To formalize the CAS-6 framework, we begin by defining the relevant categories and the functor that encodes the layered structure. Let \(\mathbf{Var}\) denote the category whose objects are smooth projective varieties over \(\mathbb{C}\), and whose morphisms are proper morphisms of varieties. This choice ensures compatibility with cohomological functors, as proper morphisms induce well-defined maps on cohomology via pushforward.
We introduce the target category \(\mathbf{LayeredVect}\), a layered category of \(\mathbb{Q}\)-vector spaces. Objects in \(\mathbf{LayeredVect}\) are 6-tuples \((V_L, V_C, V_W, V_P, V_S, V_O)\), where each \(V_\bullet\) is a finite-dimensional \(\mathbb{Q}\)-vector space equipped with additional structure: \(V_L\) and \(V_C\) carry grading for degrees and decompositions; \(V_W\) and \(V_P\) include bilinear forms for coefficients and alignments; \(V_S\) and \(V_O\) incorporate endomorphisms for invariants and realizations. Morphisms in \(\mathbf{LayeredVect}\) are 6-tuples of \(\mathbb{Q}\)-linear maps preserving these structures, ensuring layer-wise compatibility.
The CAS-6 framework is realized as a functor \(\mathcal{F}: \mathbf{Var} \to \mathbf{LayeredVect}\), assigning to each variety \(X\) its layered cohomological data. Specifically, \(\mathcal{F}(X) = (L(X), C(X), W(X), P(X), S(X), O(X))\), where each component is derived from Hodge theory. For a morphism \(f: X \to Y\), \(\mathcal{F}(f)\) consists of induced maps on each layer, such as pushforwards on cohomology for \(L\) and \(C\), and compatible transformations on subsequent layers. This functor is contravariant in nature, reflecting the pullback-oriented aspects of Hodge decompositions, but we incorporate pushforwards for cycle classes to maintain duality.
This setup draws from categorical formulations in algebraic geometry, such as those in motivic categories or derived categories of sheaves, ensuring that \(\mathcal{F}\) preserves products via the Knneth theorem0bc585. Recent mechanizations in Lean formalize similar functors for schemes, providing a blueprint for verification8389e6.
B. Precise Definitions of Six Layers with Axioms
We now define each layer of \(\mathcal{F}(X)\) precisely, along with axioms that govern their interactions.
Interaction Level \(L(X)\): This is the graded \(\mathbb{Q}\)-vector space \(\bigoplus_p H^{2p}(X, \mathbb{Q})\), capturing the cohomological degrees relevant to HC. Axioms: Graded commutativity under cup product, and functoriality under proper maps via Gysin pushforward.
Interaction Configuration \(C(X)\): For each level \(p\), \(C(X)_p = \bigoplus_{r+s=2p} H^{r,s}(X) \cap H^{2p}(X, \mathbb{Q}) \otimes \mathbb{C}\), the Hodge decomposition restricted to rational classes. Axioms: Orthogonality with respect to the Hodge metric, and closure under Knneth tensor products for products of varieties: \(C(X \times Y)_p \cong \bigoplus_{i+j=p} C(X)_i \otimes C(Y)_j\).
Interaction Weights \(W(X)\): The \(\mathbb{Q}\)-span of cycle classes, \(W(X)_p = \im(\cl_p: CH^p(X) \otimes \mathbb{Q} \to H^{2p}(X, \mathbb{Q}))\), equipped with a bilinear intersection form. Axioms: Compatibility with rational coefficients (weights in \(\mathbb{Q}\)), and multiplicativity under products: \(W(X \times Y)_p \supseteq \bigoplus_{i+j=p} W(X)_i \boxtimes W(Y)_j\).
Interaction Probabilities \(P(X)\): A probabilistic layer defined as the ratio vector space with metric \(P(X)_p = [0,1] \times (H^{p,p}(X) \cap H^{2p}(X, \mathbb{Q})) / \sim\), where the value is \(\dim W(X)_p / \dim (H^{p,p}(X) \cap H^{2p}(X, \mathbb{Q}))\), interpreted as "alignment probability." Axioms: Monotonicity under inclusions, and submultiplicativity for products: \(P(X \times Y)_p \leq \min_{i+j=p} P(X)_i \cdot P(Y)_j\).
Interaction Stability \(S(X)\): The subspace of deformation-invariant classes, \(S(X)_p = \{ \gamma \in W(X)_p \mid \gamma\) persists under small deformations in the moduli space }), formalized via the Hodge locus in the period domainb24c72. Axioms: Invariance under automorphisms of \(X\), and closure under tensor products in families.
