import numberdb.sage as numberdb
from itertools import combinations
from sage.rings.real_arb import RealBallField
R = RealBallField(numberdb.bits(50, losing=64))
def coulomb_energy(points):
# Project the supplied nonzero Cartesian vectors to the unit sphere.
unit = []
for point in points:
v = [R(x) for x in point]
norm = sum(x*x for x in v).sqrt()
unit.append([x/norm for x in v])
return sum((1 / sum((x-y)**2 for x,y in zip(a,b)).sqrt()
for a,b in combinations(unit, 2)), R(0))
coulomb_energy([(0, 0, 1), (0, 0, -1)]) # 1/2
The values for $n=3,4,5,6,12$ are computed from exact radical formulas in Sage real ball arithmetic at 397 bits and stored to 100 significant digits; $E_2=1/2$ is exact. Each formula was independently compared with a pair-distance sum on exact coordinates at 729 bits. The 397-bit results retained at least 393 relative accuracy bits. The four closed-form rows covered by Deeter's table also agree with its printed decimals.
For the remaining rows, Deeter's 13-significant-digit energies are rounded to 12 significant digits. Every source configuration for $4\leq n\leq100$ was projected radially to the unit sphere before recomputing its energy. Pair-distance sums at 264 bits agree with inner-product sums at 363 bits to better than $10^{-50}$ in absolute value. All 93 stored numerical values contain their recomputed coordinate energies within one unit in the last printed place; the largest discrepancy is less than $0.629$ of that unit. The 91 configurations with $10\leq n\leq100$ also agree with the Cambridge table within its printed last place.
The supplied coordinates are rounded: their largest squared-norm error is less than $1.582\cdot10^{-12}$. At $n=22,44$, projecting them to the unit sphere moves the computed energy outside the interval implied by Deeter's final printed digit, by discrepancies of $1.076$ and $1.281$ units of that last place. This is why the numerical rows use 12 significant digits. These comparisons check numerical agreement, not global optimality or a rigorous error bound for an unknown minimiser. The convexity bound and radius scaling were also checked throughout their applicable stored ranges.
The numerical rows use Wayne Deeter's table and Cartesian coordinates [4], retrieved on 25-Oct-2026. Its index had HTTP Last-Modified date 10-Oct-2026. The Cambridge Cluster Database [3], which cites Deeter's updates, was retrieved on the same day and had Last-Modified date 13-Aug-2026. The attached source snapshot preserves the energies and coordinates used here.