{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:measures-on-partial-orders",
  "title": "Measures on Partial Orders",
  "canonical_statement": "We determine the measures (in the sense of Harman–Snowden) on the Fraïssé class of partially ordered sets: the space of measures is a union of a plane, eight lines, and 15 isolated points. This is the first case where the space is not equidimensional, and the first primitive case in which it has dimension at least two.",
  "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:measures-on-partial-orders:event",
      "problem_id": "vibemathed:measures-on-partial-orders",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-09-01T00:00:00.000Z",
      "title": "arXiv",
      "summary": "For the Fraïssé class $\\mathfrak P$ of finite partially ordered sets, every measure $\\nu$ is completely determined by the nine values\n$$\n\\nu_{p,q}=\\nu(J_{p,q},*),\\qquad 0\\le p,q\\le2.\n$$\nThe possible $3\\times3$ matrices $(\\nu_{p,q})$ are classified explicitly. Consequently, the reduced space of all measures is exactly the union of one plane, eight lines, and 15 isolated points, with the plane and three lines meeting at one point and no other intersections.\n\nThe paper also completely classifies weak measures and supports, proves there is no regular measure on all finite posets, classifies characteristic-zero measures for the automorphism group of the universal homogeneous poset, and determines the Knop-like measures.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:measures-on-partial-orders:attempt"
      ],
      "method_family_ids": [
        "vibemathed:measures-on-partial-orders:method"
      ],
      "source_assertion_ids": [
        "vibemathed:measures-on-partial-orders:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:measures-on-partial-orders:attempt",
      "problem_id": "vibemathed:measures-on-partial-orders",
      "model": "ChatGPT 5.6 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-09-01T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Andrew Snowden states that ChatGPT Pro 5.6 produced most of the mathematical arguments. It first computationally discovered the correct classification by generating and solving the defining equations for posets of size at most seven. It then found the key universal formulas involving twins and cuts, proved that the resulting functions are weak measures, supplied existence proofs for several families of measures, identified reductions between the remaining cases, discovered the stabilization relation needed to show that a 3×3 matrix determines every measure, and assisted with many of the paper's additional results. Snowden wrote the paper himself and checked/reworked the arguments.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.02021v1",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Measures on Partial Orders",
          "url": "https://vibemathed.com/problem/measures-on-partial-orders",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:measures-on-partial-orders:verification",
      "problem_id": "vibemathed:measures-on-partial-orders",
      "solution_event_id": "vibemathed:measures-on-partial-orders:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The paper contains complete conventional mathematical proofs written and checked by Andrew Snowden. ChatGPT generated most of the mathematical arguments, but Snowden is also the author publishing the result, so this is author verification rather than independent expert verification. Johannes Flake is thanked for helpful discussions, but the paper does not state that he or another external expert independently verified the proofs. No formal proof-assistant verification or referee report is reported.",
      "sources": [
        {
          "label": "VibeMathed: Measures on Partial Orders",
          "url": "https://vibemathed.com/problem/measures-on-partial-orders",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.02021v1",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2609.02021v1",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Measures on Partial Orders",
      "url": "https://vibemathed.com/problem/measures-on-partial-orders",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:measures-on-partial-orders:signal:watch_only",
      "problem_id": "vibemathed:measures-on-partial-orders",
      "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"
}