Polarization and operator audit

The degree-four polynomial has one scalar slot and three metric slots. Full coefficient recovery therefore uses one scalar derivative and three metric derivatives, with the symmetric metric factor $3!$.

In the slot-isolation calculation, setting $k_{00}=0$ leaves two surviving middle-slot permutations. This gives the exact factor $-2$ in (eq:isolation); it does not change the inherited two-edge assignment factor.

The identity

uses compact support away from the initial surface and uniqueness of the zero-data Cauchy problem. It neither assumes that $w$ and $h_{00}$ are independent fields nor imposes an equation of motion on the configuration.