OpenAI · 166-01 · 2026-09-23 · Combinatorics

The higher-dimensional Erdős distinct-distances conjecture

preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026

Family 166. The higher-dimensional Erdős distinct-distances conjecture.

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

For every fixed $d \geq 3$ , any $n \geq 2$ distinct points in $R^{d}$ determine at least $c_{d} n^{2/ d}$ distinct distances, with $c_{d} > 0$ depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝd determines at least $c_{d} n^{2/ d}$ distinct distances, where $c_{d} > 0$ depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.

Preprint source and Lean scope

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