Terminal Moving-Face and Boundary-Normal Reduction for the Rational Hodge Conjecture: The TEBAC Hodge Program V

This preprint presents the terminal module \( \mathrm{HODGE\text{-}V} \) of the TEBAC Hodge program for the rational Hodge conjecture. The preceding modules reduce the problem, for a smooth projective complex variety \(X/\mathbb C\) and codimension \(p\), to the terminal quotient\[V_X^p = K_X^p/U_X^p,\]where\[K_X^p = H^{2p}(X,\mathbb Q)\cap H^{p,p}(X)\]is the rational Hodge carrier and \(U_X^p\) contains the already certified algebraic, Lefschetz, total-space, incidence, and specialization contributions. The goal of \( \mathrm{HODGE\text{-}V} \) is to prove terminal moving-face exhaustion\[W_{X,p}^{\mathrm{mf}} = V_X^p,\]or equivalently that every nonzero terminal detector\[0\neq \lambda\in (V_X^p)^\vee\]is nonzero on some admissible moving-face Hilbert--Chow specialization class. The manuscript proves the unconditional formal reduction of this terminal problem to an explicit local detector-active boundary-normal condition. The final local packet criterion is expressed as the existence of an admissible boundary component \(E\) and a boundary-normal class \(\bar e\) satisfying\[\partial_E(\bar e)=0,\qquad\Theta_E(\bar e)=0,\qquad\Lambda_E(\bar e)\neq 0.\]Equivalently, the detector-active amplification criterion is the rank jump\[\operatorname{rank}\begin{pmatrix}\partial_E\\\Theta_E\\\Lambda_E\end{pmatrix}>\operatorname{rank}\begin{pmatrix}\partial_E\\\Theta_E\end{pmatrix}.\] The paper deliberately does not assert that this final component-existence theorem is proved in general. Its main contribution is the non-circular Annals-style terminal reduction: if the detector-active amplification component theorem is established by classical Hilbert--Chow geometry, normal-sheaf deformation theory, semistable degeneration, and limiting mixed Hodge structure methods, then\[W_{X,p}^{\mathrm{mf}}=V_X^p,\qquadV_X^p=0,\qquadK_X^p=A^p(X).\]Thus the rational Hodge conjecture for \((X,p)\) follows conditionally on the final local detector-active boundary-normal construction. The manuscript includes a non-circularity ledger excluding the Hodge conjecture itself, the generalized Hodge conjecture, standard conjectures, Bloch--Beilinson/Murre filtrations, motivic full faithfulness, and automatic algebraicity of rational Hodge classes as hidden inputs.