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

This manuscript presents the current terminal module \ (HODGE-V\) of the TEBAC Hodge program toward the rational Hodge conjecture. For a smooth projective complex variety \ (X/C\) and codimension \ (p\), the target is the equality \ (KXᵖ = Aᵖ (X) \), where \ (KXᵖ = H^2p (X, Q) H^p, p (X) \) is the rational Hodge carrier and \ (Aᵖ (X) = spanₐ\Z: Z X algebraic of codimension p\\) is the rational subspace generated by algebraic cycle classes. The manuscript works with the terminal quotient \ (VXᵖ = KXᵖ/UXᵖ\), where \ (UXᵖ\) 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 (VXᵖ) ^ \, such that () 0\), where \ (\) 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 \ (U1 U2 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 \ (V_^resp = VXᵖ\), \ (Qₗ, ₏^U1 = 0\), and \ (Bₗ, ₏^U1 = \), and further to a finite determinant certificate \ (ₗ, ₏^U1 0\). The U3 corrected detector survival problem is reduced to a finite correction-kernel and rank-jump condition \ (L, ^det RowSpan\! (C, ^corr) \). The R3 packet-exhaustivity gate is reduced to the quotient-defect condition \ (Q, ^R3: = S, ^raw/im\! (R, ^raw) = 0\). The R1 boundedness gate is reduced to a finite local ledger certificate \ (, ^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 \ (V_^corr-resp: = spanₐ\R, ^{V (u): u K, ^corr\} VXᵖ\) and the compatibility defect \ (Qₗ, ₏^U1/U3: = VXᵖ/V_^corr-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.