Planar dual-midpoint conjecture for complementary geodesic maps
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On a connected open planar domain, consider the dual torsion-free connections D+KC and D−KC determined by a symmetric cubic tensor C. Does complementarity of their canonical local geodesic midpoint maps near the diagonal force C to be constant?
Why it matters and what counts as progress
Why it matters: The question tests whether a midpoint identity characterizes a rigid geometric structure in dimension two.
What counts as progress: A necessity proof using actual short geodesic branches, or a nonconstant cubic tensor satisfying the local midpoint condition.
How the problem developed
- 7 Sept 2026 · Conjecture [1]
Frank Nielsen and Kazuki Okamura Formulated a Euclidean dual-pair conjecture that invariance of the complementary midpoint composition forces the cubic form to be constant; proved the one-dimensional case and constant-cubic examples.
JD Jones and GPT-5.2 Pro Reported a Lean proof of the dimension-two necessity statement: complementary canonical local midpoints force the symmetric cubic tensor to be constant on a connected open planar domain.
AI contribution
GPT-assisted Lean proof settles the planar case of a dual-midpoint conjecture
On a connected open subset of the Euclidean plane, if the canonical local geodesic midpoint maps of the dual torsion-free connections D+KC and D−KC are complementary near every diagonal point, then their totally symmetric cubic tensor field C is constant. The formal proof constructs the short geodesic branches and also proves a differential obstruction criterion and a five-direction test.
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GPT-assisted Lean proof settles the planar case of a dual-midpoint conjecture