First derivatives through second order

For the dynamical evaluation, the explicit formulas (eq:scalarsolution) define a polynomial extension to arbitrary smooth ${\varphi }_0$. Its first derivatives also have smooth compact representatives: differentiate the displayed finite compositions, transpose each ${\mathsf{E}}_m$, and integrate the local differential operators by parts. Every dynamical vertex carries the compact interior factor $g$; coordinate transpositions are those already proved. Restriction to the free solution subspace is therefore legitimate. This does not assert a unique kernel for variations parametrized instead by initial Cauchy data.

Smooth first derivatives at fixed configuration do not prove a joint wavefront condition for every higher derivative, a quantum admission theorem, or existence of any singular contraction.