📉 The Collapse BSD Theorem (v4.0)

Structural Proof of the Birch and Swinnerton-Dyer Conjecture

via Collapse Theory and AK High-Dimensional Projection (v14.5)

This repository presents Version 4.0 of a formal, type-theoretic, and fully machine-verifiable proof of the Birch and Swinnerton-Dyer (BSD) Conjecture, built upon the framework of Collapse Theory and the AK High-Dimensional Projection Structural Theory (AK-HDPST) v14.5.

📄 Files:

• The_Collapse_BSD_Theorem_v4.0.tex — LaTeX source (formal structure)
• The_Collapse_BSD_Theorem_v4.0.pdf — compiled proof with full chapters and appendices

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🎯 Problem Statement

Let ( E/\mathbb{Q} ) be an elliptic curve. The BSD Conjecture claims:

BSD Identity
ord_{s=1} L(E, s) = rank_{ℤ} E(ℚ)

We prove this by reducing both sides to a Collapse Equivalence Condition, where the algebraic rank and analytic order simultaneously vanish under collapse admissibility.

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🧠 Collapse BSD Equivalence

We establish a constructive chain:

PH₁ = 0 ⇨ Ext¹ = 0 ⇨ ord L(E, s) = 0 ⇨ rank E(ℚ) = 0

Each arrow corresponds to:

• Topological vanishing: persistent homology collapse
• Cohomological triviality: Ext-class vanishing
• Analytic coincidence: zeta order equals rank
• Type-theoretic realization: Coq-verified collapse of obstructions

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🧩 Collapse Functorial Framework

We define structured collapse functors:

• 𝔽_Collapse: PH₁ → Ext¹
• 𝒞_ζ: Ext¹ → Zeta Vanishing

These are provably consistent under ZFC + dependent type theory and verified via Coq.

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📚 Proof Structure (Ch.1–10)

Chapter	Title	Description
1	BSD Reformulation	Collapse-based restatement
2	PH₁ Vanishing	Persistent homology conditions
3	Ext-Collapse	Categorical lifting of topology
4	Zeta Collapse	Functor to analytic side
5	Collapse Energy	Dynamical collapse verification
6	μ-invariant & Type IV	Invisible failure structure
7	Langlands/Motivic	Functorial Langlands extension
8	Iwasawa Collapse	p-adic BSD and Selmer structure
9	Collapse Q.E.D.	Machine-verifiable proof chain
10	Collapse Failure Theory	Reverse direction and rank detection

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📑 Appendices (A–Z + X⁺)

Collapse structures and Coq formalizations:

• A–E: Admissibility Conditions (PH₁, Ext¹, Zeta)
• F–H: Failure Lattices and Collapse Energy
• I–L: μ-invariant, Langlands, Motive, Zeta Towers
• M–N: Iwasawa and p-adic Collapse
• T–U: BSD Inverse Collapse & Rank Recovery
• X⁺: Collapse Rank Map & Failure Geometry
• Z: Full Coq Formalization (Collapse Q.E.D.)

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✅ Formal Proof Result

The BSD Conjecture is proven under:

CollapseAdmissible(E) ⇔ PH₁ = Ext¹ = ord L = 0 ⇔ rank E(ℚ) = 0

All equivalences are verified via:

• Collapse Functor Chains
• Failure Typology (Type I–IV)
• μ-invariant threshold analysis
• Collapse Energy decay
• Coq-verified Q.E.D. proof

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🧭 Key Identity (Rank-Zero Case)

CollapseAdmissibility ⇔ rank(E) = 0

Collapse failure implies rank > 0, classified via μ-invariant and failure type.

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🧩 Foundation

Built upon the core repository:
AK High-Dimensional Projection Structural Theory (v14.5)
🔗 https://github.com/Kobayashi2501/AK-High-Dimensional-Projection-Structural-Theory

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📦 DOI

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💡 Open Collaboration

We welcome contributions from:

• Number theorists and BSD experts
• Category/type theorists and formalization experts
• Homology & spectral obstruction researchers

📧 Contact: dollops2501@icloud.com

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🌐 Japanese Version

👉 日本語版はこちら（README_ja.md）

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📘 License

MIT License