TEBAC Hodge Program IV Split C: Isolated Fiber Certificate Theorem

This preprint is Split C of HODGE-IV in the TEBAC Hodge program. It isolates the most difficult residual case in the Hilbert--Chow reconstruction attack on the rational Hodge conjecture: the construction of separator certificates in the isolated fiber regime. The module is a theorem-target and obstruction-isolation 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 those separator problems not resolved by direct Chow certificates, relative total-space certificates, moving Hodge-locus certificates, or Lefschetz-type certificates. For a nonzero detector\0 (KXᵖ) ^, isolated certificate domain is denoted\ D^iso_ (X, p). non-emptiness means that there exists an actual codimension-\ (p\) algebraic cycle \ (Zₜ X\), represented by a Chow or Hilbert point \ (t\), such that\ (cl (Zₜ) ) 0 the certificate is not obtained from the previously available direct, moving, total-space, or Lefschetz regimes. The central theorem target is the Isolated Fiber Certificate Theorem: \\, 0₏ₑ₈₌ (K^pₗ, ₏ₑ₈₌/₍₎₍₋₄₅) ^, D^iso_ (X, p). target addresses primitive, non-Lefschetz Hodge detectors and asks for genuine algebraic cycle separators in the isolated fiber case. The module develops the primitive/non-Lefschetz detector quotient, the isolated Hilbert--Chow fiber operator, a protocol for testing primitive classes, failure modes, a non-circularity ledger, and acceptance tests. It explicitly excludes hidden use of the Hodge conjecture, the standard conjectures, motivic full faithfulness, analytic approximation, or monodromy-only control as substitutes for actual algebraic cycles. Thus Split C is Zenodo-ready as an isolated-fiber certificate theorem-target module. The next front is the constructive production of primitive non-Lefschetz separator cycles.