Closure Metrics: Estimate \(P(X)_n\) to identify scaling of transcendental classes, potentially linking to Voisin's results on higher-dimensional obstructions.
These extensions would test CAS-6's scalability and its ability to handle varieties where HC remains open, potentially revealing patterns in cycle deficiencies.
B. Integration with Derived Categories and Lean Formalization
To enhance CAS-6's rigor and applicability, integration with derived categories and formal verification tools like Lean is a natural next step.
Derived Categories and Fourier-Mukai Transforms: The \(K3 \times K3\) experiment suggested Fourier-Mukai (FM) kernels as candidate cycles to address transcendental gaps. Derived categories \(D^b(X)\) provide a framework to formalize this:
Define FM transforms as functors between \(D^b(K3 \times K3)\) and \(D^b(X)\), mapping coherent sheaves to cycles via Chern classes.
Extend CAS-6's output layer (\(O\)) to include derived objects, where \(O(X)_p\) incorporates classes induced by FM kernels, potentially increasing \(\dim W(X)_p\).
Computationally, use software like Macaulay2 to simulate FM transforms, testing whether they close gaps (e.g., by computing ranks of induced cycle maps).
Recent work on derived categories for K3 surfaces suggests that FM transforms can generate non-divisorial cycles, which CAS-6 could quantify via updated closure probabilities.
Lean Formalization: Lean's mathlib library supports formalizing algebraic geometry, including schemes, cohomology, and cycle class maps. CAS-6's functorial structure is ideal for Lean:
Formalize the Functor: Define \(\mathcal{F}: \mathbf{Var} \to \mathbf{LayeredVect}\) in Lean, encoding axioms like tensor closure and functoriality.
