Not evaluated
Counterexample to Kaplansky's zero-divisor conjecture (family 196)
The gist
OpenAI claims, in an announcement of October 6, 2026, a counterexample to Kaplansky's zero-divisor conjecture.
DiscoverIA status on October 11, 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 11, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 11, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-196-kaplansky-zero-divisor-r1
@misc{discoveria-oai-fam-196-kaplansky-zero-divisor-r1,
author = {{DiscoverIA}},
title = {Counterexample to Kaplansky's zero-divisor conjecture (family 196)},
howpublished = {DiscoverIA registry, entry oai-fam-196-kaplansky-zero-divisor},
note = {Status “Not evaluated”, snapshot 1, recorded on October 11, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-196-kaplansky-zero-divisor#releve-1},
urldate = {2026-10-11}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
If the result held, there would be a torsion-free group whose group algebra over a field has zero divisors, which the conjecture rules out. A group is torsion-free if none of its elements other than the identity has finite order, meaning that none of its positive powers is the identity. In the group algebra (combinations of group elements that can be added and multiplied), a zero divisor is a non-zero element whose product with another non-zero element is zero.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) is supplied by the claimant itself, and the file formalization.yaml marks it “status: unchecked”. The match between this Lean and the announced result is an estimate, and no third-party check is recorded.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · raw.githubusercontent.com
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.comlink only, reading not recorded
Method
The result is announced with a preprint. A Lean file is listed at formalization.yaml:197 under the title “A Torsion-Free Group Algebra with Zero Divisors” (config TorsionFreeZeroDivisors). DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 11, 2026 from github.com (page read, quotation checked word for word: “status: unchecked”).
Limits
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
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (group theory / algebra)
- Checked by
- claimant
- Sources
- 1 : 1 link only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
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 190 of 372, by family number
- Not evaluated : Serre's intersection-multiplicity conjecture (family 193)
- Not evaluated : Lech's multiplicity conjecture (family 194)
- Not evaluated : A counterexample to the small Cohen–Macaulay module conjecture (family 195)
- Not evaluated : Counterexample to Kaplansky's zero-divisor conjecture (family 196)
- Not evaluated : Torsion-free group algebras that are not directly finite (family 197)
- Not evaluated : A counterexample to the little finitistic-dimension conjecture (family 198)
- Not evaluated : Counterexamples to conjectures of Auslander–Reiten, Tachikawa, and Nakayama (family 199)
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