{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:ellipsoid-fitting-conjecture",
  "title": "The Ellipsoid Fitting Conjecture",
  "canonical_statement": "Given $n$ independent standard Gaussian vectors in $\\mathbb{R}^d$, an ellipsoid fit is a positive semidefinite matrix $S$ with $x_i' S x_i = d$ for every $i$. Saunderson, Parrilo and Willsky conjectured that this semidefinite feasibility problem has a sharp threshold at $n \\sim \\frac{d^2}{4}$. Proved: below the threshold a fit exists with probability tending to one, above it none does.",
  "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:ellipsoid-fitting-conjecture:event",
      "problem_id": "vibemathed:ellipsoid-fitting-conjecture",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-10T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:ellipsoid-fitting-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:ellipsoid-fitting-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:ellipsoid-fitting-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:ellipsoid-fitting-conjecture:attempt",
      "problem_id": "vibemathed:ellipsoid-fitting-conjecture",
      "model": "GPT-5.6",
      "provider": "OpenAI",
      "attempted_at": "2026-08-10T00:00:00.000Z",
      "human_collaborators": [
        "Theodor Misiakiewicz",
        "Garrett G. Wen"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The approach is the authors' own - they say so, and trace it to the dual formulation of Bandeira and Maillard. What the model did is named step by step: ChatGPT 5.4 and 5.5 were used \"to explore several possible proof strategies\", and then, \"Given an earlier draft, GPT 5.6 helped repair and complete several arguments, including the tightened head-tail decomposition in Lemma 3.5 and the decomposition used in the proof of Proposition 4.4, which ultimately led to the completion of the proofs.\"",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10184",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Ellipsoid Fitting Conjecture",
          "url": "https://vibemathed.com/problem/ellipsoid-fitting-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:ellipsoid-fitting-conjecture:verification",
      "problem_id": "vibemathed:ellipsoid-fitting-conjecture",
      "solution_event_id": "vibemathed:ellipsoid-fitting-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint days old, with no independent review.",
      "sources": [
        {
          "label": "VibeMathed: The Ellipsoid Fitting Conjecture",
          "url": "https://vibemathed.com/problem/ellipsoid-fitting-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10184",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.10184",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Ellipsoid Fitting Conjecture",
      "url": "https://vibemathed.com/problem/ellipsoid-fitting-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:ellipsoid-fitting-conjecture:signal:watch_only",
      "problem_id": "vibemathed:ellipsoid-fitting-conjecture",
      "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"
}