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:\[E_D^{\mathrm{ver}}=E_B^{\mathrm{ver}}=E_T^{\mathrm{ver}}=0.\] 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\[\Sigma^\Theta:\mathcal P^\Theta_{E,\ell,w}(v,\eta)\longrightarrow\bigl(\mathcal F^\Theta_{E,\ell,w}(v,\eta)\bigr)^\vee .\] 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\[R_{\mathrm{BSDIV}}^{\mathrm{ver}}(E)=(0,0,0,0,0).\]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 \(\Theta^{\mathrm{spec}}_{E,\ell}\), a finite primitive full-rank frame, equivalently a primitive coordinate-symbol right inverse\[\mathfrak C^\Theta:(\mathcal F^\Theta)^\vee\longrightarrow\mathcal P^\Theta\quad\text{with}\quad\Sigma^\Theta\circ\mathfrak C^\Theta=\operatorname{id}.\] 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:\[q^{\mathrm{raw}}_{E,\mathrm{spec}}=B_E^{\mathrm{arith}\,*}\langle\, ,\,\rangle_{\mathrm{NT}}.\] 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.