BSD IV: Arithmetic Bridge, Exact Local Detection, and Residual-Existence Reductions (Audited Status Version)

This preprint records the current advanced state of Module BSD-IV in the TEBAC Birch--Swinnerton--Dyer program. BSD-IV is the arithmetic bridge module: its role is to connect the central spectral kernel produced upstream with the Mordell--Weil free part of the elliptic curve, and then to identify the induced spectral height pairing with the classical Neron--Tate height pairing. The present version contains the internally closed BSD-IV chain and a detailed external verification ledger. The internal TEBAC BSD-IV fronts are organized around central-blind admissibility, support-boundary reciprocity, terminal neutral exactness, generator-complete central realization, and height-scale normalization. These internal reductions produce a formal bridge package from the central spectral kernel to the expected arithmetic Mordell--Weil target. Externally, the manuscript isolates the verification residues\[RBSDIVᵛᵉʳ(E)=(0,0,0,E_Mᵛᵉʳ,E_Hᵛᵉʳ),\]after verifying the detector comparison, support-boundary reciprocity comparison, and terminal Selmer/Sha-channel routing. In particular, the current version makes clear that the terminal residual channel is correctly identified with the standard Selmer/Sha channel, without assuming \((E)=0\). The main remaining arithmetic burden is the Mordell--Weil bridge residue\[E_Mᵛᵉʳ=0.\]This version reduces that problem to a precise local primitive response theorem: the primitive symbols generated by the spectral extractor must span the full dual of each local spectral footprint space. Equivalently, the remaining theorem-level obstruction is the primitive extractor generation statement\[ImresΘ,=( F^Θ)^.\]If this primitive dual-basis availability theorem is proved, the Mordell--Weil bridge residue closes and the module advances to the final height-comparison front. The manuscript therefore should be read as a substantially reduced and verification-stable BSD-IV module, not yet as a final unconditional proof of BSD-IV or of the Birch--Swinnerton--Dyer conjecture. Its main contribution in the present version is to isolate the last non-formal arithmetic obstruction in a clean, referee-readable theorem-level form.