Abstract. This article establishes the Ontological Taxonomy of Geometric Objects (TOOG) as an autonomous, formally self-sufficient, and mathematically necessary formal ontology. Against standard classifications by dimension, curvature, or algebraic invariants —which produce what we term ontological blindness (OB): the systematic failure of treating as formally equivalent objects whose mode of existence is heterogeneous— TOOG articulates a system of five deductively necessary levels. The taxonomy is generated by two Boolean predicates: the Predicate of Material Anchoring (Φ) and the Predicate of Irreversible Temporal Constitution (Ψ). Their Cartesian product over the formally defined universe 𝒰 = ω | ω = ⟨S, G⟩ yields four logical quadrants, one provably empty by formal logical-topological incompatibility, and two admitting exactly one non-arbitrary sub-split each. The result is precisely five non-redundant ontological levels. The sub-splits are determined by formally non-isomorphic metric structures: the Kähler manifold with the Fubini-Study metric for formal-projective objects (Level 3A), and the Fisher-Riemannian manifold with the Amari Information Tensor for formal-computational objects (Level 3B). The Kähler condition ∇J = 0 is satisfied by P (ℋ) but not by the Fisher information manifold, rendering the sub-split mathematically necessary. TOOG constitutes a demarcation advance over Ontic Structural Realism (OSR) as formulated by Ladyman & Ross (2007) and French (2014), supplying the criterion that OSR systematically lacks. Three canonical inter-level transition operators translate the ontological distinctions into mathematically specified mechanisms. TOOG stands as an autonomous foundational matrix: particular disciplines (quantum physics, thermodynamics, artificial intelligence, morphogenetic biology) are regional instantiations of categories that differential geometry and operator theory already establish with logical priority.
Cristhian Mauricio Beltrán Calderón (2026) studied this question.