The Ill-Posedness of P versus NP

Ontological Commitments, Mutual Exclusivity, and the Limits of Real-World Applicability

Author: Patrick David Aoun

Date: August 2, 2026

Abstract

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.

I. Introduction

The question of whether P equals NP stands among the most prominent open problems in theoretical computer science and mathematics. Its origins lie not in pure abstraction but in concrete combinatorial tasks that arise repeatedly in physical and engineering contexts: the traveling salesman problem, job-shop scheduling, integer linear programming, Boolean satisfiability, and related search and optimization challenges. In the early 1970s these practical difficulties were formalized into a precise decision problem concerning the relationship between efficient verifiability and efficient solvability. The resulting question has resisted every major proof technique for more than five decades and carries a Clay Millennium Prize.

The classical expectation is that the question admits a definitive yes-or-no answer within the standard non-constructive language of complexity theory. We contend that this expectation is misplaced. The formulation of P versus NP relies on an ontological commitment to a global, mind-independent domain of languages, machines, and certificates whose existence is independent of epistemic acknowledgment. When that same commitment is examined under the austere constraints required by relativistic quantum field theory—strict localization of actualized states, frame-dependent simultaneity, and ontological minimalism—it generates infinite regress and category error. The same posit that produces persistent metaphysical difficulties (the hard problem of consciousness, the mind-body divide, and related paradoxes) reappears inside the formal statement of the complexity-theoretic question.

Because the motivating instances of P versus NP are physical, any solution that claims to settle the question must in principle be applicable to real-world situations. Under a framework that identifies epistemic structure with ontic structure—Mutual Exclusivity together with Phenomenological Absolutism—no such applicable solution is conceivable. A resolution that remains inside the classical ontology fails to correspond to actualized physical reality; whereas a resolution that corrects the ontology no longer answers the classical question. We therefore contend that the apparent unsolvability of P versus NP must be diagnostic of foundational ill-posedness rather than ordinary technical difficulty.

The paper proceeds as follows. Section II presents the ontological framework. Section III examines the non-constructive commitments embedded in the classical definition of NP. Section IV shows how those commitments inherit the same metaphysical difficulties already identified in the physical setting. Section V recalls the real-world origin of the problem and the consequent demand of applicability. Section VI develops the resulting dilemma. Section VII states the predictive consequences of the analysis. Section VIII addresses principal objections. Section IX concludes.

II. Ontological Framework

We begin from the requirement that existence presupposes conceivability. An entity or fact can be said to exist only if it can be distinguished from what it is not and can, at least in principle, be acknowledged within an actualized configuration. This criterion, developed at length in our critique of actual infinity (Aoun, 2025a), rules out completed totalities that by definition exceed every finite or local grasp. Actual infinity, understood as a static, fully determinate collection containing infinitely many elements at once, fails the criterion and generates logical contradiction once the demand for conceivability is enforced.

The same demand acquires concrete physical content inside relativistic quantum field theory. Observables are associated with local algebras of operators defined over bounded spacetime regions. Actualization—what counts as physically real rather than merely potential—must therefore occur locally. Special relativity further eliminates any objective, universe-wide hypersurface of simultaneity. Consequently the only candidate for physical actuality is a localized field configuration together with its causal past light cone (Aoun, 2026a).

From localized actualization it follows that successive configurations are mutually exclusive. When one configuration appears to give way to another, the prior configuration ceases to be actual. There is no persisting global four-dimensional manifold in which all configurations coexist ontologically; such a manifold would require a preferred foliation or an external embedding structure forbidden by both relativity and ontological austerity. We term this mutual exclusion of phenomenologically successive actualizations Mutual Exclusivity (Aoun, 2025b).

Within each actualized configuration the epistemic capacity for acknowledgment is not a secondary or emergent property. The precise field pattern that realizes a moment of conscious experience is that experience from the inside. Epistemic structure and ontic structure therefore coincide. We call this identification Phenomenological Absolutism. Any appeal to a mind-independent global substrate or to entities that stand outside the present configuration reduces to further representational content internal to the sole actualized is-ness (Aoun, 2026a).

The attempt to index or coordinate multiple actualizations immediately produces a vicious regress. A meta-vantage capable of relating distinct configurations cannot itself be merely another localized actualization, for it would then fail to perform the coordinating role. It must be posited as a global or trans-realization structure, which in turn requires its own indexing, and so on without end. Ontological inflation and violation of locality result (Aoun, 2026b). The only coherent alternative is to treat each actualized configuration as a self-contained absolute is-ness. This concludes the ontological framework employed throughout the remainder of the paper.

III. Ontological Commitments of Classical Complexity Theory

The classical definition of the class NP rests on a non-constructive existential quantifier. A language L is placed in NP precisely when there exists a deterministic polynomial-time verifier V such that an input x belongs to L if and only if there exists a certificate c of length bounded by a polynomial in |x| for which V(x,c) accepts. The existential claim is understood classically: the certificate is treated as an objective mathematical object whose existence does not depend on whether any agent can construct, exhibit, or even acknowledge it.

The same non-constructive stance extends to the broader domain over which the quantifiers of complexity theory range. That domain is taken to comprise all languages over finite alphabets, all Turing machines, all possible certificates, and all asymptotic complexity classes. These entities are regarded as forming a completed, mind-independent mathematical universe. Membership relations, asymptotic bounds, and the existence or non-existence of polynomial-time algorithms are asserted as facts about this universe irrespective of any particular epistemic situation.

The parallel with the global substrate rejected in Section II is exact. Just as a persisting four-dimensional manifold or a trans-realization indexing structure would stand outside every localized actualization, the completed mathematical universe of complexity theory is posited as existing independently of acknowledgment. Both posits introduce entities whose ontic status is claimed to exceed the epistemic limits of any absolute is-ness. In the physical setting such a posit generates infinite regress; in the mathematical setting it supplies the background ontology against which the question whether P equals NP is formulated.

Thus the classical theory does not merely employ a convenient formal language. It embeds a substantive ontological commitment to existence beyond epistemic limits—the very commitment that Mutual Exclusivity and Phenomenological Absolutism rule out on pain of regress and category error.

IV. From Metaphysical Quagmires to Mathematical Ill-Posedness

The global-substrate posit examined in Section II is not confined to physics and metaphysics. The same posit reappears, under a different vocabulary, inside the formal statement of P versus NP. Once existence is allowed to outrun acknowledgment, the hard problem of consciousness, the mind-body divide, and the infinite regress of indexing structures become inevitable. The identical license, when granted inside complexity theory, renders the decision problem ill-posed.

The category error is straightforward. Classical complexity theory treats statements of the form “there exists a short certificate” or “there exists a deterministic polynomial-time algorithm” as objective facts about a completed mathematical universe. Under Mutual Exclusivity and Phenomenological Absolutism those statements either collapse into internal representational content within some absolute is-ness or else assert an ontic status that no actualized configuration can underwrite. In the latter case they inherit the same incoherence already diagnosed for actual infinity and for global spacetime manifolds.

The defect is systematic. Every major classical approach to P versus NP operates inside the non-constructive quantifier structure identified in Section III. Diagonalization arguments quantify over all Turing machines; circuit lower bounds quantify over all Boolean circuits of a given size; natural-proof barriers and algebrization results likewise presuppose a completed domain of combinatorial objects whose existence is independent of acknowledgment. Because each technique inherits the illicit existential commitment, none can escape the category error. The resistance of the problem to resolution is therefore not an accident of incomplete technique; it is the predictable consequence of asking a question whose ontological presuppositions are incoherent under the austere constraints of localized actualization.

V. Real-World Origin and the Applicability Requirement

P versus NP did not originate as an arbitrary formal exercise. Its canonical problems—the traveling salesman problem, job-shop scheduling, integer programming, Boolean satisfiability, and related combinatorial tasks—are direct abstractions of concrete physical and engineering situations. The theory therefore inherits an implicit demand of applicability: any solution that claims to settle the question must, in principle, transfer back to those real-world situations.

Under Mutual Exclusivity and Phenomenological Absolutism every physical situation is realized only as a localized field configuration, an absolute is-ness. There is no global, mind-independent search space containing all possible tours, schedules, or assignments. The completed domain over which the classical quantifiers range therefore lacks independent ontic status. What appears as a global collection of instances is either internal representational content within some actualized configuration or an illicit external posit of the kind already ruled out by the regress argument.

Consequently the applicability requirement cannot be satisfied by a solution that remains inside the classical ontology. The mathematical problem quantifies over a domain that, on the framework developed in Section II, does not correspond to anything that is ever actualized. The real-world origin of the problem, far from rescuing the classical formulation, supplies an additional reason for regarding that formulation as mismatched to the physical situations it purports to describe.

VI. The Dilemma of Applicability

The applicability requirement established in Section V generates a dilemma for any prospective solution to the classical P versus NP question.

Horn 1. Suppose a proof remains inside the non-constructive language of classical complexity theory. Such a proof quantifies over a completed domain of languages, machines, and certificates whose existence is independent of acknowledgment. Under Mutual Exclusivity and Phenomenological Absolutism that domain does not correspond to anything that is ever actualized. The proof therefore answers a question whose ontology fails to match real-world situations as characterized by the framework. Transfer of the result to physical and engineering practice is blocked at the foundational level.

Horn 2. Suppose instead that a proof restricts existence claims to what can be acknowledged inside absolute is-nesses, thereby correcting the ontological defect. The resulting statement no longer concerns the classical classes P and NP as standardly defined. It answers a different question—one formulated inside the framework of localized actualization—and consequently does not resolve the original decision problem.

In either case no resolution can simultaneously satisfy the classical formulation and remain applicable to the real world under the premises of the framework. Applicability becomes inconceivable. Because the demand of applicability is internal to the problem’s own real-world origin, the classical question is not merely unsolved; it is ill-posed. The apparent resistance of P versus NP to decisive settlement is the predictable symptom of this structural mismatch rather than a temporary gap in combinatorial insight.

VII. Predictive Consequences

The analysis yields two linked consequences, one internal to the framework and one predictive with respect to classical practice.

Internally, once Mutual Exclusivity and Phenomenological Absolutism are adopted, the classical decision problem admits no well-formed yes-or-no answer. Its quantifiers range over a domain that cannot be granted ontic status without reintroducing the global substrate and the associated regress. The question is therefore dissolved rather than answered: it rests on a category error parallel to the error that dissolves the demand for an external explanation of why there is something rather than nothing (Aoun, 2026b).

Externally, the framework predicts that the classical community will continue to find the problem unsettled. The prolonged absence of a decisive proof is not evidence of missing combinatorial insight or insufficient technical ingenuity. It is the expected manifestation of working inside an ontological frame whose central posit—existence independent of acknowledgment—is incoherent under the austere constraints of localized actualization. No refinement of technique that remains within non-constructive existential language can remove the underlying category error.

The prediction does not impugn the practical value of algorithmic research. Heuristics, approximation algorithms, average-case analyses, and finite-instance methods continue to deliver genuine engineering progress. What the framework denies is only the expectation that a definitive classical settlement of P versus NP can both remain faithful to the original formulation and transfer cleanly to real-world situations once the global-substrate assumption is rejected.

VIII. Objections and Replies

Several objections may be raised against the claim that P versus NP is ill-posed.

First, one may insist that mathematics is merely a formal game whose internal rules are independent of physical ontology. On this view the non-constructive quantifiers of complexity theory require no metaphysical defense; they are simply syntactic operations within an axiomatic system such as ZFC. We reply twofold. First, this objection underestimates the problem’s explicit physical origins: because P versus NP was formulated as an abstraction of concrete physical tasks and any solution is expected to apply back to those tasks, the ontological commitments of the formal language cannot be insulated from the physical situations they purport to describe. Second, and more fundamentally, the formalist retreat to “pure syntax” fails on its own physicalist premises. Syntax is not an ungrounded entity floating outside localized actualization. Under austere physical realism, formal symbols, rules, Turing-machine descriptions, and proofs exist exclusively insofar as they are physically instantiated—whether as silicon gate configurations, ink on paper, or localized cognitive configurations. Positing a purely formal game whose syntax exists independently of localized physical actualization simply reintroduces the same unactualized global substrate under a mathematical alias. A purely formal settlement that both relies on an unactualized syntactic substrate and fails the applicability requirement does not resolve the question that was originally asked.

Second, it may be observed that many successful mathematical theories rested on imperfect or later-revised ontologies yet still produced decisive internal results. Early calculus with infinitesimals and naïve set theory before the paradoxes are familiar examples. We grant the historical point. The present argument, however, does not claim that every theory with a defective ontology is barren. It claims only that a theory whose central question quantifies over a completed domain that is simultaneously required to map onto physical reality cannot deliver a solution that satisfies both the formal statement and the applicability demand once that domain is rejected. The parallel with successful but later-corrected theories therefore does not restore the expectation of a classical settlement.

Third, the framework of Mutual Exclusivity and Phenomenological Absolutism may itself be charged with being untestable or anti-scientific. We answer that the framework is offered as an immanent critique. It begins from the strongest commitments of austere physical realism—localized actualization in relativistic quantum field theory, frame-dependent simultaneity, and ontological parsimony—and follows those commitments without compromise. The resulting identification of epistemic and ontic structure is not an external imposition upon science; it is the endpoint reached when the realist’s own rules are applied consistently (Aoun, 2026a). Predictive success of the physical formalism is fully retained; only the gratuitous global substrate is excised.

Fourth, one may suggest that applicability can be recovered at the level of finite instances or practical computation, rendering the global formulation irrelevant. We agree that finite-instance algorithms and heuristics remain valuable. The classical P versus NP question, however, is not a claim about any finite collection of instances. It is a universal claim about asymptotic classes defined over a completed domain. Restricting attention to finite or practical cases changes the subject; it does not answer the question whose ill-posedness is at issue.

In each case the objection either concedes the applicability requirement and thereby inherits the dilemma of Section VI, or else abandons the requirement and thereby abandons the original motivation of the problem. The diagnosis of ill-posedness stands.

IX. Conclusion

We have argued that the classical P versus NP question is ill-posed. The argument proceeds from a single chain of commitments. A global-substrate ontology that posits existence independently of epistemic acknowledgment generates persistent metaphysical difficulties—infinite regress under strict localization, the hard problem of consciousness, and the mind-body divide—and stands in tension with the austere constraints of relativistic quantum field theory. The same ontology is embedded in the non-constructive quantifiers of complexity theory. Because the motivating instances of P versus NP arise from physical and engineering tasks, any solution is required to apply back to real-world situations. Under Mutual Exclusivity and Phenomenological Absolutism those situations are realized only as localized absolute is-nesses; the completed domain presupposed by the classical question lacks corresponding ontic status. A resolution that remains inside the classical language therefore fails the applicability requirement, while a resolution that corrects the ontology ceases to answer the original question. Applicability becomes inconceivable, and the decision problem is revealed as resting on a category error.

The contribution of the analysis is diagnostic. It relocates the long-standing resistance of P versus NP from the realm of incomplete combinatorial technique to the realm of foundational ill-posedness. Practical algorithmic progress is left untouched; only the expectation of a definitive classical settlement that both preserves the original formulation and transfers cleanly to physical reality is withdrawn.

Further work remains. Complexity theory may examine whether restricted, constructive, or average-case variants of the question can be reformulated without the global-substrate posit. Philosophy of mathematics may explore the consequences of enforcing conceivability as a condition of existence across other non-constructive domains. The interpretation of physical computation may investigate how localized actualization reshapes the relationship between formal models and the absolute is-nesses in which computation is realized. In each direction the present diagnosis supplies a coherent starting point: once existence is no longer permitted to outrun acknowledgment, the classical P versus NP question dissolves rather than awaits solution.

References

Aoun, P. D. (2025a). The Irrationality of Infinity: A Philosophical and Mathematical Critique. Independent publication. Amazon.

Aoun, P. D. (2025b). Mutual Exclusivity: A New Compass for Reality. Independent publication. https://www.mutual-exclusivity.com

Aoun, P. D. (2026a). Phenomenological absolutism as the endpoint of austere physical realism: Localized actualization in relativistic quantum field theory. PhilArchive. https://philarchive.org/rec/AOUPAA

Aoun, P. D. (2026b). Why is there something rather than nothing? A dissolution under mutual exclusivity, phenomenological absolutism, and austere relativistic quantum field theory. PhilArchive. https://philarchive.org/archive/AOUWIT

Arora, S., & Barak, B. (2009). Computational complexity: A modern approach. Cambridge University Press.

Cook, S. A. (1971). The complexity of theorem-proving procedures. In Proceedings of the third annual ACM symposium on Theory of computing (pp. 151–158). ACM.

Karp, R. M. (1972). Reducibility among combinatorial problems. In R. E. Miller & J. W. Thatcher (Eds.), Complexity of computer computations (pp. 85–103). Plenum Press.

Levin, L. A. (1973). Universal sequential search problems. Problems of Information Transmission, 9(3), 265–266.

Rovelli, C. (1996). Relational quantum mechanics. International Journal of Theoretical Physics, 35(8), 1637–1678.

Wallace, D. (2012). The emergent multiverse: Quantum theory according to the Everett interpretation. Oxford University Press.

Esfeld, M. (2017). A minimalist ontology of the natural world. Routledge.

Brouwer, L. E. J. (1912). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20(2), 81–96.