Not evaluated
The optimal order of convex-body covering density (family 092)
The gist
OpenAI claims, in two manuscripts announced on October 6, 2026, covering-density bounds of order n log n for convex bodies.
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-092-optimal-order-convex-body-covering-density-r1
@misc{discoveria-oai-fam-092-optimal-order-convex-body-covering-density-r1,
author = {{DiscoverIA}},
title = {The optimal order of convex-body covering density (family 092)},
howpublished = {DiscoverIA registry, entry oai-fam-092-optimal-order-convex-body-covering-density},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-092-optimal-order-convex-body-covering-density#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 20 s reading time, estimated
What it changes
The two manuscripts concern covering space by copies of a single convex body. One uses a single lattice (a regular grid of positions), the other translations; both titles state an order of n log n. If these manuscripts held, the density of these coverings would have a bound of that order.
What remains unclear
The repository file formalization.yaml marks the Lean (a language in which a computer checks every step of a proof) supplied by OpenAI as “review: status: unchecked”. No third-party check has been read, so it is not known whether the manuscripts hold.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 092.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 · A-single-lattice-covering…link only, reading not recorded
- Repositorygithub.com · Translative-covering…link only, reading not recorded
- Lean proofgithub.com · 092.mdpage read word for word
- Lean proofgithub.com · CoveringDensity.leanlink only, reading not recorded
- Lean proofgithub.com · SingleLatticeCovering.leanlink only, reading not recorded
Method
A manuscript dated September 23, 2026: “A single-lattice covering bound of order n log n”. A second one, of the same date: “Translative covering densities of order n log n”. The repository contains lean/docs/092.md and 2 Comparator statements. 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 092, section “Convex and metric geometry”, 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 89 of 372, by family number
- 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)
- Not evaluated : Subpolynomial dimension reduction in L_p (family 094)
- Not evaluated : Hyperbolicity cones without semidefinite lifts (family 095)
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