TEBAC BSD-IV: Primitive Coordinate-Symbol Right Inverse and Finite Frame Attack

This preprint is Split in the TEBAC BSD-IV series. Its purpose is to attack the raw external Mordell--Weil bridge residue in the BSD-IV arithmetic module by reducing it to a finite and explicitly auditable primitive-frame construction. The central object of the manuscript is the primitive response map\[Σ^Θ: P^ΘE,,w(v,η)( F^ΘE,,w(v,η))^ .\]The main Path A objective is to construct a primitive coordinate-symbol right inverse\[ C^Θ:( F^Θ)^ P^ΘΣ^Θ∘ C^Θ=id.\] The manuscript develops a full sequence of equivalent and progressively sharper criteria for this right-inverse problem: finite primitive full-rank frames, Gram nondegeneracy, atom separation, triangular primitive coordinate atoms, residual scalar pivots, quotient-coordinate lifts, source-coordinate completeness, coefficient-packet surjectivity, actual source-generator full rank, and finally actual coordinate-probe admission. The resulting terminal statement is that the raw Mordell--Weil verification residue closes if the actual local coefficient calculus of the primitive spectral extractor contains the coordinate-probe constructor\[ΩΘ,act:( F^Θ)^ WΘ,act,\]with\[ EΘ,act(ΘˢᵖᵉᶜE,(ΩΘ,act(φ)))=φ∀ φ∈( F^Θ)^ .\] Under this actual coordinate-probe admission rule, the response matrix for the coordinate probes is the identity matrix:\[M^Θ=(δᵢⱼ), M^Θ=1≠0.\]Consequently,\[ E^Θ=( F^Θ)^,\]and the raw Mordell--Weil bridge residue satisfies\[E_Mᵛᵉʳ=0.\] The manuscript is deliberately explicit about its status. It does not claim that BSD-IV is already fully and unconditionally closed in the raw external sense. Instead, it gives a terminal and referee-readable reduction of the raw Mordell--Weil bridge obstruction to the actual coordinate-probe admission theorem for the primitive spectral extractor. In the source-completed model, the Mordell--Weil residue closes. For absolute raw closure, the remaining upstream task is to discharge actual coordinate-probe admission from the original coefficient calculus of\[ΘˢᵖᵉᶜE,.\] Thus this upload should be read as a Zenodo-polished progress manuscript and terminal reduction preprint for the Path A attack on BSD-IV, not as a completed proof of the full Birch--Swinnerton--Dyer conjecture.