Theorem: One matched direction is not whole-algebra closure

The difference $I_{\mathrm{loc}}$ is not invariant under all directions of ${\mathfrak{g}}_U$. Indeed,

More generally, if a $C^1$ function $F:{\mathbb{R}}^2\to V$, with finite-dimensional real target $V$, has zero variation under every relative direction at every ${\varphi }_{\mathbf{b}}$, then $F$ is constant.

Proof. The first assertion follows from (eq:chargematrix). For the second, the two directions give

The connectedness of ${\mathbb{R}}^2$ proves constancy.