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\[K_X^p := H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X),\]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 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\[0\neq\lambda\in(K_X^p)^\vee,\]the isolated certificate domain is denoted\[\mathfrak D^{\mathrm{iso}}_\lambda(X,p).\]Its non-emptiness means that there exists an actual codimension-\(p\) algebraic cycle \(Z_t\subset X\), represented by a Chow or Hilbert point \(t\), such that\[\lambda(\operatorname{cl}(Z_t))\neq0\]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:\[\forall\,0\neq\lambda_{\mathrm{prim}}\in(K^{p}_{X,\mathrm{prim/nonLef}})^\vee,\qquad\mathfrak D^{\mathrm{iso}}_{\lambda}(X,p)\neq\varnothing.\]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.