Positive-temperature Bose–Einstein condensation and exact quantum depletion
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit. (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 the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Six other possible matches are listed below the card, among them Theorem 1.1 in A density-uniform condensate bound for dilute Bose gases and Theorem 1.1(Condensation in every ground state) in Ground-state condensation in the dilute hard-sphere gas.
For every a>0, there is \rho_*(a)>0 such that for each fixed 0<\rho<\rho_*(a) there is a fixed temperature T=T(a,\rho)>0 for which
We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.
We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass \(8/(3\sqrt\pi)\), giving the depletion fraction \(\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})\) for density ρ and hard-sphere exclusion distance a.
We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is \(\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})\) as \(\rho\downarrow0\). The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.
For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.
We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.
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.
There are absolute constants \varepsilon_0,c_0>0 with the following property. Fix a,\rho>0 satisfying \rho a^3<\varepsilon_0. For every sequence of particle numbers N_k and lengths L_k such that
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 a weaker statement than the advertised result (their term: “weaker-statement”).
Their note:condensation_with_mixed : ∃ ε₀ c₀ > 0, ∀ a ρ, ρ a^3 < ε₀ → ∀ thermodynamic sequences (N_k, L_k), (∀ pure ground vectors Ψ_k, c₀ ≤ liminf occupation Ψ_k) ∧ (∀ density operators supported on the ground space, c₀ ≤ liminf mixedOccupation) is ZERO-temperature ground-state condensation of the hard-sphere gas with a non-sharp lower bound c₀ (docs: ‘concerns ground states, without a positive-temperature assertion’). The summary’s headline claims are not stated: BEC for the exact canonical Gibbs state at a strictly positive volume-independent temperature, and the sharp Bogoliubov leading quantum-depletion law (thermodynamic limit before dilute limit) for hard spheres and bounded finite-range potentials.
Definitions the group asks readers to compare with the paper:Hand-built hard-sphere gas: Dirichlet form domain dirichletDomain N L a := closure (range SmoothTest.jet) (smooth compactly supported functions vanishing on d(x_i,x_j) ≤ a on the torus AddCircle L ^3), energy u := Σ ‖∇_p u‖² (no 1/2), bosonic, IsGroundVector, occupation ψ := L^{-3} ∫_Y |∫_x ψ(x,Y)|² and a custom trace-class Density.Operator; the zero-mode occupation fraction matches the standard condensate fraction, but all objects are hand-written.
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker 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:Neither headline claim is stated. What is stated — ground-state condensation with a uniform positive lower bound c₀ on the condensate fraction in the dilute hard-sphere gas (the ‘Ground-state condensation’ manuscript) — is weaker than the zero-temperature depletion law (sentence 2) and says nothing at positive temperature (sentence 1). The positive-temperature omission is declared; the depletion-law omission (hard spheres and bounded potentials) is not.
“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 ground-state Bose–Einstein condensation in the dilute hard-sphere gas. There are absolute constants $\varepsilon_0,c_0>0$ such that, whenever the density $\rho$ and hard-sphere radius $a$ satisfy $\rho a^3<\varepsilon_0$, the condensate occupation fraction has limit inferior at least $c_0$ along every thermodynamic sequence with $N/L^3\to\rho$.
The bound covers every pure ground state, every mixed state supported on the ground space, and the normalized ground-space projection. It is uniform in the gas parameter within this range and concerns ground states, without a positive-temperature assertion.
AI-drafted comparisons, with no recorded human review
Uniform ground-state condensation in the dilute hard-sphere gasHardSphere
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem and its corollary: ground-state condensation with fraction at least c₀ in the dilute hard-sphere gas for every ground state and every state on the ground space. The positive-temperature result of the family’s later paper is 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.
Uniform ground-state condensation in the dilute hard-sphere gasHardSphere
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.
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.
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.
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 267The 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
The challenge statements do not imply the headline claim: they state a nontrivially weaker 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.