A Formalized Systems-Theoretic Framework for the Hodge Conjecture: The CAS-6 Model and Its Categorical Structure
Abstract
The Hodge Conjecture (HC) asserts that every rational Hodge class on a smooth projective complex variety is algebraic. While rigorous proofs remain elusive, this paper introduces a formalized version of the CAS-6 framework---a six-layered model inspired by complex adaptive systems theory---to provide a structured, quantifiable heuristic for analyzing HC. We define CAS-6 categorically as a functor from the category of smooth projective varieties to a layered category of vector spaces and morphisms, with layers corresponding to interaction level (cohomological degree), configuration (decompositions), weights (rational coefficients), probabilities (dimensional alignments), stability (deformation invariants), and outputs (algebraic cycles). This formalization enables precise mappings: topology to levels/configurations, algebra to weights/probabilities, and geometry to stability/outputs.
Through rigorous experiments on elliptic curve products (E E and E) and K3 surface products (K3 K3), we demonstrate CAS-6's utility: full closure in low-complexity cases aligns with known theorems (e.g., Lefschetz (1,1)-theorem), while a dimensional gap in K3 K3 (404 vs. 400) is quantified as incomplete probabilistic alignment. Computational validations using symbolic algebra confirm these metrics, and we propose extensions to Calabi-Yau varieties via Fourier-Mukai transforms. While not a proof, this framework offers a novel, testable lens for HC, bridging algebraic geometry with formalized heuristics.42c76c5e28ce
Novelty and Significance Statement
Novelty
This work presents the first categorical formalization of a systems-theoretic heuristic for the Hodge Conjecture, transforming the ad hoc CAS-6 model into a precise functorial structure. Unlike traditional approaches in Hodge theory (e.g., motivic or arithmetic heuristics), we integrate complex adaptive systems principles with algebraic geometry via quantifiable metrics, such as closure probabilities defined as ratios of algebraic spans to Hodge dimensions. This enables computational testing (e.g., via SymPy for basis computations) and links to formal verification tools like Lean, drawing from recent mechanization efforts in algebraic geometry.be000503b594 The framework's novelty lies in its layered categorical mapping, which reframes transcendental obstructions as categorical incompleteness, and its application to recent HC-inspired methods like spectral analysis of cycles.eb74e3
Significance
By formalizing heuristics, this framework advances the study of HC beyond intuitive analogies, providing a diagnostic tool for identifying structural gaps (e.g., in K3 products) and guiding targeted constructions (e.g., via derived categories). It has interdisciplinary significance: in algebraic geometry, it offers quantifiable predictions for cycle existence; in systems theory, it applies adaptive models to pure math conjectures. Computationally, it facilitates experiments that could inspire new proofs or counterexamples, aligning with ongoing mechanization trends.059f05fa2200 Ultimately, it contributes to "post-rigorous" mathematics by bridging informal insights with formal structures, potentially accelerating progress on Millennium Prize Problems like HC.c4df51
Highlights
Categorical Formalization of CAS-6: Defines CAS-6 as a functor from varieties to a layered category, enabling rigorous mappings between Hodge theory domains and systems layers.
