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  "problem_id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy",
  "title": "Polylogarithmic Full-Chord Buffon Discrepancy",
  "canonical_statement": "Steinerberger introduced the Buffon discrepancy problem, asking how accurately a one-dimensional set of length $L$ in a convex body $\\Omega$ can match the Crofton-predicted line-intersection counts, and proved an $O(L^{1/3})$ upper bound via a Steinhaus longimeter construction. His third open question asks whether restricting to sets built from full chords - intersections of lines with $\\Omega$, the class containing every Steinhaus set - fundamentally changes the problem.\n\nIt does. Using the Aistleitner-Bilyk-Nikolov star-discrepancy theorem for arbitrary measures, full-chord constructions with discrepancy $O\\left((\\log L)^{3/2}\\right)$ are shown to exist for every compact convex body with finite piecewise $C^2$ boundary. In the disk, every full-chord construction is shown to have discrepancy at least $\\Omega(\\log L)$, via Schmidt's two-dimensional rectangle lower bound - where Steinerberger's concentric-circle construction, which is not full-chord, achieves discrepancy at most $100$.",
  "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:polylogarithmic-full-chord-buffon-discrepancy:event",
      "problem_id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-05-18T00:00:00.000Z",
      "title": "arXiv",
      "summary": "This settles Steinerberger's third open question and separates the two models: in the disk, full chords cost you a factor growing like $\\log L$ over what is achievable without the restriction. It also improves the Steinhaus-type $O(L^{1/3})$ to polylogarithmic within the full-chord class.\n\nIt does not settle the Buffon discrepancy problem itself. Steinerberger's first question - whether every convex body admits a set of discrepancy $O(1)$, and if not what the truth is - is untouched, and the paper's closing line names it as the natural next question. Inside the full-chord model the order is pinned only between $\\Omega(\\log L)$ and $O\\left((\\log L)^{3/2}\\right)$, and the lower bound is proved for the disk alone. The paper says the exponents are unlikely to be sharp.\n\nThe upper bound is an existence statement: it inherits the Aistleitner-Bilyk-Nikolov theorem, which is proved by transference and supplies no explicit construction.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:polylogarithmic-full-chord-buffon-discrepancy:attempt"
      ],
      "method_family_ids": [
        "vibemathed:polylogarithmic-full-chord-buffon-discrepancy:method"
      ],
      "source_assertion_ids": [
        "vibemathed:polylogarithmic-full-chord-buffon-discrepancy:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy:attempt",
      "problem_id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy",
      "model": "GPT-5.5",
      "provider": "OpenAI",
      "attempted_at": "2026-05-18T00:00:00.000Z",
      "human_collaborators": [
        "Samuel Korsky"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The published acknowledgement credits GPT-5.5 with \"assistance in performing the detailed computations and preparing an initial draft of this preprint\", and reserves the proof idea and the direction of the argument for the author.\n\nThe author sharpened that for this record after review. He told the model that the Aistleitner-Bilyk-Nikolov star-discrepancy bound and Schmidt's rectangle lower bound could likely be applied in the weighted forms the paper needs, and the model did the technical work of making those intuitions work. Those weighted adaptations are named results in the note - the support form of Aistleitner-Bilyk-Nikolov, the weighted Schmidt rectangle lower bound, and the Hardy-Krause variation bound on the chord-length function that lets Koksma-Hlawka control the length. The strategy is the author's, the machinery that realizes it is the model's, which is what this site means by co-developed.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Steinerberger - Buffon discrepancy and the Steinhaus longimeter (the problem, open question 3)",
          "url": "https://arxiv.org/abs/2603.27807",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Korsky - Randomly shifted Steinhaus longimeters and Buffon discrepancy (the earlier attempt)",
          "url": "https://arxiv.org/abs/2605.10096",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Aistleitner, Bilyk, Nikolov - the arbitrary-measure star-discrepancy theorem behind the upper bound",
          "url": "https://arxiv.org/abs/1703.06127",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2605.23020",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Polylogarithmic Full-Chord Buffon Discrepancy",
          "url": "https://vibemathed.com/problem/polylogarithmic-full-chord-buffon-discrepancy",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "cost_usd": null,
      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy:verification",
      "problem_id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy",
      "solution_event_id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint by a single author, not refereed and not endorsed by anyone independent, so this stays Unreviewed.\n\nThis site checked the reduction the note rests on. Lemma 3.1 says the chords crossing a test line form a union of two rectangles in endpoint-pair space, of measure $2\\mathcal{H}^1(\\ell \\cap \\Omega)/\\Lambda_\\Omega$ - the identity that turns a Buffon problem into a two-dimensional rectangle discrepancy problem. It was confirmed exactly for the disk by quadrature at five arc widths (agreement to $10^{-9}$), and for an ellipse by sampling the kinematic measure in $(p,\\theta)$ coordinates, which know nothing about endpoint pairs, giving agreement within 0.2% and an implied $\\Lambda_\\Omega$ of 4.6012 against a perimeter of 4.6026. The Aistleitner-Bilyk-Nikolov bound is quoted faithfully: their $(\\log N)^{d-1/2}/N$ at $d=2$ is $(\\log N)^{3/2}/N$. An independent exact-supremum harness reproduces both known growth rates: $L^{0.289}$ for Steinhaus-type constructions against the proved $L^{1/3}$, and $L^{0.511}$ for i.i.d. chords against the square root.\n\nNeither theorem itself was checked. The upper bound rests on an existence result with no explicit construction, and the closest thing this site could build - a Halton set pushed through the Rosenblatt transform of $\\mu_\\Omega$ - fits $L^{0.346}$, no better than Steinhaus. That is a limitation of the proxy, not evidence against the theorem. The $\\Omega(\\log L)$ lower bound is below the resolution of any feasible experiment.",
      "sources": [
        {
          "label": "VibeMathed: Polylogarithmic Full-Chord Buffon Discrepancy",
          "url": "https://vibemathed.com/problem/polylogarithmic-full-chord-buffon-discrepancy",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2605.23020",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2605.23020",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Polylogarithmic Full-Chord Buffon Discrepancy",
      "url": "https://vibemathed.com/problem/polylogarithmic-full-chord-buffon-discrepancy",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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  "opportunity_signals": [
    {
      "id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy:signal:recent_partial_result",
      "problem_id": "vibemathed:polylogarithmic-full-chord-buffon-discrepancy",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
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  "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"
}