Stable blowup for the defocusing Schrödinger equation
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in \(H^k(\mathbb T^{12})\), with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime. (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 two independent assessments disagree on how much of the result described above is expressed by the Lean statements.Read the assessments
belongs to H^k almost surely and has positive probability of finite-time blowup. In particular, the following assertion is false: for every energy-supercritical pair (d,p) there is a finite \alpha_0(d,p) such that, for every \alpha>\alpha_0(d,p) and every
d/2+2<m<\alpha-d/2,
the local H^m solution with the Gaussian datum extends almost surely to C(\mathbb R;H^m(\mathbb T^d)).
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There exist an odd integer p\geq3, a finite integer k>8, and a nonempty open set \mathcal U\subset H^k(\mathbb T^{12};\mathbb C) with the following property. For every u_0\in\mathcal U, the unique local solution of (1.1) extends to
u\in C([0,T);H^k)\cap C^1([0,T);H^{k-2})
for some finite T>0, but has no continuous H^k extension through T. More precisely, there are x_*\in\mathbb T^{12} and c_0>0 such that
\lim_{t\uparrow T}(T-t)^{1/(p-1)}|u(t,x_*)|=c_0.
The power can be chosen sufficiently large, and k is then fixed after that choice.
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
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We prove stable self-similar finite-time blowup for a supercritical defocusing nonlinear Schrödinger equation on the twelve-dimensional torus. For a sufficiently large odd power, the blowup initial data contain a nonempty open set in a high Sobolev space. As a consequence, Gaussian Fourier initial data of arbitrarily high Sobolev regularity can have positive probability of finite-time blowup.
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.
DefocusingNLS
The paper states
Corollary 1.2(Failure of universal Gaussian global existence)
belongs to H^k almost surely and has positive probability of finite-time blowup. In particular, the following assertion is false: for every energy-supercritical pair (d,p) there is a finite \alpha_0(d,p) such that, for every \alpha>\alpha_0(d,p) and every
d/2+2<m<\alpha-d/2,
the local H^m solution with the Gaussian datum extends almost surely to C(\mathbb R;H^m(\mathbb T^d)).
The words they share: almost, blowup, existence, finite, gaussian, global, index, initial, particular, positive, power, probability, surely, theorem, time.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DefocusingNLS.lean, the release lists it as a supporting statement.
theoremstable_blowup (p₀ : ℕ) : ∃ p k : ℕ, p₀ ≤ p ∧ Odd p ∧ 3 ≤ p ∧ ∃ hk : 8 < (k : ℝ), ∃ U : Set FourierL2, U.Nonempty ∧ IsOpen U ∧ ∀ f ∈ U, IsClassicalDefocusingFlow k (by linarith) ((p - 1) / 2) f ∧ HasFiniteTimeSelfSimilarBlowup k (by linarith) ((p - 1) / 2) (1 / ((p : ℝ) - 1)) f := 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.
Three definitions are compared by name and type only: sobolevProduct, schrodingerFlow, sobolevOddPower.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
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:stable_blowup: for every p₀ there are an odd p ≥ max(p₀,3), an integer k > 8 and a nonempty open U ⊆ H^k(𝕋¹²) (as weighted ℓ² Fourier data) such that every f ∈ U has a maximal classical solution of i∂ₜu = -Δu + |u|^(p-1)u (interaction-picture ODE with sobolevOddPower) that blows up self-similarly at a finite time T, ‖u(t)(x)‖ ~ c(T-t)^(-1/(p-1)); the quantifier gives arbitrarily large odd powers rather than visibly all large ones, and the Gaussian-data corollaries are extra. The solution cone uses the external package FixedPointTheorems.
Definitions the group asks readers to compare with the paper:The challenge file contains by sorry inside a definition: sobolevProduct k _hk f g : FourierL2 := ⟨fun n => weight * convolution coefficient, by sorry⟩ (the ℓ² membership proof of the Sobolev-algebra product; it is a Prop field so it cannot change the function, but it is a definition-level sorry); The PDE is encoded in Fourier space: lowerSobolevInclusion/lowerSobolevGenerator contraction multipliers, maximalSobolevInteractionFlow defined by Classical.choose of a patch with fallback 0 and maximalSobolevInteractionDomain as a union of patches, HasFiniteTimeSelfSimilarBlowup (pointwise value at a torus point via sobolevTorusFunction, sup of the domain = T, discontinuity at T); about a dozen statement-level run_cmd Lean.modifyEnv... auxLemmasExt meta commands and attribute [local irreducible]
“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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name:Arbitrarily large odd powers (∀ p₀ ∃ p ≥ p₀) instead of every sufficiently large odd power; for each such p one natural Sobolev index k > 8 is selected.
“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.
The formalization constructs stable self-similar finite-time blowup for defocusing nonlinear Schrödinger equations on the twelve-dimensional torus. For every prescribed lower bound on the power, it selects an odd power at least that large and a Sobolev index $k>8$ for which a nonempty open set of $H^k$ initial data has the stated classical blowup behavior. The selected quantifier gives arbitrarily large odd powers, rather than every sufficiently large odd power.
For each such construction, Gaussian Fourier data with decay exponent $\alpha>k+6$ assign positive probability to the blowup set, and almost-sure global existence fails. The exponent $\alpha$ may be arbitrarily large.
AI-drafted comparisons, with no recorded human review
Stable self-similar blowup and positive-probability Gaussian blowupDefocusingNLS
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (an open set of H^k data with stable self-similar blowup for the defocusing equation on the twelve-torus, for arbitrarily large odd powers) and the Gaussian corollary. The Schrödinger flow and two Sobolev operations are Comparator holes filled from OpenAI’s library.
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.
Stable self-similar blowup and positive-probability Gaussian blowupDefocusingNLS
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.
Jack McCarthy, OpenAI math: Comparator fixes (mirror of openai/math at fd4aeeb), 9 October 2026. Source
DefocusingNLS challenge repaired in the fork and reported passing Comparator.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
Other proofs and exact calculations
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Appendix A of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus, 24 September 2026
Reported by
cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every finite comparison of Appendix cert:arithmetic (integer box, boundary-form positivity, winding sign table and margins, high-angular polynomial, profile enclosures and degree comparisons, exterior positivity), and the low-angular counts 2,1,0,0 for P_l by Routh-Hurwitz; NOT Proposition free:main itself nor the blowup theorem, which rest on the analytic lemmas of Section sec:free”
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
Jack McCarthy, OpenAI math: Comparator fixes (mirror of openai/math at fd4aeeb), 9 October 2026. Source
DefocusingNLS challenge contained a sorry inside a definition; fixed by promoting the proof to a theorem target.
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 construction and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
From github.com/BenKnill/openai-math-hol-cards, 8 October 2026. Source
Not re-run, but the configuration review notes that the hole sobolevProduct contains a sorry membership proof inside an otherwise written definition, and Comparator’s hole comparison leaves all three hole bodies uncompared.
Questions awaiting review
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
Read the definitions the solution supplies
OpenAI’s proof file supplies some definitions. Atlas records no comparison confirming that they mean the same objects as the paper’s definitions.sobolevProduct, schrodingerFlow, sobolevOddPowerOpen the form
Identify the theorem the catalogue describes
Family 371No 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
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 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; 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 with Lean’s kernel: accepted (the check took 1174 s; peak memory 7.6 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Appendix A of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus, 24 September 2026
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every finite comparison of Appendix cert:arithmetic (integer box, boundary-form positivity, winding sign table and margins, high-angular polynomial, profile enclosures and degree comparisons, exterior positivity), and the low-angular counts 2,1,0,0 for P_l by Routh-Hurwitz; NOT Proposition free:main itself nor the blowup theorem, which rest on the analytic lemmas of Section sec:free”