A Proof of the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds via Exceptional Group Constraints

Abstract: This paper provides a complete proof of the (1,1)-type Hodge conjecture for a special class of 5-dimensional Calabi-Yau manifolds ($h^{1,1}=56$), while establishing and implementing a novel methodology based on exceptional group constraints. The core idea of this methodology is to replace the "geometric construction" that traditional methods rely on -- which becomes infeasible in high-dimensional cases -- with the "algebraic rigidity" of Lie algebra representation theory. We prove that by introducing the symmetry of the $E_7$ group, we can generate a "dimension tension" in the Spencer cohomology framework we developed, which forcibly makes the dimension of the Spencer kernel precisely match the Hodge numbers of the manifold, thereby bypassing concrete algebraic geometric calculations. This dimension matching result is a key step in satisfying the "Spencer-Hodge verification criterion" we previously established. Its completion not only solves a non-trivial special case of the Hodge conjecture, but more importantly, provides the first strong evidence for this new paradigm of connecting representation theory with Hodge theory to solve core problems in algebraic geometry.