NON-TRIVIALITY OF T-DFT YANG–MILLS THEORY: THE PROJECTED FOUR-VERTEX, CONNECTED WIGHTMAN FUNCTIONS, AND THE NON-TRIVIAL S-MATRIX IN Hᴳ

Non-Triviality of the Quantum Yang–Mills Theory This document resolves the final open question in the T-DFT constructive programme: the non-triviality of the quantum Yang–Mills theory constructed in Companions C1–C3 and O1–O3. Non-triviality means that the physical S-matrix S ≠ I, i.e., glueball–glueball scattering does not reduce to a free particle evolution. The proof proceeds in four steps that draw exclusively on results already established in the T-DFT package: (i) UV seed (asymptotic freedom): 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) · Γ(4)singlet, with fSU(3) = 1/64 ≠ 0 (Theorem II), so the projected four-vertex Γ(4)eff ≠ 0 (Lemma 4.1). (iii) The 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. Consequently, Γ(4)eff(k → 0) ≈ Γ(4)eff(Mgb) ≠ 0 (Proposition 5.1). (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). Postulating the standard LSZ interface for asymptotic confined states as Axiom O4.A1, the LSZ reduction formula then yields S ≠ I (Corollary 6.4). Fundamental Observation: The very mass gap whose existence is the object of the Millennium Prize problem 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.