What question did this study set out to answer?

This research aims to establish the P ≠ NP conjecture as a necessary condition arising from the physical constraints governed by the Metric Lag Potential framework.

July 12, 2026Open Access

P vs. NP as a Computational Boundary: A Derivation from the Metric Lag Potential

Key Points

This research aims to establish the P ≠ NP conjecture as a necessary condition arising from the physical constraints governed by the Metric Lag Potential framework.
Developed a derivation based on first principles of Metric Lag Potential.
Identified mass as computational overhead within computational manifolds.
Formulated postulates outlining the relationship between operational complexity and physical realizability.
Demonstrated that NP-problems exceed the physical limits of computational manifolds, requiring Φ_L > 1.
Identified a hypothetical algorithm solving NP-problems in polynomial time leads to impossible conditions.
Provided an information-theoretic proof concept for P ≠ NP without reliance on circuit complexity or relativization barriers.

Abstract

We propose a novel derivation of the P ≠ NP conjecture from first principles of the MetricLag Potential framework (Trautwein 2026). Building on the identification of mass ascomputational overhead and the physical prohibition of ΦL > 1 in any finite-capacitycomputational manifold, we demonstrate that P ≠ NP is not merely a mathematicalconjecture but a physical necessity of the universal substrate. Specifically, we identify: - P-problems as computational processes whose operational complexity ΣOpsscales polynomially — maintaining ΦL 1, which we show is physically forbidden by the same constraint that producesquark confinement in Yang-Mills theory. This provides the first information-theoreticproof concept for P ≠ NP, requiring no circuit complexity arguments and no relativizationbarriers. The result follows from three postulates, is internally consistent with the fullTrautwein framework, and suggests a deep connection between computationalcomplexity, physical realizability, and quantum consciousness.

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