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\[K_X^p := H²ᵖ(X,Q)∩ Hp,p(X),\]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 module focuses on the construction of explicit algebraic cycle separators for nonzero rational Hodge detectors. For a nonzero detector\[0≠λ∈(K_X^p)^,\]a separator certificate is a tuple\[Cert(λ)=(t,Z_t,cl(Z_t),λ(cl(Z_t)))\]such that \(t^p(X)\), \(Z_t⊂ X\) is the corresponding codimension-\(p\) algebraic cycle, and\[λ(cl(Z_t))≠0.\] The certificate domain is\[ D_λ(X,p)=\{\,t∈ Chow^p(X):λ(cl(Z_t))≠0\,\}.\]Thus a detector admits an algebraic separator certificate exactly when\[ D_λ(X,p)≠.\] 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:\[∀\,0≠λ∈(K_X^p)^, D_λ(X,p)≠.\]A proof of this target would imply detector no-loss,\[Ann(K_X^p)^A^p(X)=0,\]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.