Interaction Outputs \(O(X)\): The geometric realization space, \(O(X)_p = CH^p(X) \otimes \mathbb{Q}\), with morphisms to \(W(X)_p\) via \(\cl_p\). Axioms: Surjectivity conjecture (equivalent to HC), and emergence from prior layers: \(O(X)_p\) is generated by stable outputs from \(S(X)_p\).
Global Axioms for CAS-6:
Closure Axiom: The composition \(O \circ S \circ P \circ W \circ C \circ L\) is surjective onto rational Hodge classes.
Tensor Closure: For products, \(\mathcal{F}(X \times Y) \cong \mathcal{F}(X) \otimes \mathcal{F}(Y)\), preserving layers.
Functoriality: \(\mathcal{F}\) commutes with pullbacks and pushforwards in mixed Hodge structures.
These definitions ensure CAS-6 is a rigorous model, amenable to computational checks (e.g., via SymPy for dimension ratios).
C. Mappings to HC Domains: Topology (Levels/Configurations), Algebra (Weights/Probabilities), Geometry (Stability/Outputs)
The HC domains map categorically to CAS-6 layers as follows:
Topology to Levels/Configurations (\(L/C\)): Topology provides the cohomological skeleton. Formally, the functor restricts to \(L(X) \oplus C(X)\), isomorphic to the rational Hodge filtration. For HC, this maps the decomposition \(H^{2p}(X, \mathbb{Q}) \cap H^{p,p}(X)\) to configurable nodes, with Knneth ensuring product decompositions2e7fa1.
Algebra to Weights/Probabilities (\(W/P\)): Algebra handles rational structures. The map embeds \(W(X)\) into the span of cycle classes, with \(P(X)\) quantifying surjectivity via the probability metric. In HC terms, full alignment (\(P=1\)) implies the conjecture holds, as in codimension 1.
Geometry to Stability/Outputs (\(S/O\)): Geometry realizes classes algebraically. The map projects to \(S(X) \to O(X)\), where stability ensures persistence (e.g., via Noether-Lefschetz loci3d8598), and outputs confirm existence.
