{"paper":{"title":"From 12 to 6: Sharpening the Three-Charge Bound in Maxwell's Problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"A.Gabrielov, B.Shapiro, Dm.Novikov, T.Novikov","submitted_at":"2026-07-30T19:17:04Z","abstract_excerpt":"In \\cite{GNS} we proved that, for every $\\alpha>0$, the potential of three positive point charges has at most $12$ nondegenerate equilibrium points. We also observed that the same method would give the sharper bound $6$ if a certain auxiliary polynomial system $Q=R=0$ had at least four solutions, counted with multiplicity, in each open quadrant of the $(f,g)$-plane. Here we prove this four-solution statement. The main new ingredient is a separation argument at the unique saddle point of a separated-variable first integral. Consequently, the upper bound for three charges improves from $12$ to $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28785","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.28785/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}