Closure probability: 1
Stability rank: 6
The closure probability \(P = 1\) confirms HC, and the stability rank matches, indicating all classes are invariant. For a rigorous stability check, we could model monodromy via a deformation matrix, but for \(E^4\), the abelian structure ensures full invariance.
This experiment validates CAS-6's metrics, aligning with known results and demonstrating computational tractability for higher-dimensional cases.
VI. Experiment C: K3 Surface Products (\(K3 \times K3\))
A. Dimensional Analysis (404 vs. 400)
We apply the CAS-6 framework to the product of two K3 surfaces, \(X = K3 \times K3\), a 4-dimensional variety, to test the Hodge Conjecture (HC) in a context where transcendental obstructions are known to complicate matters. A K3 surface is a smooth projective surface with trivial canonical bundle and \(H^1(K3, \mathbb{Q}) = 0\). Its cohomology is: \(H^0(K3, \mathbb{Q}) \cong H^4(K3, \mathbb{Q}) \cong \mathbb{Q}\), \(H^2(K3, \mathbb{Q}) \cong \mathbb{Q}^{22}\), with Hodge numbers \(h^{2,0} = h^{0,2} = 1\), \(h^{1,1} = 20\).
For \(X = K3 \times K3\), we focus on codimension-2 cycles (\(p=2\)), corresponding to degree-4 cohomology \(H^4(X, \mathbb{Q})\). The Knneth decomposition gives:
\[
H^4(X, \mathbb{Q}) \cong (H^0 \otimes H^4) \oplus (H^2 \otimes H^2) \oplus (H^4 \otimes H^0).
\]
