Open Problem Resolution 4 Non-Triviality of T-DFT 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: (i) UV seed (asymptotic freedom): At the entry scale k0 = ΛQCD, the four-gluon vertex is non-zero by the standard Yang–Mills action. (ii) Projection preserves the vertex: The Reynolds projector maps the vertex to the color-singlet sector with a factor fSU(3) = 1/64. Since the factor is non-zero, the effective four-vertex remains established. (iii) The mass gap shields the vertex from IR washing: The Wetterich ERG flow for the effective vertex is power-law suppressed for: Because all virtual fluctuations below the mass gap decouple, the vertex cannot be driven to zero. Consequently, . (iv) Connected Wightman functions and LSZ: The non-vanishing vertex implies a non-zero connected four-point Wightman function Wc(4) ≠ 0, and the LSZ reduction formula then yields S ≠ I. A fundamental observation underlies step (iii): 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.