{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:YDE5DRN757BN4GNKENPWNAPKII","short_pith_number":"pith:YDE5DRN7","schema_version":"1.0","canonical_sha256":"c0c9d1c5bfefc2de19aa235f6681ea421bff3bbe85ff20bd4a943bf66c5f2f25","source":{"kind":"arxiv","id":"2607.28785","version":1},"attestation_state":"computed","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 $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.28785","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-30T19:17:04Z","cross_cats_sorted":[],"title_canon_sha256":"7ce203421b9857bfed8a9f0be99673d204b15aee085b4ccc9d8d7ea6045be092","abstract_canon_sha256":"e055d823fbd9d1aec2d63ba8db3ef1063220a8424e33027af2298f1c1375d2ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-03T00:12:52.578428Z","signature_b64":"0eFpNtruke6WLWqNE/5R/zUs+0WBIHKHJQMHF4VhYGz/DYbSidT0gPM0U//0IDRbr8Oe7bRVTyeVVmyc3QzTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0c9d1c5bfefc2de19aa235f6681ea421bff3bbe85ff20bd4a943bf66c5f2f25","last_reissued_at":"2026-08-03T00:12:52.576173Z","signature_status":"signed_v1","first_computed_at":"2026-08-03T00:12:52.576173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.28785","created_at":"2026-08-03T00:12:52.577719+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.28785v1","created_at":"2026-08-03T00:12:52.577719+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28785","created_at":"2026-08-03T00:12:52.577719+00:00"},{"alias_kind":"pith_short_12","alias_value":"YDE5DRN757BN","created_at":"2026-08-03T00:12:52.577719+00:00"},{"alias_kind":"pith_short_16","alias_value":"YDE5DRN757BN4GNK","created_at":"2026-08-03T00:12:52.577719+00:00"},{"alias_kind":"pith_short_8","alias_value":"YDE5DRN7","created_at":"2026-08-03T00:12:52.577719+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII","json":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII.json","graph_json":"https://pith.science/api/pith-number/YDE5DRN757BN4GNKENPWNAPKII/graph.json","events_json":"https://pith.science/api/pith-number/YDE5DRN757BN4GNKENPWNAPKII/events.json","paper":"https://pith.science/paper/YDE5DRN7"},"agent_actions":{"view_html":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII","download_json":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII.json","view_paper":"https://pith.science/paper/YDE5DRN7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.28785&json=true","fetch_graph":"https://pith.science/api/pith-number/YDE5DRN757BN4GNKENPWNAPKII/graph.json","fetch_events":"https://pith.science/api/pith-number/YDE5DRN757BN4GNKENPWNAPKII/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII/action/storage_attestation","attest_author":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII/action/author_attestation","sign_citation":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII/action/citation_signature","submit_replication":"https://pith.science/pith/YDE5DRN757BN4GNKENPWNAPKII/action/replication_record"}},"created_at":"2026-08-03T00:12:52.577719+00:00","updated_at":"2026-08-03T00:12:52.577719+00:00"}