TEBAC BSD-IV: Primitive Coordinate-Symbol Right Inverse and Finite Frame Attack
Tosho Lazarov Karadzhov · Zenodo (CERN European Organization for Nuclear Research) · 2026
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\[\Sigma^\Theta:\mathcal P^\Theta_{E,\ell,w}(v,\eta)\longrightarrow\bigl(\mathcal F^\Theta_{E,\ell,w}(v,\eta)\bigr)^\vee .\]The main Path A objective is to construct a primitive coordinate-symbol right inverse\[\mathfrak C^\Theta:(\mathcal F^\Theta)^\vee\longrightarrow\mathcal P^\Theta\quad\text{with}\quad\Sigma^\Theta\circ\mathfrak C^\Theta=\operatorname{id}.\] 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\[\Omega^{\Theta,\mathrm{act}}:(\mathcal F^\Theta)^\vee\longrightarrow\mathfrak W^{\Theta,\mathrm{act}},\]with\[\mathsf E^{\Theta,\mathrm{act}}\Bigl(\Theta^{\mathrm{spec}}_{E,\ell}(\Omega^{\Theta,\mathrm{act}}(\varphi))\Bigr)=\varphi\qquad\forall \varphi\in(\mathcal F^\Theta)^\vee .\] Under this actual coordinate-probe admission rule, the response matrix for the coordinate probes is the identity matrix:\[M^\Theta=(\delta_{ij}),\qquad\det M^\Theta=1 e0.\]Consequently,\[\Im \mathsf E^\Theta=(\mathcal F^\Theta)^\vee,\]and the raw Mordell--Weil bridge residue satisfies\[E_M^{\mathrm{ver}}=0.\] 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\[\Theta^{\mathrm{spec}}_{E,\ell}.\] 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.