Lean statement

Global boundedness and persistence for weakly reversible mass-action systems

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.Problem326.global_bounded_persistent_solution

Lean statement file: MassAction.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 149 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem global_bounded_persistent_solution :    ∀ (d : ℕ) (N : ReactionNetwork d) (κ : Reaction N → ℝ),      0 < d → WeaklyReversible N → (∀ e, 0 < κ e) →      ∀ x0 : Fin d → ℝ, PositiveState x0 →      ∃ ε : ℝ, 0 < ε ∧ ε < 1 ∧        (∃ x : ℝ → (Fin d → ℝ), IsGlobalForwardSolution N κ x0 x) ∧        ∀ x : ℝ → (Fin d → ℝ), IsGlobalForwardSolution N κ x0 x →        ∀ t : ℝ, 0 ≤ t → ∀ i : Fin d, ε ≤ x t i ∧ x t i ≤ ε⁻¹ := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

structure OAI.Problem326.ReactionNetwork line 9

structure ReactionNetwork (d : ℕ) where  complexes : Finset (Fin d → ℕ)  reactions : Finset ((Fin d → ℕ) × (Fin d → ℕ))  source_mem : ∀ e ∈ reactions, e.1 ∈ complexes  target_mem : ∀ e ∈ reactions, e.2 ∈ complexes  source_ne_target : ∀ e ∈ reactions, e.1 ≠ e.2

abbreviation OAI.Problem326.Reaction line 16

abbrev Reaction {d : ℕ} (N : ReactionNetwork d) :=  {e : (Fin d → ℕ) × (Fin d → ℕ) // e ∈ N.reactions}

definition OAI.Problem326.HasReaction line 19

def HasReaction {d : ℕ} (N : ReactionNetwork d)    (y z : Fin d → ℕ) : Prop :=  (y, z) ∈ N.reactions

definition OAI.Problem326.WeaklyReversible line 23

def WeaklyReversible {d : ℕ} (N : ReactionNetwork d) : Prop :=  ∀ e : Reaction N,    Relation.TransGen (HasReaction N) e.val.2 e.val.1

definition OAI.Problem326.monomial line 27

def monomial {d : ℕ} (x : Fin d → ℝ) (y : Fin d → ℕ) : ℝ :=  Finset.univ.prod (fun i => x i ^ y i)

definition OAI.Problem326.massAction line 30

def massAction {d : ℕ} (N : ReactionNetwork d)    (κ : Reaction N → ℝ) (x : Fin d → ℝ) : Fin d → ℝ :=  fun i => Finset.univ.sum (fun e : Reaction N =>    κ e * monomial x e.val.1 *      ((e.val.2 i : ℝ) - (e.val.1 i : ℝ)))

definition OAI.Problem326.PositiveState line 36

def PositiveState {d : ℕ} (x : Fin d → ℝ) : Prop :=  ∀ i, 0 < x i

definition OAI.Problem326.IsGlobalForwardSolution line 39

def IsGlobalForwardSolution {d : ℕ} (N : ReactionNetwork d)    (κ : Reaction N → ℝ) (x0 : Fin d → ℝ)    (x : ℝ → (Fin d → ℝ)) : Prop :=  x 0 = x0 ∧    ∀ t : ℝ, 0 ≤ t →      HasDerivAt x (massAction N κ (x t)) t

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: reaction, integer, solution, network, action, mass, reversible, weakly and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon, depending, rates, finite and constants.

Statements in Boundedness and persistence of weakly reversible mass-action systems come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Boundedness and persistence of weakly reversible mass-action systems

Theorem

Let d\ge1, let \mathcal C\subset\mathbb Z_{\ge0}^d be finite, let \mathcal R be weakly reversible, and let all reaction rates be positive constants. For every x^0\in\R_{>0}^d, the solution of (eq:mass-action) with x(0)=x^0 exists for every t\ge0. There is a number \varepsilon\in(0,1), depending on the network, the rates, and x^0, such that

\varepsilon\le x_i(t)\le\varepsilon^{-1} \qquad(t\ge0,\ 1\le i\le d).

Section 1, Introduction Read it in the paper TeX source, line 40

Show the TeX

Let $d\ge1$, let $\mathcal C\subset\mathbb Z_{\ge0}^d$ be finite, let $\mathcal R$ be
weakly reversible, and let all reaction rates be positive constants.
For every $x^0\in\R_{>0}^d$, the solution of \eqref{eq:mass-action} with
$x(0)=x^0$ exists for every $t\ge0$. There is a number
$\varepsilon\in(0,1)$, depending on the network, the rates, and $x^0$, such
that
\begin{equation}
 \varepsilon\le x_i(t)\le\varepsilon^{-1}
 \qquad(t\ge0,\ 1\le i\le d).
\end{equation}

Other possible paper matches, closest wording first

Proposition 3.3 of Boundedness and persistence of weakly reversible mass-action systems. Words of the Lean statement, its names split into words, that also appear in the theorem: forward, solution, network, action, mass, reversible and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: rates, finite and constant.
Theorem 1.1 of Uniform Permanence in Weakly Reversible Mass-Action Systems. Words of the Lean statement, its names split into words, that also appear in the theorem: global, forward, solution, network, action, mass and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: times, initial, depending, rates, finite and constants.
Corollary 3.4 of Boundedness and persistence of weakly reversible mass-action systems. Words of the Lean statement, its names split into words, that also appear in the theorem: network, action, mass, reversible, weakly and positive. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: rates, finite and constant.
Corollary 2.2 of Boundedness and persistence of weakly reversible mass-action systems. Words of the Lean statement, its names split into words, that also appear in the theorem: bound. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Corollary 2.2 of Uniform Permanence in Weakly Reversible Mass-Action Systems. Words of the Lean statement, its names split into words, that also appear in the theorem: bound.
Proposition 3.1 of Uniform Permanence in Weakly Reversible Mass-Action Systems. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

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 (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 149’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: Lean gives, for each positive initial state x0, an ε = ε(N, κ, x0) with ε ≤ x_i(t) ≤ 1/ε for all t ≥ 0 (boundedness and persistence of each trajectory). The summary's permanence (a common compact convex forward-invariant absorbing set for every positive stoichiometric class, uniform over its trajectories) is not stated; docs admit 'the bound may depend on ... the initial state'.
Definitions the group asks readers to compare with the paper: IsGlobalForwardSolution N κ x0 x := x 0 = x0 ∧ ∀ t ≥ 0, HasDerivAt x (massAction N κ (x t)) t (two-sided derivative at t = 0, harmless for forward solutions of a polynomial field)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The headline claims classwise permanence (October 5 manuscript 'Uniform Permanence in Weakly Reversible Mass-Action Systems'); the Comparator statement gives only boundedness and persistence with initial-state-dependent ε (September 25 manuscript). The scope note states the initial-state dependence but never says permanence is omitted.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves this for every finite weakly reversible reaction network with positive constant reaction rates and every strictly positive initial state. A global forward solution exists, and one $\varepsilon\in(0,1)$ bounds every concentration of every global forward solution between $\varepsilon$ and $\varepsilon^{-1}$ for all nonnegative times. The bound may depend on the network, rates, and initial state.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

60 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 18 to its own declarations, 23 to Mathlib’s documentation. 9 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves global_bounded_persistent_solution exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 8 other declarations of the file; with the theorem itself they span 38 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.Problem326 are listed without it.

structure ReactionNetworkline 9in the statementlibrary
abbrev Reactionline 16in the statementlibrary
def HasReactionline 19library
def WeaklyReversibleline 23in the statementlibrary
def monomialline 27library
def massActionline 30library
def PositiveStateline 36in the statementlibrary
def IsGlobalForwardSolutionline 39in the statementlibrary
theorem global_bounded_persistent_solutionline 46selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionnamespace Problem326structure ReactionNetwork (d : ℕ) where  complexes : Finset (Fin d → ℕ)  reactions : Finset ((Fin d → ℕ) × (Fin d → ℕ))  source_mem : ∀ e ∈ reactions, e.1 ∈ complexes  target_mem : ∀ e ∈ reactions, e.2 ∈ complexes  source_ne_target : ∀ e ∈ reactions, e.1 ≠ e.2abbrev Reaction {d : ℕ} (N : ReactionNetwork d) :=  {e : (Fin d → ℕ) × (Fin d → ℕ) // e ∈ N.reactions}def HasReaction {d : ℕ} (N : ReactionNetwork d)    (y z : Fin d → ℕ) : Prop :=  (y, z) ∈ N.reactionsdef WeaklyReversible {d : ℕ} (N : ReactionNetwork d) : Prop :=  ∀ e : Reaction N,    Relation.TransGen (HasReaction N) e.val.2 e.val.1def monomial {d : ℕ} (x : Fin d → ℝ) (y : Fin d → ℕ) : ℝ :=  Finset.univ.prod (fun i => x i ^ y i)def massAction {d : ℕ} (N : ReactionNetwork d)    (κ : Reaction N → ℝ) (x : Fin d → ℝ) : Fin d → ℝ :=  fun i => Finset.univ.sum (fun e : Reaction N =>    κ e * monomial x e.val.1 *      ((e.val.2 i : ℝ) - (e.val.1 i : ℝ)))def PositiveState {d : ℕ} (x : Fin d → ℝ) : Prop :=  ∀ i, 0 < x idef IsGlobalForwardSolution {d : ℕ} (N : ReactionNetwork d)    (κ : Reaction N → ℝ) (x0 : Fin d → ℝ)    (x : ℝ → (Fin d → ℝ)) : Prop :=  x 0 = x0 ∧    ∀ t : ℝ, 0 ≤ t →      HasDerivAt x (massAction N κ (x t)) ttheorem global_bounded_persistent_solution :    ∀ (d : ℕ) (N : ReactionNetwork d) (κ : Reaction N → ℝ),      0 < d → WeaklyReversible N → (∀ e, 0 < κ e) →      ∀ x0 : Fin d → ℝ, PositiveState x0 →      ∃ ε : ℝ, 0 < ε ∧ ε < 1 ∧        (∃ x : ℝ → (Fin d → ℝ), IsGlobalForwardSolution N κ x0 x) ∧        ∀ x : ℝ → (Fin d → ℝ), IsGlobalForwardSolution N κ x0 x →        ∀ t : ℝ, 0 ≤ t → ∀ i : Fin d, ε ≤ x t i ∧ x t i ≤ ε⁻¹ := by  sorryend Problem326endend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Boundedness and persistence of weakly reversible mass-action systems

Theorem

\varepsilon\le x_i(t)\le\varepsilon^{-1} \qquad(t\ge0,\ 1\le i\le d).

Section 1, Introduction Read it in the paper TeX source, line 40

Show the TeX

Let $d\ge1$, let $\mathcal C\subset\mathbb Z_{\ge0}^d$ be finite, let $\mathcal R$ be
weakly reversible, and let all reaction rates be positive constants.
For every $x^0\in\R_{>0}^d$, the solution of \eqref{eq:mass-action} with
$x(0)=x^0$ exists for every $t\ge0$. There is a number
$\varepsilon\in(0,1)$, depending on the network, the rates, and $x^0$, such
that
\begin{equation}
 \varepsilon\le x_i(t)\le\varepsilon^{-1}
 \qquad(t\ge0,\ 1\le i\le d).
\end{equation}

Configuration and file facts

File: lean/ComparatorChallenges/MassAction.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 149 link it.
Solution module: OAI.Analysis.MassAction.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 8 local vocabulary declarations
Lines of Lean, without comments or blank lines: 38
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas