{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:kothe-conjecture",
  "title": "Köthe Conjecture",
  "canonical_statement": "Köthe's conjecture asks whether the sum of two nil left ideals of a ring is always nil. Equivalently, by Krempa's 1972 formulation, if $I$ is a nil two-sided ideal of a ring $R$, then the matrix ideal $M_n(I)$ should be nil for every finite $n$, already for $n=2$.\n\nThe conjecture is false: there exists a ring $R$, a nil two-sided ideal $I\\subseteq R$, and a $2\\times 2$ matrix with entries in $I$ that is not nilpotent.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:kothe-conjecture:event",
      "problem_id": "vibemathed:kothe-conjecture",
      "type": "disproved",
      "status": "candidate",
      "occurred_at": "2026-09-03T00:00:00.000Z",
      "title": "Github",
      "summary": "GPT-6 Astra constructs a unital algebra $R=k\\oplus A$ over the countable field\n$$\nk=\\overline{\\mathbb F_2},\n$$\nwith $I=A$ a nil two-sided ideal, together with a matrix\n$$\nW\\in M_2(I)\n$$\nthat is not nilpotent.\n\nThe algebra $A$ is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of $A$ is nilpotent. At the same time, a suitable polynomial combination of the shifts fixes a nonzero vector; this yields a companion-type matrix with a nonzero eigenvalue, and hence a nonnilpotent matrix whose entries lie in $I$.\n\nThis formally disproves Krempa's matrix formulation of Köthe's conjecture.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:kothe-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:kothe-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:kothe-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:kothe-conjecture:attempt",
      "problem_id": "vibemathed:kothe-conjecture",
      "model": "GPT-6 Astra (pre-release)",
      "provider": "OpenAI",
      "attempted_at": "2026-09-03T00:00:00.000Z",
      "human_collaborators": [
        "Tom Adamczewski"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A pre-release GPT-6 Astra autonomously attempted the open Formal Conjectures benchmark statement, with no human steering during the run, and constructed both the mathematical counterexample and its Lean proof. It builds a nil algebra from three weighted backward shifts over $\\overline{\\mathbb F_2}$, arranges a universal mortality property for all elements, and simultaneously constructs a $2\\times2$ matrix over the resulting nil ideal that has a nonzero eigenvalue and hence is not nilpotent. Claude was later used to generate repository documentation from the completed proof; it was not the mathematical solver.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "X Announcement",
          "url": "https://x.com/tmkadamcz/status/2095785941193404457",
          "kind": "discussion",
          "license": null
        },
        {
          "label": "Challenge.lean: the compared statement, copied from Formal Conjectures",
          "url": "https://github.com/tadamcz/koethe/blob/main/Challenge.lean",
          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "Solution.lean and the Lean development",
          "url": "https://github.com/tadamcz/koethe/blob/main/Solution.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Formal Conjectures source statement (Google DeepMind)",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/Koethe.lean",
          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "Epoch AI LeanOpenProblems, the evaluation the run was part of",
          "url": "https://github.com/epoch-research/LeanOpenProblems",
          "kind": "other",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/koethe",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Köthe Conjecture",
          "url": "https://vibemathed.com/problem/kothe-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": 432,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:kothe-conjecture:verification",
      "problem_id": "vibemathed:kothe-conjecture",
      "solution_event_id": "vibemathed:kothe-conjecture:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean-verified on this site's ladder: kernel-checked, and the statement was written independently of the prover. Checked here on 5 September 2026 from a clone at b052755: 3,331 lines of Lean, zero `sorry` outside the Challenge.lean stub, zero `axiom` declarations, no `native_decide`, `unsafe` or `implemented_by` (two uses of `decide` on small numerals), mathlib pinned, Comparator in CI with the three standard axioms. The compared statement is byte-identical to Formal Conjectures' `KotherConjecture.variants.general_matrix` at 9cbe1d3c, and its one nontrivial ingredient, mathlib's `TwoSidedIdeal.matrix`, is the ideal of matrices whose every entry lies in I, so the statement is Krempa's matrix form as intended. What the kernel has certified is therefore: a ring with a nil ideal I and a non-nilpotent matrix in M_2(I). That Köthe's original statement implies the matrix form is an elementary argument (M_n(I) is a sum of n nil column left ideals) stated in the repository and not formalized; Formal Conjectures opened a PR on 4 September relating the formulations. Candidate because no ring theorist has read it yet, and the machine-generated proof account is unaudited.",
      "sources": [
        {
          "label": "VibeMathed: Köthe Conjecture",
          "url": "https://vibemathed.com/problem/kothe-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/koethe",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Github",
      "url": "https://github.com/tadamcz/koethe",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Köthe Conjecture",
      "url": "https://vibemathed.com/problem/kothe-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:kothe-conjecture:signal:verify_now",
      "problem_id": "vibemathed:kothe-conjecture",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "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"
}