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  "problem_id": "vibemathed:minimal-distance-problem-sharp-exponent",
  "title": "The Minimal Distance Problem",
  "canonical_statement": "How well separated can a family of point-line pairs in the unit square be? For every $\\varepsilon > 0$ there are arbitrarily large families $(x_1,\\ell_1),\\ldots,(x_n,\\ell_n)$ in $[0,1]^2$ with $x_i \\in \\ell_i$ and $\\mathrm{dist}(x_i,\\ell_j) \\ge n^{-2/3-\\varepsilon}$ for all $i \\ne j$. Combined with earlier work of Cohen, Pohoata and Zakharov this settles the problem at the sharp exponent $2/3$. The same construction disproves a conjecture of Hunter, Pohoata, Verstraete and Zhang about induced point-line matchings over finite fields.",
  "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.",
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      "type": "proved",
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      "occurred_at": "2026-07-22T00:00:00.000Z",
      "title": "arXiv:2607.20422 - The sharp exponent for the minimal distance problem",
      "summary": "also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang",
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      "attempted_at": "2026-07-22T00:00:00.000Z",
      "human_collaborators": [
        "Cosmin Pohoata"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The acknowledgement draws the line precisely. The author's own plan was to use a high-degree number field analogue of the Hunter-Pohoata-Verstraete-Zhang construction to reach the Ruzsa endpoint. In his words, the decisive new idea of using the codimension-one, square-difference-free, trace-zero lattice in place of a Ruzsa-like set, which is what upgrades the exponent to the sharp one, is entirely due to GPT-5.6 Pro.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
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