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 1.1 (March 2026) No substantive changes. Recompiled with latest cross-references. 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 1.3 (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.