Let EEE be a complex elliptic curve and X=E4X=E^4X=E4. Denote by 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) the codimension-2 cycle-class map. Then
cl2(spanQ{[Zi,j]}1i<j4)=H2,2(X)H4(X,Q),\operatorname{cl}_2\big(\operatorname{span}_\mathbb{Q}\{[Z_{i,j}]\}_{1\le i<j\le4}\big) \;=\; H^{2,2}(X)\cap H^4(X,\mathbb{Q}),cl2(spanQ{[Zi,j]}1i<j4)=H2,2(X)H4(X,Q),
where Zi,j=E1{pi}{pj}E4Z_{i,j}=E_1\times\cdots\times\{p_i\}\times\cdots\times\{p_j\}\times\cdots\times E_4Zi,j=E1{pi}{pj}E4 denotes the external product cycle given by fixing points piEi,pjEjp_i\in E_i,\,p_j\in E_jpiEi,pjEj and taking full factors elsewhere. In particular, the six classes cl2([Zi,j])\operatorname{cl}_2([Z_{i,j}])cl2([Zi,j]) form a Q\mathbb{Q}Q-basis of H2,2(X)H4(X,Q)H^{2,2}(X)\cap H^4(X,\mathbb{Q})H2,2(X)H4(X,Q).
2. Proof sketch
a. Knneth description of H2,2(X)H^{2,2}(X)H2,2(X).
By the Knneth theorem for singular cohomology with Q\mathbb{Q}Q-coefficients and the Hodge decomposition on each factor, every class in H2,2(X)H^{2,2}(X)H2,2(X) is a C\mathbb{C}C-linear combination of pure tensors of the form
ijk,\omega_{i}\wedge\omega_{j}\otimes\overline\omega_{k}\wedge\overline\omega_{\ell},ijk,
where mH1,0(Em)\omega_m\in H^{1,0}(E_m)mH1,0(Em) and {i,j,k,}={1,2,3,4}\{i,j,k,\ell\}=\{1,2,3,4\}{i,j,k,}={1,2,3,4}. Unordered choices of the two factors {i,j}\{i,j\}{i,j} contributing (1,0)(1,0)(1,0)-pieces parametrize a canonical six-element basis of H2,2(X)H^{2,2}(X)H2,2(X) (up to nonzero scalars).
b. Algebraic cycles from external products.
For each unordered pair {i,j}\{i,j\}{i,j} with 1i<j41\le i<j\le41i<j4, choose points piEip_i\in E_ipiEi and pjEjp_j\in E_jpjEj. The subvariety Zi,jXZ_{i,j}\subset XZi,jX obtained by fixing pi,pjp_i,p_jpi,pj and letting the remaining two coordinates vary is a smooth algebraic cycle of codimension 222; its class [Zi,j]CH2(X)[Z_{i,j}]\in\mathrm{CH}^2(X)[Zi,j]CH2(X) is the external product of point-classes on factors iii and jjj and of the fundamental classes on the other factors. The associated cohomology class cl2([Zi,j])\operatorname{cl}_2([Z_{i,j}])cl2([Zi,j]) is therefore an explicit element of H2,2(X)H4(X,Q)H^{2,2}(X)\cap H^4(X,\mathbb{Q})H2,2(X)H4(X,Q).
c. Nondegeneracy and dimension count.
The Knneth decomposition furnishes dimCH2,2(X)=6\dim_\mathbb{C}H^{2,2}(X)=6dimCH2,2(X)=6. The six algebraic classes cl2([Zi,j])\operatorname{cl}_2([Z_{i,j}])cl2([Zi,j]) are linearly independent over Q\mathbb{Q}Q (indeed, they occupy distinct Knneth summands up to scalar), hence their Q\mathbb{Q}Q-span has dimension 666. Because the cycle-class map takes them into H2,2(X)H^{2,2}(X)H2,2(X), the image of this span is a 666-dimensional rational subspace of H2,2(X)H^{2,2}(X)H2,2(X). By the dimension equality, this image equals the entire rational Hodge subspace H2,2(X)H4(X,Q)H^{2,2}(X)\cap H^4(X,\mathbb{Q})H2,2(X)H4(X,Q).
d. Conclusion.
Therefore every rational (2,2)(2,2)(2,2)-class on XXX is a rational linear combination of the product cycles [Zi,j][Z_{i,j}][Zi,j], and the cycle-class map is surjective onto H2,2(X)H4(X,Q)H^{2,2}(X)\cap H^4(X,\mathbb{Q})H2,2(X)H4(X,Q) when restricted to the span of these product cycles. \square
3. Remarks and clarifying comments
The argument depends crucially on the tensor-product nature of both the cohomology and the Chow (external product) constructions for product varieties. For curve factors, codimension-1 cycles are point-classes; external products of two such classes produce canonical codimension-2 cycles whose cycle classes realize the corresponding Knneth summands.
The statement is stronger (and elementary) in this context than a generic existence assertion: for products of curves the Chow ring is generated (under external product) by the point classes and the fundamental classes of the factors, so the combinatorial Knneth basis has immediate algebraic representatives. This combinatorial simplicity is what makes E4E^4E4 a transparent test of the CAS-6 heuristic.
Caveat: the above relies on choosing generic points on the factors in order to avoid accidental algebraic relations among cycles; nevertheless, the independence of the six classes up to rational linear relations is a structural feature coming from the Knneth decomposition and does not depend on special choices.
4. CAS-6 perspective
Viewed through CAS-6, the exhaustion by products of divisors corresponds to an exact match between the topological configurations (choices of two factors among four) and the algebraic generators (external products of point/divisor classes). The algebraic weights are simply the rational coefficients of a combination of those six generators; the probabilistic alignment PPP is maximal since dimensions coincide, and geometric outputs OOO exist concretely. In CAS-6 terms the system closes without residue for this instance of level L=4L=4L=4 (codimension p=2p=2p=2).
C. CAS-6 Perspective: Dimension Closure, No Gap, and Stability of Interaction Cycles
In Sections IV.A--IV.B we showed that for the fourfold X=E4X=E^4X=E4 the Knneth decomposition yields h2,2(X)=6h^{2,2}(X)=6h2,2(X)=6 and that six explicit algebraic cycles --- the external products of point-classes on pairs of factors --- produce a Q\mathbb{Q}Q-basis of H2,2(X)H4(X,Q)H^{2,2}(X)\cap H^4(X,\mathbb{Q})H2,2(X)H4(X,Q). We now restate and interpret this fact within the CAS-6 lexicon, emphasizing three interrelated features: dimension closure, absence of a transcendental gap, and stability of the interaction cycles.
1. Dimension closure: a precise algebraic-topological match
Let T:=H2,2(X)H4(X,Q)T:=H^{2,2}(X)\cap H^4(X,\mathbb{Q})T:=H2,2(X)H4(X,Q) denote the rational Hodge subspace of bi-degree (2,2)(2,2)(2,2). Let ACH2(X)QA\subset \mathrm{CH}^2(X)\otimes\mathbb{Q}ACH2(X)Q be the Q\mathbb{Q}Q-linear span of the six external-product cycles {[Zi,j]}1i<j4\{[Z_{i,j}]\}_{1\le i<j\le4}{[Zi,j]}1i<j4. The cycle class map
