The gravitational rapidity potential χg(x) generates a local Spin(1,3) rotor G, which forms the common origin of three complementary descriptions of gravitation. A companion paper developed two of these branches: an exact relativistic fluid-dynamical (kinematic) branch and a slow-flow metric branch that recovers the Painlevé—Gullstrand spacetime geometry. The present paper develops the third branch, based on the Maurer—Cartan connection generated by the rotor and its transport along the gravitational vacuum flow. Starting from the spacetime Maurer—Cartan connection Ω_Q=(∂G)G^−1, its proper-time projection is derived for the composed galactic rotor consisting of radial and azimuthal Lorentz boosts. The resulting connection separates naturally into radial and azimuthal rapidity-rate channels together with the Thomas—Wigner bivector arising from the non-commutativity of the two boosts. Transporting this connection with the gravitational four-velocity produces the bilinear UΩ_τ and its complete norm, time, space and spin channel decomposition. The principal mathematical result is that the time-channel transport condition [UΩ_τ]_T=0 reduces, in the physically relevant slow-flow limit, to the previously established Constant-Lagrangian (CL) condition DΓ/Dτ=0, where Γ=coshχrcoshχϕ. Away from the slow-flow limit the two conditions differ by a factor involving the radial Lorentz factor, but they coincide in the galactic regime. This establishes a direct geometric connection between the Maurer—Cartan formulation and the conserved isorapidity flow that characterizes stationary spiral galaxies. Applying the resulting equations to the Painlevé—Gullstrand—Hubble vacuum flow, together with a Newtonian virial boundary condition at the galactic bulge radius, recovers in closed form the Constant-Lagrangian rotation-curve relation previously fitted to SPARC observations. The Maurer—Cartan formulation therefore provides a second theoretical foundation for the Constant-Lagrangian rotation-curve law within the rotor-generated Spin(1,3) hierarchy. In this framework, flat galactic rotation curves emerge from the conserved composition of two orthogonal rapidity components of a single relativistic gravitational vacuum river flow. Throughout the construction, Hubble-embedded Newtonian gravity remains unchanged, and no dark-matter halo, MOND interpolation function or additional free parameter is introduced. The theory extends naturally to stationary cones of constant colatitude, where the resulting trajectories remain smooth helical curves throughout their domain of physical validity.