TEBAC BSD-IV Split: Actual Coefficient Calculus for the Primitive Spectral Extractor

This preprint continues the BSD-IV split sequence by supplying the actual coefficient-calculus construction needed after the primitive-frame reduction. It constructs a source-complete actual coefficient calculus for the primitive spectral extractor \(Θ^{}E,\), admits coordinate probes on each local footprint space, and obtains the identity response matrix for the corresponding local response functionals. Consequently, the raw Mordell--Weil verification residue closes for the strengthened source-complete actual extractor:\[E_M^{}=0.\] The manuscript is status-explicit. It should be read as a Mordell--Weil bridge gate closure for the source-complete actual extractor, not as an unconditional closure of BSD-IV and not as a completed proof of the full Birch--Swinnerton--Dyer conjecture. The remaining BSD-IV front after this upload is the raw height comparison\[q^{}E,=B_E,*\, ,\,_{}.\]