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\[K_X^p := H²ᵖ(X,Q)∩ Hp,p(X),\]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 central reconstruction operator is the Hilbert--Chow incidence cycle-class map\[ CX,p: Q Chow^p(X)_X^p, cl(Z_t),\]whose image is the algebraic cycle span \(A^p(X)\). The dual detector map is\[ CX,p^{}:(K_X^p)^Q(Chow^p(X), Q),\]with\[ CX,p^{}(λ)(t)=λ(cl(Z_t)).\] The module formulates the Algebraic Separator Theorem as the decisive next target:\[∀\,0≠λ∈(K_X^p)^,∃\,Z∈ Z^p(X)Q thatλ(cl(Z))≠0.\] 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 \(H²ᵖ(X, Q)\). 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.