{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras",
  "title": "Non-MF groups and non-finite full group C*-algebras",
  "canonical_statement": "Let $\\Gamma$ be a property (T) group admitting an injective, non-surjective endomorphism and let $G$ be the associated ascending HNN-extension. Let\n$$\nW=\\left(\\bigoplus_{G/\\Gamma}\\mathbb Z/2\\mathbb Z\\right)\\rtimes G.\n$$\nWe show that $W$ is not an MF group and that $C^*(G)$ is not a finite $C^*$-algebra. The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments in a hopefully more palatable form.",
  "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:non-mf-groups-and-non-finite-full-group-c-algebras:event",
      "problem_id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-28T00:00:00.000Z",
      "title": "arXiv",
      "summary": "If $\\Gamma$ is a non-coHopfian property-(T) group and $G$ is its ascending HNN extension, then the full group algebra $C^*(G)$ is not finite. Indeed, Kazhdan projections $p_H<p_\\Gamma$ become unitarily equivalent under the stable letter, which cannot occur inside a finite $C^*$-algebra.\n\nFor\n$$\nW=\\left(\\bigoplus_{G/\\Gamma}\\mathbb Z/2\\mathbb Z\\right)\\rtimes G,\n$$\nthe paper proves more strongly that every homomorphism\n$$\nW\\to U\\!\\left(\\prod M_{d_n}/\\bigoplus M_{d_n}\\right)\n$$\nkills an explicit nonidentity element $b_\\gamma$. Hence $W$ is not MF. Consequently $C_r^*(W)$ is an explicit stably finite but non-MF $C^*$-algebra.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras:attempt"
      ],
      "method_family_ids": [
        "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras:method"
      ],
      "source_assertion_ids": [
        "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras:attempt",
      "problem_id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras",
      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-28T00:00:00.000Z",
      "human_collaborators": [
        "Caleb Eckhardt"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Caleb Eckhardt states that the mathematical ideas and proofs were generated by ChatGPT 5.6 Sol. He prompted Sol to seek constructions starting from the non-sofic examples of Kun and Thom, themselves building on recent OpenAI examples. Sol developed the core arguments, including the property-(T)/Kazhdan-projection mechanism and a workaround using a rescaled Hilbert-Schmidt representation and a nontrivial $1$-cocycle to prove the generalized wreath-product group is non-MF. Eckhardt internalized, refined, and rewrote the arguments.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.28772",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Non-MF groups and non-finite full group C*-algebras",
          "url": "https://vibemathed.com/problem/non-mf-groups-and-non-finite-full-group-c-algebras",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras:verification",
      "problem_id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras",
      "solution_event_id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Unreviewed. arXiv 2608.28772 (seven pages) read here: \"The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments\"; Eckhardt internalised and rewrote them and takes responsibility. Willett, Fournier-Facio, Dogon and Shulman are thanked for input and one consequence, which is comment rather than a check of the complete proof; not refereed; Sauer's Lean work on related examples is separate.",
      "sources": [
        {
          "label": "VibeMathed: Non-MF groups and non-finite full group C*-algebras",
          "url": "https://vibemathed.com/problem/non-mf-groups-and-non-finite-full-group-c-algebras",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.28772",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.28772",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Non-MF groups and non-finite full group C*-algebras",
      "url": "https://vibemathed.com/problem/non-mf-groups-and-non-finite-full-group-c-algebras",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras:signal:watch_only",
      "problem_id": "vibemathed:non-mf-groups-and-non-finite-full-group-c-algebras",
      "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"
}