Coulomb energy and spherical configurations

Thomson problem: general case

Collaboration beta

For every number of identical charges on a sphere, determine the least Coulomb energy and every configuration attaining it. The source develops general reductions and sharp restricted seven-point comparisons, while the unrestricted seven-point and all-cardinality targets remain open.

E_{N} = \min_{X \in (S^{2})^{N}} \sum_{i < j} | x_{i} - x_{j} |^{- 1}, argmin E / (O (3) \times S_{N})

Known results and sources

Illustration: Charges lie on the unit sphere and interact through inverse distance. The arrangement is illustrative.

Research problem

Exact mathematical statement

For every integer $N\ge1$ , let $X=(x_1,…,x_N)\in(S^2)^N$ , where $S^2\subset\mathbb R^3$ is the unit sphere. Define the unordered Coulomb pair energy

E(X)=\sum_{1\le i<j\le N}\frac1{\|x_i-x_j\|},\qquad E_N=\min_{X\in(S^2)^N}E(X),

with collisions assigned energy $+\infty$ . Determine $E_N$ and all classes

\{X:E(X)=E_N\}/\bigl(O(3)\times\mathfrak S_N\bigr).

Reflections and permutations are included. A stationary configuration, a strict local minimum, a symmetry-restricted solution, or an asymptotic distribution does not determine every global minimizing shape. The cases $N=1,2$ are separate from geometric results requiring $N\ge3$ .

Problem infographic

Problem at a glance

Illustration: Seven illustrative points on a unit sphere accompany the Coulomb energy formula and the two questions of the minimum value and all minimizing shapes for arbitrary N. — Illustration: The general Thomson problem asks for both the least Coulomb energy and every minimizing shape, up to orthogonal transformations and relabelling.

Current mathematical picture

Where work on Thomson problem: general case stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureUsing separate pair moments as full geometric realizability

This is a formal pair measure rather than a realizable seven-point configuration. It refutes the relaxation that forgets shared geometry, not Coulomb optimality or the rank-constrained H7 target. Retain rank, shared inner products, conditioned contact remainders, and genuine multi-point constraints.

Route status · Narrowed route

Main reductionCurrent reduction

General geometric and algebraic reductions preserve exact realizability and the physical branch. For N=7, the certified antipodal, near-antipodal, and affine-five-point classes reduce any non-bipyramid configuration with E≤e_B to the stated residual domain. It must be handled for both lower and tied energies; stationary-only restrictions do not apply to every sublevel point.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeExclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.Task status · Ready to work on

Work mapped so far

Thomson problem: general case in numbers

1.2kretained lines of mathematical investigation1,160 in the current working snapshot

Argument development: 642 · 55%
Explored or eliminated routes: 73 · 6%
Computational analysis: 142 · 12%
Open obligations: 135 · 12%
Definitions and setup: 168 · 14%

9selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byJ. J. ThomsonProposed1904Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Exclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.

Suggested move: Suggested next step: Prove the selected-pair comparison on 1/400<s<√(27/200) with the precise distance and moment restrictions, and test any truncation against the source's actual geometric failed-bound witness.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeUsing separate pair moments as full geometric realizability

Route status · Narrowed route

Narrowed routeDropping the contact remainder from the selected-pair identity

The source encloses L between −5237/10^6 and −5236/10^6 while E−e_B lies between 4667/100000 and 4668/100000; the omitted positive remainder is essential. Retain enough contact remainder, improve the conditioned kernel, or prove stronger geometric constraints; the existing farthest-pair and coarse restrictions do not fix the truncation.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Exclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.Suggested move: Suggested next step: Prove the selected-pair comparison on 1/400<s<√(27/200) with the precise distance and moment restrictions, and test any truncation against the source's actual geometric failed-bound witness.

Ready to work on

Exclude or classify residual seven-point minimizers tying e_B outside the pentagonal-bipyramid orbit.Suggested move: Suggested next step: After optimality of the value is proved, retain the complete O(3)-and-permutation orbit-defect condition on the equality fiber and derive an exact real infeasibility certificate or identify every additional tied orbit.

Ready to work on

Construct an exact global comparison that preserves shared geometry for the polynomial, rational, or algebraic route.Suggested move: Suggested next step: Choose the rank-at-most-three H7 target or a concrete real certificate problem, prove chart coverage and physical feasibility, and keep finite-shape assumptions separate from finite stationary-energy values.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

For arbitrary N, the selected sources leave open the global minimum of the pairwise Coulomb energy Σ(i<j) 1/|xᵢ−xⱼ| for distinct points on the unit sphere. Solved small cases and numerical configurations do not provide an all-N classification.

[1][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025-06-10
PreprintAmore and collaborators numerically explored minima for up to 150 charges and stationary configurations for up to 24. These searches do not classify all global minima.[3]
2013
Peer reviewedSchwartz proved, by a rigorous computer-assisted argument, that the triangular bipyramid is the minimizing five-point configuration for Coulomb energy on the sphere.[1][2]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma4 of 94
equivalence1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementWhich sphere configurations minimize Coulomb energy for each number of charges?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementUnique minimum within each Möbius classintermediate
retained route statementEnergy values and shape certificates are distinctintermediate
retained route statementFinite rational sharpness requires finite shape classesintermediate
retained route statementSharp antipodal and near-antipodal comparisonintermediate
retained route statementAffine five-point and great-circle restrictionsintermediate
retained route statementResidual lower and tied energies both remainintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureUsing separate pair moments as full geometric realizabilityreported failure
Useful failureDropping the contact remainder from the selected-pair identityreported failure
Research targetExclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.open
Research targetExclude or classify residual seven-point minimizers tying e_B outside the pentagonal-bipyramid orbit.open
Research targetConstruct an exact global comparison that preserves shared geometry for the polynomial, rational, or algebraic route.open
Narrowed routeUsing separate pair moments as full geometric realizabilityThis is a formal pair measure rather than a realizable seven-point configuration. It refutes the relaxation that forgets shared geometry, not Coulomb optimality or the rank-constrained H7 target. Retain rank, shared inner products, conditioned contact remainders, and genuine multi-point constraints.
Narrowed routeDropping the contact remainder from the selected-pair identityThe source encloses L between −5237/10^6 and −5236/10^6 while E−e_B lies between 4667/100000 and 4668/100000; the omitted positive remainder is essential. Retain enough contact remainder, improve the conditioned kernel, or prove stronger geometric constraints; the existing farthest-pair and coarse restrictions do not fix the truncation.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeExclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointExclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.

Thomson problem: general case · ready to start

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Sources and references5 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
The 5-Electron Case of Thomson's Problempeer reviewed result · Richard Evan Schwartz · Experimental Mathematics 22(2), 157–186; author manuscript · 2013 · accessed Sep 7, 2026
2
Energy suite: spherical energy minimization papers and programsauthoritative webpage · Richard Evan Schwartz · accessed Sep 7, 2026
3
Exploring the energy landscape of the Thomson problem: local minima and stationary statespreprint · Paolo Amore, Victor Figueroa, Enrique Diaz, Jorge A. López, Trevor Vincent · arXiv · 2025-06-10 · ARXIV 2506.08398 · accessed Sep 7, 2026
4
Lowest minima located for the Thomson problemsoftware or dataset · Cambridge Cluster Database · accessed Sep 7, 2026
5
Mathematical Problems for the Next Century, Problem 7original source · Steve Smale · The Mathematical Intelligencer 20, 7–15 · 1998 · DOI 10.1007/BF03025291 · accessed Sep 7, 2026

Important qualifications

This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.
The 1904 attribution is reported by a later primary research paper; Thomson's original 1904 publication was not inspected.
No claim about all seven-charge configurations, lower-energy alternatives or ties is imported from the private packet. Later unpublished claims on Schwartz's broader energy page are not treated as the published five-electron Coulomb result.

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Thomson problem: general case

Exact mathematical statement

Problem at a glance

Where work on Thomson problem: general case stands

Thomson problem: general case in numbers

Exclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

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