Resolution of Open Problem O1: Topological Collapse of the Bethe–salpeter Equation and the Exact Glueball Mass Eigenvalue

We resolve Open Problem O1 of the T-DFT framework by embedding the Bethe–Salpeter equation (BSE) for the two-gluon bound state within the topological machinery of Theorems I and II. The strategy proceeds in five steps: Step 1: We write the standard 2PI-kernel BSE for the scalar glueball channel 0++. Step 2: We apply the Reynolds projector to both sides, proving that the 64 × 64 color-matrix BSE collapses to a single scalar equation whose effective kernel is amplified by the topological factor 1/fSU(3) = 64. Step 3: We insert the restricted T-DFT gluon propagator DGμν(q) = fSU(3) Dfreeμν(q), whose use inside the loop is justified by the Null Commutator Theorem proven in companion document O3. Step 4: We invoke Theorem I to replace the standard infrared-divergent loop integral by a holographically regulated integral bounded below by q2 ≥ Λ2QCD. Step 5: We extract the bound-state pole condition. The unique self-consistent solution of the projected gap equation yields the exact eigenvalue: giving the final scalar glueball mass: in agreement with Lattice QCD at the 0.35% level. The mass gap Δ > 0 emerges as a deterministic topological eigenvalue, not a perturbative artifact.