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 P̂G 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: fSU(2)fund = 1/4, fSU(2)adj = 1/9, fSU(3)adj⊗adj = 1/64, and fZ2 = 1/2. These purely algebraic results have been machine-verified with zero global sorry in Lean 4/Mathlib4. The HVE framework maps these fractions onto observable physical suppression via the Vacuum Suppression Law (VSL): Oobs = Oideal · exp(−σG Ω3 fG) where σG = α/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 fSU(3)adj⊗adj = 1/64 yields the pure-glue mass gap Mgb = 8 Λ(0)QCD ≈ 1904 MeV (quenched scale), lying within the quenched lattice range 1710–1980 MeV. We discuss how the same algebraic mechanism reproduces the proton magnetic moment to 0.047% accuracy and motivates predictions in hadronic spectroscopy, nuclear decay, and satellite navigation.