TEBAC BSD-III: Central Kernel and Analytic Rank for Elliptic Curves over Q

This preprint presents the BSD-III module in the five-module TEBAC program toward the Birch--Swinnerton--Dyer conjecture for elliptic curves over Q. Its scope is deliberately narrow and referee-facing: assuming the BSD-I centered operator package and the discharged BSD-II completed determinant/comparison package, the paper proves the analytic center theoremₒ=₁ (E, s) = \! (D₁ₒ₃-1), consequentlyₒ=₁L (E, s) = \! (D₁ₒ₃-1). \ The manuscript is designed as an analytic-rank consequence module only. It does not introduce a new determinant mechanism, does not construct a Selmer/cohomological interface, does not prove a Mordell--Weil bridge, does not identify algebraic rank, and does not attempt the final Birch--Swinnerton--Dyer leading-term assembly. Those tasks belong downstream, principally to BSD-IV and then BSD-V. The purpose of this paper is to isolate and discharge the analytic center theorem as a clean theorem-level consequence of the finished BSD-I/BSD-II import package, while keeping the analytic-rank statement strictly separate from later arithmetic bridge semantics.