Bimetric Gravity from the Cartan Conformal Connection in the Geometric Algebra Cl(4, 2)

We show that the Cartan conformal connection in the geometric algebra Cl(4, 2)
naturally encodes a bimetric gravitational theory. The 15-parameter connection 1-
form decomposes into a Lorentz spin connection (6 parameters), two independent
vierbeins (4+4 parameters) identied with the translational and special conformal
gauge elds, and a dilatation scalar (1 parameter). The two vierbeins dene two
Lorentzian metrics on a single manifold, coupled through torsion constraints. The
curvature of the conformal connection, projected onto the two null directions of R4,2 ,
yields a pair of Einstein-like eld equations with a characteristic sign asymmetry:
the cross-sector source terms enter with opposite signs in the two equations. This
asymmetry is not postulated but arises geometrically from the conformal inversion
I : n ↔ n̄, a discrete Z2 element of O(4, 2) not connected to the identity. The
vanishing of the dilatation curvature imposes a dynamical constraint linking the two
metrics through the dilatation eld. The resulting eld equations coincide with those
of the bimetric cosmological model of Petit et al., providing an algebraic foundation
for that framework and connecting it to the well-established gauge theory of gravity
programme of Lasenby, Doran, and Gull. We discuss the relation to the bimetric
theories of Damour, Hossenfelder, and Kogan, and argue that the conformal algebraic
formulation is more restrictive: the sign asymmetry and the inter-metric coupling
are derived, not chosen.