The Ill-Posedness of P versus NP: Ontological Commitments, Mutual Exclusivity, and the Limits of Real-World Applicability

In this paper, we argue that the classical P versus NP question is ill-posed. Its formulation relies on a non-constructive, global-substrate ontology that posits mathematical existence independently of epistemic acknowledgment. When this ontology is subjected to the same austere constraints demanded by relativistic quantum field theory—localized actualization, frame-dependent simultaneity, and ontological minimalism—it generates infinite regress and category error. Because the motivating instances of P versus NP arise from physical and engineering problems, any genuine solution must be applicable to real-world situations. Under a framework that identifies epistemic and ontic structure (Mutual Exclusivity together with the principle of Phenomenological Absolutism), no such applicable solution is conceivable: a resolution that remains inside the classical ontology fails to match actualized physical reality, while a resolution that corrects the ontology ceases to answer the classical question. The apparent unsolvability of P versus NP is therefore diagnostic of foundational ill-posedness rather than ordinary technical difficulty. We aim to develop the argument systematically in this paper, address objections, and indicate consequences for complexity theory and the philosophy of mathematics.