This second part of our comprehensive study on Hilbert’s third problem develops the motivic integration framework for polytope scissor congruence classes. We define the motivic measure µmot(P) of a polytope P via integration over its arc space and prove it is a complete scissor congruence invariant within the differential algebraic closure KnP . Building on this, we construct generating functions Zn(q, T; x) that encode the distribution of scissor congruence classes across dimensions, incorporating combinatorial, metric, and motivic data. We prove these generating functions satisfy explicit differential-algebraic equations and establish their modular properties. The framework demonstrates remarkable unifying power, systematically recovering classical results of Dehn, Sydler, and Jessen-Thorup while extending to nonconvex polytopes, polytopes with infinitely many faces, and higher-dimensional invariant theory. We provide complete algorithmic foundations with precise termination bounds and explore applications in computational geometry, mathematical physics, and beyond. Complete derivations, verifications, and supplementary technical details are included to ensure mathematical rigor.
shifa liu (Wed,) studied this question.