CAT'S THEORY: The Triune Constraint - A Formal Argument for Trinitarian Metaphysics via the P×I×Pr Invariant

This paper presents a novel eliminative argument for Trinitarian metaphysics grounded in empirical validation of a structural invariant across all coherent systems. Beginning with a recognition on November 2, 2024, that toroidal geometry maps precisely onto Gospel narrative structure, the work culminates in a systematic demonstration that every coherent system instantiates three irreducible factors in multiplicative dependence: Pattern (lawful structure), Intent (directed selection), and Presence (actualized instantiation). Through contemporary grounding theory, modal logic, and the Principle of Sufficient Reason, the argument demonstrates that monadic grounds cannot maintain independent variation, dyadic grounds introduce explanatory asymmetry, and impersonal grounds cannot ground genuine teleology. The only remaining candidate is a triune personal ground, uniquely satisfied by Christian Trinitarian doctrine. The paper includes formal rebuttals to Platonist, naturalist, and Buddhist Madhyamaka alternatives, establishes the discovery timeline for intellectual property priority (initial synthesis November 2, 2024; multiplicative invariant articulated May 25, 2025), and demonstrates implications for philosophy of religion, analytic metaphysics, artificial intelligence alignment, and the structural requirements of coherent agency. Keywords: Grounding theory Metaphysical necessity Trinitarian theology Structural invariants P×I×Pr framework Principle of sufficient reason Triadic ontology Toroidal geometry Transcendental argument Modal logic Fine-Schaffer grounding Structural realism Coherence theory Teleology Aseity