SWhat 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.
Scientific 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
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.
Contributors
- The named authors in the research record · SCIENCE: Source-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.
Provisional WRS assessment, edition 2026-10-23.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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