Hilbert’s sixteenth problem: uniform bounds for limit cycles
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Resolves the uniform boundedness assertion in Hilbert’s sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
For each integer d\geq1, there is a finite nonnegative integer B(d) such that every real planar polynomial vector field of degree at most d has at most B(d) limit cycles in the whole plane.
No Lean statement in the release is paired with this theorem.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
If \deg F\le5, the system (1.1) has at most two limit cycles in \R^2. Some members of this class have two limit cycles. Thus its exact maximum is two.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert’s sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.
Every classical Liénard system \(\dot x=y-F(x)\), \(\dot y=-x\), with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.
One Lean statement in this family
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the images of nonconstant periodic solutions are in one-to-one correspondence with the zeros of
\Delta(r)=M_+(0,r)-M_-(0,r)
on the single open interval \mathcal I=\mathcal I_+\cap\mathcal I_-. If this interval is empty, there are no such periodic solutions. Each periodic orbit has exactly one positive and one negative intersection with the y-axis. Under the correspondence, isolated periodic orbits are exactly isolated zeros of \Delta.
The words they share: orbits, periodic, solutions, exactly, images, nonconstant, isolated, system.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
QuinticLienard.lean, the release lists it as a main result.
theoremmain : (∀ F : Polynomial ℝ, F.degree ≤ 5 → {C : Set Plane | IsLimitCycle F C}.encard ≤ 2) ∧ (∃ F : Polynomial ℝ, F.degree ≤ 5 ∧ {C : Set Plane | IsLimitCycle F C}.encard = 2) := 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.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
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 quintic Lienard system x' = y - F(x), y' = -x (at most 2 limit cycles, exactly 2 attained); the general uniform boundedness theorem for planar polynomial vector fields of degree n is not stated.
Definitions the group asks readers to compare with the paper:IsLimitCycle F C := IsPeriodicOrbit F C ∧ ∃ U open ⊇ C, ∀ periodic orbits C' ⊆ U, C' = C (isolated nonconstant periodic orbit, standard); counts via Set.encard
“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:The headline’s main clause — the Hilbert’s-sixteenth uniform bound on limit cycles of every real planar polynomial vector field by a constant depending only on the degree — is not stated; only the quintic Liénard count is.
“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.
For the quintic Liénard system $x'=y-F(x)$, $y'=-x$, the formalized result proves that every real polynomial $F$ of degree at most five yields at most two limit cycles, and that some such $F$ yields exactly two. Limit cycles are isolated images of nonconstant periodic solutions. No sign, parity, hyperbolicity, or amplitude restriction is imposed.
AI-drafted comparisons, with no recorded human review
Exact two-cycle bound for quintic Liénard systemsQuinticLienard
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: quintic Liénard systems have at most two limit cycles, and some have exactly two.
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.
Exact two-cycle bound for quintic Liénard systemsQuinticLienard
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.
Other people’s accounts of running proof-verification software, with links to their reports.
AscendFoam, AscendFoam/hierarchical-melnikov-lean: Lean 4 formalisation of the hierarchical Melnikov lower bound for Hilbert numbers, 7 October 2026. Source
Repository’s Lean CI (lake build and axiom audit) green on 6 of 6 runs, latest 7 Oct 19:15 UTC.
Problems reported by others
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
Lean community, OpenAI releases solutions to hundreds of problems (117 messages, 6-10 Oct 2026), 6 October 2026. Source
Robert Joseph’s audit: the Lean covers only the quintic Lienard case of the Hilbert 16th result although the catalogue links it as formalised.
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 this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
Follow-up work by others
AscendFoam, AscendFoam/hierarchical-melnikov-lean: Lean 4 formalisation of the hierarchical Melnikov lower bound for Hilbert numbers, 7 October 2026. Source
AscendFoam reports a Lean formalisation of a lower bound of order d^2 log^2 d for the number of hyperbolic limit cycles.
References outside the release
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Explain a result, compare statements or propose a correction
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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 “Landmark of the century”. The page says that nothing on it is verified.
Chaoyang Qin, Xiaoming Sun, Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds, 8 October 2026. Source
Qin and Sun note that OpenAI’s draft claiming finiteness of the Hilbert number H(d) appeared after their own lower-bound work, and that their order d^2 (log d)^2 lower bound is independent of it.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The family headline says it resolves the uniform boundedness assertion in Hilbert’s sixteenth problem, but only quintic Lienard systems are formalised.
Questions awaiting review
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 143The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
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 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.