{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1160",
  "content_hash": null,
  "slug": "support-three-zero-divisor-f2-group-ring-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over \\(\\mathbb F_2\\) as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources. For a positive answer, give a finite or recursive presentation of a torsion-free group \\(G\\), explicit finite supports and coefficients for nonzero \\(\\alpha,\\beta\\in\\mathbb F_2[G]\\), and verify \\(\\alpha\\beta=0\\) in the group algebra.",
  "relevance": "Defines the dated frontier and the exact remainder that research on this problem must resolve.",
  "relevance_source": "recorded",
  "body": "UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over \\(\\mathbb F_2\\) as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.\n\nA complete resolution must satisfy this condition: For a positive answer, give a finite or recursive presentation of a torsion-free group \\(G\\), explicit finite supports and coefficients for nonzero \\(\\alpha,\\beta\\in\\mathbb F_2[G]\\), and verify \\(\\alpha\\beta=0\\) in the group algebra.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/62548/zero-divisor-conjecture-for-finite-fields",
      "locator": "See dataset.references[0] for the exact external source and locator."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/62548/zero-divisor-conjecture-for-finite-fields",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "relations": [
    {
      "slug": "R1159",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "support-three-zero-divisor-f2-group-ring",
      "title": "support three zero divisor f2 group ring",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}