Power savings for intersective polynomial differences and prime arguments
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of \(\{1,\ldots,N\}\) avoiding nonzero values \(h(1),h(2),\ldots\) as differences has size \(O_h(N^{1-c_k})\), with \(c_k\gt 0\) depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every integer k\ge2, there is a constant c_k\in(0,1) with the following property. For every intersective polynomial h\in\Z[x] of degree k with positive leading coefficient, there are a constant C_h\ge1 and an integer N_h\ge1 such that, whenever N\ge N_h and A\subseteq[N] satisfy
(A-A)\cap h(\N_+)\subseteq\{0\},
one has
|A|\le C_h N^{1-c_k}.
No Lean statement in the release is paired with this theorem.
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let h\in\Z[x] be prime-intersective, of degree at least two and with positive leading coefficient. There exist constants c_h>0 and C_h\ge1 such that, for every positive integer N and every A\subseteq[N],
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
One more possible match
Theorem 1.1in A power saving for square-difference-free sets.
Reported proof verification
One Lean statement in the release; one independent group reports that Comparator accepted a run for this file at the version shown here.
An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent \(c_k\gt 0\) such that every set \(A\subseteq\{1,\ldots,N\}\) whose differences avoid all nonzero values \(h(1),h(2),\ldots\), where h is an intersective polynomial of degree k with positive leading coefficient, satisfies \(|A|=O_h(N^{1-c_k})\). The implied constant may depend on h, but the power-saving exponent depends only on its degree.
Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set \(A\subseteq\{1,\ldots,N\}\) whose differences avoid all nonzero values \(h(p)\) at primes satisfies \(|A|\le C_hN^{1-c_h}\), where \(c_h\gt 0\) and \(C_h\ge1\) depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L-functions to obtain the required prime-distribution estimates.
We prove that there are absolute constants c > 0 and C < ∞ such that every set \(A\subseteq\{1,\ldots,N\}\) with no nonzero square difference satisfies \(|A|\le C N^{1-c}\). This answers the fixed-power question posed by Green and Sawhney.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
SquareDifference.lean, the release lists it as a supporting statement.
theorempower_saving : ∃ c C : ℝ, 0 < c ∧ ∀ N : ℕ, 1 ≤ N → ∀ A : Finset ℤ, A ⊆ Finset.Icc 1 (N : ℤ) → SquareDifference.IsSquareDifferenceFree A → (A.card : ℝ) ≤ C * (N : ℝ) ^ (1 - c) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:Only the square case h(x) = x^2 is stated: IsSquareDifferenceFree A := ∀ a ∈ A, ∀ b ∈ A, ∀ m ≥ 1, a - b ≠ m^2 and ∃ c C, 0 < c ∧ ∀ N ≥ 1, ∀ A ⊆ Icc 1 N,... → A.card ≤ C * N^(1-c) (docs: ‘a fixed power saving for sets with no nonzero square difference’). The summary’s general intersective polynomials of degree k >= 2 (with c_k depending only on degree) and the prime-argument result are not formalized.
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:Every intersective h of every degree k ≥ 2 (with c_k depending only on k) and the prime-argument version; only the square case (Green–Sawhney’s fixed-power question) is formalized.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization proves a fixed power saving for sets with no nonzero square difference. There are absolute constants $c>0$ and $C$ such that every $A\subseteq\{1,\ldots,N\}$ satisfying $a-b\ne m^2$ for all $a,b\in A$ and integers $m\ge1$ has $|A|\le C N^{1-c}$. The bound is uniform for every integer $N\ge1$.
AI-drafted comparisons, with no recorded human review
Power saving for square-difference-free setsSquareDifference
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: square-difference-free subsets of {1, …, N} have at most CN^{1−c} elements. The polynomial and prime-argument generalisations of the family’s later papers are not part of it.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Power saving for square-difference-free setsSquareDifference
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“REFUSED”
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 182No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Compare the proposed paper and Lean matches
Family 182Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In A power saving for intersective polynomial differences with an exponent depending only on the degree, 5 October 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Why this passage is shown
Theorem 1.1
In A Power Saving for Polynomial Differences at Prime Arguments, 5 October 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
coefficient, degree, integer, intersective, leading, positive, prime
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Why this passage is shown
Theorem 1.1
In A power saving for square-difference-free sets
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
integer, positive, subset
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“partial”
What their term means
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.