Ordinary two-point correlations and the corrected Elliott conjecture
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Proves the ordinary two-point Chowla conjecture, with a bound \(O(X/(\log X)^c)\) for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times \(n^{it}\) for \(|t|\le X\). (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
What the papers state
Theorem 1.1(A logarithmic saving for affine Liouville correlations)
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There is an absolute constant c>0 such that, for every fixed choice of integers a_1,a_2\ge1 and b_1,b_2\ge0 with a_1b_2-a_2b_1\ne0, there is a constant C_{a_1,a_2,b_1,b_2} for which
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let f_1,f_2:\N\to\mathbb D be multiplicative, and suppose that at least one satisfies (1.2). For every fixed pair of distinct nonnegative integers h_1,h_2,
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let f_1,f_2:\N\to\mathbb D be multiplicative, with at least one satisfying (1.2). For all fixed integers a_1,a_2\ge1 and b_1,b_2\ge0 with a_1b_2-a_2b_1\ne0,
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.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.
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Let f_1,f_2:\N\to\mathbb D be multiplicative, and suppose that at least one satisfies (1.2). For every fixed pair of distinct nonnegative integers h_1,h_2,
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.
There is an absolute constant c>0 such that, for every fixed choice of integers a_1,a_2\ge1 and b_1,b_2\ge0 with a_1b_2-a_2b_1\ne0, there is a constant C_{a_1,a_2,b_1,b_2} for which
OrdinaryTwoPointCorrelations.lean, the release lists it as a supporting statement.
theoremliouville_log_saving : ∃ c : ℝ, 0 < c ∧ ∀ a₁ a₂ b₁ b₂ : ℕ,0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ → ∃ C : ℝ, 0 < C ∧ ∀ X : ℝ, 3 ≤ X → ‖affineSum liouville liouville a₁ a₂ b₁ b₂ ⌊X⌋₊‖ ≤ C * X / Real.rpow (Real.log X) 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?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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 or imply the whole advertised result (their term: “full”).
Their note:None found; affine forms take natural-number coefficients (a₁,a₂>0, b₁,b₂≥0), which loses nothing up to re-indexing; custom non-pretentiousness definition checked against the standard one.
Definitions the group asks readers to compare with the paper:def UniformlyNonpretentious (f : ℕ → ℂ) : Prop := ∀ (q : ℕ), 0 < q → ∀ (χ : DirichletCharacter ℂ q) (K : ℝ), ∀ᶠ N : ℕ in atTop, ∀ t : ℝ, |t| ≤ (N : ℝ) → K ≤ squaredDistance f (characterTwist χ t) N (custom but matches the standard definition; two files use different but equivalent encodings)
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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.
The formalization proves ordinary two-point cancellation for multiplicative functions. For the Liouville function on every fixed pair of nonproportional affine forms, the correlation sum up to $X$ is $O(X/(\log X)^c)$ for an absolute $c>0$, with the implied constant depending on the forms.
For two one-bounded multiplicative functions, if at least one is uniformly nonpretentious against all Dirichlet-character twists with frequency $|t|\le N$, then their shifted and nonproportional affine correlation sums divided by $N$ tend to zero. These are ordinary averages, with no logarithmic averaging.
AI-drafted comparisons, with no recorded human review
Ordinary Elliott cancellation under uniform nonpretentiousnessOrdinaryElliott
Unreviewed AI draft: states a paper’s theorem as the paper does. States the binary corrected Elliott theorem for ordinary averages under uniform nonpretentiousness of one factor, one of the paper’s two main theorems.
Ordinary Liouville and multiplicative two-point correlationsOrdinaryTwoPointCorrelations
Unreviewed AI draft: states a paper’s theorem as the paper does. States the logarithmic saving for Liouville correlations on nonproportional affine forms, together with the binary corrected Elliott theorem and its affine corollary: the three theorem-like statements of the paper’s introduction.
Who ran the proof-verification program?
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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.
Ordinary Elliott cancellation under uniform nonpretentiousnessOrdinaryElliott
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
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Papers elsewhere in the release that cite this family’s papers
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References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
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Identify the theorem the catalogue describes
Family 007No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Address a question raised in an independent assessment
Family 007Atlas 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 (A logarithmic saving for affine Liouville correlations)
In Ordinary two-point correlations of multiplicative functions, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
absolute, affine, fixed
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 1.2 (Binary corrected Elliott)
In Ordinary two-point correlations of multiplicative functions, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
elliott, fixed, multiplicative
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
Corollary 1.3 (Affine correlations)
In Ordinary two-point correlations of multiplicative functions, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
affine, fixed, multiplicative
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
“full”
What their term means
The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 the headline as OpenAI wrote it.
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 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.