VII. On The Law of Conservation of Energy

Kaoru Inoue, Masayuki Yamazaki

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.

A. Lees

Kawasaki, Hideyo

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.