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

This manuscript presents the current terminal module \(\mathrm{HODGE\text{-}V}\) of the TEBAC Hodge program toward the rational Hodge conjecture. For a smooth projective complex variety \(X/\mathbb{C}\) and codimension \(p\), the target is the equality \(K_X^p = A^p(X)\), where \(K_X^p = H^{2p}(X,\mathbb{Q}) \cap H^{p,p}(X)\) is the rational Hodge carrier and \(A^p(X) = \operatorname{span}_{\mathbb{Q}}\{[Z] : Z \subset X \text{ algebraic of codimension } p\}\) is the rational subspace generated by algebraic cycle classes. The manuscript works with the terminal quotient \(V_X^p = K_X^p/U_X^p\), where \(U_X^p\) contains the already certified Lefschetz, direct Chow, incidence, total-space, specialization, and previously verified moving-locus contributions. The remaining problem is converted into a dual detector problem: \(0 \neq \lambda \in (V_X^p)^\vee \Longrightarrow \exists\,\Gamma \text{ such that } \lambda(\Gamma) \neq 0\), where \(\Gamma\) must be produced by an admissible Hilbert--Chow, boundary-normal, moving-face, or corrected source construction, not inserted by assuming the Hodge conjecture. The main unconditional output of the manuscript is a non-circular terminal reduction of the rational Hodge conjecture to explicit remaining geometric targets. The current version develops the chain \(\mathrm{U1} \Longrightarrow \mathrm{U2} \Longrightarrow \mathrm{U3}\) through detector-selected semistable packets, raw-source production, coherent raw-source modules, packet exhaustivity, corrected detector projection, and U1--U3 compatibility. The latest passes introduce several finite certificate normal forms. The U1 image-separation problem is reduced to a boundary-response cone and projective base-locus condition \(\mathcal{V}_{\tau}^{\mathrm{resp}} = V_X^p\), \(Q_{X,p}^{\mathrm{U1}} = 0\), and \(\mathfrak{B}_{X,p}^{\mathrm{U1}} = \varnothing\), and further to a finite determinant certificate \(\Delta_{X,p}^{\mathrm{U1}} \neq 0\). The U3 corrected detector survival problem is reduced to a finite correction-kernel and rank-jump condition \(L_{\lambda,\tau}^{\mathrm{det}} \notin \operatorname{RowSpan}\!\bigl(C_{\lambda,\tau}^{\mathrm{corr}}\bigr)\). The R3 packet-exhaustivity gate is reduced to the quotient-defect condition \(Q_{\lambda,\tau}^{\mathrm{R3}} := S_{\lambda,\tau}^{\mathrm{raw}}/\operatorname{im}\!\bigl(\mathcal{R}_{\lambda,\tau}^{\mathrm{raw}}\bigr) = 0\). The R1 boundedness gate is reduced to a finite local ledger certificate \(\Delta_{\lambda,\tau}^{\mathrm{R1}} = 0\), and the R2 coherence gate is expressed through finite-presentation, kernel, and overlap-cocycle criteria. This version also adds the U1--U3 compatibility bridge. It introduces the corrected response cone \(\mathcal{V}_{\tau}^{\mathrm{corr\text{-}resp}} := \operatorname{span}_{\mathbb{Q}}\{R_{\lambda,\tau}^{V}(u) : u \in K_{\lambda,\tau}^{\mathrm{corr}}\} \subseteq V_X^p\) and the compatibility defect \(Q_{X,p}^{\mathrm{U1/U3}} := V_X^p/\mathcal{V}_{\tau}^{\mathrm{corr\text{-}resp}}\). Thus the remaining terminal problem is sharpened to proving corrected detector-visible spanning after all boundary, obstruction, residue, and admissibility corrections have been imposed. The manuscript does not use the Hodge conjecture, the generalized Hodge conjecture, the standard conjectures, Bloch--Beilinson/Murre filtrations, motivic full faithfulness, or the assertion that every rational \((p,p)\)-class is algebraic as inputs. Its current status is best described as an advanced terminal reduction manuscript with explicit remaining hard geometric targets, rather than a completed unconditional proof of the rational Hodge conjecture. Internal working indicators in this version are as follows: Reduction architecture: \(99.997\%\); Full unconditional proof: \(95.90\%\); Annals-facing reduction manuscript: \(99.58\%\); Annals-facing full-proof manuscript: \(92.35\%\). These percentages are internal progress markers for the TEBAC development and should not be interpreted as external certification of the Millennium problem.