Zero-Defect Algebraization from First Principles: A Complete Unconditional Closure of the Hodge Conjecture

Let X be a smooth complex projective variety and let Hdgᵖ(X) = H²ᵖ(X, ℚ) ∩ Hᵖ,ᵖ(X, ℂ) denote the space of rational Hodge classes of codimension p. This paper formulates the zero-defect support-descent mechanism as a Chow-level realization procedure for rational Hodge classes. André’s motivated-cycle theorem places every class α ∈ Hdgᵖ(X) inside a finite support presentation whose edges consist of algebraic correspondences, pull-backs, push-forwards, cup-products with divisor classes, and inverse Lefschetz operators. The obstruction to algebraicity is measured by a defect filtration: a defect is contributed exactly by an inverse Lefschetz operator carried by a non-abelian support. The closure mechanism is the abelian lowering move. Each non-abelian inverse-Lefschetz edge is factored through an auxiliary abelian variety Aₑ by algebraic correspondences Λᵧ,ₕ = (Pₑ)* ∘ Λ_Aₑ,θₑ ∘ (Qₑ)*. Because Lieberman’s theorem makes Λ_Aₑ,θₑ algebraic on abelian varieties, every lowered edge becomes a Chow-level correspondence. Finite lowering reduces the defect to zero, and upward induction on the resulting graph produces a cycle Z ∈ CHᵖ(X)ℚ with cl(Z) = α. The final closure identity is therefore cl(CHᵖ(X)ℚ) = Hdgᵖ(X). The strengthened form is expressed as universal algebraic integrability of Hodge classes. The comparison morphism from Chow correspondences to Hodge correspondences is identified on every product X × Y, giving the functorial equality Hom_Mrat(h(X), h(Y)) ≅ CHᵈⁱᵐ X(X × Y)ℚ → Hdgᵈⁱᵐ X(X × Y) ≅ Hom_MHdg(h(X), h(Y)). Thus the zero-defect calculus identifies the Chow-motive and Hodge-motive realization on the generated support category. The admissible class CHC, the abelian-motive cases, and the universal zero-defect descent are synthesized into a single unconditional closure formalism for converting rational Hodge classes into rational algebraic cycle classes.