Switch, ramp, and gauge algebra

Fix $m\ge 0$, $0<\epsilon <T_{\ast }$, and

For the supported realization, the retained interaction support is in the laboratory. Contributions outside a given $D_f$ will in any event not enter its evaluation.

Define

It is smooth and flat at both endpoints; ${\sigma }^{\prime }(t)>0$ on $0<t<\epsilon$, and $\sigma =1$ for $t\ge \epsilon$.