1) V1 Every planar point set of at least five points has two occurring distances represented by at most n unordered pairs, and the minimum number of such distances tends to infinity with n.
open, filed Tue Aug 25 2026 06:25:14 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Six-role fleet passed. Independent ordered-pair transcription is proved equivalent; offDiag excludes zero distance; filter cardinality bounds multiplicity; good-distance membership forces occurrence and the exact 2n threshold; false-premise control preflights red/restatement; negation remains both open claims; literature and final Jig dedupe agree. Whole attacks tested diameter-graph iteration, total-pair accounting, Guth-Katz distinct-distance bounds, rich-distance incidence bounds, convex/nonconvex decomposition, and counterexamples from grids and clustered circles. None forces a second sparse distance in general, much less uniform divergence; no cosmetic partial filed.
Scope. Finite sets of distinct points in the Euclidean plane. Lean counts ordered off-diagonal pairs, so the source threshold n is represented exactly as 2n. The asymptotic quantifier is uniform over every n-point set.