YuanXian Theory: A Constructive Proof of the Hodge Conjecture

The Hodge Conjecture, one of the seven Millennium Prize Problems, asserts that every rational Hodge class on a non-singular complex projective variety is a rational linear combination of algebraic cycles. This paper proposes a constructive proof based on the YD-T^64 framework and True-Circle Self-Consistency (TCSC) axioms of YuanXian Theory (YXT). By constructing the Hodge-Laplacian operator (H) on the 64-dimensional torus (T^64) with TCSC involution symmetry, we establish a spectral correspondence between the kernel of the operator and Hodge classes. We further develop a bidirectional verification pipeline between SageMath and Lean 4. The core conclusion is that the Hodge Conjecture emerges as a natural corollary of the spectral theory on the TCSC-symmetric (T^64). 霍奇猜想是千禧年七大数学难题之一, 断言复射影代数簇上每一个有理霍奇类均可表示为代数闭链的有理线性组合。本文基于元宪理论 (YXT) 的 YD-T^64 框架与 TCSC 公理体系, 提出了一种构造性证明路径。通过在 64 维环面 (T^64) 上构造带 TCSC 对合对称性的霍奇-拉普拉斯算子 (H), 我们建立了该算子零特征空间与霍奇类群之间的谱对应, 并构建了 SageMath 与 Lean 4 的双向验证流水线。核心结论为: 霍奇猜想是 (T^64) 上霍奇-拉普拉斯算子谱理论的必然推论。