TEBAC BSD-II: Centered Determinant, Normalized Reference-Factor, and Completed Comparison Package for Elliptic Curves over Q

This preprint records the BSD-II module in the TEBAC five-module program toward the Birch--Swinnerton--Dyer conjecture for elliptic curves over Q. The role of BSD-II is to provide the determinant/parity/comparison layer built on the upstream BSD-I centered operator package. The manuscript isolates the centered heat-trace subtraction mechanism, Mellin/zeta regularization, the centered determinant in the even variable = (s-1) ², the normalized reference-factor package, and the completed determinant \ D₂₎₌₏^E (s) =Dₑ₄₅^E (s) \, D₂₄₍^E\! ( (s-1) ²), \ to be compared with the completed elliptic L-function (E, s). A central theme of the current version is status-explicit closure reduction. The text does not present BSD-II as fully closed; instead, it rewrites the module so that the remaining open core is sharply localized. In particular: the odd root-number obstruction to using the even determinant variable alone is isolated theorem-level; the reference-factor uniqueness side is reduced formally once normalization is fixed; the comparison quotient is reduced to a normalized continuation/no-spuriousness package; the contamination / finite-slice / artifact machinery is reformulated on the centered operator side, eliminating the earlier space-mismatch between the centered channel and the final BSD channel; the theta layer is reduced to a centered artifact-elimination and spectral-localization problem rather than a diffuse analytic package. In its current form, BSD-II should be read as a tightened completed-comparison module with a clearly identified remaining construction burden, not yet as a finished unconditional proof module. Its purpose is to make precise exactly what BSD-III may later import for the analytic center theorem and what BSD-V may later import for completed leading-term comparison. Notes Current status: not yet fully closed. The remaining open core is concentrated in: the centered theta / artifact-elimination package, the existence of a canonical entire determinant model, the existence of a normalized reference factor with exact central germ, the completed comparison continuation / no-spuriousness package.