The Equivalence Theorem: T-DFT Yang Mills as the Physical Sector of Canonical Yang Mills

This document establishes the Equivalence Theorem: the T-DFT Yang–Mills theory constructed in Companions C1–C3 and O1–O4 is not a distinct theory but the physical sector of canonical SU(3) Yang–Mills theory, rigorously identified. Four independent results converge to this conclusion: (i) BRST–Reynolds Isomorphism (Section 2): The Reynolds projector extracts exactly the same equivalence class as the BRST charge . Formally: The T-DFT theory and canonical BRST quantisation of Yang–Mills are isomorphic on the physical Hilbert space. (ii) Weak Measure Equivalence via FMS (Section 3): For every gauge-invariant observable , the expectation values computed with the standard Yang-Mills measure and the T-DFT measure coincide: This is the Fröhlich–Morchio–Strocchi (FMS) mechanism formalised within the T-DFT framework. (iii) Dimensional Transmutation and the D = 4 Origin of the Gap (Section 4): The algebraic invariant fSU(3) = 1/64 sets the ratio Mgb/ΛQCD = 8, but the strict positivity Mgb > 0 is dynamically generated exclusively in D = 4 by the conformal anomaly . In any other dimension D ≠ 4, either ΛQCD = 0 (no transmutation) or the mechanism fails. (iv) The Gribov Horizon as Exact Saddle Point (Section 5): In the thermodynamic limit V → ∞, the Gribov parameter equation ceases to be a mean-field approximation and becomes an exact identity by the Gärtner–Ellis large deviations theorem. The T-DFT identification: is therefore exact in the continuum limit, not an approximation. Taken together, these four results close every objection in categories R1–R3 raised by formal reviewers of the Clay submission, and complete the logical chain from the T-DFT constructive programme to the Jaffe–Witten problem statement.