Operator support weight is invariant under wire permutation

Put $S_{\sigma }:={\sigma }^{-1}(S)$, the target positions occupied by source factors from $S$. Let ${\alpha }_S$ reorder those source factors into their target order, and let ${\alpha }_{S^c}$ reorder the complementary factors likewise; these are wire bijections by [F2]. On each elementary tensor, both ${\mathrm{\Pi }}_{S_{\sigma }}^{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}{\mathrm{\Pi }}_{\sigma }$ and $({\alpha }_S\otimes {\alpha }_{S^c}){\mathrm{\Pi }}_S^{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}$ place each factor vector in the same target order. Elementary tensors span by [F4], and the maps agree on them by [F1], so ${\mathrm{\Pi }}_{S_{\sigma }}^{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}{\mathrm{\Pi }}_{\sigma }=({\alpha }_S\otimes {\alpha }_{S^c}){\mathrm{\Pi }}_S^{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}$.

If $E$ is supported in $S$, [L1] gives $A_S$ with ${\mathrm{\Pi }}_S^{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}E({\mathrm{\Pi }}_S^{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}{)}^{-1}=A_S\otimes I$. Conjugating by step 1.1 gives ${\mathrm{\Pi }}_{S_{\sigma }}^{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}{E}^{\mathrm{\prime }}({\mathrm{\Pi }}_{S_{\sigma }}^{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}{)}^{-1}=({\alpha }_S\otimes {\alpha }_{S^c})(A_S\otimes I)({\alpha }_S\otimes {\alpha }_{S^c}{)}^{-1}$. On $y\otimes z$ the right side is ${\alpha }_S{A}_S{\alpha }_S^{-1}y\otimes z$, hence equals $({\alpha }_S{A}_S{\alpha }_S^{-1})\otimes I$ by [F3]; thus $E^{\mathrm{\prime }}$ is supported in $S_{\sigma }$.

Apply the same implication to $E^{\mathrm{\prime }}$ and the inverse wire permutation. It gives the converse: if $E^{\mathrm{\prime }}$ is supported in $S_{\sigma }$, then $E$ is supported in $S$. Thus $S\mapsto {\sigma }^{-1}(S)$ bijects the support sets of $E$ and $E^{\mathrm{\prime }}$, preserving cardinality; their minimum support sizes agree.