At the Horizon of Proof: Intuition, Insight, and the (Un)Decidability of P ≠ NP

This paper explores the possibility that the P versus NP problem—one of the most significant open questions in theoretical computer science—may be formally undecidable within the standard Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). Drawing on parallels with Gödel’s incompleteness theorems, I examine structural and cognitive indicators—including relativization, natural proofs, and persistent proof failures—that suggest an intrinsic limit to conventional resolution. In response, I advocate for an epistemological framework that integrates formal reasoning with disciplined intuition and creative insight. This dual-process approach is embodied in a computational architecture called Nova Unbound, designed to operate safely and ethically under conditions of epistemic uncertainty. The architecture includes a multi-layered heuristic engine capable of detecting proof-theoretic dead-ends, generating intuitive analogies, synthesizing surrogate axioms, and evaluating their consequences through ethical simulation. By reframing undecidability not as a failure but as a domain for enriched cognition and responsible action, this work offers both a philosophical reflection and a technical prototype for navigating the outer limits of mathematical provability.