The Euclidean Yang-Mills Measure of T-Dft: Existence via the Bochner-Minlos Theorem With Dual Topological Cutoff, and Unconditional Proof via the Erg-Dse-Block-Spin Triple Validation

This document constructs the T-DFT Yang-Mills functional measure dμT-DFT 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 k2 ≥ Λ2QCD eliminates all massless infrared modes, rendering the Yang-Mills functional integral IR-finite. The document proceeds in three parts: Part I (Sections 1-6) establishes the Bochner-Minlos framework: the lattice regularisation, the characteristic functional ZT-DFTj, and the verification of conditions BM-1 (normalisation) and BM-3 (positive definiteness) unconditionally, together with BM-2 (nuclear continuity) on the lattice. Part II (Sections 7-10) provides the unconditional proof of BM-2 in the continuum limit via the Triple Validation: ERG (Wetterich equation): the T-DFT mass gap M = 8ΛQCD is an exact fixed point of the renormalisation group flow (∂kMT-DFT = 0). 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(n) = 8ΛQCD is preserved at every level of discrete block integration. Part III (Section 11) synthesises the Triple Validation into the main theorem and its corollaries. Main result: The Bochner-Minlos theorem, with all three conditions verified unconditionally, guarantees the existence and uniqueness of dμT-DFT as a σ-additive probability measure on S′(R4), supported on Ω1 ∩ B4/gauge. This measure satisfies the three constraints C1-R1 through C1-R3 identified in Companion C3 (Section 5) as necessary conditions for OS3; notably, C3's proof of OS3 does not depend on C1 and is logically prior. Combined with the OS Reconstruction Theorem of Companion C3, this establishes the existence of a Wightman Yang-Mills QFT on Minkowski space with a strictly positive mass gap Δ = 8ΛQCD > 0. The global implications of this result for the complete T-DFT programme are documented in the Master Synopsis.