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\[K_X^p := H²ᵖ(X,Q)∩ Hp,p(X),\]the algebraic cycle span\[A^p(X):=spanQ\{\,cl(Z):Z⊂ X algebraic of codimension p\,\},\]and the residual obstruction\[R_X^p:=K_X^p/A^p(X),\]the paper studies the dual Hilbert--Chow detector map\[ CX,p^{}:(K_X^p)^Q(Chow^p(X), Q),\]defined by\[ CX,p^{}(λ)(t)=λ(cl(Z_t)).\] The central target is the Algebraic Separator Theorem:\[∀\,0≠λ∈(K_X^p)^,∃\,Z∈ Z^p(X)Q thatλ(cl(Z))≠0.\] The module translates this target into detector-matrix rank closure. Given detectors\[λ_1,,λ_N∈(K_X^p)^\]and algebraic cycles\[Z_1,,Z_M∈ Z^p(X)Q,\]the detector matrix is\[Mᵢⱼ:=λ_i(cl(Z_j)).\]Full rank closure requires\[rank M=QK_X^p.\] 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.