Néron Height and Regulator Calibration in the Determinant--Selmer Bridge

This preprint is Core V of the TEBAC BSD program. It isolates the N\'eron height and regulator calibration layer in the determinant--Selmer approach to the Birch--Swinnerton--Dyer problem over \(\mathbb{Q}\). The module identifies the relevant height pairing through the N\'eron--Tate expression \(\bigl\langle B_E^{\mathrm{arith}}v, B_E^{\mathrm{arith}}w \bigr\rangle_{\mathrm{NT}}\). Consequently, under integral lattice compatibility, the spectral regulator agrees with the classical N\'eron--Tate regulator: \(\operatorname{Reg}_{\mathrm{spec}}(E)=\operatorname{Reg}(E)\). The preprint also records the finite-index variant: if the image of the spectral lattice under \(B_E^{\mathrm{arith}}\) is a sublattice of index \(I_E\), then \(\operatorname{Reg}_{\mathrm{spec}}(E)=I_E^2\operatorname{Reg}(E)\). This separates the regulator comparison from the rank bridge and from the finite Sha/Tamagawa/torsion ledger. This is a claim-safe preprint module. It does not assert a standalone proof of the Birch--Swinnerton--Dyer conjecture. Its role is to isolate the height and regulator calibration layer required after the Selmer--Kummer bridge and before the final leading-coefficient assembly.