Coupling expansion versus a scalar background

The metric equations in this definition are unswitched at the retained order; the scalar equation is switched. On $P_g$ the scalar operator equals the unswitched curved operator, so the tuple satisfies the Einstein–scalar equations there through the stated orders.

This definition does not assert a nonlinear completion of the mixed switched system outside the plateau, nor does it construct a switched, gauge-fixed gravitational action. Those are separate requirements for a later quantum model.