RESOLUTION OF OPEN PROBLEM O3: PROOF OF THE SLAVNOV–TAYLOR IDENTITY UNDER THE REYNOLDS PROJECTOR

This document provides a rigorous proof that the Slavnov-Taylor Ward identity of pure SU(3) Yang-Mills theory is preserved under the action of the Reynolds Projector (P̂_G). This is a foundational step in the T-DFT framework, ensuring that topological confinement does not conflict with gauge invariance.

The central result is the Null Commutator Theorem: [P̂_G, δ_BRST] = 0. This theorem is established in five rigorous steps:

1. Axiomatic definition: Establishment of the BRST symmetry within the T-DFT context.
2. Integral construction: Definition of the Reynolds Projector via the Haar measure on the SU(3) group manifold.
3. Null Commutator Proof: Demonstration of the commutation via the color-singlet nature of Q_BRST and the invariance of the Haar measure.
4. Transversality Derivation: Proof that the projected vacuum polarization tensor remains transverse to all orders.
5. Topological Connection: Linking the result to Theorem I of the T-DFT framework via the exactness of the vacuum differential form.

Impact: The proof resolves a critical structural concern: it demonstrates that restricting the gluon phase-space to the color-singlet fraction f = 1/64 does not anomalously break gauge invariance. It establishes that the topological projection and the quantum gauge structure are geometrically independent and mutually transparent.

This document serves as Companion O3 in the T-DFT series, providing the gauge-theoretic justification for the mass gap derivations presented in Theorem IV and the Bochner-Minlos construction (Companion C1).