Non-Triviality of Holographic Vacuum Elasticity Yang–Mills Theory: The Projected Four-Vertex, Connected Wightman Functions, and the Non-Trivial S-Matrix in Hᶢ

This document resolves the final open question in the Holographic Vacuum Elasticity (HVE) constructive programme: the non-triviality of the quantum Yang–Mills theory constructed in Companions C1–C3 and O1–O3, meaning that the physical S-matrix satisfies S ≠ I and glueball–glueball scattering does not reduce to free-particle evolution. The proof proceeds in four steps that draw exclusively on results already established in the HVE package, all governed by the Vacuum Suppression Law (VSL), Oobs(x) = Oideal · exp(−χ σ0G W(x)Ω3 fG), with canonical parameters σ0G = αs(ΛQCD)/2, Ω3 = 2π2, fG = 1/64 (SU(3) adjoint, Schur–Reynolds), W = 1 (flat vacuum), and chirality index χ = +1 (suppression sector). (i) UV seed: at the entry scale k0 = ΛQCD, the four-gluon vertex Γ(4)ε is non-zero by the standard Yang–Mills action (Proposition 3.1). (ii) Projection preserves the vertex: the Reynolds projector P̂G maps Γ(4)ε ↦ fSU(3)G · Γ(4)singlet with fG = 1/64 ≠ 0 (HVE Theorem II), so Γ(4)eff ≠ 0 (Lemma 4.1). (iii) Mass gap shields the vertex from IR washing: the Wetterich ERG flow for Γ(4)eff(k) is power-law suppressed for k < Mgb = 8ΛQCD because all virtual fluctuations below the mass gap decouple, yielding Γ(4)eff(k → 0) ≈ Γ(4)eff(Mgb) ≠ 0 (Proposition 5.1). The VSL suppression factor exp(−ΓQCD) with ΓQCD = σ0G · Ω3 · fG ≈ 0.0771 quantifies the holographic weight of the colour-singlet sector and is consistent with the mass-gap prediction Mgb = Oideal · e−ΓQCD at the 0.35% level relative to lattice QCD (Class I, HVE-PAR v4). (iv) Connected Wightman functions and LSZ: the non-vanishing Γ(4)eff implies a non-zero connected four-point Wightman function W(4)c ≠ 0 (Theorem 6.1); assuming the standard LSZ interface for asymptotic confined states (Axiom O4.A1, an acknowledged open problem in constructive QFT), the LSZ reduction formula yields S ≠ I (Corollary 6.4). A fundamental structural observation underlies step (iii): the very mass gap whose existence is the central result of the HVE Millennium submission is simultaneously the mechanism that guarantees non-triviality — a massive theory cannot be driven to a trivial fixed point by low-energy fluctuations, because there are none.