Not evaluated
A counterexample to periodic tiling in dimension three (family 155)
The gist
In its release of October 6, 2026, OpenAI claims, in a manuscript, a translational tile with no fully periodic tiling in dimension three.
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-155-counterexample-periodic-tiling-dimension-r1
@misc{discoveria-oai-fam-155-counterexample-periodic-tiling-dimension-r1,
author = {{DiscoverIA}},
title = {A counterexample to periodic tiling in dimension three (family 155)},
howpublished = {DiscoverIA registry, entry oai-fam-155-counterexample-periodic-tiling-dimension},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-155-counterexample-periodic-tiling-dimension#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 the manuscript held, it would provide, in dimension three, a translational tile that admits no fully periodic tiling. A tiling is fully periodic when its pattern repeats at regular intervals in every direction of space, and a translation is a simple shift.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI itself: no third-party check is recorded, and formalization.yaml carries the mention “unchecked”. It is not known whether the Lean statement faithfully matches the manuscript's.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 155.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-translational-tile-with…link only, reading not recorded
- Lean proofgithub.com · 155.mdpage read word for word
- Lean proofgithub.com · PeriodicTilingThree.leanlink only, reading not recorded
Method
The openai/math repository holds the manuscript “A translational tile with no fully periodic tiling in dimension three”, dated September 23, 2026. It also holds the Lean file lean/ComparatorChallenges/PeriodicTilingThree.lean. The page lean/docs/155.md describes the scope of the Lean formalisation and lists 1 Comparator statement; history.md reports no revision of this manuscript. 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 (Combinatorics)
- In openai/math
- family 155, section “Combinatorics”, 1 manuscript 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
- 4 : 1 read word for word, 3 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 151 of 372, by family number
- Not evaluated : Zero entropy does not guarantee a smooth positive-volume model (family 152)
- Not evaluated : Arithmetic classification of Bernoulli convolutions (family 153)
- Not evaluated : Pointwise multiple ergodic averages for mixing transformations (family 154)
- Not evaluated : A counterexample to periodic tiling in dimension three (family 155)
- Not evaluated : Borsuk's conjecture fails in dimension nine (family 156)
- Not evaluated : The Euclidean plane cannot be colored with five colors (family 158)
- Not evaluated : Erdős's reciprocal-sum conjecture and quasipolynomial Szemerédi bounds (family 159)
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