TEBAC Hodge Program IV Split A: Algebraic Separator Theorem and Detector--Matrix Rank Closure

This preprint is a Split A module of the TEBAC Hodge Program IV. It sharpens the Hilbert--Chow reconstruction architecture of HODGE-IV into a detector-matrix rank-closure framework for the Algebraic Separator Theorem. The module is a theorem-target and rank-closure formalism; it does not claim a completed proof of the full Hodge conjecture. Starting from the rational Hodge carrier\[K_X^p := H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X),\]the algebraic cycle span\[A^p(X):=\operatorname{span}_{\mathbb{Q}}\{\,\operatorname{cl}(Z):Z\subset X\text{ algebraic of codimension }p\,\},\]and the residual obstruction\[R_X^p:=K_X^p/A^p(X),\]the paper studies the dual Hilbert--Chow detector map\[\mathcal C_{X,p}^{\vee}:(K_X^p)^\vee\longrightarrow\operatorname{Fun}_{\mathbb Q}(\operatorname{Chow}^p(X),\mathbb Q),\]defined by\[\mathcal C_{X,p}^{\vee}(\lambda)(t)=\lambda(\operatorname{cl}(Z_t)).\] The central target is the Algebraic Separator Theorem:\[\forall\,0\neq\lambda\in(K_X^p)^\vee,\quad\exists\,Z\in Z^p(X)_{\mathbb Q}\quad\text{such that}\quad\lambda(\operatorname{cl}(Z))\neq0.\] The module translates this target into detector-matrix rank closure. Given detectors\[\lambda_1,\dots,\lambda_N\in(K_X^p)^\vee\]and algebraic cycles\[Z_1,\dots,Z_M\in Z^p(X)_{\mathbb Q},\]the detector matrix is\[M_{ij}:=\lambda_i(\operatorname{cl}(Z_j)).\]Full rank closure requires\[\operatorname{rank} M=\dim_{\mathbb Q}K_X^p.\] The paper proves the formal equivalences between detector separation, injectivity of the dual incidence map, full-rank detector matrices, finite primitive cycle frames, and vanishing of the residual obstruction. It also formulates the remaining constructive gap: producing an actual algebraic cycle separator for each nonzero detector, without replacing exact rational cohomology by analytic approximation, currents, limits, or hidden motivic assumptions. Thus Split A is Zenodo-ready as a detector-matrix rank-closure and algebraic-separator theorem-target module. The subsequent front is the construction of explicit separator certificates.