs. sufficient settings.</p> </li> <li> <p>Added the “terminal valuation inheritance” lemma to carry bounds to actual trajectories.</p> </li> <li> <p>Clarified the proof narrative (from sufficient window to monotone envelope to termination).</p> </li> <li> <p>Standardized references (label-based) and cleaned up formatting.</p> </li> <li> <p>Curated artifacts: CSV certificate, verifier script, and published SHA256 for integrity.</p> </li> </ul> <p><strong>Update 19/Aug/2025(v94.7.20)</strong></p> <ul> <li> <p><strong>Title</strong>: <em>Short-Window Anchors and a Finite-Cover Certificate for Collatz (Unconditional)</em></p> </li> <li> <p><strong>Authors</strong>: Yoshihito Kawanishi</p> </li> <li> <p><strong>Version</strong>: V94-7-20</p> </li> <li> <p><strong>Description</strong>:<br>“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.”</p> </li> </ul> <p><strong>Update August 2025(v93.2):</strong></p> <p>This update strengthens the structural proof with:</p> <ul> <li> <p>Clear formulation of <strong>strong induction</strong> for all odd integers (Theorem 7)</p> </li> <li> <p>Improved <strong>theorem structure</strong> and logical flow</p> </li> <li> <p>Enhanced <strong>visual diagrams</strong> (e.g., <em>Z = 27 compression tree</em>)</p> </li> <li> <p>Expanded <strong>glossary</strong> and refined definitions</p> </li> <li> <p>Sharper <strong>ZFC-based loop exclusion</strong></p> </li> </ul> <p>This version replaces V92.3 with a clearer and more rigorous presentation.</p> <p><strong>---</strong></p> <p><strong>Update July 2025 (v92.3):</strong><br> This version presents the proof in a clearer and more reader-friendly structure, aligned with the expectations of top mathematical journals.</p> <p>---</p> <p>[Update July 2025] The updated version (v92.1.2) is titled: <br>**"A Structural Proof of the Collatz Conjecture via Recursive Compression"** <br>This reflects the refined formulation and educational emphasis in the final version.- Added full section numbering for journal submission<br>- Improved Glossary (Appendix E) with alphabetized terms<br>- Enhanced structural clarity and notation consistency</p> <p>---</p> <p>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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17421484 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Machine-Checkable Weak Covering for the Collatz (3x+1) Problem Kawanishi, Yoshihito Collatz Conjecture Structural Proof Recursive Compression Arithmetic Logic Induction Number Theory finite-cover certificate Lyapunov automated reasoning certificate formal verification <h2>Update 23 Oct 2025 (v94.8-4)</h2> <p><strong>Highlights</strong></p> <ul> <li> <p><strong>Paper ⇄ Artifact fully synchronized.</strong> Strict examples no longer show <code>--sha</code>; checksum is performed <strong>separately</strong> via <code>sha256.txt</code>. README, Sec. 8, and the 6-file ZIP now match.</p> </li> <li> <p><strong>“Unconditional” clarified.</strong> A one-paragraph definition was added in Sec. 8: <em>unconditional</em> 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).</p> </li> <li> <p><strong>References and labels corrected.</strong> All “see Theorem 6.12” instances fixed to <strong>Corollary 6.12</strong>; “see Corollary 8.1” unified; cross-refs audited.</p> </li> <li> <p><strong>Interface clarification.</strong> <code>--mode</code> applies <strong>only</strong> to <code>verify_v2.py</code> (<code>branch|minimal</code>). The strict checker <code>verify_v2_branch_strict.py</code> accepts <strong>no</strong> <code>--mode</code> or <code>--sha</code> flags.</p> </li> <li> <p><strong>Typesetting hygiene.</strong> Long monospaced strings use <code>{\ttfamily\nolinkurl{…}}</code> to avoid overfull boxes; code snippets use a single <code>\begin{code} … \end{code}</code> environment; bibliography wrapped with <code>\begingroup\sloppy … \endgroup</code> to curb underfull boxes; all link colors set to <strong>black</strong>.</p> </li> <li> <p><strong>Sec. 14.5 refreshed.</strong> File names and CSV mentions updated to match the shipped bundle (<code>verify_v2_branch_strict.py</code>, etc.).</p> </li> <li> <p><strong>Reproducibility simplified.</strong> Two steps only: checksum, then one-command strict acceptance. Logs behavior documented: empty when stdout isn’t redirected; contains <code>[ACCEPT]</code> when piped/tee’d.</p> </li> <li> </li> </ul> <p><strong>Update 25/Sep/2025 (v94.8-3)</strong></p> <ul> <li>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.</li> <li>Short-window thresholds consolidated: Theta2 = 5, Theta3 = 6 with end-window anchor t_k >= 2; macros aligned across statements and figures.</li> <li>Checker ↔ Theory alignment table placed immediately after Theorem (checker soundness) and before the certified corollary.</li> <li>Typesetting hygiene: T1 + lmodern + microtype, xurl, pdfstring safe commands; all overfull/underfull resolved; first occurrence of 2^m set as 2^{\,m}.</li> <li>Artifacts curated (filenames unchanged): cert.txt, sha256.txt, verify.py (stdlib-only), optional cert_branchsafe.txt; README includes one-command verification.</li> <li>Data/Code Availability updated to point to this version DOI; keywords adjusted for automated reasoning / formal verification.</li> </ul> <p>One-command verification:<br>python3 verify.py --cert cert.txt --sha sha256.txt --m 13 --H 30</p> <p><strong>Update 29/Aug/2025(v94.8-2)</strong></p> <ul> <li> <p>Unified notation: accelerated map as “C”, compression map as “D”.</p> </li> <li> <p>Aligned short-window thresholds and documented the minimal vs. sufficient settings.</p> </li> <li> <p>Added the “terminal valuation inheritance” lemma to carry bounds to actual trajectories.</p> </li> <li> <p>Clarified the proof narrative (from sufficient window to monotone envelope to termination).</p> </li> <li> <p>Standardized references (label-based) and cleaned up formatting.</p> </li> <li> <p>Curated artifacts: CSV certificate, verifier script, and published SHA256 for integrity.</p> </li> </ul> <p><strong>Update 19/Aug/2025(v94.7.20)</strong></p> <ul> <li> <p><strong>Title</strong>: <em>Short-Window Anchors and a Finite-Cover Certificate for Collatz (Unconditional)</em></p> </li> <li> <p><strong>Authors</strong>: Yoshihito Kawanishi</p> </li> <li> <p><strong>Version</strong>: V94-7-20</p> </li> <li> <p><strong>Description</strong>:<br>“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.”</p> </li> </ul> <p><strong>Update August 2025(v93.2):</strong></p> <p>This update strengthens the structural proof with:</p> <ul> <li> <p>Clear formulation of <strong>strong induction</strong> for all odd integers (Theorem 7)</p> </li> <li> <p>Improved <strong>theorem structure</strong> and logical flow</p> </li> <li> <p>Enhanced <strong>visual diagrams</strong> (e.g., <em>Z = 27 compression tree</em>)</p> </li> <li> <p>Expanded <strong>glossary</strong> and refined definitions</p> </li> <li> <p>Sharper <strong>ZFC-based loop exclusion</strong></p> </li> </ul> <p>This version replaces V92.3 with a clearer and more rigorous presentation.</p> <p><strong>---</strong></p> <p><strong>Update July 2025 (v92.3):</strong><br> This version presents the proof in a clearer and more reader-friendly structure, aligned with the expectations of top mathematical journals.</p> <p>---</p> <p>[Update July 2025] The updated version (v92.1.2) is titled: <br>**"A Structural Proof of the Collatz Conjecture via Recursive Compression"** <br>This reflects the refined formulation and educational emphasis in the final version.- Added full section numbering for journal submission<br>- Improved Glossary (Appendix E) with alphabetized terms<br>- Enhanced structural clarity and notation consistency</p> <p>---</p> <p>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.</p> |
| title | Machine-Checkable Weak Covering for the Collatz (3x+1) Problem |
| topic | Collatz Conjecture Structural Proof Recursive Compression Arithmetic Logic Induction Number Theory finite-cover certificate Lyapunov automated reasoning certificate formal verification |
| url | https://doi.org/10.5281/zenodo.17421484 |