Not evaluated
Positive lower density of large prime gaps (family 026)
The gist
In its release of October 6, 2026, OpenAI claims, using an unreleased internal model, that large gaps between primes have positive lower density.
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-026-positive-lower-density-large-prime-gaps-r1
@misc{discoveria-oai-fam-026-positive-lower-density-large-prime-gaps-r1,
author = {{DiscoverIA}},
title = {Positive lower density of large prime gaps (family 026)},
howpublished = {DiscoverIA registry, entry oai-fam-026-positive-lower-density-large-prime-gaps},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-026-positive-lower-density-large-prime-gaps#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 were correct, large gaps between consecutive primes (the difference between two successive prime numbers) would have a strictly positive lower density: the proportion they represent would not shrink to zero.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) supplied by OpenAI itself is marked as unchecked in its tracking file, and no third-party verification is recorded. The sources read do not say where the question stood before the announcement.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 026.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 · Positive-lower-density-of…link only, reading not recorded
- Lean proofgithub.com · 026.mdpage read word for word
- Lean proofgithub.com · PrimeGaps.leanlink only, reading not recorded
Method
The openai/math repository contains the manuscript “Positive lower density of large prime gaps”, dated September 25, 2026, and the file lean/docs/026.md, which describes the Lean formalisation with one Comparator statement (lean/ComparatorChallenges/PrimeGaps.lean). The family is not among the sources of formalization.yaml; 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 (Number theory)
- In openai/math
- family 026, section “Number theory”, 1 manuscript dated Sep 25, 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 26 of 372, by family number
- Not evaluated : Patterson’s first moment for cubic Gauss sums (family 023)
- Not evaluated : An asymptotic formula for the number of totients (family 024)
- Not evaluated : Erdős’s short Egyptian-fraction conjecture (family 025)
- Not evaluated : Positive lower density of large prime gaps (family 026)
- Not evaluated : Integral density on curve character varieties (family 027)
- Not evaluated : The Gaussian moat conjecture (family 028)
- Not evaluated : Artin's primitive root conjecture: infinitude for every base (family 029)
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