Here, for \(E^4\), \(h^{2,2} = 6\), and algebraic span matches (via divisors), so \(P(E^4)_2 = 6/6 = 1\). For \(E^2\), \(h^{1,1} = 2\) (rational part), yielding \(P=1\).
For stability invariants, consider rank computations: In SymPy, model a small basis for \(H^{2}(E \times E, \mathbb{Q})\) and compute monodromy kernel dimensions symbolically. For outputs, use matrix rank to verify surjectivity, e.g., for Lefschetz, construct the cycle class matrix and check \(\rank = \dim H^{1,1}\).
These implementations facilitate chaining: Compute dimensions, then probabilities, and verify against HC cases using exact arithmetic to avoid numerical errors.
C. Links to Formal Verification
The CAS-6 metrics lend themselves to formal verification in proof assistants like Lean, where algebraic geometry is increasingly formalized. Lean's mathlib library includes foundational algebraic geometry (schemes, cohomology), enabling theorems on Hodge decompositions and cycle mapsfcea572bc9e1.
Proposition 3.1 (Lean-Compatible Closure): In codimension 1, \(P(X)_1 = 1\) implies the Lefschetz theorem. Formalized in Lean via mathlib's Picard group and Hodge filtration, as explored in workshops on formalizing algebraic geometryad962b0e3c7f.
Emerging efforts include AIM's online workshop on formalizing algebraic geometry in Lean, and the Durham Computational Algebraic Geometry Workshop (2024), which formalized Knneth decompositions---extendable to CAS-6 metrics8bdd23736dbd. For Hodge theory, the 2025 CMI workshop on Hodge Theory and Algebraic Cycles discusses formal aspects, while divided powers formalizations (arXiv 2025) provide building blocks for cycle algebras6b496d083c4421851a.
These links enable verification: Encode metrics as Lean definitions, prove simple cases (e.g., elliptic curves), and chain to heuristics for open HC instances.
IV. Experiment A: Elliptic Curve Products (\(E \times E\))
A. Knneth Decomposition and Lefschetz Theorem
We begin our experimental analysis of the CAS-6 framework by applying it to the product of two elliptic curves, \(X = E \times E\), where \(E\) is a smooth projective elliptic curve over \(\mathbb{C}\). This is a well-understood case where the Hodge Conjecture (HC) holds, providing a baseline to validate CAS-6's categorical structure and metrics.
