{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate",
  "title": "Nikolov-Ullman Pure-DP Query Release Conjecture",
  "canonical_statement": "Nikolov and Ullman asked, as Open Problem 1 on DifferentialPrivacy.org, whether $k$ statistical queries over a universe of size $T$ can be released under pure differential privacy at the square-root error rate that the known lower bounds suggest, rather than the cube-root rate of the classical small-database method. They can: for every $n$ and $\\varepsilon > 0$ there is an $\\varepsilon$-differentially private mechanism with expected error $O(\\min\\{1, \\sqrt{\\log(2T)\\log(2k)/(\\varepsilon n)}\\})$.",
  "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:nikolov-ullman-pure-dp-square-root-rate:event",
      "problem_id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-22T00:00:00.000Z",
      "title": "arXiv:2607.20418 - Pure-DP Statistical Query Release at the Conjectured Square-Root Rate",
      "summary": "information-theoretic; a polynomial-time implementation remains open",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:nikolov-ullman-pure-dp-square-root-rate:attempt"
      ],
      "method_family_ids": [
        "vibemathed:nikolov-ullman-pure-dp-square-root-rate:method"
      ],
      "source_assertion_ids": [
        "vibemathed:nikolov-ullman-pure-dp-square-root-rate:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate:attempt",
      "problem_id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate",
      "model": "Codex, Harmonic Aristotle",
      "provider": "OpenAI / Harmonic",
      "attempted_at": "2026-07-22T00:00:00.000Z",
      "human_collaborators": [
        "Jack Fitzsimons"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The generative-AI disclosure states that Codex and Aristotle were used in connection with Lean formalization and proof search, and that Codex also gave editorial feedback on clarity and organization. Proof search is a mathematical contribution, but the disclosure does not say which steps came from where.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.20418 - Pure-DP Statistical Query Release at the Conjectured Square-Root Rate",
          "url": "https://arxiv.org/abs/2607.20418",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Nikolov-Ullman Pure-DP Query Release Conjecture",
          "url": "https://vibemathed.com/problem/nikolov-ullman-pure-dp-square-root-rate",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate:verification",
      "problem_id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate",
      "solution_event_id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Single-author arXiv preprint with a companion Lean 4 development that the paper says machine-checks the finite construction, pure privacy after deterministic decoding, and the all-regimes upper bound, with an axiom audit and a paper-to-Lean crosswalk in the artifact. We have not compiled it. Not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Nikolov-Ullman Pure-DP Query Release Conjecture",
          "url": "https://vibemathed.com/problem/nikolov-ullman-pure-dp-square-root-rate",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.20418 - Pure-DP Statistical Query Release at the Conjectured Square-Root Rate",
          "url": "https://arxiv.org/abs/2607.20418",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.20418 - Pure-DP Statistical Query Release at the Conjectured Square-Root Rate",
      "url": "https://arxiv.org/abs/2607.20418",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Nikolov-Ullman Pure-DP Query Release Conjecture",
      "url": "https://vibemathed.com/problem/nikolov-ullman-pure-dp-square-root-rate",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate:signal:watch_only",
      "problem_id": "vibemathed:nikolov-ullman-pure-dp-square-root-rate",
      "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"
}