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

We provide 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 . The central result is the Null Commutator Theorem:

This theorem is established in five steps:

• Step 1: Axiomatic definition of BRST symmetry;
• Step 2: Integral construction of the Reynolds Projector via the Haar measure on SU(3);
• Step 3: Proof of the null commutator via the color-singlet nature of the BRST charge and the invariance of the Haar measure;
• Step 4: Derivation of the transversality of the projected vacuum polarization tensor; and
• Step 5: Connection to Theorem I of the T-DFT framework via the exactness of the vacuum differential form.

The proof resolves the structural concern that restricting the gluon phase-space to the color-singlet fraction f = 1/64 might anomalously break gauge invariance. It establishes that the topological projection and the quantum gauge structure are geometrically independent and mutually transparent.