The third branch of the rotor hierarchy differentiates the rotor itself. At the Dirac level the ordered rotor Qg = Qg,φQg,r, Qg,x = e(χx/2)αx, has the Maurer–Cartan connection Ωτ = (DQg/Dτ)Qg−1, a pure bivector with a Thomas–Wigner rotation channel. Transported with the four-velocity Ug, its β0 channel is the rate of the composed Lorentz factor:
The second identity is what the lift adds: the vector grade of the transport product is the kinematic acceleration of the previous model, so the Maurer–Cartan route and the kinematic route are one object. Isorapidity, DΓ/Dτ = 0, removes the β0 channel; the virial closure at the bulge radius R then gives the CL rotation law. The BQ-Pauli version is the Maurer–Cartan route. Choose the rapidities.
Units: c = 1, rs = 2GM/c² = 1. Paper chart: χr = ψN − ψH = √(rs/r) − Hr/c, so vr = c tanh χr ≈ √(2GM/r) − Hr in slow flow. The galaxy-scale preset takes M ≈ 10¹¹ M⊙, R ≈ 2 kpc and a local Hz chosen so that rc ≈ 20R.
Weyl representation, following de Haas (2026), A Spin(1,3) Rapidity Theory for Galactic Rotation Curves, §4–5, lifted from §3. Orthonormal legs βr, βθ, βφ = βI, βJ, βK; αx = β0βx; Σθ = βφβr (cyclic); β̃θ = χβθ. For a stationary, axisymmetric field D/Dτ = ur∂r with ur = c sinh χr.
Rotor, Maurer–Cartan connection, transport product and the rotation law (eqs. 43, 67–96) from de Haas (2026), A Spin(1,3) Rapidity Theory for Galactic Rotation Curves, lifted to the Dirac level; the BQ-Pauli version is the Maurer–Cartan route, the kinematic route is step 11. Every identity is checked exactly in bq_dirac_mc_connection.py. Part of the BQ algebra at the Dirac level.