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

This preprint records the current expanded verification 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 imported from the upstream analytic-rank layer with the Mordell--Weil free part of the elliptic curve, and then to compare the induced spectral height pairing with the classical Neron--Tate height pairing. The present version contains a consolidated and expanded proof-chain audit of the BSD-IV external verification ledger. The detector comparison, support-boundary reciprocity comparison, and terminal Selmer/Sha-channel routing have been stabilized as verified residues: D^ver=EB^ver=ET^ver=0. \ The remaining Mordell--Weil bridge residue has been reduced through a sequence of equivalent local formulations: primitive dual-basis availability, primitive response-cokernel vanishing, absence of silent local footprints, primitive observation injectivity, primitive hyperplane escape, projective primitive escape-defect vanishing, and finally the construction of a primitive coordinate-symbol right inverse to the response map\^: P^₄, , ₖ (v, ) (F^₄, , ₖ (v, ) ) ^. \ In normalized verification form, BSD-IV closes by passing to the primitive-observable footprint convention and by using the Neron-normalized pullback height convention. Under these conventions the external verification ledger becomes₁ₒ₃₈ₕ^ver (E) = (0, 0, 0, 0, 0). means that the detector, boundary, terminal residual, normalized Mordell--Weil bridge, and normalized height residues all vanish. The manuscript is deliberately explicit that this is not yet the same as an absolute raw unconditional closure of BSD-IV. The remaining raw obstruction is now sharply isolated: one must construct, directly from the primitive spectral extractor \ (^spec₄, \), a finite primitive full-rank frame, equivalently a primitive coordinate-symbol right inverse\ C^: (F^) ^ P^^ C^=id. \ If this construction is supplied, then the Mordell--Weil verification residue closes in the raw external sense. If an independent raw spectral height pairing is retained, one must also prove its exact comparison with the Neron--Tate height: ^raw₄, ₒ₏₄₂=BE^arith\, *\, , \, ₍ₓ. \ Thus this version should be read as a substantially advanced BSD-IV reduction and normalized-closure manuscript, not as a completed unconditional proof of the full Birch--Swinnerton--Dyer conjecture. Its main contribution is to isolate the last raw BSD-IV obstruction in a precise finite-frame construction theorem for the primitive spectral extractor.