Not evaluated
Universal optimality of the triangular lattice (family 090)
The gist
On October 6, 2026, OpenAI announced, from an unreleased internal model, four manuscripts around the universal optimality of the triangular lattice, with Lean files.
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-090-universal-optimality-triangular-lattice-r1
@misc{discoveria-oai-fam-090-universal-optimality-triangular-lattice-r1,
author = {{DiscoverIA}},
title = {Universal optimality of the triangular lattice (family 090)},
howpublished = {DiscoverIA registry, entry oai-fam-090-universal-optimality-triangular-lattice},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-090-universal-optimality-triangular-lattice#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 the manuscripts were correct, the triangular lattice would be optimal for a whole family of energies at once, which is what “universal optimality” means. According to their titles, one manuscript concerns circle packing in the plane and another the planar Coulomb renormalized energy.
What remains unclear
The Lean files (Lean is a language in which a computer checks every step of a proof) are supplied by OpenAI itself, and the repository marks them “unchecked”. No third-party verification is recorded.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 090.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 · An-atomic-certificate-for…link only, reading not recorded
- Repositorygithub.com · Universal-optimality-of…link only, reading not recorded
- Repositorygithub.com · A-sharp-Fourier…link only, reading not recorded
- Repositorygithub.com · Triangular-minimality-for…link only, reading not recorded
- Lean proofgithub.com · 090.mdpage read word for word
- Lean proofgithub.com · AtomicGaussian.leanlink only, reading not recorded
- Lean proofgithub.com · PlanarPacking.leanlink only, reading not recorded
- Lean proofgithub.com · TriangularEnergy.leanlink only, reading not recorded
- Lean proofgithub.com · TriangularGaussian.leanlink only, reading not recorded
Method
According to CONTENTS.md, family 090 comprises four manuscripts. Two are dated September 23, 2026: “Universal optimality of the triangular lattice” and “A sharp Fourier certificate for planar circle packing”. A third, dated the same day, is “Triangular minimality for planar Coulomb renormalized energy”. The fourth, “An atomic certificate for triangular-lattice universal optimality”, is dated September 26, 2026. The file lean/docs/090.md describes the scope of the formalisation, with four Comparator statements. Two files are lean/ComparatorChallenges/AtomicGaussian.lean and lean/ComparatorChallenges/PlanarPacking.lean. The other two are lean/ComparatorChallenges/TriangularEnergy.lean and lean/ComparatorChallenges/TriangularGaussian.lean. The family is not among the sources of formalization.yaml; history.md reports no revision of these manuscripts. 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 (Convex and metric geometry)
- In openai/math
- family 090, section “Convex and metric geometry”, 4 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
- 10 : 1 read word for word, 9 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 87 of 372, by family number
- Not evaluated : The Mahler conjectures and symplectic width (family 087)
- Not evaluated : Petty's projection-volume conjecture and simplex counterexamples (family 088)
- Not evaluated : Bounded-distortion L_1 embeddings of planar and bounded-treewidth graphs (family 089)
- Not evaluated : Universal optimality of the triangular lattice (family 090)
- Not evaluated : The logarithmic Brunn–Minkowski conjecture (family 091)
- Not evaluated : The optimal order of convex-body covering density (family 092)
- Not evaluated : Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures (family 093)
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