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 carrierXᵖ: = H^2p (X, Q) H^p, p (X), algebraic cycle spanᵖ (X): =spanₐ\\, cl (Z): Z X algebraic of codimension p\, \, the residual obstructionXᵖ: =KXᵖ/Aᵖ (X), paper studies the dual Hilbert--Chow detector map\ Cₗ, ₏^: (KXᵖ) ^ ₐ (Chowᵖ (X), Q), by\ Cₗ, ₏^ () (t) = (cl (Zₜ) ). \ The central target is the Algebraic Separator Theorem: \\, 0 (KXᵖ) ^, \, Z Zᵖ (X) ₐ that (cl (Z) ) 0. \ The module translates this target into detector-matrix rank closure. Given detectors\₁, , N (KXᵖ) ^ algebraic cycles₁, , ZM Zᵖ (X) ₐ, detector matrix is₈₉: =ᵢ (cl (Zⱼ) ). rank closure requires M= ₐKXᵖ. \ 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.