dimQH2(X,Q)=6,dimCH1,1(X)=4,\dim_{\mathbb Q} H^2(X,\mathbb{Q}) = 6,\qquad \dim_{\mathbb C} H^{1,1}(X)=4,dimQH2(X,Q)=6,dimCH1,1(X)=4,
and the Lefschetz theorem gives
H1,1(X)H2(X,Q)=cl1(CH1(X)Q),H^{1,1}(X)\cap H^2(X,\mathbb{Q})=\operatorname{cl}_1\big(\mathrm{CH}^1(X)\otimes\mathbb{Q}\big),H1,1(X)H2(X,Q)=cl1(CH1(X)Q),
i.e. the rational (1,1)(1,1)(1,1)-classes are exactly the images of divisor classes under the cycle-class map cl1\operatorname{cl}_1cl1.
A concrete rational basis may be chosen from the divisor classes
=[E{p}], =[{q}E], =[]\alpha=[E\times\{p\}],\ \beta=[\{q\}\times E],\ \delta=[\Delta]=[E{p}], =[{q}E], =[] together with a suitable linear combination completing the basis. These classes span the entire rational (1,1)(1,1)(1,1)-space.
2. CAS-6 assignment for XXX
We map the six CAS-6 components (L,C,W,P,S,O)(L,C,W,P,S,O)(L,C,W,P,S,O) to the mathematical data of XXX as follows.
Interaction Level LLL. The cohomological degree under consideration is 222 (i.e. 2p=22p=22p=2 with p=1p=1p=1). Thus LLL = degree 222 determines the codimension relevant to the conjecture.
Interaction Configuration CCC. The Knneth factors and Hodge decomposition determine configuration: the relevant summands contributing to H1,1H^{1,1}H1,1 are H1,0H0,1H^{1,0}\otimes H^{0,1}H1,0H0,1 and its conjugate, as well as the summands arising from H2H0H^2\otimes H^0H2H0 and H0H2H^0\otimes H^2H0H2. These configurations specify the topological skeleton of possible classes.
Interaction Weights WWW. The admissible coefficients for linear combinations of cycle classes are rational numbers. In practice, the Nron--Severi lattice furnishes integral generators whose Q\mathbb{Q}Q-linear span yields the rational Hodge classes; thus WWW is realized concretely by integer/rational weights on the divisor generators.
Interaction Probabilities PPP. Interpreted heuristically, PPP measures the expected compatibility (or density) between the topological skeleton and the algebraic span. For XXX one has a dimension match: the algebraic span (divisor classes) fills the rational (1,1)(1,1)(1,1)-space. Hence PPP attains its maximal heuristic value (certainty of alignment) in this instance.
Interaction Stability SSS. Divisor classes on a smooth projective surface are deformation-stable: under small complex deformations that preserve projectivity, divisor classes persist (modulo the behavior of Picard rank). Therefore the geometric realizations corresponding to the algebraic weights are robust; the system exhibits high SSS.
Interaction Outputs OOO. The outputs are the actual algebraic cycles (the divisors themselves). Each rational (1,1)(1,1)(1,1)-class is realized by a concrete geometric object in XXX.
3. Structural closure and its mathematical meaning
With the assignment above, the CAS-6 "closure" condition---namely that topological nodes and configurations admit algebraic weights and probabilities which produce stable geometric outputs---is satisfied exactly for XXX in codimension 111. Formally:
The topological subspace T:=H1,1(X)H2(X,Q)T:=H^{1,1}(X)\cap H^2(X,\mathbb{Q})T:=H1,1(X)H2(X,Q) equals the image A:=cl1(CH1(X)Q)A:=\operatorname{cl}_1(\mathrm{CH}^1(X)\otimes\mathbb{Q})A:=cl1(CH1(X)Q). Thus the map
cl1:CH1(X)QT\operatorname{cl}_1 : \mathrm{CH}^1(X)\otimes\mathbb{Q} \longrightarrow Tcl1:CH1(X)QT
is surjective, which, in the CAS-6 vocabulary, is the algebraic closure GA=IdTG\circ A = \mathrm{Id}_TGA=IdT.
The algebraic coefficients (weights) required to express an arbitrary element of TTT as a linear combination of divisor classes lie in Q\mathbb{Q}Q and are computable in principle, manifesting WWW. Because these combinations produce bona fide divisors, the outputs OOO are realized and stable, manifesting SSS.
Consequently, XXX is a canonical example where the CAS-6 system reaches a fixed point of closure: topology \to algebra \to geometry without residue.
4. Heuristic implications and lessons
