Not evaluated
Power savings for polynomial-difference-free sets (family 182)
The gist
On October 6, 2026, OpenAI announced 3 manuscripts on power savings for square-difference-free sets and for polynomial differences.
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-182-power-savings-polynomial-difference-free-r1
@misc{discoveria-oai-fam-182-power-savings-polynomial-difference-free-r1,
author = {{DiscoverIA}},
title = {Power savings for polynomial-difference-free sets (family 182)},
howpublished = {DiscoverIA registry, entry oai-fam-182-power-savings-polynomial-difference-free},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-182-power-savings-polynomial-difference-free#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
A square-difference-free set is a set of integers no two distinct elements of which ever differ by a perfect square. A power saving means that, among the integers up to a large bound, such a set is smaller than the full set by a factor that is a power of that bound. If the manuscripts were correct, they would give such a saving for these sets and, according to their titles, for intersective polynomial differences and for polynomial differences at prime arguments.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI itself and marked “unchecked” in formalization.yaml. No third-party verification is recorded. A single Comparator statement accompanies the 3 manuscripts, and the sources read do not say whether it covers the polynomial differences.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 182.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-power-saving-for…link only, reading not recorded
- Repositorygithub.com · A-Power-Saving-for…link only, reading not recorded
- Repositorygithub.com · A-power-saving-for-square…link only, reading not recorded
- Lean proofgithub.com · 182.mdpage read word for word
- Lean proofgithub.com · SquareDifference.leanlink only, reading not recorded
Method
The openai/math repository contains 3 manuscripts. The manuscript “A power saving for square-difference-free sets” is dated September 24, 2026. The manuscript “A power saving for intersective polynomial differences with an exponent depending only on the degree” is dated October 5, 2026. So is the manuscript “A Power Saving for Polynomial Differences at Prime Arguments”. OpenAI attaches lean/docs/182.md and lean/ComparatorChallenges/SquareDifference.lean. According to history.md, no revision is reported. 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 182, section “Combinatorics”, 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 176 of 372, by family number
- Not evaluated : The circulant Hadamard conjecture (family 179)
- Not evaluated : Barnette's Hamiltonian-cycle conjecture (family 180)
- Not evaluated : The Erdős–Gallai cycle-decomposition conjecture (family 181)
- Not evaluated : Power savings for polynomial-difference-free sets (family 182)
- Not evaluated : Power savings for planar halving lines and k-sets (family 183)
- Not evaluated : Coloring and independence in graphs with forbidden subgraphs (family 184)
- Not evaluated : Counterexamples to infinite matroid intersection and packing/covering (family 185)
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