Gauge Theory on the Metric Bundle of a Spin 4-Manifold, III: BRST Structure, Anomaly Cancellation, and the Effective Four-Dimensional Theory

We develop the BRST quantization of the gauge theory on the metric bundle Y¹⁴ = Met(X^4) introduced in Parts I–II. The inhomogeneous gauge group G = H ⋉ N requires ghost fields for both the "rotational" (H) and "translational" (N) sectors; we construct the BRST operator, prove its nilpotency Q^2 = 0 on all fields, and derive the complete gauge-fixed action. We analyze the anomaly structure: the 4D gauge anomalies cancel via the Pati–Salam embedding, while the 14D perturbative anomalies vanish because the fermions live in the full (non-chiral) real 128-dimensional spinor representation of Cl(0,14), for which the anomaly polynomial I₁₆ is identically zero. The global anomaly is absent because the mod-2 index of the real Dirac operator vanishes. We construct the effective 4-dimensional theory by background-field expansion and fiber integration, recovering the Standard Model field content at tree level with higher-dimensional operators suppressed by the fiber compactification scale. We formulate an asymptotic safety conjecture for the full 14-dimensional theory.