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 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 central reconstruction operator is the Hilbert--Chow incidence cycle-class map\[\mathcal C_{X,p}:\mathbb Q\langle \operatorname{Chow}^p(X)\rangle\longrightarrowK_X^p,\qquadt\longmapsto \operatorname{cl}(Z_t),\]whose image is the algebraic cycle span \(A^p(X)\). The dual detector map is\[\mathcal C_{X,p}^{\vee}:(K_X^p)^\vee\longrightarrow\operatorname{Fun}_{\mathbb Q}(\operatorname{Chow}^p(X),\mathbb Q),\]with\[\mathcal C_{X,p}^{\vee}(\lambda)(t)=\lambda(\operatorname{cl}(Z_t)).\] The module formulates the Algebraic Separator Theorem as the decisive next target:\[\forall\,0\neq\lambda\in(K_X^p)^\vee,\quad\exists\,Z\in Z^p(X)_{\mathbb Q}\quad\text{such that}\quad\lambda(\operatorname{cl}(Z))\neq0.\] 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 \(H^{2p}(X,\mathbb Q)\). 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.