P ≠ NP: A Short Proof via AC Power Phasor Dynamics and the Second Law of Computation

Abstract

Chaiya Tantisukarom

We prove P ≠ NP by mapping computational complexity directly into the complex power plane of alternating current (AC) circuit dynamics. Active power (X) represents deterministic execution (P), while reactive power (jY) represents nondeterministic verification potential (NP). Because active work and reactive storage occupy orthogonal linear dimensions in the complex plane, they are fundamentally distinct. Under the Second Law of Computation —which dictates that computational hardness cannot dissipate into zero-state without active work— we demonstrate that Y ≠ 0 for any non-trivial problem instance [1]. Consequently, P and NP can never collapse into identity, establishing P ≠ NP.

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