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Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L1 and satisfy exact local conservation of mass, momentum and kinetic energy. (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

What the papers state

Theorem 1.1

In Nonuniqueness for the periodic hard-sphere Boltzmann equation, 23 September 2026

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.

There is a nonnegative density F_0 on \T^3\times\R^3, with bounded velocity support and

\int_{\T^3\times\R^3} F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,

for which the hard-sphere Boltzmann equation has two distinct global admissible solutions, in the sense of Definition 2.1, with initial datum F_0. On some common initial interval both solutions are strongly continuous in L^1, and their collision gains and losses are integrable in space-time-velocity.

Theorem 1.2

In Nonuniqueness with local conservation for the hard-sphere Boltzmann equation, 5 October 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

There exist R<\infty and a nonnegative datum F_0 with

\supp_vF_0\subset B_R(0),\qquad \int_\Omega F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,

for which (1.1) has two global entropy solutions with local conservation in the sense of Definition 1.1, differing on a set of positive (t,x,v)-measure. Both solutions belong to C([0,\infty);L^1(\Omega)). Moreover, for every finite T and every k\ge0, they satisfy

\int_0^T\!\!\int_\Omega \langle v\rangle^k\bigl[Q^+(F,F)+Q^-(F,F)\bigr] \dd x\dd v\dd t<\infty, \qquad \langle v\rangle=(1+|v|^2)^{1/2}.

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; one independent group reports that Comparator accepted a run for this file at the version shown here.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

5 October 2026, current edition.

Read it herePDF

We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy that gives rise to two distinct such solutions. Both are strongly continuous in L1, and their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval.

Nonuniqueness for the periodic hard-sphere Boltzmann equation

23 September 2026, current edition.

Read it herePDF

We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on \(\mathbb T^3\times\mathbb R^3\). This density has bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities. On a common initial interval, they are strongly continuous in L1, and their collision gains and losses are integrable.

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.

BoltzmannNonuniqueness

The paper states

Theorem 1.1

In Nonuniqueness for the periodic hard-sphere Boltzmann equation, 23 September 2026

There is a nonnegative density F_0 on \T^3\times\R^3, with bounded velocity support and

\int_{\T^3\times\R^3} F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,

for which the hard-sphere Boltzmann equation has two distinct global admissible solutions, in the sense of Definition 2.1, with initial datum F_0. On some common initial interval both solutions are strongly continuous in L^1, and their collision gains and losses are integrable in space-time-velocity.

The words they share: space, sphere, strongly, log, support, definition, datum, gains, common, losses, continuous, integrable, admissible, collision, global, density, nonnegative, initial, velocity, time, times, bounded, interval, periodic, nonuniqueness, distinct, boltzmann, two, hard, solutions.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Two periodic hard-sphere Boltzmann solutions with the same data

BoltzmannNonuniqueness.lean, the release lists it as a supporting statement.

theorem nonuniqueness :    ∃ f₀ : Density, BoundedVelocitySupport f₀ ∧      ∃ (F G : Evolution) (T : ℝ), AdmissibleGlobal f₀ F ∧ AdmissibleGlobal f₀ G ∧        StrongEarly T F ∧ StrongEarly T G ∧        ∃ t ∈ Set.Icc 0 T, ¬(F t =ᵐ[MeasureTheory.volume] G t) :=  sorry

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.

One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.

Read the complete Lean statement

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: nonuniqueness gives bounded-velocity-support data f₀ and two AdmissibleGlobal renormalized solutions F, G (all renormalizations β, gain/loss integrability, weak continuity, local mass conservation as a weak identity, global momentum conservation, global energy inequality ≤ and entropy-dissipation inequality) that are strongly L¹-continuous with integrable gain/loss on a common [0,T] only and differ at some t ≤ T. The summary’s exact local conservation of momentum and kinetic energy, and strong L¹ continuity for all times, are not stated (docs: ‘later local-conservation refinements are outside it’).

Definitions the group asks readers to compare with the paper: Hand-built kinetic setup: gain/loss as Bochner integrals over ℝ³×S² with sphereArea := volume.toSphere (the collision_fibers hypotheses prevent silent zero values of non-integrable integrals), extendedEntropyProduction (lower semicontinuous extension of (A-B)log(A/B)), entropyDissipation, relativeEntropy against a normalized Maxwellian; a statement-level run_cmd Lean.modifyEnv... auxLemmasExt meta command

“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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).

The gap they name: The October refinement in the headline is not stated: exact LOCAL conservation of momentum and kinetic energy, and strong L1 continuity for all times. The Lean states the September result: local mass, total momentum, the energy and entropy inequalities, and strong L1 continuity on [0,T].

“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization proves nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions on $\mathbb T^3\times\mathbb R^3$ with the same nonnegative initial density. The data have bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities.

On a common initial interval they are strongly continuous in $L^1$, with integrable collision gains and losses. These are the conditions of the selected periodic result; later local-conservation refinements are outside it.

AI-drafted comparisons, with no recorded human review

  • Two periodic hard-sphere Boltzmann solutions with the same data BoltzmannNonuniqueness

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: two distinct global renormalised solutions of the periodic hard-sphere Boltzmann equation with the same initial density. The local-conservation refinements of the family’s later paper are outside 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.

Two periodic hard-sphere Boltzmann solutions with the same data BoltzmannNonuniqueness

One report, at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal 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.10 October 2026Original report

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.

Within this family

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

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Explain a result, compare statements or propose a correction

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What people have said

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 counterexample and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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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 363 The 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

Compare the proposed paper and Lean matches

  • Family 363 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form