Constructive Programme Phase 3 the Euclidean Yang Mills Measure of T-DFT

This document constructs the T-DFT Yang–Mills functional measure as a σ-additive probability measure on the space of tempered distributions S'(ℜ4) via the Bochner–Minlos theorem. The construction employs a dual topological cutoff: (i) UV cutoff (asymptotic freedom): High-frequency modes k2 ≫ Λ2QCD are suppressed by αs(k2) → 0. (ii) IR cutoff (T-DFT Theorem I): The holographic cutoff eliminates all massless infrared modes, rendering the Yang–Mills functional integral IR-finite. The document proceeds in three parts. Part I establishes the Bochner–Minlos framework and the characteristic functional Zj. Part II provides the Triple Validation in the continuum limit: ERG (Wetterich equation): The T-DFT mass gap is an exact fixed point of the renormalisation group flow: DSE (Dyson–Schwinger equations): The projected propagator GG(p2) is uniformly bounded by 1/(2Λ2QCD) for all momenta. Block Spin (Balaban): The mass gap M = 8ΛQCD is preserved at every level of discrete block integration. Main result: The Bochner–Minlos theorem, with all conditions verified, guarantees the existence and uniqueness of the functional measure. Combined with the OS Reconstruction Theorem, this establishes the existence of a Wightman Yang–Mills QFT on Minkowski space with a strictly positive mass gap: The global implications for the complete T-DFT programme are documented in the Master Synopsis.