OpenAI · 183-01 · 2026-09-25 · Combinatorics
A power saving for planar halving lines
preprints/A-power-saving-for-planar-halving-lines-September-25-2026
Family 183. Power savings for planar halving lines and -sets.
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
Improves the planar halving-line bound to for sets with no three collinear and an absolute . More generally, an -point set with no three collinear has strictly separable -subsets for , with an absolute . The constants and positive exponents are nonquantitative.
There are absolute constants ε > 0 and C such that every sufficiently large even n-point set in the plane with no three collinear has at most unordered halving pairs. This gives a power saving over the classical bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 183: Power savings for planar halving lines and <i>k</i>-sets. 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.