Configurations in the Euclidean space related to the 3D genome reconstruction problem from partially phased data

A motivation for studying the following problems comes from applications to Biology; see \cite{cifuentes20233d}. In the $3$-dimensional Euclidean space ${\bf{E}}^3$, fix six pairwise distinct points \begin{equation*} \label{eqA} \begin{array}{ccc} A=(a_1,a_2,a_3), & B=(b_1,b_2,b_3), & C=(c_1,c_2,c_3), \\ D=(d_1,d_2,d_3), & E=(e_1,e_2,e_3), & F=(f_1,f_2,f_3) \end{array} \end{equation*} together with two further points $X^*=(x_1^*,x_2^*,x_3^*)$ and $Y^*=(y_1^*,y_2^*,y_3^*)$ in $\mathbf{E}^3$. We aim to show that System $(*)$ consisting of the following six equations in the unknowns $X=(x_1,x_2,x_3)$ and $Y=(y_1,y_2,y_3)$ \begin{equation} \label{egy} \frac{1}{\|X-T\|^2} +\frac{1}{\|Y-T\|^2}=\frac{1}{\|X^*-T\|^2} +\frac{1}{\|Y^*-T\|^2}, \quad T\in\{A,B,\ldots,F\}. \end{equation} has only finitely many solutions provided that both of the following two conditions are satisfied: (i) no four of the fixed points $A,B,C,D,E,F$ are coplanar; (ii) no four of the six spheres of center $T$ and radius $1/\sqrt{k_T}$ with \begin{equation} \label{kxy} k_T=\frac{1}{\|X^*-T\|^2} +\frac{1}{\|Y^*-T\|^2} \end{equation} share a common point in ${\bf{E}}^3$. Furthermore, we exhibit configurations $ABCDEFX^*Y^*$, showing that (i) is also necessary. This result is an improvement on \cite[Theorem 1]{cifuentes20233d} where the finiteness of solutions of System $(*)$ is only ensured for sufficiently generic choices of the points $A,B,\ldots,F,X^*,Y^*.$ We also show if System $(*)$ has finitely many solutions and System $(*)$ extended with $T\in\{A,B,\ldots,F,G\}$ has some solutions other than $(X^*,Y^*)$ and $(Y^*,X^*)$ then $G$ lies on an explicitly given affine variety $W\subsetneqq \mathbb{R}^3$ only depending on $\{A,B,\ldots,F\}$. This result proves the \cite[Conjecture 1]{cifuentes20233d}.