:=[E{p}],:=[{q}E],:=[]\alpha := [E\times\{p\}],\qquad \beta := [\{q\}\times E],\qquad \delta:= [\Delta]:=[E{p}],:=[{q}E],:=[]
are elements of CH1(X)\mathrm{CH}^1(X)CH1(X). The cohomology classes cl1(),cl1(),cl1()\operatorname{cl}_1(\alpha),\operatorname{cl}_1(\beta),\operatorname{cl}_1(\delta)cl1(),cl1(),cl1() lie in H1,1(X)H2(X,Q)H^{1,1}(X)\cap H^2(X,\mathbb{Q})H1,1(X)H2(X,Q). Together with the class u1+1uu\otimes 1 + 1\otimes uu1+1u (or suitable linear combinations to obtain an integral basis), one obtains a Q\mathbb{Q}Q-basis for the rational Hodge classes of type (1,1)(1,1)(1,1).
The cycle class map in this instance is surjective onto the rational (1,1)(1,1)(1,1)-classes:
cl1(CH1(X)Q)=H1,1(X)H2(X,Q).\operatorname{cl}_1\big(\mathrm{CH}^1(X)\otimes\mathbb{Q}\big) \;=\; H^{1,1}(X)\cap H^2(X,\mathbb{Q}).cl1(CH1(X)Q)=H1,1(X)H2(X,Q).
This surjectivity is exactly the Lefschetz theorem in this context and furnishes the prototypical example where the CAS-6 heuristic predicts closure between topology, algebra, and geometry.
4. CAS-6 interpretation for EEE\times EEE
Under the CAS-6 dictionary:
LLL and CCC (level and configuration) correspond to passing to degree 222 cohomology and to the Knneth configurations H1,0H0,1H^{1,0}\otimes H^{0,1}H1,0H0,1 etc.; this yields the topological skeleton H1,1(X)H^{1,1}(X)H1,1(X).
WWW and PPP (weights and probabilities) correspond to rational linear combinations of divisor classes; here admissible weights are rational and the expected dimension of the algebraic span equals the Hodge dimension, so probabilistic alignment is maximal.
SSS and OOO (stability and output) correspond to the deformation stability of divisors and the actual algebraic realization of each rational Hodge class as a divisor; both are satisfied in this case.
Hence the CAS-6 system for X=EEX=E\times EX=EE is closed: the topology yields a finite set of Hodge types; the algebraic layer supplies rational weights (divisor coefficients) that completely span the rational Hodge subspace; and the geometric layer realizes each such class as a genuine algebraic cycle. This closure exemplifies the CAS-6 heuristic claim that when level/configuration (topology) and weight/probability (algebra) match in dimension and rational structure, the geometric output will exist and be stable.
5. Remarks
The EEE\times EEE computation is a model case: codimension 111 is governed by the Lefschetz theorem, so no transcendental obstruction arises. The situation for p2p\ge 2p2 (higher codimension) is fundamentally different and constitutes the genuine challenge of the Hodge Conjecture; such instances will be the subject of subsequent experiments.
Although the above is classical and well-known, it plays a crucial role in validating the CAS-6 heuristic: a nontrivial framework should recover known positive cases before being used to probe the frontier where the conjecture remains open.
B. Divisors and the Lefschetz (1,1)(1,1)(1,1)-Theorem
