Certificates of Global Optimality for Thomson Configurations

Definitions and mathematical statement

Let $N$ be a positive integer and let \[ S^2=\{x\in{\mathbb{R}}^3:\lVert x\rVert_2=1\},\qquad \mathcal C_N=\{(x_1,\ldots,x_N)\in(S^2)^N:x_i\ne x_j\text{ whenever }i\ne j\}. \] For $X=(x_1,\ldots,x_N)\in\mathcal C_N$, define the Thomson energy by \[ E_{\mathrm T}(X)=\sum_{1\le i<j\le N}\frac{1}{\lVert x_i-x_j\rVert_2}. \] Distances are Euclidean chordal distances on the unit sphere. For $N=1$, the sum is empty and equals zero.

Problem. Develop rigorous certificates establishing global optimality of candidate Thomson configurations. More precisely, given $N$ and a candidate $X^*\in\mathcal C_N$, seek a certificate proving \[ \forall X\in\mathcal C_N,\qquad E_{\mathrm T}(X)\ge E_{\mathrm T}(X^*). \] The requested certificate must establish the inequality over the entire configuration space, rather than merely local minimality or numerical evidence. The question does not assert that every candidate is optimal, and the supplied record specifies neither a certificate format nor a terminating certification algorithm.

The source also requests corresponding certificates for Riesz-energy variants. If an inverse-power variant with specified exponent $s$ is intended, its unnormalized energy has the form \[ E_s(X)=\sum_{1\le i<j\le N}\lVert x_i-x_j\rVert_2^{-s}, \] and the corresponding target is \[ \forall X\in\mathcal C_N,\qquad E_s(X)\ge E_s(X^*). \] However, the supplied record does not specify the permitted exponents, normalization, treatment of logarithmic limiting energies, or whether any sphere other than $S^2$ is intended. Thus this inverse-power formula describes a conditional interpretation; the full scope of the Riesz extension remains unresolved.

Short English statement

For a proposed configuration of $N$ distinct points on the unit sphere, develop a rigorous certificate that its total reciprocal-distance energy is no larger than that of any competing configuration. The source also asks for Riesz-energy analogues, but leaves their precise parameter range and conventions unspecified.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30003437$, source problem code catalogue entry, entitled Certificates of Global Optimality for Thomson Configurations; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied.
• Supplied report citation: Real Algebraic Geometry With a View Toward Moment Problems and Optimization (2018), Oberwolfach Reports, pp. 771--862, DOI 10.4171/owr/2017/14. https://doi.org/10.4171/owr/2017/14. The record gives proposed year 2017, while its citation labels the report 2018; these supplied dates have not been reconciled.
• Supplied literature reference: de Laat, moment methods in energy minimization. https://arxiv.org/abs/1503.07228.
• Supplied literature reference: Oberwolfach energy-minimization report. https://doi.org/10.4171/owr/2017/14.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that semidefinite hierarchies and high-precision dual certificates prove or strongly certify several small-particle cases, while a general method proving global optimality for arbitrary Thomson or Riesz configurations is not known. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied editorial label does not establish mathematical correctness or independent source verification.