Theorem: Scalar relation tests

The calibrated scalar assignments

descend exactly when

The permitted real set is empty, a singleton, or all of $\mathbb{R}$.

Proof. The descent criterion gives precisely those linear relation equations, using injectivity of the remaining sharp argument. Each equation is affine in $c$. If two distinct real values solve every equation, their difference forces $\beta =0$, and then $\alpha =0$; all real values solve. If there are not two solutions, the solution set is empty or a singleton.