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:\[E_Dᵛᵉʳ=E_Bᵛᵉʳ=E_Tᵛᵉʳ=0.\] 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\[Σ^Θ: P^ΘE,,w(v,η)( F^ΘE,,w(v,η))^ .\] 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 becomes\[RBSDIVᵛᵉʳ(E)=(0,0,0,0,0).\]This 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 \(ΘˢᵖᵉᶜE,\), a finite primitive full-rank frame, equivalently a primitive coordinate-symbol right inverse\[ C^Θ:( F^Θ)^ P^ΘΣ^Θ∘ C^Θ=id.\] 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:\[qʳᵃʷE,spec=B_Earith\,*\, ,\,NT.\] 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.