{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees",
  "title": "Odifreddi's Problem 3 on Irreducible m-Degrees",
  "canonical_statement": "Odifreddi asked, as Problem 3 in his surveys \"Strong Reducibilities\" (1981) and \"Reducibilities\" (1999), whether every computably enumerable $tt$-degree contains a c.e. irreducible $m$-degree, meaning an $m$-degree consisting of a single $1$-degree. Answered negatively: there is a c.e. $tt$-degree containing no c.e. irreducible $m$-degree. This also shows Jockusch's 1969 theorem, which produces an irreducible $m$-degree inside every c.e. $tt$-degree, is strictly optimal and cannot be strengthened to make that degree c.e.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees:event",
      "problem_id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-05-04T00:00:00.000Z",
      "title": "arXiv:2605.03066 - A Computably Enumerable tt-Degree Without Computably Enumerable Irreducible m-Degrees",
      "summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:odifreddi-problem-3-irreducible-m-degrees:attempt"
      ],
      "method_family_ids": [
        "vibemathed:odifreddi-problem-3-irreducible-m-degrees:method"
      ],
      "source_assertion_ids": [
        "vibemathed:odifreddi-problem-3-irreducible-m-degrees:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees:attempt",
      "problem_id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees",
      "model": "Gemini Deep Think",
      "provider": "Google DeepMind",
      "attempted_at": "2026-05-04T00:00:00.000Z",
      "human_collaborators": [
        "Patrizio Cintioli"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Credited at the level of the paper rather than the lemma. The author describes the work as the result of an extended human-AI interaction in which several structural ideas and technical arguments emerged from exploratory sessions with Gemini Deep Think, after which he fully reworked and verified all arguments and takes sole responsibility for their correctness. Nothing is attributed step by step, so the contribution is real but unitemised.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.03066 - A Computably Enumerable tt-Degree Without Computably Enumerable Irreducible m-Degrees",
          "url": "https://arxiv.org/abs/2605.03066",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Odifreddi's Problem 3 on Irreducible m-Degrees",
          "url": "https://vibemathed.com/problem/odifreddi-problem-3-irreducible-m-degrees",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees:verification",
      "problem_id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees",
      "solution_event_id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A five-page arXiv preprint, not peer-reviewed. The construction rests on Degtev's c.e. semirecursive sets with rigid complement, so it is short enough to check by hand, but no independent check is on record.",
      "sources": [
        {
          "label": "VibeMathed: Odifreddi's Problem 3 on Irreducible m-Degrees",
          "url": "https://vibemathed.com/problem/odifreddi-problem-3-irreducible-m-degrees",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.03066 - A Computably Enumerable tt-Degree Without Computably Enumerable Irreducible m-Degrees",
          "url": "https://arxiv.org/abs/2605.03066",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2605.03066 - A Computably Enumerable tt-Degree Without Computably Enumerable Irreducible m-Degrees",
      "url": "https://arxiv.org/abs/2605.03066",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Odifreddi's Problem 3 on Irreducible m-Degrees",
      "url": "https://vibemathed.com/problem/odifreddi-problem-3-irreducible-m-degrees",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees:signal:watch_only",
      "problem_id": "vibemathed:odifreddi-problem-3-irreducible-m-degrees",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}