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 PG 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 . Evaluating the Clebsch–Gordan character integrals for the fundamental symmetry groups SU(2) and Z2, we establish that the admissible fractions: are immutable algebraic constants. Two corollaries extend the framework to SU(3): Corollary II-A (Algebraic Additivity) derives the D=3 vacuum weight fG = 2 for the SU(3) adjoint algebra via generator summation over color-flux tubes; Corollary II-B derives the D=4 spacetime confinement fraction for the gluon–gluon vacuum via the full Clebsch–Gordan decomposition of 8 ⊗ 8: These two SU(3) fractions play distinct roles in the T-DFT framework and in Theorem IV (Yang–Mills). Any effective vacuum metric deformation must respect these discrete symmetry-locked projections, precluding the existence of continuous phenomenological fitting parameters.