Quantifiable Heuristics: Introduces metrics like closure probability (e.g., algebraic span/Hodge dimension ratio) for diagnosing HC validity in specific varieties.
Validated Experiments: Computes explicit bases and dimensions for elliptic products (full closure via Lefschetz) and K3 K3 (gap of 4, signaling incompleteness), with symbolic code validations.
Integration with Modern Tools: Links to Fourier-Mukai transforms and Lean formalization for testing transcendental classes, inspired by recent spectral and deformation approaches.9f95b70e32a2
Cross-Disciplinary Insights: Reframes HC as categorical systemic closure, offering a bridge between algebraic geometry and complex adaptive systems for heuristic-guided research.
Outline
Introduction
Background on the Hodge Conjecture and Millennium Problems. Motivation for formalizing heuristics in algebraic geometry. Overview of CAS-6's categorical structure and contributions.
Formal Definition of the CAS-6 Framework
Categorical setup: Functors from varieties to layered vector spaces. Precise definitions of six layers with axioms (e.g., closure under tensor products). Mappings to HC domains: Topology (levels/configurations), algebra (weights/probabilities), geometry (stability/outputs). Rationale for formal heuristics, with propositions on compatibility with known theorems.
Quantitative Metrics and Computational Tools
Definitions of closure probability, stability invariants, and output realizations. Implementation in symbolic algebra (e.g., SymPy examples for dimension computations). Links to formal verification (e.g., Lean theorems for simple cases).
