Exact factor for spherical E1a extensions

Prove that every 1-extension of type E1a applied to a minimally rigid graph multiplies the number of complex spherical realizations by exactly two.

Background

The plane and sphere have the same class of minimally rigid graphs, but their realization counts differ. Computations show that spherical E1a extensions have multiplier two through 13 vertices, motivating the authors’ conjecture that this holds for all minimally rigid graphs.

References

\begin{conjecture} Let $G$ be a minimally rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E1a. Then $\frac{2{G'}{2{G}=2$. \end{conjecture}

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 7.2, subsection “Sphere,” immediately before the spherical E1a conjecture