What was already known
Spencer, Szemerédi and Trotter established the universal O(n^(4/3)) upper bound. Earlier gains did not give a fixed smaller exponent.
AI involvement
The OpenAI-authored manuscript belongs to its internal-model research collection; individual-session autonomy is undocumented.
Institutional AI authorship and collection provenance support AI-led work.
Importance
Claims a fixed exponent improvement over the classical planar unit-distance upper bound.
Limitations
This upper bound does not prove an n^(1+o(1)) upper bound. It is distinct from later point-set constructions with more than n^(1+ε) unit-distance pairs. The formalization note contains one contradictory scope sentence; its theorem file states the claimed bound.
Technical verification
The repository's Lean main theorem states the same universal fixed-power bound with an actual proof, and its formalization page maps the comparator to it. One sentence in that page says the separate unit-distance theorem is not included while its next paragraph and comparator link say it is; the theorem file supports the latter reading.
Formal checking
Uniform Cn^β bound with 1≤β<4/3 for all finite planar sets. formal proof
- Checker
- Lean kernel via OAI.PlanarUnitDistances.main
- Scope
- Uniform Cn^β bound with 1≤β<4/3 for all finite planar sets
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Default: no named human explanation of the incidence, information, height, and algebraic obstruction chain is documented.
Correction and publication history
Claims a universal fixed-power improvement over the n^(4/3) bound.
Read this published revision