C. CAS-6 Metrics: Dimension Match Implying Probability 1
We apply the CAS-6 functor \(\mathcal{F}: \mathbf{Var} \to \mathbf{LayeredVect}\), computing metrics for \(X = E^4\).
Interaction Level \(L(X)\): \(L(X) = H^4(X, \mathbb{Q})\), dimension 38, with focus on \(p=2\).
Interaction Configuration \(C(X)\): \(C(X)_2 = H^{2,2}(X) \cap H^4(X, \mathbb{Q})\), dimension 6, from Knneth.
Interaction Weights \(W(X)\): \(W(X)_2 = \im(\cl_2)\), generated by divisor products (e.g., \(E \times \{pt\} \times E \times \{pt\}\)). Dimension is 6, as cycles span \(H^{2,2}\).
Interaction Probabilities \(P(X)\): Compute
  \[
  P(X)_2 = \frac{\dim W(X)_2}{\dim (H^{2,2}(X) \cap H^4(X, \mathbb{Q}))} = \frac{6}{6} = 1.
  \]
This indicates full closure, consistent with HC.
Interaction Stability \(S(X)\): Stability measures deformation invariance. For abelian varieties, \(H^{2,2}(E^4)\) is invariant under the moduli space of \(E^4\), as cycle classes persist (via the Picard group and endomorphisms). Thus, \(S(X)_2 = \dim W(X)_2 = 6\).
