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Planar Euclidean midpoint rigidity

JD Jones

Palomar Registrypalomar/PALOMAR-2026-10-09-000016Oct 9, 2026math.DGscore 0

A complete Lean proof of the dimension-two case of Nielsen–Okamura Conjecture 9.1. On any connected open planar Euclidean domain, complementarity of the actual canonical local affine-parameter midpoints of a smooth pair of dual torsion-free connections forces the symmetric cubic tensor to be constant. The proof constructs and identifies the short geodesic germs and derives the fourth-order obstruction, including the exceptional rank-deficient tensors. A second theorem needs only everywhere Frechet differentiability and vanishing of that obstruction; five fixed direction tests at every point…

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