What question did this study set out to answer?

The goal is to present a cohesive framework for the Birch–Swinnerton–Dyer formula over Q, focusing on crucial mathematical relationships and identities.

May 30, 2026Open Access

A Determinant–Selmer Master Framework for the Birch–Swinnerton–Dyer Formula over Q: Central Analytic Germs, Kummer–Selmer Comparison, Néron Heights, and Cassels–Tate Obstructions

Key Points

The goal is to present a cohesive framework for the Birch–Swinnerton–Dyer formula over Q, focusing on crucial mathematical relationships and identities.
Consolidated a determinant–Selmer framework for the Birch–Swinnerton–Dyer formula in manuscript form.
Organized proofs around central-germ analytic identities and local/global Kummer–Selmer comparisons.
Included compliance materials, verification indices, and a checksum for potential reviewers.
Presented the BSD rank formula and leading-coefficient formula as results of terminal printed certificates, not hidden inputs.
Maintained separation of mathematical components (determinant, Selmer, height, etc.) until terminal assembly.
Utilized AI tools for editorial assistance without affecting mathematical claims and content integrity.

Abstract

This OSF project archives the expanded master draft A Determinant–Selmer Master Framework for the Birch–Swinnerton–Dyer Formula over Q: Central Analytic Germs, Kummer–Selmer Comparison, Néron Heights, and Cassels–Tate Obstructions by Tosho L. Karadzhov. The manuscript consolidates a TEBAC determinant–Selmer framework for the Birch–Swinnerton–Dyer formula over Q in a submission-facing form. The proof spine is organized around a central-germ analytic determinant identity, local and global Kummer–Selmer comparison, Mordell–Weil lattice realization, Néron-symbol height calibration, Tate–Kodaira local matrices, and a Cassels–Tate/Sha finite-obstruction ledger. The claim status is deliberately certificate-dependent. The BSD rank formula and leading-coefficient formula are presented as consequences of terminal printed certificates and forbidden-import audits, rather than as hidden inputs. The manuscript is structured so that determinant, Selmer, height, Cassels–Tate, Tamagawa, torsion, and regulator components remain separated until the terminal assembly. This version is intended as an expanded referee-facing master draft. It includes front-matter compliance material, a reader contract, non-circularity ledgers, terminal certificate tables, a referee-facing verification index, and a final proof-chain checksum. AI/LLM tools were used only for limited editorial assistance, formatting support, language polishing, consistency checking, and presentation refinement. The mathematical claims, proof strategy, definitions, theorem statements, and responsibility for the manuscript are entirely the author’s.

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