Exact factor for three-dimensional E1s63 extensions

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

Background

In three dimensions, the authors investigate selected rigidity-preserving extension subclasses using probabilistic Gröbner-basis computations. Type E1s63 extensions doubled the realization count in all experiments through nine vertices, leading to an explicit conjecture for arbitrary minimally 3-rigid graphs.

References

\begin{conjecture} Let $G$ be a minimally 3-rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E1s63. Then $\frac{3{G'}{3{G}=2$. \end{conjecture}

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 7.3, subsection “Space,” immediately before the E1s63 conjecture

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 Conjecture, Section 8.3.1, Extension constructions