Not evaluated
Pinned distances and a power saving for planar unit distances
Family 167
The gist
In its release of October 6, 2026, OpenAI claims, in two manuscripts, a weak pinned planar distance theorem and a power saving for planar unit distances.
DiscoverIA status on October 12, 2026: not evaluated, meaning no third-party evaluation found.
AI draft, not reviewedWhat has not been reviewed?
Text written by an AI (DiscoverIA pipeline), on . The status and its sources come from the registry; only the explanatory text has not been reviewed. A reported error is corrected, then the text is reviewed.
Cite this entry
A single status snapshot since Oct 12, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-167-pinned-distances-power-saving-planar-unit-r1
@misc{discoveria-oai-fam-167-pinned-distances-power-saving-planar-unit-r1,
author = {{DiscoverIA}},
title = {Pinned distances and a power saving for planar unit distances (family 167)},
howpublished = {DiscoverIA registry, entry oai-fam-167-pinned-distances-power-saving-planar-unit},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-167-pinned-distances-power-saving-planar-unit#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 20 s reading time, estimated
What it changes
If these manuscripts held, they would concern pairs of points in the plane at unit distance, and distances measured from a fixed point. A power saving is expressed as a power of the problem size.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI itself: no third-party check is recorded, and formalization.yaml carries the mention “unchecked”. It is not known whether the two Lean statements faithfully match those of the manuscripts. The sources read do not say what bound the saving is measured against.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 167.md
Step 1 of 4 on the scale: not evaluated. Next step: under review.
- Announced
- Recordednot evaluated
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
The result is declared neither right nor wrong.
To move to “Under review”: a public review under way. To “Peer reviewed”: a journal publication or a review by named experts.
Sources
- Main sourcegithub.com · CONTENTS.mdlink only, reading not recorded
- Repositorygithub.com · The-weak-pinned-planar…link only, reading not recorded
- Repositorygithub.com · A-power-saving-for-planar…link only, reading not recorded
- Lean proofgithub.com · 167.mdpage read word for word
- Lean proofgithub.com · PinnedDistances.leanlink only, reading not recorded
- Lean proofgithub.com · PlanarUnitDistances.leanlink only, reading not recorded
Method
The openai/math repository holds two manuscripts dated September 23, 2026: “The weak pinned planar distance theorem” and “A power saving for planar unit distances”. It also holds two Lean files, lean/ComparatorChallenges/PinnedDistances.lean and lean/ComparatorChallenges/PlanarUnitDistances.lean. The page lean/docs/167.md describes them and lists 2 Comparator statements; history.md reports no revision. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 12, 2026 from github.com (page read, quotation checked word for word: “describes the scope of the Lean formalization”).
Limits
No named third party: the recorded checks come from the claimant or from a platform tool. The claimant supplies a Lean proof; as of this date, DiscoverIA has found no independent verification of this proof or of the result.
Key facts
- Claimed by
- OpenAI (unreleased internal model)
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (Combinatorics)
- In openai/math
- family 167, section “Combinatorics”, 2 manuscripts dated Sep 23, 2026
- Lean proof
- supplied by the claimant, marked “unchecked” (not reviewed) in its repository
- Revisions
- none reported by the claimant
- Checked by
- claimant (Lean supplied, not checked by a third party)
- Sources
- 6 : 1 read word for word, 5 links only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
Family with a Lean proof supplied by the claimant. No third-party check read; the source marks this proof as not reviewed (“unchecked”).
Source: github.comread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 161 of 372, by family number
- Not evaluated : Hindman's finite sums and products conjecture (family 164)
- Not evaluated : Exact crossing numbers of complete and complete bipartite graphs (family 165)
- Not evaluated : The higher-dimensional Erdős distinct-distances conjecture (family 166)
- Not evaluated : Pinned distances and a power saving for planar unit distances (family 167)
- Not evaluated : Combinatorial invariance of Kazhdan–Lusztig polynomials (family 168)
- Not evaluated : The Shareshian–Wachs e-positivity conjecture (family 169)
- Not evaluated : Sharp logarithmic exponents for off-diagonal Ramsey numbers (family 170)
Related frontiers
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