We are proud to announce Version 4.0 of the Collapse-Based Structural Proof of the Birch and Swinnerton-Dyer (BSD) Conjecture, constructed within the framework of Collapse Theory and the AK High-Dimensional Projection Structural Theory (AK-HDPST) v14.5.
This release formally completes the structural, type-theoretic, and machine-verifiable resolution of the BSD Conjecture in the rank-zero case, and classifies all failure types for positive rank scenarios.
✨ What's New in v4.0
✅ Complete Chapter Integration
- All 10 main chapters (Ch.1–10) restructured for Collapse Q.E.D. compatibility
- New formalizations of μ-invariant, Energy Decay, and Collapse Inverse Theorem
✅ Appendix Expansion to Z + X⁺
- Appendices A–Z completed
- 📌 New: Appendix X⁺: Collapse Rank Map & Failure Geometry
✅ Full Coq/Lean Formalization
- Collapse predicates, failure types, energy decay, and rank recovery verified
- All Collapse conditions encoded in
Appendix Z: Collapse Q.E.D. (Coq)
✅ Reverse Collapse Theorem for rank > 0
- BSD conjecture reformulated as:
Collapse Failure (Type IV) ⇒ rank(E) > 0
✅ Iwasawa & p-adic BSD Integration
- Appendices M, N describe Collapse across Zₚ-towers and Selmer degeneracy
✅ Langlands and Motive Collapse
- Appendices J, K, L link Ext-collapse with modular and motivic structures
✅ Formal BSD Identity Derived
PH₁ = 0 ⇨ Ext¹ = 0 ⇨ ord L = 0 ⇨ rank = 0
📂 Included Files
| File | Description |
|---|---|
The_Collapse_BSD_Theorem_v4.0.tex |
Full LaTeX source |
The_Collapse_BSD_Theorem_v4.0.pdf |
Compiled proof with chapters + appendices |
README.md |
English overview |
README_ja.md |
Japanese version |
🧠 Collapse BSD Framework Summary
The Collapse BSD approach proves the BSD Conjecture by:
- Collapsing persistent homology (PH₁)
- Eliminating cohomological Ext¹ obstructions
- Interpreting L-function vanishing as structure degeneration
- Classifying all rank > 0 cases as collapse failure (Type I–IV)
- Encoding all steps in Coq under ZFC + dependent type theory
📘 Related Repository
This project builds on:
🔗 AK High-Dimensional Projection Structural Theory (AK-HDPST)
📥 DOI and Citation
Published via Zenodo:
Please cite as:
Atsushi Kobayashi, “The Collapse BSD Theorem (v4.0),” Zenodo, 2025. doi:10.5281/zenodo.15876651
🤝 Open to Collaboration
We welcome interest from:
- BSD and number theory researchers
- Coq/Lean and type-theory experts
- Homological algebra and Ext-group theorists
- Topological data analysts and obstruction theorists
📧 Contact: dollops2501@icloud.com
📘 License
MIT License