TEBAC Birch--Swinnerton--Dyer Program: Unified Roadmap

We present a unified conceptual roadmap for a five-module TEBAC program toward the Birch--Swinnerton--Dyer conjecture for elliptic curves over Q. The manuscript is organized so that a referee can inspect the intended argument module by module: the canonical arithmetic/operator setup (BSD-I), the spectral determinant and functional-equation package for the completed L-function (BSD-II), the center-kernel/analytic-rank module (BSD-III), the independent bridge from the central spectral kernel to the Mordell–Weil rank (BSD-IV), and the final assembly of the rank statement and leading-coefficient formula (BSD-V). Тhe overarching aim of the TEBAC program is to replace direct analytic root-finding for L-functions by dimension counts of canonical spectral kernels. Instead of studying zeros of L (E, s) directly, each module isolates a Hilbert-space/Dirac-type package whose central eigenspace is intended to encode the relevant arithmetic information. In this way, questions about orders of vanishing are reformulated as spectral statements about finite-dimensional kernels of distinguished operators. This record is a programmatic master outline, not yet a completed proof manuscript. Its purpose is to document the modular architecture, notation, target theorems, and dependency structure of the proposed BSD program in a referee-readable form. In particular, BSD-III and BSD-IV are treated here as target proof modules whose detailed arguments are intended to appear in separate self-contained preprints before any final proof-level BSD claim is made.