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    "summary": "Every set of N distinct points in the Euclidean plane, for N at least two, determines at least cN/logN distinct nonzero distances for a universal positive constant c. The proof combines the Elekes-Sharir rigid-motion framework with three-dimensional line-incidence bounds, polynomial partitioning and ruled-surface geometry.",
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        "description": "The result obtains the sharp power exponent but leaves a logarithmic gap from the square-grid order N/sqrt(logN); it is not the exact planar extremal formula. Plane and regulus concentration conditions belong to the intermediate incidence theorem and cannot be omitted in an arbitrary line-arrangement claim.",
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      "statement": "# Guth and Katz: a near-linear lower bound for planar distinct distances\n\nEvery set of N distinct points in the Euclidean plane, for N at least two, determines at least cN/logN distinct nonzero distances for a universal positive constant c. The proof combines the Elekes-Sharir rigid-motion framework with three-dimensional line-incidence bounds, polynomial partitioning and ruled-surface geometry.\n\n## Scientific scope\n\nThe result obtains the sharp power exponent but leaves a logarithmic gap from the square-grid order N/sqrt(logN); it is not the exact planar extremal formula. Plane and regulus concentration conditions belong to the intermediate incidence theorem and cannot be omitted in an arbitrary line-arrangement claim.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nLarry Guth; Nets Hawk Katz\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [On the Erdos distinct distance problem in the plane](https://arxiv.org/abs/1011.4105v3)\n",
      "summary": "Every set of N distinct points in the Euclidean plane, for N at least two, determines at least cN/logN distinct nonzero distances for a universal positive constant c. The proof combines the Elekes-Sharir rigid-motion framework with three-dimensional line-incidence bounds, polynomial partitioning and ruled-surface geometry.",
      "title": "Guth and Katz: a near-linear lower bound for planar distinct distances"
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    "title": "Guth and Katz: a near-linear lower bound for planar distinct distances",
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    "what_changed": "Every set of N distinct points in the Euclidean plane, for N at least two, determines at least cN/logN distinct nonzero distances for a universal positive constant c. The proof combines the Elekes-Sharir rigid-motion framework with three-dimensional line-incidence bounds, polynomial partitioning and ruled-surface geometry.",
    "scope": "Every set of N distinct points in the Euclidean plane, for N at least two, determines at least cN/logN distinct nonzero distances for a universal positive constant c. The proof combines the Elekes-Sharir rigid-motion framework with three-dimensional line-incidence bounds, polynomial partitioning and ruled-surface geometry. The result obtains the sharp power exponent but leaves a logarithmic gap from the square-grid order N/sqrt(logN); it is not the exact planar extremal formula. Plane and regulus concentration conditions belong to the intermediate incidence theorem and cannot be omitted in an arbitrary line-arrangement claim.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The result obtains the sharp power exponent but leaves a logarithmic gap from the square-grid order N/sqrt(logN); it is not the exact planar extremal formula. Plane and regulus concentration conditions belong to the intermediate incidence theorem and cannot be omitted in an arbitrary line-arrangement claim.",
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