Gauge Fields from the Gauss Equation of the Metric Bundle

Paper 3 of 6 in the Metric Bundle Programme. In earlier work, we showed that the DeWitt supermetric on the space of Lorentzian metrics over a four-dimensional spacetime X has signature (6, 4), yielding the Pati-Salam gauge group from the maximal compact subgroup of (6, 4). In this paper, we carry out the Gauss-Codazzi-Ricci decomposition of the scalar curvature of the fourteen-dimensional metric bundle Y^14 = (X) restricted to a metric section g X ↪ Y. We show that the Einstein-Hilbert, Yang-Mills, and extrinsic curvature terms emerge with the correct relative signs: (i) +RX for gravity, (ii) -|II|² for the extrinsic curvature (torsion), and (iii) - (h/4) |F|² with h > 0 for the Yang-Mills gauge fields. The fibre ^+ (4, ) / (4) is a symmetric space of non-compact type with scalar curvature R₅₈₁ₑ₄ = -36. We compute the gauge kinetic metric h₀₁ for the (4) = (2) L × (2) R gauge sector via the Kaluza-Klein mechanism and find hL = hR = 6 I₃, confirming left-right symmetry gL = gR at the unification scale and the absence of ghosts in the gauge sector. The non-vanishing commutators of the shape operators, Aₘ, Aₙ ≠ 0 for 24 of the 45 normal-direction pairs, provide the non-abelian field strength via the Ricci equation. Part of a six-paper series deriving the Pati-Salam gauge group, fermion content, gauge dynamics, anomaly cancellation, and the three-generation structure from the geometry of the metric bundle Y14 = Met (X4).