Not evaluated
Hyperbolicity cones without semidefinite lifts (family 095)
The gist
In its release of October 6, 2026, OpenAI claims that hyperbolicity cones without semidefinite lifts exist.
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-095-hyperbolicity-cones-without-semidefinite-r1
@misc{discoveria-oai-fam-095-hyperbolicity-cones-without-semidefinite-r1,
author = {{DiscoverIA}},
title = {Hyperbolicity cones without semidefinite lifts (family 095)},
howpublished = {DiscoverIA registry, entry oai-fam-095-hyperbolicity-cones-without-semidefinite},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-095-hyperbolicity-cones-without-semidefinite#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
If this result held, some hyperbolicity cones would have no semidefinite lift. A hyperbolicity cone is a convex shape defined by a polynomial whose roots stay real along every line parallel to a fixed direction. A semidefinite lift describes it as the projection of a region lying in a space of higher dimension, defined by a positive semidefinite matrix constraint.
What remains unclear
No independent assessment is recorded. The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI and marked “review: status: unchecked” in formalization.yaml. The sources read do not say how the three manuscripts fit together; one is titled “An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone”.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 095.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 · Hyperbolicity-Cones…link only, reading not recorded
- Repositorygithub.com · A-Nonspectrahedral…link only, reading not recorded
- Repositorygithub.com · An-Exact-Semidefinite…link only, reading not recorded
- Lean proofgithub.com · 095.mdpage read word for word
- Lean proofgithub.com · HyperbolicCones.leanlink only, reading not recorded
Method
OpenAI publishes three manuscripts, including “A nonspectrahedral hyperbolicity cone” (September 24, 2026) and “Hyperbolicity Cones Without Semidefinite Lifts” (October 5, 2026). The third is also dated October 5, 2026. The repository also holds lean/docs/095.md, which describes the scope of the Lean formalisation (1 Comparator statement), and lean/ComparatorChallenges/HyperbolicCones.lean. 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 095, section “Convex and metric geometry”, 3 manuscripts dated Sep 24, 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 92 of 372, by family number
- 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)
- Not evaluated : The Gaussian propeller conjecture (family 096)
- Not evaluated : The Euclidean Steinitz–Bergström conjecture (family 097)
- Not evaluated : A negative answer to the Lang–Plaut problem (family 098)
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