TEBAC Hodge Program IV Split B: Separator Certificate Construction

This preprint is Split B of HODGE-IV in the TEBAC Hodge program. It develops the separator-certificate construction calculus for the Algebraic Separator Theorem in the rational Hodge conjecture. The module is a certificate-construction and theorem-target paper; it does not claim a completed proof of the full Hodge conjecture. Starting from 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 module focuses on the construction of explicit algebraic cycle separators for nonzero rational Hodge detectors. For a nonzero detector\[0\neq\lambda\in(K_X^p)^\vee,\]a separator certificate is a tuple\[\mathsf{Cert}(\lambda)=(t,Z_t,\operatorname{cl}(Z_t),\lambda(\operatorname{cl}(Z_t)))\]such that \(t\in\operatorname{Chow}^p(X)\), \(Z_t\subset X\) is the corresponding codimension-\(p\) algebraic cycle, and\[\lambda(\operatorname{cl}(Z_t))\neq0.\] The certificate domain is\[\mathfrak D_\lambda(X,p)=\{\,t\in \operatorname{Chow}^p(X):\lambda(\operatorname{cl}(Z_t))\neq0\,\}.\]Thus a detector admits an algebraic separator certificate exactly when\[\mathfrak D_\lambda(X,p)\neq\varnothing.\] The module organizes the admissible certificate regimes: direct Chow certificates, relative total-space certificates, moving Hodge-locus certificates, hyperplane/Lefschetz certificates, and isolated fiber certificates. It also records the failure modes where monodromy control, Hodge-locus control, currents, limits, or analytic approximation do not produce actual algebraic cycle classes. The remaining theorem-level target is the Separator Certificate Existence Theorem:\[\forall\,0\neq\lambda\in(K_X^p)^\vee,\qquad\mathfrak D_\lambda(X,p)\neq\varnothing.\]A proof of this target would imply detector no-loss,\[\operatorname{Ann}_{(K_X^p)^\vee}A^p(X)=0,\]and hence\[R_X^p=0.\] Thus Split B is Zenodo-ready as a separator-certificate construction module. The next unresolved front is the isolated-fiber certificate theorem, where one must construct actual algebraic cycle separators not obtained from direct Chow, total-space, moving-locus, or Lefschetz regimes.