The Four-Dimensional Gauge Theory of the Metric Bundle: Dynamics and Anomaly Cancellation

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 \SO(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} = \Met(X) restricted to a metric section g\colon X ↪ Y. We show that the Einstein-Hilbert, Yang-Mills, and extrinsic curvature terms emerge with the correct relative signs: (i) +R_X 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 \GL^+(4,\R)/\SO(4) is a symmetric space of non-compact type with scalar curvature R_{\mathrm{fibre}} = -36. We compute the gauge kinetic metric h_{ab} for the \SO(4) = \SU(2)_L × \SU(2)_R gauge sector via the Kaluza-Klein mechanism and find h_L = h_R = 6\cdot I_3, confirming left-right symmetry g_L = g_R at the unification scale and the absence of ghosts in the gauge sector. The non-vanishing commutators of the shape operators, [A_m, A_n] ≠ 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). v3 changes (March 2026): v3: Recompiled with updated cross-references. v4 changes (March 2026): v4: Recompiled with updated cross-references. Dynamics+anomalies merge (May 2026): May 2026: Expanded and retitled. The four-dimensional anomaly verification formerly in the standalone Paper IV is folded into this paper as a new Anomaly Cancellation section (PS-level cubic and mixed gauge-gravitational anomalies, the Witten global SU(2) anomaly, and all six Standard Model conditions after breaking), so this paper now establishes that the derived 4D gauge theory is sign-correct, ghost-free, AND anomaly-free. The generation number is routed to the higher-dimensional index theorem of Paper V (no D=4 mod-24 constraint). Title changed from "Gauge Fields from the Gauss Equation of the Metric Bundle." Referee-driven revision (May 2026): May 2026 (post referee simulation): corrected the printed su(2)_L commutator orientation; clarified that the sign structure follows from the pointwise Gauss-Codazzi-Ricci identity (unconditional) while its identification with the four-dimensional effective-action coefficients additionally assumes the section condition; rescoped the SU(4) kinetic-metric normalisation (the load-bearing claim for unification is the Dynkin-index equality of Paper I); clarified that R_fibre=-30 is computed in Paper VIII and is not used here; added the Percacci-Nesti GraviGUT prior-art citation. All computations independently re-derived and confirmed in the referee simulation. Second referee pass (May 2026): May 2026: fixed the Percacci-Nesti citation metadata (now the 2010 "Chirality in unified theories of gravity"); tightened the section-condition caveat in abstract/intro (the sign structure is unconditional from the pointwise Gauss identity, while its identification with effective-action coefficients assumes the section condition); made the U(1)_Y-gravitational condition explicit (sum of hypercharges = 0); updated stale LaTeX header; added original Kaluza-Klein references (Witten 1981, Cho-Freund 1975); qualified the loose use of "torsion". A second independent referee simulation re-derived every load-bearing computation and found no errors; its verdict was framing-only, expecting acceptance on a revised version.