The results align with Lefschetz and Knneth, demonstrating CAS-6's ability to formalize HC's success in this case via categorical and computational tools.
V. Experiment B: Higher Elliptic Products (\(E^4\))
A. Higher-Degree Cohomology
We extend our analysis to the fourfold product of an elliptic curve, \(X = E^4 = E \times E \times E \times E\), where \(E\) is a smooth projective elliptic curve over \(\mathbb{C}\). This experiment tests the CAS-6 framework on a higher-dimensional variety where the Hodge Conjecture (HC) is known to hold in certain degrees, particularly for codimension 2 (\(p=2\)), due to the algebraic nature of cycles in abelian varieties. We focus on the degree-4 cohomology, \(H^4(X, \mathbb{Q})\), as it contains the \((2,2)\)-classes relevant to HC.
For an elliptic curve \(E\), the cohomology groups are \(H^0(E, \mathbb{Q}) \cong H^2(E, \mathbb{Q}) \cong \mathbb{Q}\) and \(H^1(E, \mathbb{Q}) \cong \mathbb{Q}^2\). The Knneth decomposition for \(X = E^4\) gives:
\[
H^4(X, \mathbb{Q}) \cong \bigoplus_{i_1 + i_2 + i_3 + i_4 = 4} H^{i_1}(E, \mathbb{Q}) \otimes H^{i_2}(E, \mathbb{Q}) \otimes H^{i_3}(E, \mathbb{Q}) \otimes H^{i_4}(E, \mathbb{Q}).
\]
The terms contributing to degree 4 are:
\((1,1,1,1)\): \(H^1 \otimes H^1 \otimes H^1 \otimes H^1\), dimension \(2 \times 2 \times 2 \times 2 = 16\),
\((2,1,1,0)\): \(H^2 \otimes H^1 \otimes H^1 \otimes H^0\), dimension \(1 \times 2 \times 2 \times 1 = 4\) (and permutations, 4 terms),
