The BSD Conjecture as a Positivity Normal-Form Theorem

We reformulate the Birch and Swinnerton-Dyer conjecture as a positivity normal-form theorem. Four axioms (finite capacity, height positivity, Iwasawa control, Euler composition) characterise BSD as the unique rank-compatible L-function theory. For rank ≤ 1, all axioms are verified via Gross–Zagier and Kolyvagin. For rank ≥ 2, the Height Saturation Conjecture provides a thermodynamic completion route. We identify the missing higher Gross–Zagier formula as the central obstruction. Published in English and German versions. Part of the Functional Stability Theory (FST) programme.