A compact regular comparison algebra

Let ${\mathcal{P}}_{\mathrm{reg}}$ be the algebra of finite polynomials in compactly smeared linear variables

Declare $M_a$ central and use, through first order,

All pairings exist on this regular algebra. This is a bosonic comparison subalgebra; it neither discards interacting ghosts nor represents the native nonlocal lapse automatically FR.

For notational clarity, $A_{\xi }W$ means that the differential operator $A_{\xi }$ acts in the first field slot. Formal adjoints refer to $(\cdot ,\cdot {)}_0$.