Holographic Vacuum Elasticity and the Derivation of the Vacuum Suppression Law from First Principles

We present a self-contained derivation of the Vacuum Suppression Law (VSL), Oobs = Oideal · exp(−σΩ3fG), within the Holographic Vacuum Elasticity (HVE) framework. The framework rests on six analytically distinct but mutually reinforcing pillars: (I) holographic reduction of the bulk vacuum action to the boundary hypersurface ∂M ≅ S3 via the generalised Stokes theorem and BRST symmetry; (II) algebraic rigidity of gauge-carrier confinement fractions through the Schur lemma and the Reynolds projector, yielding the rational invariants ffundSU(2) = 1/4, fgrav = 1/9, and fadjSU(3) = 1/64; (III) derivation of the vacuum elasticity constant σ = α/2 from the functional determinant of the Dirac operator and charge-conjugation symmetry (ZC2) via the Atiyah–Singer index theorem; (IV) Euclidean continuation and the isomorphism SO(4) ≅ [SU(2)L × SU(2)R]/Z2, fixing the gravitational sector; (V) proof that the VSL produces a positive, strictly sub-Planckian, and renormalisation-group-stable vacuum energy density; and (VI) topological stability of the proton as a Skyrmion soliton, giving τ1/2p ∼ 1066 yr. Together these steps establish the VSL as a unique exponential suppression law derived from first principles, with no free parameters beyond the Standard Model constants α and ΛQCD. Applied to four independent observables, the VSL yields: the pure-glue mass gap M = 8ΛQCD = 1704 MeV (within 0.35% of the Lucini–Teper quenched lattice result); the proton magnetic moment μp = 3e−απ2μN ≈ 2.7915 μN (0.047% from experiment); the 7Be@C60 electron-capture enhancement of 0.83% (residual < 0.01%); and an 80.2% variance reduction in the Hardy et al. 198Au decay dataset. All six pillars admit machine-verified proofs in the Lean 4/Mathlib companion package with zero global sorry.