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 \(K_X^p = A^p(X)\), where \(K_X^p = H²ᵖ(X,Q) ∩ Hp,p(X)\) is the rational Hodge carrier and \(A^p(X) = spanQ\{[Z] : Z ⊂ X algebraic of codimension p\}\) 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: \(0 ≠ λ ∈ (V_X^p)^ ∃\,Γ such that λ(Γ) ≠ 0\), where \(Γ\) 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 \(Vτʳᵉˢᵖ = V_X^p\), \(QX,pU1 = 0\), and \(BX,pU1 = \), and further to a finite determinant certificate \(ΔX,pU1 ≠ 0\). The U3 corrected detector survival problem is reduced to a finite correction-kernel and rank-jump condition \(Lλ,τᵈᵉᵗ ∉ RowSpan\!(Cλ,τᶜᵒʳʳ)\). The R3 packet-exhaustivity gate is reduced to the quotient-defect condition \(Qλ,τR3 := Sλ,τʳᵃʷ/im\!(Rλ,τʳᵃʷ) = 0\). The R1 boundedness gate is reduced to a finite local ledger certificate \(Δλ,τR1 = 0\), 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 \(Vτ^{{corr-resp}} := spanQ\{Rλ,τV(u) : u ∈ Kλ,τᶜᵒʳʳ\} ⊆ V_X^p\) and the compatibility defect \(QX,pU1/U3 := V_X^p/Vτ^{{corr-resp}}\). 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.