Conservative Motion Theory – PNP VIII Grand Unification of Computation: The Physical Law Behind P, NP, and BQP

Kawasaki, Hideyo

Published Nov 20, 2025• 0 citations

Abstract

This final paper of the PNP series unifies all previous results into a single physicalprinciple governing computation. Within the Conservative Motion Theory (CMT), P, NP,and BQP correspond to three distinct dynamical regimes:P = strictly conservative, BQP = unitarily conservative, NP = maximally non-conservative.Combining PNP I–VII, we derive the Grand Unification of Computation:All physically realizable computation is exactly BQP,and no stronger computation iscompatible with conservation laws.Thus computability is not an abstract mathematical accident but a direct consequenceof the physical laws encoded in CMT: entropy conservation, spectral invariance, Fredholmmass conservation, and unitary dynamics.

DOI: 10.5281/zenodo.17659034

Related papers

Conservative Motion Theory – PNP VII Physical Maximality of BQP: Why Quantum Computation is the Final Computation Allowed by Nature

Kawasaki, Hideyo

This paper completes the PNP series within Conservative Motion Theory (CMT). Weshow that BQP is not merely a complexity class but the physical upper bound of computationimposed by conservation laws of entropy, energy, Fredholm mass, and spectral rank.Combining PNP I–VI, we obtain:BQP is the maximal computation permitted by the laws of physics.Any attempt to extend quantum computation beyond BQP (PostBQP, QMA, #P,nonlinear quantum mechanics, superposition counting) requires superconservative operatorsthat increase Hilbert–Schmidt mass or violate entropy conservation—dynamics forbidden byCMT.Thus computability is a direct physical consequence: Nature allows exactly BQP andnothing stronger.

Nov 20, 20250 citations

Conservative Motion Theory – PNP VI The Quantum–Superconservative Barrier: Why BQP Cannot Be Extended Beyond BQP

Kawasaki, Hideyo

We prove that quantum computation, modeled as unitary conservative motion withinConservative Motion Theory (CMT), has an intrinsic and non-extendable upper boundary.Any attempt to enlarge BQP—such as PostBQP, QMA, or hypothetical super-quantummodels—requires superconservative operators with Fredholm mass strictly exceeding that ofany unitary evolution.Such operators violate conservation of entropy, rank spectrum, and Hilbert–Schmidtinvariants, and therefore cannot arise from any physically realizable quantum dynamics.BQP is the maximal physically possible class.Thus the quantum–superconservative barrier is absolute: quantum computation cannotbe extended into the non-conservative or superconservative regimes without breakingfundamental physical laws.

Nov 20, 20250 citations

Conservative Motion Theory – PNP V The Fredholm–Quantum Barrier: Why BQP Cannot Simulate NP Search

Kawasaki, Hideyo

Within Conservative Motion Theory (CMT), quantum computation corresponds tounitary conservative motion with strictly preserved Fredholm mass and entropy.We prove that NP search operators possess an intrinsic Fredholm–entropy and non–conservativemass that cannot be reproduced, compressed, or interferometrically encoded by any polynomialsizeunitary quantum circuit.Thus BQP, the class of polynomial-time quantum computations, is strictly containedin the conservative sector and cannot simulate the exponential non-conservative branchingrequired for NP search.BQP does not contain NP.This establishes the Fredholm–Quantum Barrier: quantum interference preserves conservativerank and entropy and therefore cannot emulate NP search dynamics.

Nov 20, 20250 citations

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

Kawasaki, Hideyo

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.

Nov 9, 20250 citations

Conservative Motion Theory – PNP III The Fredholm Separation Principle: A Non–Conservative Rank Barrier for Deterministic Computation

Kawasaki, Hideyo

Within the Conservative Motion Theory (CMT), deterministic polynomial–time computationscorrespond to reflection–positive Fredholm contractions T satisfying ∥T∥op ≤ 1 andpossessing finite conservative rank. Nondeterministic computations correspond to maximallynon–conservative branching operators B whose Fredholm mass grows exponentially withinput size n.We prove the Fredholm Separation Principle: no reflection–positive contraction T cansimulate a maximally non–conservative branching operator B of an NP–complete computation.Equivalently, the conservative rank of T is strictly sub–exponential while that of B is Θ(2n).Thus P and NP correspond to two disjoint components in Fredholm–operator space,yielding a structural proof of P ̸= NP within CMT.

Nov 20, 20250 citations

Conservative Motion Theory – PNP IV The Fredholm–Entropy Barrier: Irreducibility of NP Search Under Deterministic Conservation

Kawasaki, Hideyo

In addition to the exponential non–conservative mass established in PNP I–III, we showthat NP search problems possess an intrinsic Fredholm–entropy which cannot be reduced byany reflection–positive deterministic operator.For an NP search operator Bn (e.g. for SAT), the normalized singular-value distributionconverges to an entropy-maximizing measure supported on Θ(2n) independent eigenmodes.Deterministic polynomial–time operators Tn preserve entropy and cannot compress thismeasure into the polynomial-rank conservative sector.Thus NP search is irreducible under conservative motion:SearchNP ̸≤CMT P.Combined with the Fredholm Separation Principle of PNP III, this yields the full CMTseparationP ̸= NP and SearchNP cannot be simulated by P.

Nov 20, 20250 citations