{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1627",
  "content_hash": null,
  "slug": "l1z-convolution-open-map-status-packet-quality-20260801",
  "type": "claim",
  "title": "Dated status and exact unresolved remainder",
  "summary": "Unresolved in this packet after the dated source check. Strongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair. Exact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.",
  "relevance": "For Openness of convolution on l1 of the integers, this successor gives readable dated status prose and the exact remaining research boundary.",
  "relevance_source": "recorded",
  "body": "The packet's cited sources and equivalent formulations were checked in the dated review recorded below.\n\nStrongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair.\n\nExact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1112/jlms/jdt004",
      "locator": "main norm-controlled inversion inequalities for smooth Banach subalgebras"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1112/jlms/jdt004",
    "locator": "main norm-controlled inversion inequalities for smooth Banach subalgebras"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1334",
      "title": "Current checked status and unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing",
      "metadata": {
        "reason": "Replaces unreadable status prose with the dated review from 2026-08-01."
      }
    },
    {
      "slug": "l1z-convolution-open-map",
      "title": "l1z convolution open map",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}