Resolution of Open Problem O2: Verification of the Wightman Axioms in the T-DFT Reynolds-Projected Hilbert Space

We verify that the physical Hilbert space of the T-DFT framework—the Reynolds-invariant subspace of the two-gluon color-singlet sector—satisfies all five Wightman axioms of constructive quantum field theory 2, 3. The proof proceeds axiom by axiom, drawing on the four established theorems of the T-DFT framework (Theorems I–IV) and on the companion resolutions of Open Problems O1 19 (Bethe-Salpeter equation) and O3 18 (Slavnov-Taylor identity under Reynolds projection). Axiom W1 (Poincaré covariance): The Reynolds Projector commutes with the full Poincaré group by the independence of color and spacetime degrees of freedom. Axiom W2 (Spectral condition): The mass-gap eigenvalue derived in the O1 companion provides the required lower bound on the physical spectrum: Axiom W3 (Locality): The Slavnov-Taylor Ward identity preserved by (proven in the O3 companion) guarantees that contains only gauge-invariant, causally local states. Axiom W4 (Completeness): The Peter-Weyl decomposition of Theorem II 14 proves that is the closed, complete subspace spanned by the unique color-singlet sector of . Axiom W5 (Vacuum uniqueness): Schur's Lemma applied to the compact group SU(3), combined with the unique critical point of the Morse function in Theorem I 13, guarantees a non-degenerate, unique vacuum state in . The Open Problem O2 is thereby resolved.