A Canonical Factorization Theorem with Universality for Admissible Physical and Analytic Realizations: The Combinatorial Equilibrium-Selection Law

Physical, spectral, arithmetic, and quantum-channel realizations are formulated on different target spaces and need not a priori descend through a common mediating quotient. The present theorem shows that, for the admissible completion-locked class, such a quotient exists canonically and governs the limiting realized data. More precisely, we prove a canonical factorization theorem for the combinatorial equilibrium-selection law on countable history spaces equipped with a partially defined composition law, an additive rigidity functional R, a positive prior μ, a rigidity-compatible equivalence relation, a finite equilibrium quotient Σₑq = 𝓜 / ∼, and a strictly positive rigidity gap. Here a completion lock means that the completion, normalization, and regularization data are fixed in advance. For every admissible Borel realization map Π: Γ → B into a Polish target that is constant on equilibrium classes, the theorem proves the initiality of the equilibrium quotient, an exact decomposition of the realized pushforward measure into equilibrium and off-equilibrium contributions, exponential concentration and total-variation control, quantitative factorization for bounded observables, and canonical quotient-mediated limits for extracted invariants continuous at the canonical limiting measure. The principal corollaries verify that the same theorem governs the mass–energy identity, the spin-amplitude weighted paths quantum channel law, four-dimensional gravitational vacuum selection, and analytic Hilbert–Pólya / Riemann-hypothesis operator extraction. A quantum-channel refinement further shows that the spin-amplitude weighted paths channel is an admissible realization whose barycentric extraction reproduces the full Gibbs-weighted channel family at every temperature and recovers the canonical equilibrium channel in the strong-selection limit. The theorem is exact at the level of realized measure decomposition and asymptotic at the level of extracted-invariant factorization. Synthesis. Within the completion-locked admissible class, admissibility reduces physical, spectral, arithmetic, and quantum-channel multiplicity to a common quotient law: realizations factor through the same finite equilibrium quotient, and their canonical limiting data are determined by the same equilibrium weights. This quotient mechanism is precisely the combinatorial equilibrium-selection law. License note: Distributed under CC BY-NC-ND 4. 0.