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

This upload contains the publication-clean master version of TEBAC Hodge Program V: Terminal Moving-Face and Boundary-Normal Reduction for the Rational Hodge Conjecture. 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 in codimension \(p\), and \(U_X^p\) contains the source already certified by the admissible TEBAC construction. The terminal obstruction is tested by nonzero detectors\[0 \neq \lambda \in (V_X^p)^\vee.\] The goal of \(\mathrm{HODGE\text{-}V}\) is to prove that every such detector is nonzero on some admissible terminal moving-face, Hilbert--Chow, boundary-normal, or semistable packet class. Equivalently, the terminal quotient vanishes once no detector obstruction survives:\[V_X^p = 0.\] The manuscript organizes the terminal reduction around the chain\[\begin{aligned}\text{detector}&\longrightarrow \text{semistable packet}\longrightarrow \text{boundary-normal tangent image} \\&\longrightarrow \text{raw source production}\longrightarrow \text{coherent source module} \\&\longrightarrow \text{corrected detector projection}.\end{aligned}\] In this formulation, the decisive terminal task is to produce admissible packet data whose terminal image escapes the detector hyperplane\[\ker(\lambda) \subset V_X^p.\] The paper deliberately separates unconditional formal implications from the remaining theorem targets. It proves the non-circular terminal reduction architecture and records the exact local, finite-packet, coherence, boundedness, exhaustivity, and corrected-projection gates required to complete the unconditional chain. In particular, the remaining terminal assertions are formulated as explicit mathematical targets rather than hidden assumptions. The publication-clean version removes internal construction labels, sequential working language, and pass-by-pass drafting markers. The manuscript is now organized in an Annals-style mathematical form through definitions, lemmas, theorem targets, terminal implications, and non-circularity ledgers. The technical appendix assembles the terminal reduction lemmas in a clean order, including moving-face reduction, detector hyperplane formalism, boundary-normal image separation, raw-source finite generation, finite Hilbert--Chow packet-envelope boundedness, coherent module criteria, and finite-presentation conditions for raw source constraints. The reduction is explicitly non-circular. The manuscript does not assume the identities\[K_X^p = A^p(X),\qquadV_X^p = 0,\]nor does it invoke the Hodge conjecture, the generalized Hodge conjecture, the standard conjectures, Bloch--Beilinson/Murre filtrations, motivic full faithfulness, or automatic algebraicity of rational Hodge classes as hidden inputs. Thus the manuscript records the current terminal \(\mathrm{HODGE\text{-}V}\) reduction architecture in a clean publication-facing form. Its purpose is twofold: to present the terminal TEBAC Hodge V reduction as a consolidated theorem-target manuscript, and to isolate the precise mathematical gates whose proof would complete the unconditional terminal chain toward the rational Hodge conjecture. This version supersedes earlier internal working versions by removing sequential construction language and by reorganizing the manuscript into a cleaner Annals-style mathematical structure.