{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1301",
  "content_hash": null,
  "slug": "exponential-square-free-group-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The sole answer proposes a consequence of Schanuel's conjecture. Comments identify gaps in how reduced words enter the argument and ask whether Ax-Schanuel yields an unconditional proof; the thread contains no settled unconditional resolution.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. The MathOverflow question, answer, and all comments were checked on 2026-07-27. The visible argument is conditional on Schanuel's conjecture, and commenters dispute whether its algebraic-independence step handles every reduced word. Ax's 1971 theorem on Schanuel-type differential results was checked because the thread asks whether it applies. The audit did not find a published specialization proving this exact homeomorphism-group statement. Searches for the two generators, their inverse maps, and free subgroups of Homeo_+(R) found general constructions but no direct unconditional treatment of this pair. A local TheoremDB search for exponential-square generators and the exact functional words found no duplicate.\n\nA complete resolution must satisfy: Prove that every nonempty freely reduced word in f^{+/-1} and g^{+/-1} differs from the identity at some x>0, or exhibit a nonempty freely reduced word that is the identity on all x>0. A conditional proof must state its hypothesis explicitly and does not settle the unconditional target.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/499819/do-x-mapsto-ex-1-and-x-mapsto-x2-generate-a-free-group-on-mathbbr-to",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/499819/do-x-mapsto-ex-1-and-x-mapsto-x2-generate-a-free-group-on-mathbbr-to",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "relations": [
    {
      "slug": "R1300",
      "title": "Complete the stated acceptance conditions",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "R1524",
      "title": "Dated status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming"
    },
    {
      "slug": "exponential-square-free-group",
      "title": "exponential square free group",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}