cl2:CH2(X)Q H4(X,Q)\operatorname{cl}_2:\; \mathrm{CH}^2(X)\otimes\mathbb{Q}\ \longrightarrow\ H^4(X,\mathbb{Q})cl2:CH2(X)Q H4(X,Q)
restricts to a linear map cl2A:AT\operatorname{cl}_2|_{A}:A\to Tcl2A:AT. The calculations of IV.A--IV.B yield
dimQA=dimQT=6,andcl2(A)=T.\dim_{\mathbb Q}A \;=\; \dim_{\mathbb Q}T \;=\; 6, \qquad\text{and}\qquad \operatorname{cl}_2(A)=T.dimQA=dimQT=6,andcl2(A)=T.
Thus the algebraic span AAA and the topological skeleton TTT coincide as rational vector spaces: there is exact dimension closure. In CAS-6 terms this is the condition that the algebraic layer (W,P)(W,P)(W,P) supplies sufficient degrees of freedom (weights/probabilities) to parametrize every topological node/configuration (L,C)(L,C)(L,C) at the level under consideration.
2. No transcendental gap: interpretation and consequence
The equality cl2(A)=T\operatorname{cl}_2(A)=Tcl2(A)=T implies the nonexistence of transcendental (2,2)(2,2)(2,2)-classes in this instance: every rational Hodge class is algebraic. From the CAS-6 viewpoint, there is no "orphan" topological configuration left unassigned an algebraic weight---no residual mode that would indicate system incompleteness. Consequently, the system at level L=4L=4L=4 (codimension p=2p=2p=2) is algebraically closed: topology \to algebra \to geometry is surjective at the algebraic stage. This provides a precise sense in which the Hodge Conjecture holds for X=E4X=E^4X=E4 at the stated degree.
3. Stability of interaction cycles: deformation-theoretic and moduli considerations
Beyond static existence, CAS-6 emphasizes stability (SSS) of outputs under perturbation. For the product variety E4E^4E4 the algebraic cycles Zi,jZ_{i,j}Zi,j are canonical external products of point-classes and fundamental classes; their classes in cohomology vary continuously in families of varieties built as products of elliptic curves, and they persist under small deformations that preserve the product structure. More precisely:
Infinitesimal stability. The classes cl2([Zi,j])\operatorname{cl}_2([Z_{i,j}])cl2([Zi,j]) are Hodge classes arising from the Knneth decomposition and therefore form part of the flat subbundle of cohomology over the parameter space of product families; their (1,1)-type is rigidly determined by the tensor structure. In deformation spaces that maintain the separable product structure, these classes do not exit the Hodge locus, reflecting high infinitesimal stability.
Family-level persistence. If one varies the complex structures of the elliptic factors within families, the Knneth-indexed classes deform compatibly; for generic deformations that do not collapse the combinatorial factorization, the dimension equality remains and the algebraic representatives persist (possibly after rational adjustments), so that the CAS-6 stability measure SSS remains high.
Robustness to perturbations of algebraic data. Because the generators are constructed from point-classes (divisors on curves), they are less susceptible to delicate transcendental phenomena typical of higher-dimensional cycles; thus they constitute robust "interaction motifs" in the CAS-6 language.
4. Synthesis and implications for the heuristic program
The conjunction of dimension closure, absence of a transcendental gap, and stability of cycles implies that, for X=E4X=E^4X=E4, the CAS-6 system attains a fixed-point of closure: topological configurations are exactly parametrized by algebraic degrees of freedom, and these in turn realize stable geometric outputs. Heuristically, this demonstrates two important principles for the CAS-6 approach to HC:
Dimension matching is a reliable heuristic indicator. When the algebraically-constructible subspace attains the same dimension as the Hodge subspace, one can expect closure (and in codimension 1 this is guaranteed by Lefschetz).
Canonical external-product constructions are high-quality candidates. In product varieties whose factors have controlled cohomology (e.g., curves, abelian varieties), external products often exhaust the relevant Hodge summands and thus serve as natural system-level motifs.
5. Limitations and caution
