Thomson $N=7$: tube rigidity of the pentagonal bipyramid for $\tau\le 1/10$

Let $P=(P_0,\dots,P_6)$ be the pentagonal bipyramid with the regular pentagon $P_k=(\cos\tfrac{2\pi k}5,\sin\tfrac{2\pi k}5,0)$ at indices $k=0,\dots,4$ and the poles $(0,0,\pm1)$ at indices $5,6$; let $\varphi(t)=(2-2t)^{-1/2}$, so that $\|x-y\|^{-1}=\varphi(\langle x,y\rangle)$ for unit vectors, and write $g_{ij}=\langle P_i,P_j\rangle$ and $E(y)=\sum_{i<j}\|y_i-y_j\|^{-1}$. Let $c_1=\cos\tfrac{2\pi}5$ and $c_2=\cos\tfrac{4\pi}5$.

Let $\tau\le\tfrac1{10}$ and let $y_0,\dots,y_6$ be unit vectors in $\mathbb R^3$ such that

1. $|\langle y_0,y_1\rangle+1|\le\tau$;
2. $|\langle y_0,y_r\rangle|\le\tau$ and $|\langle y_1,y_r\rangle|\le\tau$ for $r=2,\dots,6$;
3. for $2\le r<r'\le6$, $|\langle y_r,y_{r'}\rangle-c_1|\le\tau$ or $|\langle y_r,y_{r'}\rangle-c_2|\le\tau$.

Then there is a permutation $\sigma$ of $\{0,\dots,6\}$ with

The point is combinatorial: colouring the edges of $K_5$ on the five ring points by "near $c_1$"/"near $c_2$", no triangle can be monochromatic (a Gram-determinant argument), and the only triangle-free 2-colourings of $K_5$ are a pentagon/pentagram pair, which matches the ring of $P$ after relabelling.

Formalization Note All named objects (pentBipyramid, coulombEnergy, Base.phi, Reg.W, Reg.muP, Reg.gP, Reg.tau, Reg.Dpair, Reg.Qhess, Reg.Pen, Glue.TubeRigid) come from the platform definition ThomsonN7_core, taken verbatim from the source; R3 is EuclideanSpace ℝ (Fin 3).