Algebraic Quantization of Phase Space Measure: Topological Confinement Fractions via the Peter–Weyl Theorem

We prove that for any fundamental interaction governed by a compact Lie group G, the macroscopic phase-space measure undergoes a strict topological reduction. Constructing the Reynolds projector P̂G and applying the Peter–Weyl theorem, we show that the accessible phase-space fraction f is a deterministic topological invariant, rigidly fixed by the relative multiplicity of the trivial representation inside the unitary dual Ĝ. Evaluating the Clebsch–Gordan character integrals for the fundamental symmetry groups SU(2) and Z2, we establish that the admissible fractions belong to a discrete rational set. Extending the framework to SU(3), we introduce the extensive spatial vacuum weight WSU(3) = 2 for the adjoint algebra, and the spacetime confinement fraction fadj⊗adj = 1/64 for the gluon–gluon vacuum. These mathematical results demonstrate that the phase space of confined gauge theories is algebraically quantized, precluding the existence of continuous phenomenological fitting parameters. Under a phenomenological mass-scaling conjecture, this fraction yields a glueball mass scale of 8ΛQCD. Using the quenched Λ(0)QCD ≈ 238 MeV, consistent with the pure Yang–Mills lattice simulations, this gives Mgb ≈ 1904 MeV, lying within the quenched lattice range of 1710–1980 MeV. The core algebraic derivations establishing f = 1/64 have been completely machine-verified using the Lean 4 proof assistant.