Not evaluated
Amenability, unitarizability, and Ulam stability (family 251)
The gist
On October 6, 2026, OpenAI announced two group theory manuscripts: one linking unitarizability and amenability for discrete groups, the other on strong Ulam stability.
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-251-amenability-unitarizability-ulam-stability-r1
@misc{discoveria-oai-fam-251-amenability-unitarizability-ulam-stability-r1,
author = {{DiscoverIA}},
title = {Amenability, unitarizability, and Ulam stability (family 251)},
howpublished = {DiscoverIA registry, entry oai-fam-251-amenability-unitarizability-ulam-stability},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-251-amenability-unitarizability-ulam-stability#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
A discrete group is amenable if it admits an invariant mean, that is, a way of averaging the bounded functions on the group that stays the same when those functions are shifted by an element of the group. If these manuscripts held up, they would link this property to unitarizability and to a form of Ulam stability. The first title specifies the setting of discrete groups; the titles do not give the detailed assumptions.
What remains unclear
OpenAI itself supplies the Lean (a language in which a computer checks every step of a proof), and its review is marked “unchecked”. No third-party check or evaluation has been read.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 251.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 · Unitarizability-Implies…link only, reading not recorded
- Repositorygithub.com · Strong-Ulam-Stability…link only, reading not recorded
- Lean proofgithub.com · 251.mdpage read word for word
- Lean proofgithub.com · Dixmier.leanlink only, reading not recorded
- Lean proofgithub.com · DixmierAllDiscrete.leanlink only, reading not recorded
Method
Two manuscripts are published: “Unitarizability implies amenability for discrete groups” (September 23, 2026) and “Strong Ulam Stability Characterizes Amenability” (October 5, 2026). OpenAI supplies lean/docs/251.md and 2 Comparator statements: lean/ComparatorChallenges/Dixmier.lean and lean/ComparatorChallenges/DixmierAllDiscrete.lean. The file 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 (Group theory)
- In openai/math
- family 251, section “Group theory”, 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 245 of 372, by family number
- Not evaluated : Thompson's group F is nonamenable (family 248)
- Not evaluated : A finitely generated counterexample to Eilenberg–Ganea (family 249)
- Not evaluated : The Boone–Higman conjecture and higher finiteness (family 250)
- Not evaluated : Amenability, unitarizability, and Ulam stability (family 251)
- Not evaluated : A non-residually-finite torsion-free hyperbolic group (family 252)
- Not evaluated : An infinite finitely presented simple amenable group (family 253)
- Not evaluated : Artin's K(π,1) conjecture and CAT(0) obstructions (family 254)
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