{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R940",
  "content_hash": null,
  "slug": "cmon-cocongruences-effective-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-31. A recent answer proves effectivity when X admits a positive degree map to the natural numbers, including free commutative monoids. Its comments continue to clarify details, and the unrestricted case remains unresolved on the page. Prove that the canonical map X amalgam_U X ->Y is an isomorphism for every cocongruence p,q, or give explicit finitely presented commutative monoids and maps forming a cocongruence for which the map fails to be an isomorphism.",
  "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. A recent answer proves effectivity when X admits a positive degree map to the natural numbers, including free commutative monoids. Its comments continue to clarify details, and the unrestricted case remains unresolved on the page.\n\nA complete resolution must satisfy this condition: Prove that the canonical map X amalgam_U X ->Y is an isomorphism for every cocongruence p,q, or give explicit finitely presented commutative monoids and maps forming a cocongruence for which the map fails to be an isomorphism.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/511792/is-every-cocongruence-in-mathbfcmon-effective",
      "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/511792/is-every-cocongruence-in-mathbfcmon-effective",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "relations": [
    {
      "slug": "R939",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "cmon-cocongruences-effective",
      "title": "cmon cocongruences effective",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}