The successful closure in the EEE\times EEE case yields several instructive lessons for the CAS-6 heuristic program:
a. Dimension matching is a strong indicator. When the algebraic generators provide a linear span whose dimension equals the Hodge dimension, empirical closure is expected and, in codimension 111, guaranteed by Lefschetz. The CAS-6 notion of high PPP corresponds precisely to this dimension match.
b. Stability is crucial for interpretability. The deformation stability of divisors makes the geometric outputs meaningful beyond a single variety in a family; CAS-6 captures this by assigning high SSS.
c. Heuristic validation precedes generalization. A viable heuristic framework should recover classical positive results; the EEE\times EEE case confirms that CAS-6 is not vacuous but rather reproduces known theorems in an organizationally revealing way.
5. Transition to higher codimension
While the CAS-6 closure is complete in the EEE\times EEE example, the framework also suggests where and why difficulties arise when increasing the interaction level LLL (higher degree cohomology) or considering varieties with richer transcendental cohomology. In particular, a mismatch between the dimension of the topological skeleton and the algebraic span (dimT>dimA \dim T > \dim AdimT>dimA) signals low PPP (probabilistic misalignment) and therefore a potential failure of closure. This phenomenon underlies the subsequent experiments---most notably the case of K3K3K3\times K3K3K3---and motivates the search for nontrivial algebraic constructions (correspondences, Fourier--Mukai kernels, Kummer/Shioda--Inose transfers, etc.) that could restore CAS-6 closure in higher codimension.
IV. Heuristic Experiment B --- Higher Product E4E^4E4
A. Knneth decomposition for (2,2)(2,2)(2,2)-classes
Let EEE be a complex elliptic curve and set
X=E4=EEEE,X \;=\; E^4 \;=\; E\times E\times E\times E,X=E4=EEEE,
a smooth projective variety of complex dimension 444. We shall describe the Hodge summand H2,2(X)H^{2,2}(X)H2,2(X) via the Knneth decomposition, count its dimension, and exhibit the natural algebraic generators arising from products of divisors (points on factors). These computations make precise why the CAS-6 heuristic predicts closure in this case.
1. Cohomology of a single elliptic curve and Knneth formalism
For a single elliptic curve EEE we have (with complex coefficients)
H0(E,C)C,H1(E,C)H1,0(E)H0,1(E),H2(E,C)C,H^0(E,\mathbb C)\cong \mathbb C,\qquad H^1(E,\mathbb C) \cong H^{1,0}(E)\oplus H^{0,1}(E),\qquad H^2(E,\mathbb C)\cong \mathbb C,H0(E,C)C,H1(E,C)H1,0(E)H0,1(E),H2(E,C)C,
