Exceptional Symmetry as a Source of Algebraic Cycles: Non-Constructive Methods for the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds

Classical variational Hodge structure theory characterizes the algebraicity of Hodge classes by studying the transversality of period mappings under geometric deformations. However, when algebraic varieties lack appropriate deformation families, this method faces applicability limitations. This paper develops a non-constructive method based on exceptional Lie group constraints to handle such cases. Our main technical contribution is establishing a dimension control mechanism for Spencer cohomology theory under Lie group constraints. Specifically, we prove that when a compact K\"ahler manifold $X$ is equipped with $E_7$ group constraints, the corresponding Spencer kernel $\mathcal{K}_\lambda^{1,1}$ has complex dimension simultaneously constrained by two bounds: representation theory gives the lower bound $\dim_{\mathbb{C}}\mathcal{K}_\lambda^{1,1} \geq 56$, while the degenerate Spencer-de Rham mapping gives the upper bound $\dim_{\mathbb{C}}\mathcal{K}_\lambda^{1,1} \leq h^{1,1}(X)$. For 5-dimensional Calabi-Yau manifolds satisfying $h^{1,1}(X) = 56$, this dimension constraint becomes the equality $\dim_{\mathbb{C}}\mathcal{K}_\lambda^{1,1} = 56 = h^{1,1}(X)$. Combined with our established Spencer-calibration equivalence principle, this dimension matching result is sufficient to verify the $(1,1)$-type Hodge conjecture. Our proof completely avoids explicit algebraic cycle construction, instead achieving the goal through abstract dimensional arguments. This method demonstrates the application potential of Lie group representation theory in algebraic geometry, providing new theoretical tools for handling geometric objects where traditional deformation methods are not applicable.