The Birch-Swinnerton-Dyer Rank Equality from Source Closure and Endpoint Readout
Internal ID: CGI-RSR-000032
Author(s): Barry L. Petersen
Document Type: Research Paper
Publication Date: June 2026
Original Creation Date: June 17, 2026
Status: Public archival release
Domains: Mathematics
Research Topics: Number theory / arithmetic geometry, Clay Millennium Problems, Birch and Swinnerton-Dyer Conjecture, elliptic curves, Coherence Geometry, source closure, endpoint readout
Abstract
We formulate the rank equality in the Birch–Swinnerton-Dyer conjecture as a source-closure problem in Coherence Geometry. For a fixed elliptic curve \(E/\mathbb Q\), we construct a coherent BSD source \(S_{\mathrm{BSD}}(E)\) whose degree layers are defined by local coherence, global obstruction vanishing, and primitive terminal readout. The fixed curve determines a finite CG substrate \(\Sigma_E\). The constrained-coherence capacity mechanism inherited from the GCT–SCRC–CS-SCT framework gives a finite upper bound on independent primitive terminal coherence directions. By the Exterior–Terminal Support Law, admissible exterior source layers vanish above this capacity, hence the BSD source has a maximal nonzero terminal degree \(r\). Rank synchronization shows that any source-valid CG endpoint rank projection reads this same terminal degree: such a projection counts only source-supported rank data and is nondegenerate on the primitive terminal rank layer it is constructed to read. The arithmetic endpoint is identified with the Mordell–Weil exterior filtration, whose maximal nonzero exterior degree is \(\mathrm{rank},E(\mathbb Q)\). The analytic endpoint is identified with terminal persistence of the analytically continued germ of \(L(E,s)\) in the filtration \(\mathcal I_1^k\) of analytic germs vanishing at \(s=1\), whose maximal persistence depth is \(\mathrm{ord}_{s=1},L(E,s)\). Thus the arithmetic and analytic CG endpoint rank projections identify both public BSD rank quantities with the same terminal source degree, giving
$$
\mathrm{ord}_{s=1},L(E,s)=\mathrm{rank},E(\mathbb Q).
$$
Available Document
DOI: 10.5281/zenodo.20731232
Citation:
Petersen, B. L. (2026). The Birch–Swinnerton-Dyer Rank Equality from Source Closure and Endpoint Readout. Zenodo. https://doi.org/10.5281/zenodo.20731232
Source Code and Supporting Materials
None.
Summary and Notes
In the Coherence Geometry formulation, the Mordell–Weil rank and the analytic order of vanishing are not treated as unrelated public invariants that must be forced to agree after the fact. They are treated as endpoint readouts of a common coherent source.
The paper develops three linked components:
Endpoint recovery. The arithmetic endpoint recovers Mordell–Weil exterior rank, while the analytic endpoint recovers vanishing-filtration persistence of (L(E,s)) at (s=1).
Source closure. A fixed elliptic curve determines a finite CG substrate. Finite primitive terminal capacity implies that the associated BSD source has a maximal nonzero terminal degree.
Rank synchronization. Any source-valid endpoint rank projection reads the same terminal source degree.
Related Work
An earlier BSD-oriented paper in the Coherence Geometry research corpus CGI-RSR-000004 studied an operator bridge between arithmetic heights and analytic residues:
Petersen, B. L. (2026). An Operator Bridge Between Arithmetic Heights and Analytic Residues for Elliptic Curves. Zenodo. https://doi.org/10.5281/zenodo.19969495
The present paper uses a different proof architecture. It replaces the operator-bridge framing with a source-closure and endpoint-readout framework. The earlier operator-bridge paper remains part of the developmental record, but the current BSD rank-equality approach is represented by CGI-RSR-000032.