Proposition: Algebraic extension equivalence

The following are equivalent:

1. a preliminary linear lift exists;

2. every $m^{\circ }(k)$, $k\in {\mathcal{K}}_{\mathrm{int}}$, has some extension in $V$;

3. there exists a linear $\mathsf{M}:{\mathcal{K}}_{\mathrm{int}}\to V$ with $r\mathsf{M}=m^{\circ }$.

This equivalence imposes no continuity, locality, normalization, or wavefront condition.

Proof. For (i)$\Rightarrow$(ii), choose $A$ with $\mathsf{Rec}A=k$ and use $F_0(A)$.

For (ii)$\Rightarrow$(iii), choose a Hamel basis of ${\mathcal{K}}_{\mathrm{int}}$, choose an extension for the canonical value of each basis element, and extend linearly. Every vector is a finite linear combination, so the resulting assignment is a linear map with the required restriction.

For (iii)$\Rightarrow$(i), take $F_0=\mathsf{M}\mathsf{Rec}$.