OPERATIVE GEOMETRIC ONTOLOGY: The Dialectical Loop Among Quantum Mechanics, Generative Artificial Intelligence, and Complex Systems

This paper proposes an ontological taxonomy of geometric objects grounded in the primary criterion of the matter/formality distinction. Against standard classifications by dimension, curvature, or algebraic invariants—which produce what we term ontological blindness—we articulate a five-level system that maps the mode of existence of geometric objects from spontaneous material emergence to purely formal, immaterial relational structures. The central argument is that this taxonomy does not constitute a descriptive catalogue but an operative methodological infrastructure, demonstrated by its systematic intersection with the three scientific domains where the matter/formality tension is most radically tested: Quantum Mechanics, Generative Artificial Intelligence, and Complex Systems. Analysis reveals that the relationship among these disciplines is not arbitrary: they form a necessary coevolutionary triad—a dialectical loop in which each field corrects the methodological blindness of the others. A key finding is that the mathematical isomorphism between Tensor Networks in quantum many-body physics (Level 3A) and deep-learning latent-space compression (Level 3B) is not a formal coincidence but a structural theorem: generative AI emulates the deep relational architecture of the quantum formalism, not the macroscopic appearance of the classical world. Three concrete research lines are proposed. This framework offers contemporary science criteria of ontological demarcation for guiding AI design, complex-systems modeling, and philosophical interpretation of quantum physics—positioning the taxonomy as the cornerstone of a progressive research programme in the Lakatosian sense.