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

We present a self-contained derivation of the Vacuum Elasticity Law (VEL),

O_obs = O_ideal · exp(-σ Ω₃ f_G),

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 f^fund_SU(2) = 1/4, f_grav = 1/9, and f^adj_SU(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 (Z^C₂) 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) cosmological suppression of the vacuum energy density; and (VI) topological stability of the proton. Together these steps establish the VEL as the unique exponential suppression law governing every quantum observable in the confined phase. All six pillars admit independent machine-verified proofs (Lean 4/Mathlib companion package).