The BSD Conjecture as a Positivity Normal-Form Theorem

We reformulate the Birch and Swinnerton-Dyer conjecture as a positivity normal-form characterisation. Four axioms (finite capacity, height positivity, controlled Iwasawa growth, Euler-system coherence) identify BSD as the expected canonical product form once explicit bridge hypotheses are supplied. For rank ≤ 1, this is a normal-form reading of the Gross–Zagier/Kolyvagin theorem package. For rank ≥ 2, Height Saturation is conditional: it names the missing higher Gross–Zagier height/cycle coupling and higher-rank Selmer/Kolyvagin/Sha control, but does not prove them. Note: This is an advanced draft. Major revisions are not excluded. CHANGELOG Changes in Version 1.5 (July 2026) Maintenance and LaTeX/PDF design release following the July 2026 font vectorization, page isolation, and layout audit. This version does not upgrade the theorem status or add a new BSD proof claim. Font vectorization: Activated 100% Type-1 vector fonts (fontenc T1 + lmodern) and glyphtounicode mapping across English and German manuscripts, eliminating MiKTeX bitmap font fallbacks and restoring vector crispness and text searchability. Table of contents layout: Applied REVTeX \onecolumngrid \clearpage \tableofcontents \clearpage \twocolumngrid page isolation to position the TOC on dedicated pages (pages 2–3) and isolate Section 1 on a fresh page (page 4). Hyperref destination filtering: Added \HyperDestNameFilter (de.#1 / en.#1) to prevent PDF anchor collisions when merging bilingual documents. Layout audit: Verified zero Overfull \hbox layout warnings and confirmed table column alignment within margin boundaries. DE/EN synchronization: Rebuilt English, German, and combined PDFs (BSD_Positivity_EN.pdf, BSD_Positivity_DE.pdf, BSD_Positivity_kombi.pdf) synchronously with genuine UTF-8 umlauts verified in the German text track. Changes in Version 1.4 (June 2026) Maintenance release after the June 2026 citation, source-metadata, and English-style checks. This version does not upgrade the theorem status or add a new BSD proof claim. Sources: The Kim2024 bibliography entry was updated from forthcoming status to the published Transactions of the American Mathematical Society reference: 377(5), 3691--3725, DOI 10.1090/tran/9125. Citations: The English programme-comparison table now cites the RH companion explicitly; EN/DE bibliography widths and current Zenodo self-reference titles were synchronised. Style and claim hygiene: Remaining English register edges were softened toward journal-style normal-form language, without changing the mathematical content. Guardrails: The higher-rank part remains conditional on the named higher Gross--Zagier, Selmer/Kolyvagin, and Sha-control bridges; rank-2 numerics remain diagnostics, not proof-grade closure. DE/EN: English, German, and combined PDFs were rebuilt synchronously; the German text track was checked for real umlauts. Changes in Version 1.3 (May 2026) Maintenance: Published the post-v1.2 disclosure/style hotfix: AI-use wording was removed from thanks/acknowledgment contexts and retained only as a neutral AI Disclosure / KI-Offenlegung section. Maintenance: Updated the IUT companion reference to the stable Concept DOI 10.5281/zenodo.19960781. Minor: Added the advanced-draft note directly to the Zenodo description while preserving the v1.2 description structure and related identifiers. Changes in Version 1.2 (May 2026) Major: Added Post-Zookeeper higher-rank module framing and synchronized the German paper with the English Szpiro/theta bridge. Critical: Corrected the Szpiro/Tamagawa implication: unweighted control of Tamagawa product does not imply Szpiro without a prime-weighted local upgrade. Critical: Added bridge-hypothesis and Advice guardrails: Euler-system norm coherence is not full higher-rank Selmer control, and rank-2 numerics are reference-normalized diagnostics until advice-free channels and negative controls are supplied. Minor: Fixed German LaTeX labels and updated related self-references to Concept DOIs. Minor: Source-metadata cleanup after 2026-05-18 check: corrected the German LSZ/JEMS DOI and bibliographic data, synchronized RFEP/RH/IUT self-reference titles with current Zenodo records, added the IUT DOI, and replaced the old CRM version DOI by the current CRM concept DOI. Changes in Version 1.1 (March 2026) No substantive changes. Recompiled with latest cross-references.