Theorem: Injective partial-contraction presentations

Both $q_L:E_L\to {\mathcal{L}}_2$ and $q_R:E_R\to {\mathcal{R}}_2$ are algebraic isomorphisms onto their images.

Proof. Equation (eq:qrelation) and Lemma lem:primitive give $vK=0$. If $v$ were nonzero at a point, it would have a smooth reciprocal near that point, making $K=0$ on a nonempty open set. Lemma lem:fullsupport excludes this. Hence $v=0$, including as an algebraic tensor. Conjugation and variable exchange give the right case.