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Lean statement

Stable self-similar blowup and positive-probability Gaussian blowup

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.

Theorems selected for verification: OAI.DefocusingNLS.stable_blowup OAI.DefocusingNLS.gaussian_blowup OAI.DefocusingNLS.gaussian_not_almost_sure_global

Lean statement file: DefocusingNLS.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 371 link it.

See the candidate theoremWho ran the proof-verification programReport 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 stable_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 := by  sorrytheorem gaussian_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) ∧      ∀ α : ℝ, (k : ℝ) + 6 < α →        (∀ᵐ g ∂fourierGaussianLaw, Memℓp (weightedGaussianCoefficient k α g) 2) ∧        (∀ᵐ g ∂fourierGaussianLaw, ∀ n,          sobolevFourierCoefficient k (weightedGaussianVector k α g) n =            (1 + ‖n‖ ^ 2) ^ (-α / 2) • g n) ∧        0 < weightedGaussianLaw k α U := by  sorrytheorem gaussian_not_almost_sure_global (p₀ : ℕ) :    ∃ p k : ℕ, p₀ ≤ p ∧ Odd p ∧ 3 ≤ p ∧ ∃ hk : 8 < (k : ℝ),      ∀ α : ℝ, (k : ℝ) + 6 < α →        ¬ ∀ᵐ f ∂weightedGaussianLaw k α,          maximalSobolevInteractionDomain k (by linarith) ((p - 1) / 2) f = univ := 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.

  • definition OAI.DefocusingNLS.frequencyLattice line 22

    noncomputable def frequencyLattice : Submodule ℤ (EuclideanSpace ℝ (Fin 12)) :=  Submodule.span ℤ (Set.range (EuclideanSpace.basisFun (Fin 12) ℝ).toBasis)

  • definition OAI.DefocusingNLS.complexGaussian line 35

    noncomputable def complexGaussian (v : ℝ≥0) : Measure ℂ :=  ((gaussianReal 0 v).prod (gaussianReal 0 v)).map    Complex.equivRealProdCLM.symm

  • definition OAI.DefocusingNLS.fourierGaussianLaw line 46

    noncomputable def fourierGaussianLaw : Measure (frequencyLattice → ℂ) :=  Measure.infinitePi (fun _ : frequencyLattice => complexGaussian (1 / 2))

  • definition OAI.DefocusingNLS.sobolevVariance line 49

    noncomputable def sobolevVariance (k α : ℝ) (n : frequencyLattice) : ℝ :=  (1 + ‖n‖ ^ 2) ^ (k - α)

  • definition OAI.DefocusingNLS.weightedGaussianCoefficient line 52

    noncomputable def weightedGaussianCoefficient (k α : ℝ)    (g : frequencyLattice → ℂ) (n : frequencyLattice) : ℂ :=  Real.sqrt (sobolevVariance k α n) • g nrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • abbreviation OAI.DefocusingNLS.FourierL2 line 58

    abbrev FourierL2 := lp (fun _ : frequencyLattice => ℂ) (2 : ℝ≥0∞)

  • definition OAI.DefocusingNLS.weightedGaussianVector line 63

    noncomputable def weightedGaussianVector (k α : ℝ)    (g : frequencyLattice → ℂ) : FourierL2 :=  ∑' n, lp.single 2 n (weightedGaussianCoefficient k α g n)

  • definition OAI.DefocusingNLS.sobolevFourierCoefficient line 67

    noncomputable def sobolevFourierCoefficient (k : ℝ) (f : FourierL2)    (n : frequencyLattice) : ℂ :=  (1 + ‖n‖ ^ 2) ^ (-k / 2) • f n

  • definition OAI.DefocusingNLS.weightedGaussianLaw line 71

    noncomputable def weightedGaussianLaw (k α : ℝ) : Measure FourierL2 :=  fourierGaussianLaw.map (weightedGaussianVector k α)run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • abbreviation OAI.DefocusingNLS.SchrodingerTorus line 76

    abbrev SchrodingerTorus := Fin 12 → AddCircle (2 * Real.pi)

  • definition OAI.DefocusingNLS.frequencyCoordinates line 80

    noncomputable def frequencyCoordinates (n : frequencyLattice) : Fin 12 → ℤ :=  ((EuclideanSpace.basisFun (Fin 12) ℝ).toBasis.restrictScalars ℤ).repr n

  • definition OAI.DefocusingNLS.torusCharacter line 83

    noncomputable def torusCharacter (n : frequencyLattice) : C(SchrodingerTorus, ℂ) where  toFun x := ∏ j : Fin 12, fourier (frequencyCoordinates n j) (x j)  continuous_toFun := by fun_prop

  • definition OAI.DefocusingNLS.sobolevTorusTerm line 87

    noncomputable def sobolevTorusTerm (k : ℝ) (f : FourierL2) (n : frequencyLattice) :    C(SchrodingerTorus, ℂ) :=  sobolevFourierCoefficient k f n • torusCharacter n

  • definition OAI.DefocusingNLS.sobolevTorusFunction line 91

    noncomputable def sobolevTorusFunction (k : ℝ) (f : FourierL2) : C(SchrodingerTorus, ℂ) :=  ∑' n, sobolevTorusTerm k f nrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • definition OAI.DefocusingNLS.sobolevProductWeight line 96

    noncomputable def sobolevProductWeight (k : ℝ) (n : frequencyLattice) : ℝ :=  (1 + ‖n‖ ^ 2) ^ (k / 2)run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • definition OAI.DefocusingNLS.sobolevProductCoefficient line 101

    noncomputable def sobolevProductCoefficient (k : ℝ) (f g : FourierL2)    (n : frequencyLattice) : ℂ :=  ∑' m, sobolevFourierCoefficient k f m * sobolevFourierCoefficient k g (n - m)

  • definition OAI.DefocusingNLS.sobolevProduct line 105

    noncomputable def sobolevProduct (k : ℝ) (_hk : 6 < k) (f g : FourierL2) : FourierL2 :=  ⟨fun n => (sobolevProductWeight k n : ℂ) * sobolevProductCoefficient k f g n,    by sorry⟩run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • definition OAI.DefocusingNLS.fourierConjugate line 111

    noncomputable def fourierConjugate (f : FourierL2) : FourierL2 :=  ⟨fun n => conj (f (-n)), by    apply memℓp_gen    simpa only [Complex.norm_conj, Function.comp_def, Equiv.neg_apply] using      (Equiv.neg frequencyLattice).summable_iff.mpr        ((lp.memℓp f).summable (by norm_num : 0 < (2 : ℝ≥0∞).toReal))⟩run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • definition OAI.DefocusingNLS.sobolevOddPower line 120

    noncomputable def sobolevOddPower (k : ℝ) (hk : 6 < k) : ℕ → FourierL2 → FourierL2  | 0, f => f  | m + 1, f => sobolevProduct k hk      (sobolevProduct k hk (sobolevOddPower k hk m f) (fourierConjugate f)) frun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • definition OAI.DefocusingNLS.schrodingerMultiplier line 127

    noncomputable def schrodingerMultiplier (t : ℝ) (n : frequencyLattice) : ℂ :=  Complex.exp (-Complex.I * ((t * ‖n‖ ^ 2 : ℝ) : ℂ))

  • definition OAI.DefocusingNLS.schrodingerFlow line 130

    noncomputable def schrodingerFlow (t : ℝ) : FourierL2 →ₗᵢ[ℂ] FourierL2 where
    Show the remaining 31 lines
      toFun f := ⟨fun n => schrodingerMultiplier t n * f n, by    apply (lp.memℓp f).mono'    intro n    simp only [norm_mul, schrodingerMultiplier, Complex.norm_exp, Complex.mul_re,      Complex.neg_re, Complex.neg_im, Complex.I_re, Complex.I_im,      Complex.ofReal_re, Complex.ofReal_im, neg_zero, zero_mul, mul_zero,      sub_zero, Real.exp_zero, one_mul]    exact le_rfl⟩  map_add' f g := by    ext n    change schrodingerMultiplier t n * (f n + g n) =      schrodingerMultiplier t n * f n + schrodingerMultiplier t n * g n    exact mul_add _ _ _  map_smul' c f := by ext n; simp [mul_left_comm]  norm_map' f := by    have hm (n : frequencyLattice) : ‖schrodingerMultiplier t n‖ = 1 := by      simp only [schrodingerMultiplier, Complex.norm_exp, Complex.mul_re, Complex.neg_re,        Complex.neg_im, Complex.I_re, Complex.I_im, Complex.ofReal_re, Complex.ofReal_im,        neg_zero, zero_mul, mul_zero, sub_zero, Real.exp_zero]    apply le_antisymm    · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      intro n      change ‖schrodingerMultiplier t n * f n‖ ≤ ‖f n‖      rw [norm_mul, hm, one_mul]    · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      intro n      change ‖f n‖ ≤ ‖schrodingerMultiplier t n * f n‖      rw [norm_mul, hm, one_mul]run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • definition OAI.DefocusingNLS.fourierContraction line 163

    noncomputable def fourierContraction (m : frequencyLattice → ℂ)    (hm : ∀ n, ‖m n‖ ≤ 1) : FourierL2 →L[ℂ] FourierL2 :=
    Show the remaining 23 lines
      LinearMap.mkContinuous    { toFun := fun f => ⟨fun n => m n * f n, (lp.memℓp f).mono' (by        intro n        rw [norm_mul]        simpa using mul_le_mul_of_nonneg_right (hm n) (norm_nonneg (f n)))⟩      map_add' := by        intro f g        ext n        change m n * (f n + g n) = m n * f n + m n * g n        exact mul_add _ _ _      map_smul' := by        intro c f        ext n        change m n * (c * f n) = c * (m n * f n)        ring }    1 (by      intro f      simp only [one_mul]      apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      intro n      change ‖m n * f n‖ ≤ ‖f n‖      rw [norm_mul]      simpa using mul_le_mul_of_nonneg_right (hm n) (norm_nonneg (f n)))

  • definition OAI.DefocusingNLS.lowerSobolevInclusion line 189

    noncomputable def lowerSobolevInclusion : FourierL2 →L[ℂ] FourierL2 :=  fourierContraction (fun n => (((1 + ‖n‖ ^ 2)⁻¹ : ℝ) : ℂ)) (by    intro n    rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (by positivity)]    exact inv_le_one_of_one_le₀ (le_add_of_nonneg_right (sq_nonneg _)))

  • definition OAI.DefocusingNLS.lowerSobolevGenerator line 195

    noncomputable def lowerSobolevGenerator : FourierL2 →L[ℂ] FourierL2 :=  fourierContraction (fun n => -Complex.I * (((‖n‖ ^ 2 / (1 + ‖n‖ ^ 2)) : ℝ) : ℂ)) (by    intro n    rw [norm_mul, norm_neg, Complex.norm_I, one_mul, Complex.norm_real,      Real.norm_eq_abs, abs_of_nonneg (by positivity)]    exact (div_le_one (by positivity)).mpr (by linarith [sq_nonneg ‖n‖]))run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

7 more definitions are reached the same way; the full file lists them.

Definitions taken from OpenAI’s library 3

Comparator compares only the name and type of each definition listed here. When a theorem is configured, its proof uses the definition in OpenAI’s solution library. Read that definition as well as the one in this file when comparing the statement with the paper.

  • Comparator checks the name and type of sobolevProduct; its definition is in OpenAI’s library, in the modules that OAI.MathematicalPhysics.DefocusingNLS.MainTheorems imports.
  • Comparator checks the name and type of schrodingerFlow; its definition is in OpenAI’s library, in the modules that OAI.MathematicalPhysics.DefocusingNLS.MainTheorems imports.
  • Comparator checks the name and type of sobolevOddPower; its definition is in OpenAI’s library, in the modules that OAI.MathematicalPhysics.DefocusingNLS.MainTheorems imports.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: almost, blowup, existence, finite, gaussian, global, index, initial and seven more.

Corollary 1.2 of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus

Corollary (Failure of universal Gaussian global existence)

For the fixed power p and index k in Theorem thm:blowup, and for every \alpha>k+6, the initial datum

u_0^\omega(x)=\sum_{n\in\mathbb Z^{12}} g_n(\omega)\langle n\rangle^{-\alpha}e^{in\cdot x}

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)).

Section 1, Introduction Read it in the paper TeX source, line 93
Show the TeX
For the fixed power $p$ and index $k$ in Theorem~\ref{thm:blowup}, and
for every $\alpha>k+6$, the initial datum
\begin{equation}
 u_0^\omega(x)=\sum_{n\in\mathbb Z^{12}}
       g_n(\omega)\langle n\rangle^{-\alpha}e^{in\cdot x}
\end{equation}
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 $C(\mathbb R;H^m(\mathbb T^d))$.

Other possible paper matches, closest wording first

  • Theorem 1.1 of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus. Words of the Lean statement, its names split into words, that also appear in the theorem: eq, nonempty, data, odd, finite, power and six more. Found in the paper’s title, which its main theorem states: defocusing, similar, self, stable and torus. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sufficiently.
  • Proposition 2.1 of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus. Words of the Lean statement, its names split into words, that also appear in the theorem: zero, im, one, re, complex and real. Found in the paper’s title, which its main theorem states: defocusing, similar, self, blowup, stable and torus.
  • Proposition 5.5 of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus. Words of the Lean statement, its names split into words, that also appear in the theorem: norm, coordinate, positive, pi, maps, continuous and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: prescribed, bound, schr and sufficiently.
  • Proposition 6.2 of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus. Words of the Lean statement, its names split into words, that also appear in the theorem: functions, coordinate, initial, pi, map, continuous and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sufficiently.
  • Theorem 4.1 of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/2, generator, spans, conjugate, re, complex and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sufficiently.
  • Proposition 3.1 of Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, coefficients, function, positive, particular, space and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounded.

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; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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; the run was still in progress at this snapshot (192 accepted, 224 pending).

  • Accepted Their term: “accepted”.

    Re-run through Comparator with Lean’s kernel: accepted (the check took 1174 s; peak memory 7.6 GiB).

    Reported 10 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.

Definitions compared by name and type only (3): sobolevProduct, schrodingerFlow, sobolevOddPower. Their bodies come from OpenAI’s solution library.

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 371’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.

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express.

  • MathVet

    Finds the whole advertised result stated or implied Their term: “full”.

    The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record

    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]

  • Carlos Toledo’s cert-machine

    Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.

    The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record

    Their note: 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.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”

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

301 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: 127 to its own declarations, 136 to Mathlib’s documentation. 35 declared names also link to the same declaration in OpenAI’s library. 4 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAI/-! # Stable self-similar blowup on the twelve-dimensional torusTheorem 1.1 and Corollary 1.2 of OpenAI, *Stable self-similar blowup for asupercritical defocusing Schrödinger equation on the torus* (September 2026).The Sobolev space is represented by its weighted square-summable Fouriercoefficients. The maximal flow is the union of classical interaction solutions.-/noncomputable sectionopen Set Filter Topology MeasureTheory ProbabilityTheoryopen scoped ENNReal NNReal ComplexConjugate ContDiffattribute [local instance 2000] instPolynormableSpace  TopologicalSpace.PseudoMetrizableSpace.regularSpacenamespace DefocusingNLSnoncomputable def frequencyLattice : Submodule ℤ (EuclideanSpace ℝ (Fin 12)) :=  Submodule.span ℤ (Set.range (EuclideanSpace.basisFun (Fin 12) ℝ).toBasis)instance : DiscreteTopology frequencyLattice := by  unfold frequencyLattice  infer_instanceinstance : IsZLattice ℝ frequencyLattice := by  unfold frequencyLattice  infer_instancerun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def complexGaussian (v : ℝ≥0) : Measure ℂ :=  ((gaussianReal 0 v).prod (gaussianReal 0 v)).map    Complex.equivRealProdCLM.symminstance complexGaussian_isProbabilityMeasure (v : ℝ≥0) :    IsProbabilityMeasure (complexGaussian v) := by  unfold complexGaussian  infer_instancerun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def fourierGaussianLaw : Measure (frequencyLattice → ℂ) :=  Measure.infinitePi (fun _ : frequencyLattice => complexGaussian (1 / 2))noncomputable def sobolevVariance (k α : ℝ) (n : frequencyLattice) : ℝ :=  (1 + ‖n‖ ^ 2) ^ (k - α)noncomputable def weightedGaussianCoefficient (k α : ℝ)    (g : frequencyLattice → ℂ) (n : frequencyLattice) : ℂ :=  Real.sqrt (sobolevVariance k α n) • g nrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}abbrev FourierL2 := lp (fun _ : frequencyLattice => ℂ) (2 : ℝ≥0∞)instance : MeasurableSpace FourierL2 := borel FourierL2instance : BorelSpace FourierL2 := ⟨rfl⟩noncomputable def weightedGaussianVector (k α : ℝ)    (g : frequencyLattice → ℂ) : FourierL2 :=  ∑' n, lp.single 2 n (weightedGaussianCoefficient k α g n)noncomputable def sobolevFourierCoefficient (k : ℝ) (f : FourierL2)    (n : frequencyLattice) : ℂ :=  (1 + ‖n‖ ^ 2) ^ (-k / 2) • f nnoncomputable def weightedGaussianLaw (k α : ℝ) : Measure FourierL2 :=  fourierGaussianLaw.map (weightedGaussianVector k α)run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}abbrev SchrodingerTorus := Fin 12 → AddCircle (2 * Real.pi)instance : Fact (0 < 2 * Real.pi) := ⟨Real.two_pi_pos⟩noncomputable def frequencyCoordinates (n : frequencyLattice) : Fin 12 → ℤ :=  ((EuclideanSpace.basisFun (Fin 12) ℝ).toBasis.restrictScalars ℤ).repr nnoncomputable def torusCharacter (n : frequencyLattice) : C(SchrodingerTorus, ℂ) where  toFun x := ∏ j : Fin 12, fourier (frequencyCoordinates n j) (x j)  continuous_toFun := by fun_propnoncomputable def sobolevTorusTerm (k : ℝ) (f : FourierL2) (n : frequencyLattice) :    C(SchrodingerTorus, ℂ) :=  sobolevFourierCoefficient k f n • torusCharacter nnoncomputable def sobolevTorusFunction (k : ℝ) (f : FourierL2) : C(SchrodingerTorus, ℂ) :=  ∑' n, sobolevTorusTerm k f nrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def sobolevProductWeight (k : ℝ) (n : frequencyLattice) : ℝ :=  (1 + ‖n‖ ^ 2) ^ (k / 2)run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def sobolevProductCoefficient (k : ℝ) (f g : FourierL2)    (n : frequencyLattice) : ℂ :=  ∑' m, sobolevFourierCoefficient k f m * sobolevFourierCoefficient k g (n - m)noncomputable def sobolevProduct (k : ℝ) (_hk : 6 < k) (f g : FourierL2) : FourierL2 :=  ⟨fun n => (sobolevProductWeight k n : ℂ) * sobolevProductCoefficient k f g n,    by sorry⟩run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def fourierConjugate (f : FourierL2) : FourierL2 :=  ⟨fun n => conj (f (-n)), by    apply memℓp_gen    simpa only [Complex.norm_conj, Function.comp_def, Equiv.neg_apply] using      (Equiv.neg frequencyLattice).summable_iff.mpr        ((lp.memℓp f).summable (by norm_num : 0 < (2 : ℝ≥0∞).toReal))⟩run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def sobolevOddPower (k : ℝ) (hk : 6 < k) : ℕ → FourierL2 → FourierL2  | 0, f => f  | m + 1, f => sobolevProduct k hk      (sobolevProduct k hk (sobolevOddPower k hk m f) (fourierConjugate f)) frun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def schrodingerMultiplier (t : ℝ) (n : frequencyLattice) : ℂ :=  Complex.exp (-Complex.I * ((t * ‖n‖ ^ 2 : ℝ) : ℂ))noncomputable def schrodingerFlow (t : ℝ) : FourierL2 →ₗᵢ[ℂ] FourierL2 where  toFun f := ⟨fun n => schrodingerMultiplier t n * f n, by    apply (lp.memℓp f).mono'    intro n    simp only [norm_mul, schrodingerMultiplier, Complex.norm_exp, Complex.mul_re,      Complex.neg_re, Complex.neg_im, Complex.I_re, Complex.I_im,      Complex.ofReal_re, Complex.ofReal_im, neg_zero, zero_mul, mul_zero,      sub_zero, Real.exp_zero, one_mul]    exact le_rfl⟩  map_add' f g := by    ext n    change schrodingerMultiplier t n * (f n + g n) =      schrodingerMultiplier t n * f n + schrodingerMultiplier t n * g n    exact mul_add _ _ _  map_smul' c f := by ext n; simp [mul_left_comm]  norm_map' f := by    have hm (n : frequencyLattice) : ‖schrodingerMultiplier t n‖ = 1 := by      simp only [schrodingerMultiplier, Complex.norm_exp, Complex.mul_re, Complex.neg_re,        Complex.neg_im, Complex.I_re, Complex.I_im, Complex.ofReal_re, Complex.ofReal_im,        neg_zero, zero_mul, mul_zero, sub_zero, Real.exp_zero]    apply le_antisymm    · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      intro n      change ‖schrodingerMultiplier t n * f n‖ ≤ ‖f n‖      rw [norm_mul, hm, one_mul]    · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      intro n      change ‖f n‖ ≤ ‖schrodingerMultiplier t n * f n‖      rw [norm_mul, hm, one_mul]run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def fourierContraction (m : frequencyLattice → ℂ)    (hm : ∀ n, ‖m n‖ ≤ 1) : FourierL2 →L[ℂ] FourierL2 :=  LinearMap.mkContinuous    { toFun := fun f => ⟨fun n => m n * f n, (lp.memℓp f).mono' (by        intro n        rw [norm_mul]        simpa using mul_le_mul_of_nonneg_right (hm n) (norm_nonneg (f n)))⟩      map_add' := by        intro f g        ext n        change m n * (f n + g n) = m n * f n + m n * g n        exact mul_add _ _ _      map_smul' := by        intro c f        ext n        change m n * (c * f n) = c * (m n * f n)        ring }    1 (by      intro f      simp only [one_mul]      apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      intro n      change ‖m n * f n‖ ≤ ‖f n‖      rw [norm_mul]      simpa using mul_le_mul_of_nonneg_right (hm n) (norm_nonneg (f n)))noncomputable def lowerSobolevInclusion : FourierL2 →L[ℂ] FourierL2 :=  fourierContraction (fun n => (((1 + ‖n‖ ^ 2)⁻¹ : ℝ) : ℂ)) (by    intro n    rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (by positivity)]    exact inv_le_one_of_one_le₀ (le_add_of_nonneg_right (sq_nonneg _)))noncomputable def lowerSobolevGenerator : FourierL2 →L[ℂ] FourierL2 :=  fourierContraction (fun n => -Complex.I * (((‖n‖ ^ 2 / (1 + ‖n‖ ^ 2)) : ℝ) : ℂ)) (by    intro n    rw [norm_mul, norm_neg, Complex.norm_I, one_mul, Complex.norm_real,      Real.norm_eq_abs, abs_of_nonneg (by positivity)]    exact (div_le_one (by positivity)).mpr (by linarith [sq_nonneg ‖n‖]))run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def schrodingerInteractionField (k : ℝ) (hk : 6 < k) (m : ℕ)    (t : ℝ) (f : FourierL2) : FourierL2 :=  (-Complex.I) • schrodingerFlow (-t) (sobolevOddPower k hk m (schrodingerFlow t f))run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}structure SobolevInteractionPatch (k : ℝ) (hk : 6 < k) (m : ℕ) (f₀ : FourierL2) where  left : ℝ  right : ℝ  left_neg : left < 0  right_pos : 0 < right  curve : ℝ → FourierL2  initial : curve 0 = f₀  solves : ∀ t ∈ Ioo left right,    HasDerivAt curve (schrodingerInteractionField k hk m t (curve t)) tdef maximalSobolevInteractionDomain (k : ℝ) (hk : 6 < k) (m : ℕ) (f₀ : FourierL2) : Set ℝ :=  ⋃ P : SobolevInteractionPatch k hk m f₀, Ioo P.left P.rightnoncomputable def maximalSobolevInteractionFlow (k : ℝ) (hk : 6 < k) (m : ℕ)    (f₀ : FourierL2) (t : ℝ) : FourierL2 := by  classical  exact if ht : t ∈ maximalSobolevInteractionDomain k hk m f₀ then    (Classical.choose (mem_iUnion.mp ht)).curve t  else 0run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}noncomputable def maximalSobolevSchrodingerFlow (k : ℝ) (hk : 6 < k) (m : ℕ)    (f₀ : FourierL2) (t : ℝ) : FourierL2 :=  schrodingerFlow t (maximalSobolevInteractionFlow k hk m f₀ t)run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}def IsClassicalDefocusingFlow (k : ℝ) (hk : 6 < k) (m : ℕ) (f : FourierL2) : Prop :=  let u := maximalSobolevSchrodingerFlow k hk m f  let D := maximalSobolevInteractionDomain k hk m f  u 0 = f ∧ ContinuousOn u D ∧    ContDiffOn ℝ 1 (fun t => lowerSobolevInclusion (u t)) D ∧    ∀ t ∈ D, HasDerivAt (fun s => lowerSobolevInclusion (u s))      (lowerSobolevGenerator (u t) - Complex.I • lowerSobolevInclusion        (sobolevOddPower k hk m (u t))) trun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}def HasFiniteTimeSelfSimilarBlowup (k : ℝ) (hk : 6 < k) (m : ℕ)    (a : ℝ) (f : FourierL2) : Prop :=  ∃ T : ℝ, 0 < T ∧ ∃ x : SchrodingerTorus, ∃ c : ℝ, 0 < c ∧    Ico 0 T ⊆ maximalSobolevInteractionDomain k hk m f ∧    BddAbove (maximalSobolevInteractionDomain k hk m f) ∧    sSup (maximalSobolevInteractionDomain k hk m f) = T ∧    Tendsto (fun t => (T - t) ^ a *      ‖sobolevTorusFunction k (maximalSobolevSchrodingerFlow k hk m f t) x‖)      (𝓝[<] T) (𝓝 c) ∧    ¬ ContinuousWithinAt (maximalSobolevSchrodingerFlow k hk m f) (Iio T) Tattribute [local irreducible] HasFiniteTimeSelfSimilarBlowup IsClassicalDefocusingFlow  maximalSobolevInteractionDomain/-- Theorem 1.1: arbitrarily large odd powers admit a nonempty open family ofclassical solutions with finite-time self-similar blowup on the twelve-torus. -/theorem stable_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 := by  sorry/-- Corollary 1.2: for every permitted Gaussian Fourier decay exponent,the same open blowup family has positive Gaussian probability. -/theorem gaussian_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) ∧      ∀ α : ℝ, (k : ℝ) + 6 < α →        (∀ᵐ g ∂fourierGaussianLaw, Memℓp (weightedGaussianCoefficient k α g) 2) ∧        (∀ᵐ g ∂fourierGaussianLaw, ∀ n,          sobolevFourierCoefficient k (weightedGaussianVector k α g) n =            (1 + ‖n‖ ^ 2) ^ (-α / 2) • g n) ∧        0 < weightedGaussianLaw k α U := by  sorry/-- In particular, Gaussian data are not almost surely globally continuable. -/theorem gaussian_not_almost_sure_global (p₀ : ℕ) :    ∃ p k : ℕ, p₀ ≤ p ∧ Odd p ∧ 3 ≤ p ∧ ∃ hk : 8 < (k : ℝ),      ∀ α : ℝ, (k : ℝ) + 6 < α →        ¬ ∀ᵐ f ∂weightedGaussianLaw k α,          maximalSobolevInteractionDomain k (by linarith) ((p - 1) / 2) f = univ := by  sorryend DefocusingNLSendend 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.

Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus

Theorem (An open set of blowup data)

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 (eq:nls) 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.

Section 1, Introduction Read it in the paper TeX source, line 67
Show the TeX
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 \eqref{eq:nls} 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.
Corollary (Failure of universal Gaussian global existence)

For the fixed power p and index k in Theorem thm:blowup, and for every \alpha>k+6, the initial datum

u_0^\omega(x)=\sum_{n\in\mathbb Z^{12}} g_n(\omega)\langle n\rangle^{-\alpha}e^{in\cdot x}

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)).

Section 1, Introduction Read it in the paper TeX source, line 93
Show the TeX
For the fixed power $p$ and index $k$ in Theorem~\ref{thm:blowup}, and
for every $\alpha>k+6$, the initial datum
\begin{equation}
 u_0^\omega(x)=\sum_{n\in\mathbb Z^{12}}
       g_n(\omega)\langle n\rangle^{-\alpha}e^{in\cdot x}
\end{equation}
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 $C(\mathbb R;H^m(\mathbb T^d))$.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 371 link it.
Solution module
OAI.MathematicalPhysics.DefocusingNLS.MainTheorems
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
37 local vocabulary declarations
Lines of Lean, without comments or blank lines
216
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Stable self-similar blowup and positive-probability Gaussian blowup

Family 371: Stable blowup for the defocusing Schrödinger equation

Candidate pairing

Corollary (Failure of universal Gaussian global existence)

For the fixed power p and index k in Theorem thm:blowup, and for every \alpha>k+6, the initial datum

u_0^\omega(x)=\sum_{n\in\mathbb Z^{12}} g_n(\omega)\langle n\rangle^{-\alpha}e^{in\cdot x}

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)).

Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus

TeX label cor:gaussian, build/main.tex, line 93

The Lean statement says

theorem stable_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 := by  sorrytheorem gaussian_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) ∧      ∀ α : ℝ, (k : ℝ) + 6 < α →        (∀ᵐ g ∂fourierGaussianLaw, Memℓp (weightedGaussianCoefficient k α g) 2) ∧        (∀ᵐ g ∂fourierGaussianLaw, ∀ n,          sobolevFourierCoefficient k (weightedGaussianVector k α g) n =            (1 + ‖n‖ ^ 2) ^ (-α / 2) • g n) ∧

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.

Comparator checks only the name and type of 3 definitions, among them sobolevProduct; their bodies come from the solution.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.two independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

37 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/DefocusingNLS.html

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More about this statement

  • Commentary 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. Ben Knill, BenKnill/openai-math-hol-cards: HOL Light proof cards for eleven families, 2026-10-08 Source

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  • 3 definition holes: Comparator compares these by name and type only, so a reader checks what each solution body means.
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