  \]
This indicates full closure, consistent with HC.
Interaction Stability \(S(X)\): Stability is assessed via deformation invariance. For \(E \times E\), an abelian surface, the Picard group is deformation-invariant (since \(\Pic(E \times E)\) is discrete modulo torsion). Thus, \(S(X)_1 = \dim W(X)_1 = 4\).
Interaction Outputs \(O(X)\): Outputs are \(O(X)_1 = CH^1(X) \otimes \mathbb{Q}\), with realization index \(r(X)_1 = \dim O(X)_1 - \dim S(X)_1 = 0\), as all stable classes are realized.
Proposition 4.1 (Full Closure for \(E \times E\)): The functor \(\mathcal{F}\) induces an isomorphism on the codimension-1 layer: \(W(X)_1 \to H^{1,1}(X) \cap H^2(X, \mathbb{Q})\), with \(P(X)_1 = 1\), \(S(X)_1 = \dim W(X)_1\), and \(r(X)_1 = 0\). Proof: By the Lefschetz theorem, \(\cl_1\) is surjective, and Knneth ensures \(W(X)_1\) spans via tensor products. Stability follows from deformation theory of abelian surfaces, and outputs align since HC holds.
This isomorphism confirms CAS-6's prediction of full closure, formalized categorically.
C. Computational Verification of Bases and Metrics
To ensure rigor, we compute bases and metrics using symbolic algebra in SymPy, verifying the dimensions and closure probability.
Consider the basis for \(H^2(E \times E, \mathbb{Q})\). Let \(E\) have a basis for \(H^1(E, \mathbb{Q})\) as \(\{\omega_1, \omega_2\}\) (e.g., holomorphic and anti-holomorphic forms). Then:
\(H^1(E) \otimes H^1(E)\): Basis \(\{\omega_1 \otimes \omega_1, \omega_1 \otimes \omega_2, \omega_2 \otimes \omega_1, \omega_2 \otimes \omega_2\}\), with \(\omega_1 \otimes \omega_2, \omega_2 \otimes \omega_1 \in H^{1,1}\).
\(H^2(E) \otimes H^0(E)\): Basis \(\{[E] \otimes 1\}\), in \(H^{2,0}\).
