{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:daykin-frankl-conjecture",
  "title": "The Daykin–Frankl conjecture on convex subsets of the Boolean lattice",
  "canonical_statement": "In 1983, Daykin and Frankl conjectured that if $P$ is a convex subset of $Q_n$, then it contains at least\n$$\n|P|\\binom{n}{\\lfloor n/2\\rfloor}2^{-n}\n$$\npairwise incomparable elements. We verify and communicate an LLM-generated proof of this conjecture.",
  "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:daykin-frankl-conjecture:event",
      "problem_id": "vibemathed:daykin-frankl-conjecture",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-09-02T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Let $P\\subseteq Q_n$ be convex. Williams proves the stronger statement that for every $k\\ge0$,\n$$\nw(P\\times Q_k)\n\\ge\nw(Q_{n+k})\\,|P|\\,2^{-n},\n$$\nwhere $w$ denotes poset width.\n\nTaking $k=0$ gives\n$$\nw(P)\\ge\n|P|\\binom{n}{\\lfloor n/2\\rfloor}2^{-n},\n$$\nwhich is exactly the Daykin-Frankl conjecture.\n\nThe proof proceeds by induction on $n$, reducing the step to a structural lemma for a convex subset of $R\\times Q_1$ and carefully recombining antichains from its two layers.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:daykin-frankl-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:daykin-frankl-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:daykin-frankl-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:daykin-frankl-conjecture:attempt",
      "problem_id": "vibemathed:daykin-frankl-conjecture",
      "model": "GPT-5.6 Sol Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-09-02T00:00:00.000Z",
      "human_collaborators": [
        "Kada Williams"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Kada Williams explicitly credits ChatGPT 5.6 Sol Pro with generating the proof content of the note. The proof establishes a stronger product inequality for convex subsets of Boolean lattices and derives the original Daykin-Frankl conjecture as the case $k=0$. Williams verifies, writes up, and takes responsibility for communicating the argument.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.03087",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Daykin–Frankl conjecture on convex subsets of the Boolean lattice",
          "url": "https://vibemathed.com/problem/daykin-frankl-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:daykin-frankl-conjecture:verification",
      "problem_id": "vibemathed:daykin-frankl-conjecture",
      "solution_event_id": "vibemathed:daykin-frankl-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Unreviewed. arXiv 2609.03087 (four pages) read here: the note describes itself as verifying and communicating an LLM-generated proof, credits ChatGPT 5.6 Sol Pro, and gives the induction on dimension with the R x Q_1 convexity lemma in full. Checked by the human author, not by anyone independent; not peer reviewed; no formalization.",
      "sources": [
        {
          "label": "VibeMathed: The Daykin–Frankl conjecture on convex subsets of the Boolean lattice",
          "url": "https://vibemathed.com/problem/daykin-frankl-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.03087",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2609.03087",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Daykin–Frankl conjecture on convex subsets of the Boolean lattice",
      "url": "https://vibemathed.com/problem/daykin-frankl-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:daykin-frankl-conjecture:signal:watch_only",
      "problem_id": "vibemathed:daykin-frankl-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"
}