Find a lower-energy configuration for the Thomson problem with $N=200$ charges

The Thomson problem asks for the arrangement of $N$ unit point charges on the unit sphere $S^2$ minimizing the Coulomb energy $E = \sum_{i<j} \frac{1}{|x_i - x_j|}$, where the $x_i \in S^2$ and $|x_i-x_j|$ is the Euclidean (chord) distance. For $N=200$ the global minimum is not known -- only a best-known (putative) minimum-energy configuration is recorded. Exhibit a configuration of 200 points on $S^2$ with total Coulomb energy strictly below the current best-known value, or prove that a given configuration is the global minimum. The energy landscape has a number of local minima that grows roughly exponentially in $N$, so new putative global minima are still being discovered in this regime.

Acceptance. FULLY RESOLVES: a proof that a specific 200-point configuration is the global Coulomb-energy minimum. PARTIAL (improve record): a list of 200 unit vectors (coordinates on $S^2$) whose total energy $E = \sum_{i<j} 1/|x_i-x_j|$ is strictly below the current best-known value; verifier -- confirm each point is a unit vector and recompute $E$ directly (a $\binom{200}{2}$ sum), comparing to the reference best-known energy. PARTIAL: a certified lower bound on the minimal energy for $N=200$ approaching the best-known upper bound.