Remark: Separate independence does not give joint independence (audit 3)

Linear independence of $\{{{\sigma }_{\epsilon }}^{(k)}{\}}_k$ on an interval and of $\{{\partial }^{\rho }{\chi }_{\mathrm{I}}{\}}_{\rho }$ on an open set does not imply linear independence of the products $\{{{\sigma }_{\epsilon }}^{(k)}(t){\partial }^{\rho }{\chi }_{\mathrm{I}}(x){\}}_{k,\rho }$ on $U$. Counterexample: if locally in the slab ${\chi }_{\mathrm{I}}(x)={\sigma }_{\epsilon }(t)$ (a switch that is a function of time on $U$), then

on $U$: a nontrivial linear relation between the products ${{\sigma }_{\epsilon }}^{(1)}\cdot {\chi }_{\mathrm{I}}$ ($k=1$, $\rho =0$) and ${{\sigma }_{\epsilon }}^{(0)}\cdot {\partial }_0{\chi }_{\mathrm{I}}$ ($k=0$, $\rho =e_0$), although $\{{\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime }\}$ is independent (Lemma lem:to-indep) and so is $\{{\chi }_{\mathrm{I}},{\partial }_0{\chi }_{\mathrm{I}}\}$. The sufficient test above is therefore stated for the joint family. The actual-ramp transfer of Proposition prop:to-actual is unaffected: with a universal switch the jets of ${\chi }_{\mathrm{I}}$ at a point can be prescribed independently of ${\sigma }_{\epsilon }$, and on $\{{\chi }_{\mathrm{I}}=1{\}}^{\circ }$ only $\rho =0$ occurs.