Thomson $N=7$: the cubic remainders near the pentagonal bipyramid are $O(r)\sum\|y_i-P_i\|^2$

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 $E(y)=\sum_{i<j}\|y_i-y_j\|^{-1}$.

For $y\in(\mathbb R^3)^7$ write $h_i=y_i-P_i$, $\tau_{ij}=\langle y_i,y_j\rangle-g_{ij}$ and

If $0\le r\le\tfrac1{1000}$ and $\|h_i\|\le r$ for all $i$ (no unit-norm assumption), then

Since $\tau_{ij}=\langle P_i,h_j\rangle+\langle h_i,P_j\rangle+\langle h_i,h_j\rangle$, each $D_{ij}$ consists of terms of order at least three in $h$, and the bound makes this quantitative on the $r$-ball.

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).