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 independentvierbeins (4+4 parameters) identied with the translational and special conformalgauge elds, and a dilatation scalar (1 parameter). The two vierbeins dene twoLorentzian metrics on a single manifold, coupled through torsion constraints. Thecurvature 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. Thisasymmetry is not postulated but arises geometrically from the conformal inversionI : n ↔ n̄, a discrete Z2 element of O(4, 2) not connected to the identity. Thevanishing of the dilatation curvature imposes a dynamical constraint linking the twometrics through the dilatation eld. The resulting eld equations coincide with thoseof the bimetric cosmological model of Petit et al., providing an algebraic foundationfor that framework and connecting it to the well-established gauge theory of gravityprogramme of Lasenby, Doran, and Gull. We discuss the relation to the bimetrictheories of Damour, Hossenfelder, and Kogan, and argue that the conformal algebraicformulation is more restrictive: the sign asymmetry and the inter-metric couplingare derived, not chosen.