OpenAI · 167-02 · 2026-09-23 · Combinatorics
A power saving for planar unit distances
preprints/A-power-saving-for-planar-unit-distances-September-23-2026
Family 167. Pinned distances and a power saving for planar unit distances.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves the weak pinned Erdős distance conjecture: for every fixed , all but points of any -point planar set determine at least distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by for an absolute .
We prove a power saving for the planar unit-distance problem: for some absolute , every set of n points in the Euclidean plane determines unordered pairs at unit distance.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
This preprint is named in the Lean scope note for 167: Planar distinct distances and unit-distance bounds. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.