Prove the exact factor for three-dimensional E1s63 extensions

Prove that every rigidity-preserving 1-extension of type E1s63 applied to a minimally 3-rigid graph multiplies the number of complex three-dimensional realizations by exactly two.

Background

Randomized Gröbner-basis computations show a factor of two for every tested E1s63 extension on graphs with at most nine vertices. The paper formulates the observed pattern as a conjecture, reflecting the lack of a general combinatorial counting method in dimension three.

References

The extensions of type~E1s63 indeed did increase the number of realizations always by a factor of two in our experiments. \begin{conjecture} Let $G$ be a minimally 3-rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E1s63. Then $\frac{\lambda_3(G')}{\lambda_3(G)}=2$. \end{conjecture}

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 8, subsection “Space,” subsection “Extension constructions”