Theorem Ii: Algebraic Quantization of Phase Space Measure Topological Confinement Fractions via the Peter–weyl Theorem Including Corollaries Ii-a and Ii-B: Extensions to Su(3) Gauge Algebra

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: f ∈ 1, 1/2, 1/4, 1/8 are immutable algebraic constants. Two corollaries extend the framework to SU (3): Corollary II-A (Algebraic Additivity): Derives the D = 3 vacuum weight f (3D) G = 2 for the SU (3) adjoint algebra via generator summation over color-flux tubes. Corollary II-B: Derives the D = 4 spacetime confinement fraction fadj⊗adj = 1/64 for the gluon-gluon vacuum via the full Clebsch-Gordan decomposition of 8 ⊗ 8. These two SU (3) fractions play distinct roles in the Meta-Law 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.