TEBAC Hodge Program IV: Hilbert--Chow Reconstruction and Primitive Cycle Frames

This preprint is the fourth module of the TEBAC Hodge program. It develops the Hilbert--Chow reconstruction architecture and primitive cycle-frame formalism for a Clay-compatible modular attack on the rational Hodge conjecture. The module is a reconstruction framework and theorem-target paper; it does not claim a completed proof of the full Hodge conjecture. Starting from the earlier TEBAC Hodge modules, the paper works with 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). \ The central reconstruction operator is the Hilbert--Chow incidence cycle-class map\ Cₗ, ₏: Q Chowᵖ (X) Xᵖ, cl (Zₜ), image is the algebraic cycle span \ (Aᵖ (X) \). The dual detector map is\ Cₗ, ₏^: (KXᵖ) ^ ₐ (Chowᵖ (X), Q), \ Cₗ, ₏^ () (t) = (cl (Zₜ) ). \ The module formulates the Algebraic Separator Theorem as the decisive next target: \\, 0 (KXᵖ) ^, \, Z Zᵖ (X) ₐ that (cl (Z) ) 0. \ It also develops primitive cycle frames, rank descent, detector-matrix tests, and invariant/moving/isolated residue ledgers. The paper explicitly excludes hidden use of the Hodge conjecture, the standard conjectures, motivic full faithfulness, or analytic approximation in place of exact equality in \ (H^2p (X, Q) \). Thus HODGE-IV closes the Hilbert--Chow reconstruction architecture of the TEBAC Hodge program and prepares the sharper Split A front on detector-matrix rank closure.