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 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 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\[0≠λ∈(K_X^p)^,\]the isolated certificate domain is denoted\[ Dⁱˢᵒ_λ(X,p).\]Its non-emptiness means that there exists an actual codimension-\(p\) algebraic cycle \(Z_t⊂ X\), represented by a Chow or Hilbert point \(t\), such that\[λ(cl(Z_t))≠0\]and 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:\[∀\,0≠λₚᵣᵢₘ∈(KᵖX,prim/nonLef)^, Dⁱˢᵒλ(X,p)≠.\]This 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.