A sufficient finite-to-global criterion

No such injectivity is established for the present three labels.

Injectivity is a sufficient route, not a necessary condition for one particular identity to be true. A direct global algebraic proof can establish that identity even when the chosen evaluation is not injective.

For example, with $X=\mathbb{R}$,

evaluation is not injective on $U=\mathrm{span}\{1,t\}$, but $m=n_1$ is a global identity. The correct warning is that a finite fit alone does not provide such a proof.