Prove Propositions: Formalize results like Proposition 4.1 (full closure for \(E \times E\)) using Lean's cohomology modules, extending to partial closure for \(K3 \times K3\).
Verify Metrics: Encode closure probability \(P(X)_p\) as a Lean function, proving properties like submultiplicativity for products.
The 2025 CMI workshop on Hodge Theory and Algebraic Cycles highlights Lean's potential for formalizing conjectures, and CAS-6 could contribute by providing a heuristic layer to such efforts.
C. Computational Experiments
To scale CAS-6's applicability, we propose computational experiments to test its predictions:
SageMath Pipeline: Extend the SymPy pipeline to SageMath, which supports advanced algebraic geometry computations (e.g., intersection theory on K3 surfaces). For a Calabi-Yau threefold, compute Hodge numbers, simulate cycle classes, and estimate \(P(X)_p\).
FM Kernel Simulations: Implement FM transforms in Macaulay2 to generate candidate cycles for \(K3^n\), testing whether they increase \(\dim W(X)_p\). This could involve numerical rank computations over \(\mathbb{Q}\).
Large-Scale Varieties: Test CAS-6 on varieties like hyper-Khler manifolds or Fano varieties, using distributed computing to handle high-dimensional cohomology.
These experiments align with trends in experimental mathematics, where computational tools guide theoretical insights, as seen in recent computational Hodge theory advances.
D. Philosophical Reflections on Post-Rigorous Math
CAS-6 embodies the "post-rigorous" paradigm, where intuitive heuristics are formalized without requiring complete proofs, as articulated by Tao. Philosophically, it raises questions about the role of heuristics in tackling conjectures like HC:
