Exploration of the Proof of the Hodge Conjecture Based on Shui's 11-Dimensional 11th-Order Differentiation and High-Dimensional Prime Number Theory

As one of the seven Millennium Prize Problems proposed by the Clay Mathematics Institute, the Hodge Conjecture focuses on establishing the in-depth connection between the topological and algebraic properties of complex algebraic varieties. Traditional research is mostly confined to classical mathematical branches, making it difficult to break through the core bottlenecks. Taking Shui's 11-dimensional 11th-order differentiation theory and high-dimensional prime number theory as the core, this paper integrates the dimensional expansion capability of N-axis theory and the precision optimization effect of adding 1.6189 to the tail of π. It systematically sorts out the core connotation and preliminary foundation of the Hodge Conjecture, explores new proof ideas and phased breakthroughs, analyzes the current unsolved key problems, and looks forward to the future research directions. This paper provides cross-branch theoretical support and practical paths for the further proof of the Hodge Conjecture, and at the same time conforms to the specifications of Zenodo academic archiving to ensure the rigor of the content and standardization of the format.