Fixed inputs and the shrinking ramp

Let $m\ge 0$, and fix smooth real symmetric tensors $h,k$ on a neighbourhood of the closed slab. They have no imposed field equations or spatial decay. Set

The coupling is formal. The limit concerns its coefficient tests, not a numerical perturbation series.

Fix

Retain the same nonempty relatively compact open plateau $U$ and algebra

The group ${\mathcal{G}}_p$ is generated through order ${\kappa }^2$ by flows of $\kappa \xi +{\kappa }^2\zeta$, with $\xi ,\zeta \in {\mathfrak{g}}_p$.

Choose $\delta >0$, once and for all, such that

Only $0<\epsilon <\delta$ will be considered. Thus the same group lies above every ramp in this tail of the family. It is not enlarged as the width shrinks.

Use the predecessor profile

The proof also applies to any other fixed smooth profile with the same constant extensions and endpoint values. Constants may depend on that profile. No uniform theorem for arbitrary $\epsilon$-dependent profiles is asserted.