More formally, if cl2:CH2(X)QH4(X,Q)\operatorname{cl}_2 : \mathrm{CH}^2(X)\otimes\mathbb Q \to H^4(X,\mathbb Q)cl2:CH2(X)QH4(X,Q) denotes the cycle-class map in codimension 222, then the images cl2([Zi,j])\operatorname{cl}_2([Z_{i,j}])cl2([Zi,j]) span a six-dimensional subspace of H2,2(X)H^{2,2}(X)H2,2(X). By the dimension count above this subspace equals the whole of H2,2(X)H^{2,2}(X)H2,2(X). Therefore
cl2(spanQ{[Zi,j]})=H2,2(X)H4(X,Q).\operatorname{cl}_2\big(\mathrm{span}_\mathbb{Q}\{[Z_{i,j}]\}\big) \;=\; H^{2,2}(X)\cap H^4(X,\mathbb Q).cl2(spanQ{[Zi,j]})=H2,2(X)H4(X,Q).
Hence, for X=E4X=E^4X=E4, the codimension-2 Hodge classes are exhausted by explicit algebraic cycles given by products of point-classes on pairs of factors.
5. CAS-6 interpretation for E4E^4E4
Applying the CAS-6 dictionary:
Topology (L,CL,CL,C): The level LLL corresponds to cohomological degree 444 (codimension p=2p=2p=2); the configurations CCC correspond to the choices of two factors among four, which index the six Knneth summands of type (2,2)(2,2)(2,2).
Algebra (W,PW,PW,P): The admissible algebraic weights are rational coefficients in linear combinations of the six product cycles [Zi,j][Z_{i,j}][Zi,j]. The algebraic span has dimension exactly equal to the Hodge dimension: thus the probabilistic heuristic PPP (dimension match / expected alignment) attains its maximal value.
Geometry (S,OS,OS,O): Each algebraic class has an explicit geometric representative (a product-of-points cycle), and these representatives are deformation-meaningful in families of product varieties. Stability SSS is high in the sense that these classes persist in the family of products of elliptic curves, and the outputs OOO are realized concretely.
Conclusion
The higher-product test E4E^4E4 substantiates the CAS-6 heuristic: the topological skeleton indexed by Knneth factors is exactly filled by algebraic generators constructed as products of divisors/point-classes. In this setting the Hodge Conjecture for codimension 222 classes poses no obstruction: the combinatorial and algebraic structures are in precise agreement, and the CAS-6 system is closed.
This positive alignment contrasts with the subsequent case K3K3K3\times K3K3K3, where a small dimensional deficit appears and signals the true locus of difficulty for the Hodge Conjecture in higher codimension.
B. Exhaustion by Products of Divisors
We now formalize the assertion made in IV.A: for the fourfold X=E4X=E^4X=E4 (product of four complex elliptic curves) every Hodge class of type (2,2)(2,2)(2,2) is obtained (up to rational linear combination) from the external product of codimension-1 algebraic classes on the factors --- equivalently, by products of divisors/point-classes on pairs of factors. Below we state this as a precise proposition and give a succinct, rigorous argument (sketch) that exhibits the algebraic exhaustion.
1. Proposition
