These mappings are functorial: a diagram in \(\mathbf{Var}\) induces commutative squares in \(\mathbf{LayeredVect}\), preserving HC's inter-domain relations.
D. Rationale for Formal Heuristics, with Propositions on Compatibility with Known Theorems
Formal heuristics transform intuitive analogies into verifiable structures, aligning with "post-rigorous" mathematics where informal insights harden into formalisms0c92ee. In algebraic geometry, this is exemplified by mechanized proofs in Lean for schemes353a78, extending to heuristics for conjectures like HC. CAS-6's rationale is to diagnose obstructions (e.g., low \(P\)) and guide constructions (e.g., via \(S\)), testable computationally.
Proposition 2.1 (Compatibility with Lefschetz): For \(p=1\), CAS-6 closure holds: \(P(X)_1 = 1\) and \(S(X)_1 \to O(X)_1\) is an isomorphism, implying HC via the (1,1)-theorem. Proof: By Lefschetz, \(\dim W(X)_1 = \dim (H^{1,1} \cap H^2(X, \mathbb{Q}))\), so \(P=1\); stability follows from Picard group deformation invariance3b02e9.
Proposition 2.2 (Product Compatibility): For abelian varieties like elliptic products, tensor closure implies \(P(X \times Y)_p = P(X)_i \cdot P(Y)_j\) for decompositions, aligning with known HC cases033ee1.
These propositions validate CAS-6 against theorems, enabling heuristic predictions for open cases like K3 products.
III. Quantitative Metrics and Computational Tools
A. Definitions of Closure Probability, Stability Invariants, and Output Realizations
To operationalize the CAS-6 framework for heuristic analysis of the Hodge Conjecture (HC), we introduce quantitative metrics derived from the layered structure. These metrics transform the categorical mappings into computable invariants, enabling diagnostics of alignment across domains.
Closure Probability: For a variety \(X\) and codimension \(p\), the closure probability \(P(X)_p\) is defined as the ratio
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