Lemma: Dependence on the residual field

The map $h\mapsto L_h$ is a linear differential operator depending only on second derivatives of $h$. In particular, it is continuous on smooth configurations.

Proof. A cubic monomial of type $hh{\partial }^{2}h$ contributes only when its two undifferentiated factors are varied. Its remaining coefficient is linear in ${\partial }^{2}h$. A monomial of type $h\partial h\partial h$ has only one undifferentiated factor. Varying twice in constant directions therefore gives zero. This proves the assertion directly from the total derivative count.