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  "problem_id": "vibemathed:maxwell-three-charge-bound",
  "title": "Maxwell's Three-Charge Equilibrium Bound",
  "canonical_statement": "How many nondegenerate equilibrium points can the potential of three positive point charges have? Gabrielov, Novikov and Shapiro had proved at most $12$, and observed that their method would give $6$ if an auxiliary polynomial system had at least four solutions with multiplicity in each open quadrant. That four-solution statement holds, so the bound is $6$, and six is attained for special charge values.",
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      "occurred_at": "2026-07-30T00:00:00.000Z",
      "title": "arXiv:2607.28785 - From 12 to 6: Sharpening the Three-Charge Bound in Maxwell's Problem",
      "summary": "the sharp bound for three charges; the general Maxwell bound was separately disproved in July 2026",
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      "attempted_at": "2026-07-30T00:00:00.000Z",
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        "Dmitry Novikov",
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      "outcome": "The acknowledgement states that parts of the work, including the saddle-separation argument and the symbolic and numerical verification, were developed with the assistance of Claude, and that all results were independently verified by the authors. The abstract calls that separation argument at the unique saddle point of a separated-variable first integral the main new ingredient, so the model's contribution reaches the load-bearing step.",
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