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 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), module focuses on the construction of explicit algebraic cycle separators for nonzero rational Hodge detectors. For a nonzero detector\0 (KXᵖ) ^, separator certificate is a tuple () = (t, Zₜ, cl (Zₜ), (cl (Zₜ) ) ) that \ (tᵖ (X) \), \ (Zₜ X\) is the corresponding codimension-\ (p\) algebraic cycle, and\ (cl (Zₜ) ) 0. \ The certificate domain is\ D_ (X, p) =\\, t Chowᵖ (X): (cl (Zₜ) ) 0\, \. a detector admits an algebraic separator certificate exactly when\ D_ (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 (KXᵖ) ^, D_ (X, p). proof of this target would imply detector no-loss, (₊ₗ㵵) ^Aᵖ (X) =0, henceXᵖ=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.