Corollary: Evaluation of the parent-parameter test

For the actual presentation maps of this manuscript,

For every fixed real $c$, the calibrated left and right scalar maps exist uniquely as algebraic bilinear maps:

Proof. Theorem thm:qinjective makes the two relation kernels zero. The sharp generating map is injective by the inherited premise. Multilinearity of distributional pairing therefore defines unique bilinear maps on the actual images. Constants belong to ${\mathcal{P}}_{\Sigma }$.