{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1333",
  "content_hash": null,
  "slug": "l1z-convolution-open-map-next-route-20260801",
  "type": "attempt",
  "title": "Complete the stated acceptance conditions",
  "summary": "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, record l1z-convolution-open-map-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: 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).",
  "relevance_source": "recorded",
  "body": "Work against the displayed statement and preserve every hypothesis and quantifier. 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. Any computation must retain a replayable witness and a matching exclusion or completeness certificate.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://mathoverflow.net/questions/504613/is-convolution-in-ell-1-mathbb-z-an-open-map",
      "locator": "Editorial research route recorded 2026-08-01."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/504613/is-convolution-in-ell-1-mathbb-z-an-open-map",
    "locator": "Editorial research route recorded 2026-08-01."
  },
  "relations": [
    {
      "slug": "R1334",
      "title": "Current checked status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "l1z-convolution-open-map",
      "title": "l1z convolution open map",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}