{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:double-covers-discrete-box",
  "title": "New Bounds for Double Covers of the Discrete Box",
  "canonical_statement": "For $A=\\{0,1,2\\}^d$, write $f(d)$ for the fewest proper sub-boxes covering every point exactly twice. Leader, Miličević and Tan asked whether $f(d)\\ge 2^d$ for all $d$, as Question 4.1 of the PatternBoost paper. The paper gives new bounds on $f(d)$.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:double-covers-discrete-box:event",
      "problem_id": "vibemathed:double-covers-discrete-box",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-10T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Improved bounds rather than a settled question: the asked-for inequality is not established in general.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:double-covers-discrete-box:attempt"
      ],
      "method_family_ids": [
        "vibemathed:double-covers-discrete-box:method"
      ],
      "source_assertion_ids": [
        "vibemathed:double-covers-discrete-box:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:double-covers-discrete-box:attempt",
      "problem_id": "vibemathed:double-covers-discrete-box",
      "model": "Claude",
      "provider": "Anthropic",
      "attempted_at": "2026-07-10T00:00:00.000Z",
      "human_collaborators": [
        "Patrick White"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The author credits large language models used as reasoning engines with producing several of the key ideas, naming the modular refinement of Section 3, the crossing lemma of Section 5 and the constructions of Section 7.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.09014",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: New Bounds for Double Covers of the Discrete Box",
          "url": "https://vibemathed.com/problem/double-covers-discrete-box",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:double-covers-discrete-box:verification",
      "problem_id": "vibemathed:double-covers-discrete-box",
      "solution_event_id": "vibemathed:double-covers-discrete-box:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No independent check. The disclosure is unusually specific about which sections the model produced. Preprint, not refereed.",
      "sources": [
        {
          "label": "VibeMathed: New Bounds for Double Covers of the Discrete Box",
          "url": "https://vibemathed.com/problem/double-covers-discrete-box",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.09014",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2607.09014",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: New Bounds for Double Covers of the Discrete Box",
      "url": "https://vibemathed.com/problem/double-covers-discrete-box",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:double-covers-discrete-box:signal:recent_partial_result",
      "problem_id": "vibemathed:double-covers-discrete-box",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}