C. Limitations and Heuristic Epistemology
While CAS-6 provides a rigorous heuristic framework, it has limitations:
Non-Resolution of HC: CAS-6 is diagnostic, not resolutive. It identifies gaps (e.g., \(K3 \times K3\)) but does not construct new cycles to close them, limiting its ability to prove or disprove HC.
Computational Scalability: The SymPy pipeline is effective for low-dimensional cases but may struggle with higher K3 products or Calabi-Yau varieties due to exponential growth in cohomology dimensions.
Transcendental Challenges: The framework quantifies gaps but offers limited insight into constructing transcendental cycles, a known difficulty in HC.
Formalization Gaps: While Lean-compatible, full formalization of CAS-6's functor requires extensive development, as current mathlib libraries cover only basic schemes.
From an epistemological perspective, CAS-6 embodies "post-rigorous" mathematics, as described by Tao, where intuition is refined into formal structures without requiring full proofs. Heuristics, as seen in Euler's or Grothendieck's work, guide conjecture exploration by simplifying complex problems into testable models. CAS-6's strength lies in its ability to unify domains via metrics, but its heuristic nature means it complements, not replaces, rigorous methods. It aligns with the epistemology of experimental mathematics, where computational insights drive theoretical advances, as seen in recent computational Hodge theory efforts.
Future work could address limitations by integrating spectral methods for cycle construction, scaling computations with advanced tools (e.g., SageMath), and pursuing Lean formalization to bridge heuristic and rigorous realms.
VIII. Philosophical Foundations of the CAS-6 Framework
A. Existence Through Interaction: A Relational Ontology
The CAS-6 framework is grounded in a relational ontology inspired by the principle that "something exists if it interacts with something else; if something may exist but leaves no trace of interaction, it can be considered non-existent." This philosophy, rooted in relational perspectives from philosophy of science (e.g., Leibnizian relationalism or Rovelli's relational quantum mechanics), posits that the existence of mathematical entities, such as algebraic cycles in the context of the Hodge Conjecture, is defined by their interactions within a structured system---in this case, the cohomological and algebraic structures of a variety.
