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Conservative Motion Theory VI: Exponential Time Barrier and the Law of Conservation of Computation

Kawasaki, Hideyo

Published Nov 9, 2025• 0 citations

Abstract

This paper completes the CMT-based proof of the classical separationP ̸= NP.Where CMT III established the Fredholm–analytic distinction between deterministic (conservative)and nondeterministic (non-conservative) computation, the present volume proves theExponential Time Barrier: every non-conservative computational step induces an unavoidableexponential lower bound on computational complexity.Using the operator-theoretic law of conservation of computation, we show that: (1) theclass P corresponds to compositions of reflection-positive, self-adjoint Fredholm operators,whose powers grow polynomially; while (2) the class NP necessarily contains at least onenon-self-adjoint direction, producing exponential growth of operator powers via the Fredholmentropy barrier.The main theorem follows:P = {polynomially bounded conservative operators},NP = {operators with essential non-conservative branching}.Since non-conservative branching forces exponential spectral growth, it cannot lie in thepolynomially bounded closure. Thus P ̸= NP.Connections with physical computation and the quantum limit (PNP7–8) demonstratethat BQP is the maximal computational class permitted by the law of conservation ofcomputation.

DOI: 10.5281/zenodo.17565348

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