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The weak pinned planar distance theorem

OpenAI · September 23, 2026

Lean

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Abstract

We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but $o(n)$ points of any n-point planar set determine at least $n^{1-\varepsilon}$ distinct nonzero distances.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Planar distinct distances and unit-distance bounds (Lean scope notes) (lean/docs/167.md, October 2026). The release's scope note says the Lean development covers Theorem 1.1 (for each fixed s>0, F_n(s)→0) and the derived statement about pins, and links both comparator statements; one sentence in the note says the unit-distance theorem is "not included", while the next paragraph describes it as formalized. [family-level]
  • OpenAI, PinnedDistances.lean (comparator statement) (openai/math, October 2026). Comparator file with the target statement theorem main (s : ℝ) (hs : 0<s) : Tendsto (fun n => F n s) atTop (𝓝 0) in namespace WeakPinned, i.e. Theorem 1.1 rather than Corollary 1.2.
  • OpenAI, OAI/Geometry/PinnedDistances/Main.lean (openai/math, October 2026). Top-level file of the release's proof: OAI.WeakPinned.main is derived from mainStatement, which is proved by contradiction from imported lemmas (sequence_arithmetic_model, arithmetic_sequence_impossible).
  • cert-machine, PinnedDistances comparator run (run 38019899278). Records exit code zero and acceptance of the release's pinned-distance solution by both Lean's default kernel and nanoda.

Digests and expositions

Elsewhere

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Cite

@misc{OAI:The-weak-pinned-planar-distance-theorem-September-23-2026,
  author = {{OpenAI}},
  title = {{The weak pinned planar distance theorem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf}{OAI:The-weak-pinned-planar-distance-theorem-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/167.md. The Lean statements have not been checked by people against the paper's claims.

Family 167: Planar distinct distances and unit-distance bounds

Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $o(n)$ points of any n-point planar set determine at least $n^{1-\varepsilon}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O(n^{4/3-\delta})$ for an absolute δ > 0.

Other papers in this family: