Abstract Within the framework of generative mathematics, this paper provides a complete solution path for the Hodge Conjecture. The core proposition: Every Hodge class is a rational linear combination of algebraic cycles—this is the geometric signature of the strict equivalence of the analysis side (harmonic forms), the topology side (cohomology classes), and the algebra side (algebraic cycles) of the Axiom 4 coupling network under high-dimensional doubly periodic boundary conditions. Complete argument chain: 1. High-dimensional algebraic varieties → k ≥ 3-dimensional spiral tori: Theorem 3.1 rigorously proves that the high-dimensional generalization is a structural direct corollary of the single-variable coupling kernel of Axiom 4.2. Hodge classes → the harmonic signature of the isoperimetrically optimal steady state (analysis side).3. Cohomology classes → the integralization of the topological charge (topology side).4. Algebraic cycles → rational linear combinations of independent closed paths (algebra side).5. The three-discipline unification loop enforces equivalence: - Analysis Is Topology (Lemma 1): harmonic forms ⇔ integralized topological charge. - Topology Is Algebra (Lemma 2): integer topological charge ⇔ algebraic multiplicity. - Algebra Is Analysis (synthetic corollary): group operations and gradient flow are homologous.6. Synthesis: Hodge class ⇔ harmonic form ⇔ integralized topological charge ⇔ algebraic cycle. All steps are rigorously guaranteed by the axiom system and the L1–L2 layer theorems. The rigor of the high-dimensional generalization is intrinsically guaranteed by the single-variable coupling kernel structure of Axiom 4—cross-generator coupling is structurally absent within the Axiom 4 framework; the joint isoperimetric functional is strictly decomposable; and the joint convexity is directly guaranteed by the strict convexity of each component. Unification with the Millennium Problems: The Hodge Conjecture is the final verification of the generative three-discipline unification loop in the dimension of complex algebraic geometry. Riemann (k=1), BSD (k=2), Yang-Mills (k ≥ 2, multi-component), P vs NP (I=0 vs I ≠ 0), N-S (time-varying locking), and Hodge (three-discipline unification)—all six problems are the survivor signatures of the external cutting of Axiom 4 on different problem domains. Keywords: Hodge Conjecture; three-discipline unification; spiral torus; algebraic cycle; harmonic form; topological charge; generativism; unification of the Millennium Problems
Zhao Jun (Sun,) studied this question.