The Birch–Swinnerton-Dyer Rank Equality from Source Closure and Endpoint Readout

This paper formulates the rank equality in the Birch--Swinnerton-Dyer conjecture as a source-closure and endpoint-readout problem in Coherence Geometry. For a fixed elliptic curve E/Q, the paper constructs a coherent BSD source S_BSD(E) whose admissible degree layers are selected by local coherence, global obstruction vanishing, and primitive terminal readout. The fixed curve determines a finite CG substrate Sigma_E. A constrained-coherence capacity mechanism inherited from the GCT--SCRC--CS-SCT framework gives finite primitive terminal capacity; by exterior-terminal support, the BSD source closes at a maximal terminal source degree r. Rank synchronization then shows that source-valid endpoint rank projections read this same terminal degree. The arithmetic endpoint is identified with the Mordell--Weil exterior filtration, whose maximal nonzero exterior degree is the Mordell--Weil rank. The analytic endpoint is identified with terminal persistence of the analytically continued germ of L(E,s) through the vanishing filtration at s=1, whose maximal persistence depth is the analytic order of vanishing. Thus the arithmetic and analytic CG endpoint rank projections identify both public BSD rank quantities with the same terminal source degree. This v1.0 archival release supersedes earlier BSD-oriented notes by the author. Those earlier notes should be regarded as developmental background rather than the current statement of the author's BSD approach. Internal Reference ID: CGI-RSR-000032