TEBAC BSD-I: Canonical Arithmetic/Operator Setup and Center-Parity Package for Elliptic Curves over Q

This preprint presents BSD-I, the foundational upstream module of the TEBAC Birch--Swinnerton--Dyer program for elliptic curves over Q. Its purpose is not to solve the full Birch--Swinnerton--Dyer conjecture in isolation, but to fix, in a precise and non-circular way, the operator-theoretic and parity-theoretic package that later modules of the program are intended to import. 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. The paper begins by fixing the arithmetic center at $s=1$, introducing the centered local variable ν = s-1 and the even variable σ = (s-1)², and proving the root-number parity lemma for the completed L-function Λ(E,s). This establishes the first invariant structural constraint of the BSD program: in the odd root-number case, the centered even variable σ alone cannot retain the full parity data, so any later determinant or comparison layer must account explicitly for the odd linear factor in ν. The main body of the preprint then develops a theorem-level prototype arithmetic/operator package. It constructs normalized local arithmetic data at the archimedean place, at good finite places, and at bad primes; proves local-to-global closure of the resulting Hilbert-space assembly; performs the centered reduction from the global arithmetic Hilbert space to the centered BSD channel; and builds a distinguished centered Dirac-type prototype operator with compact resolvent and symmetry about the arithmetic center. In the resulting minimal centered model, the operator package is closed as a foundational prototype andyields a finite-dimensional central eigenspace at the spectral point $1$. Conceptually, the paper should be read as an upstream gatekeeper module. It freezes the exact operator package, centered symmetry, and parity interface that later modules of the TEBAC BSD program are expected to use. While the present model is intentionally minimal and may later admit stronger arithmetic-semantic refinements, BSD-I is already closed here as a theorem-level foundational prototype.