Proposition: Joint admission through second order

The bare coefficients $F_{c_0},F_{c_1},F_{c_2}$, regarded as polynomial functionals of $(h,k,\phi )$, have compact derivative kernels of split type. Consequently they satisfy (eq:cone).

Proof. The zeroth coefficient has a smooth derivative. The first is bilinear in $h,\phi$; its mixed kernel is a derivative of

By (eq:Lambda), this is a smooth multiplier times a derivative of a translation kernel, hence split.

The second coefficient is linear in $\phi$, quadratic in $h$, and linear in $k$. Its highest kernels are generated by:

with fixed tensors, further derivatives, and permutations of background slots suppressed. The $k$-linear term is the corresponding two-slot kernel.

For the first expression, the two coordinate factors share only $x$. For the local quadratic expression, the two delta factors share only $w$, and the resulting distribution meets the Green kernel only at $w$. In the nested expression, attach the delta factor to the inner coordinate kernel at $w$, then attach the outer Green kernel at that same single slot. Each step is covered by Lemma lem:split-calculus. No square of a propagator on identified endpoints has been introduced.

Integrate the internal variable $w$. Retarded support and the factors ${\sigma }_{\epsilon }$, together with the compact observation support, confine every relevant vertex to ${\mathcal{D}}_f$. Thus the pushforwards are proper; a smooth compact cutoff equal to one there can be inserted if desired. Lower derivatives are obtained by smearing remaining slots against smooth configurations, with the same compact localization.

All kernels are therefore split and compact. The flat ramp introduces no distributional boundary multiplier at $t=0$. Lemma lem:split-cone proves the claimed admission.