Theorem: Compactly tested nonlinear non-descent

For every fixed $m\ge 0$, the nonlinear rule

does not preserve ${\mathcal{I}}_{\mathrm{sol}}[[\kappa ]]$. It therefore does not descend to the unchanged unrestricted classical free-shell quotient.

A compactly smeared scalar equation detects the failure exactly at order $\kappa$.

Proof. Choose $(t_{\ast },{\mathbf{x}}_{\ast })\in U$ and a sufficiently small product neighborhood inside $U$. Define

Thus $u_m^{\prime \prime }+m^2{u}_m=0$, and $u_m^{\prime }>0$ on a sufficiently small time interval about $t_{\ast }$.

Take the Grassmann algebra with one odd generator $\vartheta$, and the smooth free configuration

All equations (eq:freeeq) hold.

Choose a nonnegative $f\in {C_c}^{\infty }(U)$, supported in that neighborhood, normalized by

The functional

belongs to ${\mathcal{I}}_{\mathrm{eq}}$. On the displayed free configuration,

and therefore

It follows that

Thus $s{F}_f\notin {\mathcal{I}}_{\mathrm{sol}}[[\kappa ]]$.