{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
  "title": "WOWII Conjecture 72: Two induced trees pin down tree($  G  $)",
  "canonical_statement": "For a connected graph $  G  $, let $  t=  $ tree($  G  $) (order of a largest induced tree), $  A=  $ average eccentricity, and $  L=  $ maximum independence number of a neighbourhood. Then\n$   \\lceil (A+L)/3 \\rceil \\le t.   $\n(The evenly-divided reading of the conjecture holds; a stronger reading that divides only $  L  $ by three is false.)",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:event",
      "problem_id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-07-23T00:00:00.000Z",
      "title": "X post by @4thRT (with attached diagram and ChatGPT conversation link)",
      "summary": "The evenly-divided reading of WOWII Conjecture 72 holds:\n$   \\lceil(A + L)/3\\rceil \\le t,   $\nwhere $  t =  $ tree($  G  $) (order of a largest induced tree), $  A =  $ average eccentricity and $  L =  $ maximum neighbourhood independence number.\nA stronger reading that divides only $  L  $ by three is false. The argument rests on two elementary observations (a diametral path is chordless and therefore induces a tree on $  D+1  $ vertices; a maximum independent set in a neighbourhood induces a star on $  L+1  $ vertices). The original conjecture’s precise wording is not yet pinned in a public formal repository, so statement fidelity remains to be audited.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:attempt"
      ],
      "method_family_ids": [
        "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:method"
      ],
      "source_assertion_ids": [
        "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:attempt",
      "problem_id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
      "model": "GPT 5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-07-23T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The model was given a short prompt (copied from a successful earlier run on a different WOWII conjecture) and asked to find a solution or counterexample to an open conjecture of its choosing. It selected WOWII Conjecture 72, produced the two-lemma argument (diametral path induces a tree on $  D+1  $ vertices; maximum independent neighbourhood induces a star on $  L+1  $ vertices), derived the bound $  \\lceil(A+L)/3\\rceil\\le t  $, and explicitly distinguished the evenly-divided reading (true) from the stronger reading that divides only $  L  $ by three (false). The human then posted the diagram and commentary.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "X post by @4thRT (with attached diagram and ChatGPT conversation link)",
          "url": "https://x.com/4thRT/status/2080412225232400539",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: WOWII Conjecture 72: Two induced trees pin down tree($  G  $)",
          "url": "https://vibemathed.com/problem/wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:verification",
      "problem_id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
      "solution_event_id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Re-derived in full by this site on 17 August 2026 - the argument is elementary and correct, and short enough to state: a shortest path between two vertices at maximum distance D is induced, so it induces a path (a tree) on D+1 vertices, giving tree(G) >= D+1 >= A+1 since average eccentricity is at most D; a maximum independent set in a neighbourhood plus its centre induces a star on L+1 vertices, giving tree(G) >= L+1; hence A + L <= 2*tree(G) - 2, and ceil((A+L)/3) <= tree(G) follows by integrality. The stronger reading (dividing only L by three) fails on the claimed counterexample family. What keeps this Candidate is not the mathematics but the statement: Conjecture 72's canonical wording is not publicly pinned (no formal statement exists in the Formal Conjectures repository), so which reading DeLaViña intended is unconfirmed, and the X post plus transcript is the only artifact. Site-confirmed records this site's independent re-derivation of the proved reading.",
      "sources": [
        {
          "label": "VibeMathed: WOWII Conjecture 72: Two induced trees pin down tree($  G  $)",
          "url": "https://vibemathed.com/problem/wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "X post by @4thRT (with attached diagram and ChatGPT conversation link)",
          "url": "https://x.com/4thRT/status/2080412225232400539",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "X post by @4thRT (with attached diagram and ChatGPT conversation link)",
      "url": "https://x.com/4thRT/status/2080412225232400539",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: WOWII Conjecture 72: Two induced trees pin down tree($  G  $)",
      "url": "https://vibemathed.com/problem/wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g:signal:verify_now",
      "problem_id": "vibemathed:wowii-conjecture-72-two-induced-trees-pin-down-tree-g",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "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"
}