Not evaluated
The complete Crouzeix conjecture (family 325)
The gist
On October 6, 2026, OpenAI announced two manuscripts on the complete Crouzeix conjecture, from an unreleased internal model, with Lean files supplied by OpenAI.
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-325-complete-crouzeix-conjecture-r1
@misc{discoveria-oai-fam-325-complete-crouzeix-conjecture-r1,
author = {{DiscoverIA}},
title = {The complete Crouzeix conjecture (family 325)},
howpublished = {DiscoverIA registry, entry oai-fam-325-complete-crouzeix-conjecture},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-325-complete-crouzeix-conjecture#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 were correct, one would bring a direct proof of the complete Crouzeix inequality and the other a complete Crouzeix theorem, with an optimal similarity. The repository files this family under functional analysis.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI itself; its formalization.yaml file carries “review: status: unchecked”, and no third-party verification is recorded. The sources read do not say how much of the manuscripts this Lean covers.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 325.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-direct-proof-of-the…link only, reading not recorded
- Repositorygithub.com · The-complete-Crouzeix…link only, reading not recorded
- Lean proofgithub.com · 325.mdpage read word for word
- Lean proofgithub.com · CompleteCrouzeix.leanlink only, reading not recorded
- Lean proofgithub.com · CrouzeixHilbert.leanlink only, reading not recorded
- Lean proofgithub.com · DirectCrouzeix.leanlink only, reading not recorded
- Lean proofgithub.com · HilbertCrouzeix.leanlink only, reading not recorded
- Lean proofgithub.com · StructuralCrouzeix.leanlink only, reading not recorded
Method
The openai/math repository contains two manuscripts, in the “Functional analysis” section: “The complete Crouzeix theorem: optimal similarity and a common positive boundary representation” (September 23, 2026) and “A direct proof of the complete Crouzeix inequality” (September 26, 2026). The file lean/docs/325.md, which “describes the scope of the Lean formalization”, goes with 5 Comparator statements, in lean/ComparatorChallenges: CompleteCrouzeix.lean, CrouzeixHilbert.lean, DirectCrouzeix.lean, HilbertCrouzeix.lean and StructuralCrouzeix.lean. 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).
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 (Functional analysis)
- In openai/math
- family 325, section “Functional analysis”, 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
- 9 : 1 read word for word, 8 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 319 of 372, by family number
- Not evaluated : Tingley's sphere-isometry problem (family 322)
- Not evaluated : Relative independence of the separable quotient problem (family 323)
- Not evaluated : Lipschitz equivalence without linear isomorphism (family 324)
- Not evaluated : The complete Crouzeix conjecture (family 325)
- Not evaluated : The cotype–cotype conjecture under the approximation property (family 326)
- Not evaluated : Markov type characterizes superreflexivity (family 327)
- Not evaluated : Fixed points of nonexpansive maps in reflexive Banach spaces (family 328)
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