Holographic Vacuum Elasticity and Schur-Reynolds Confinement Fractions

We develop the Holographic Vacuum Elasticity (HVE) framework, in which the macroscopic phase-space measure of a confined gauge theory is determined by a single algebraically quantized constant: the Schur-Reynolds fraction fG: = dim (HG) / dim (H). Constructing the Reynolds projector PG from the Bochner-Haar integral and applying the Peter-Weyl decomposition, we prove that fG is a positive rational number completely fixed by the multiplicity of the trivial representation in the tensor product V ⊗ V. For the Standard Model gauge groups we establish the hierarchy of Schur-Reynolds fractions: ffundSU (2) = 1/4, fadjSU (2) = 1/9, fadj⊗adjSU (3) = 1/64, and fZ2 = 1/2. These purely algebraic results have been machine-verified with zero global sorry in Lean 4/Mathlib4 (TheoremIIₐxiomfree. lean and associated files in the HVE. * namespace). The HVE framework maps these fractions onto observable physical suppression via the Vacuum Suppression Law (VSL): Oobs = Oideal · exp (−σGΩ3fG), where σG = αG/2 is the vacuum elasticity coupling and Ω3 = 2π2 is the S3 boundary volume. The physical application of this law to mass-gap estimation rests on three explicitly stated hypotheses: Infrared Gauge Ergodicity (H1), Topological Confinement Filter (H2), and a Topological Mass-Scaling Conjecture (H3), which we substantially motivate via dimensional analysis and an independent SU (N) spectral hierarchy. Under these hypotheses, the fraction fadj⊗adjSU (3) = 1/64 yields the pure-glue mass gap Mgb = 8ΛQCD ≈ 1704 MeV (0. 35% from the Lucini-Teper quenched lattice result). We further show that the same algebraic mechanism reproduces the proton magnetic moment to 0. 047% accuracy via the VSL with Oideal = 3μN (SU (6) flavour symmetry) and Γ = απ2, providing an independent cross-domain validation of the framework.