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Lean statement

Gaussian Fourier minorants with their construction

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.AtomicTriangular.sharp_gaussian_minorants_with_construction

Lean statement file: TriangularGaussian.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 090 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 sharp_gaussian_minorants_with_construction (α : ℝ) (hα : 0 < α) :    ∃ f : SchwartzMap Plane ℝ,      Radial f ∧ (∀ x, f x ≤ gaussian α x) ∧      (∀ w, (realFourier f w).im = 0 ∧ 0 ≤ (realFourier f w).re) ∧      (∀ a ∈ A, a ≠ 0 → f a = gaussian α a) ∧      (∀ w ∈ dualA, w ≠ 0 → realFourier f w = 0) ∧      (1 ≤ α → AtomicConstruction α f) := 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.

  • abbreviation OAI.AtomicTriangular.Plane line 18

    abbrev Plane := EuclideanSpace ℝ (Fin 2)

  • definition OAI.AtomicTriangular.b line 20

    def b : ℝ := Real.sqrt 3 / 2

  • definition OAI.AtomicTriangular.triangularPoint line 22

    def triangularPoint (jk : ℤ × ℤ) : Plane :=  (Real.sqrt b)⁻¹ •    (EuclideanSpace.single 0 ((jk.1 : ℝ) + (jk.2 : ℝ) / 2) +      EuclideanSpace.single 1 ((jk.2 : ℝ) * b))

  • definition OAI.AtomicTriangular.A line 27

    def A : Set Plane := Set.range triangularPoint

  • definition OAI.AtomicTriangular.dualA line 29

    def dualA : Set Plane := {w | ∀ a ∈ A, ∃ n : ℤ, inner ℝ a w = (n : ℝ)}

  • definition OAI.AtomicTriangular.gaussian line 31

    def gaussian (α : ℝ) (x : Plane) : ℝ := Real.exp (-Real.pi * α * ‖x‖ ^ 2)

  • definition OAI.AtomicTriangular.Radial line 33

    def Radial (f : Plane → ℝ) : Prop := ∀ x y, ‖x‖ = ‖y‖ → f x = f y

  • definition OAI.AtomicTriangular.realFourier line 35

    def realFourier (f : Plane → ℝ) (w : Plane) : ℂ := 𝓕 (fun x => (f x : ℂ)) w

  • definition OAI.AtomicTriangular.Construction.h line 43

    def h : ℝ := 17 / 50

  • definition OAI.AtomicTriangular.Construction.H line 45

    def H : ℝ := 27 / 50

  • definition OAI.AtomicTriangular.Construction.eta line 47

    def eta : ℝ := 1 / 5

  • definition OAI.AtomicTriangular.Construction.kappa line 49

    def kappa : ℝ := Real.pi / 36

  • definition OAI.AtomicTriangular.Construction.residues line 51

    def residues : Finset ℤ := {0,1,3,4,7,9,12,13,16,19,21,25,27,28,31}

  • definition OAI.AtomicTriangular.Construction.residue line 53

    def residue (a : Fin 15) : ℤ := ![0,1,3,4,7,9,12,13,16,19,21,25,27,28,31] a

  • definition OAI.AtomicTriangular.Construction.Node line 55

    def Node : Type := {n : ℤ // 0 < n ∧ n % 36 ∈ residues}

  • definition OAI.AtomicTriangular.Construction.FiniteNodes line 57

    def FiniteNodes : Finset ℤ := {1,3,4,7,9,12,13,16,19,21}

  • definition OAI.AtomicTriangular.Construction.sineFactor line 59

    def sineFactor (a : Fin 15) (s : ℂ) : ℂ :=  2 * Complex.sin ((kappa : ℂ) * (s - (residue a : ℂ)))

  • definition OAI.AtomicTriangular.Construction.P line 62

    def P (s : ℂ) : ℂ := ∏ a : Fin 15, (sineFactor a s)^2

  • definition OAI.AtomicTriangular.Construction.dBasis line 64

    def dBasis (n : Node) : ℂ → ℂ := dslope P (n.val : ℂ)

  • definition OAI.AtomicTriangular.Construction.cBasis line 66

    def cBasis (n : Node) : ℂ → ℂ := dslope (dBasis n) (n.val : ℂ)

  • definition OAI.AtomicTriangular.Construction.finiteC line 68

    def finiteC (a : Fin 15) (s : ℂ) : ℂ :=  4 * (kappa : ℂ)^2 * ∏ r ∈ Finset.univ.erase a, (sineFactor r s)^2

  • definition OAI.AtomicTriangular.Construction.finiteD line 71

    def finiteD (a : Fin 15) (s : ℂ) : ℂ :=  2 * (kappa : ℂ) * sineFactor a s *    Complex.cos ((kappa : ℂ) * (s - (residue a : ℂ))) *      ∏ r ∈ Finset.univ.erase a, (sineFactor r s)^2

  • definition OAI.AtomicTriangular.Construction.phase line 76

    def phase (s : ℂ) : ℂ := Complex.exp ((Real.pi : ℂ) * Complex.I * s / 18)

  • definition OAI.AtomicTriangular.Construction.atomFactor line 78

    def atomFactor (a : Fin 15) : Polynomial ℂ :=  -C (phase (residue a)) + C 2 * X - C (phase (-residue a)) * X^2

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

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: construction, finite, function, gaussian, infinite, interpolation, minorants, radial and three more.

Theorem 1.2 of An atomic certificate for triangular-lattice universal optimality

Theorem (Sharp Gaussian minorants)

For every \alpha>0 there is a real radial Schwartz function f_\alpha:\mathbb R^2\to\mathbb R such that

f_\alpha(x)\le k_\alpha(x),\qquad \widehat f_\alpha(\xi)\ge0 \quad(x,\xi\in\mathbb R^2),

and

f_\alpha(a)=k_\alpha(a)\quad(a\in A\setminus\{0\}),\qquad \widehat f_\alpha(w)=0\quad(w\in A^*\setminus\{0\}).

For \alpha\ge1, the construction uses two finite 20-by-20 interpolation blocks and an absolutely summable infinite correction.

Section 1, Introduction Read it in the paper TeX source, line 95
Show the TeX
For every $\alpha>0$ there is a real radial Schwartz function
$f_\alpha:\mathbb R^2\to\mathbb R$ such that
\[
 f_\alpha(x)\le k_\alpha(x),\qquad
 \widehat f_\alpha(\xi)\ge0
 \quad(x,\xi\in\mathbb R^2),
\]
and
\[
 f_\alpha(a)=k_\alpha(a)\quad(a\in A\setminus\{0\}),\qquad
 \widehat f_\alpha(w)=0\quad(w\in A^*\setminus\{0\}).
\]
For $\alpha\ge1$, the construction uses two finite $20$-by-$20$ interpolation
blocks and an absolutely summable infinite correction.

Other possible paper matches, closest wording first

  • Proposition 5.1 of An atomic certificate for triangular-lattice universal optimality. Words of the Lean statement, its names split into words, that also appear in the theorem: columns, construction, input, parameter, coefficients, gaussian and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: interpolation.
  • Proposition 4.1 of Universal optimality of the triangular lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: nodes, node, full, gaussian, fourier, schwartz and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: including, lattice and interpolation.
  • Theorem 2.1 of Universal optimality of the triangular lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: normalized, derivatives, gaussian, fourier, schwartz, radial and three more.
  • Proposition 2.2 of A sharp Fourier certificate for planar circle packing. Words of the Lean statement, its names split into words, that also appear in the theorem: im, derivative, gaussian, fourier, schwartz, radial and two more.
  • Theorem 1.1 of A sharp Fourier certificate for planar circle packing. Words of the Lean statement, its names split into words, that also appear in the theorem: sharp, sqrt, certificate, fourier, schwartz, radial and one more.
  • Proposition B.5 of A sharp Fourier certificate for planar circle packing. Words of the Lean statement, its names split into words, that also appear in the theorem: basis, atoms, input, node, full, coefficient and two more.

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 090’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: “partial”.

    Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record

    Their note: Universal minimality of the triangular lattice for lower-limit energy per particle is stated; the renormalized Riesz (0<s<2) and Coulomb minimality, the Sandier-Serfaty and Brauchart-Hardin-Saff (spherical log energy) claims are not.

    Definitions the group asks readers to compare with the paper: Custom energy/diskEnergy (liminf of per-particle disk energy in ℝ≥0∞, with diskPoints := if h : finite then h.toFinset else ∅), DensityOne, latticeEnergy := ∑' over nonzero lattice points; AtomicGaussian/TriangularGaussian carry a huge bespoke 'atomic construction' (explicit rational constants such as h = 17/50, H = 27/50, residue list, 20x23 jet matrices) inside the conclusion HasAtomicConstruction α f, unreadable as a statement

  • 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: Triangular minimality for unit-background renormalized Riesz energies (0<s<2) and for the planar Coulomb renormalized energy, hence the Sandier–Serfaty and two-dimensional Brauchart–Hardin–Saff conclusions. These come from two of the family's four manuscripts ('Universal optimality of the triangular lattice' and 'Triangular minimality for planar Coulomb renormalized energy'), neither of which the scope note links. PlanarPacking (Cohn–Elkies sharpness) is an extra result the headline does not mention.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“It also constructs sharp radial Schwartz minorants for every Gaussian potential: each minorant lies below the Gaussian, has nonnegative real Fourier transform, agrees with the Gaussian at nonzero triangular-lattice points, and vanishes on nonzero dual-lattice points after Fourier transformation. For the stated parameter range, the formalization includes the explicit atomic interpolation construction.”

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

275 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: 165 to its own declarations, 158 to Mathlib’s documentation. 45 declared names also link to the same declaration in OpenAI’s library. 2 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/-!# Sharp Gaussian minorants with atomic interpolationThe sharp Gaussian minorants of the atomic certificate for triangular-latticeuniversal optimality, including the construction for parameters at least one.-/noncomputable sectionopen scoped BigOperators FourierTransform SchwartzMapopen Complex Polynomialnamespace AtomicTriangularabbrev Plane := EuclideanSpace ℝ (Fin 2)def b : ℝ := Real.sqrt 3 / 2def triangularPoint (jk : ℤ × ℤ) : Plane :=  (Real.sqrt b)⁻¹ •    (EuclideanSpace.single 0 ((jk.1 : ℝ) + (jk.2 : ℝ) / 2) +      EuclideanSpace.single 1 ((jk.2 : ℝ) * b))def A : Set Plane := Set.range triangularPointdef dualA : Set Plane := {w | ∀ a ∈ A, ∃ n : ℤ, inner ℝ a w = (n : ℝ)}def gaussian (α : ℝ) (x : Plane) : ℝ := Real.exp (-Real.pi * α * ‖x‖ ^ 2)def Radial (f : Plane → ℝ) : Prop := ∀ x y, ‖x‖ = ‖y‖ → f x = f ydef realFourier (f : Plane → ℝ) (w : Plane) : ℂ := 𝓕 (fun x => (f x : ℂ)) wend AtomicTriangularnamespace AtomicTriangular.Constructiondef b : ℝ := Real.sqrt 3 / 2def h : ℝ := 17 / 50def H : ℝ := 27 / 50def eta : ℝ := 1 / 5def kappa : ℝ := Real.pi / 36def residues : Finset ℤ := {0,1,3,4,7,9,12,13,16,19,21,25,27,28,31}def residue (a : Fin 15) : ℤ := ![0,1,3,4,7,9,12,13,16,19,21,25,27,28,31] adef Node : Type := {n : ℤ // 0 < n ∧ n % 36 ∈ residues}def FiniteNodes : Finset ℤ := {1,3,4,7,9,12,13,16,19,21}def sineFactor (a : Fin 15) (s : ℂ) : ℂ :=  2 * Complex.sin ((kappa : ℂ) * (s - (residue a : ℂ)))def P (s : ℂ) : ℂ := ∏ a : Fin 15, (sineFactor a s)^2def dBasis (n : Node) : ℂ → ℂ := dslope P (n.val : ℂ)def cBasis (n : Node) : ℂ → ℂ := dslope (dBasis n) (n.val : ℂ)def finiteC (a : Fin 15) (s : ℂ) : ℂ :=  4 * (kappa : ℂ)^2 * ∏ r ∈ Finset.univ.erase a, (sineFactor r s)^2def finiteD (a : Fin 15) (s : ℂ) : ℂ :=  2 * (kappa : ℂ) * sineFactor a s *    Complex.cos ((kappa : ℂ) * (s - (residue a : ℂ))) *      ∏ r ∈ Finset.univ.erase a, (sineFactor r s)^2def phase (s : ℂ) : ℂ := Complex.exp ((Real.pi : ℂ) * Complex.I * s / 18)def atomFactor (a : Fin 15) : Polynomial ℂ :=  -C (phase (residue a)) + C 2 * X - C (phase (-residue a)) * X^2def atomPolynomial : Polynomial ℂ := ∏ a : Fin 15, atomFactor adef cAtomPolynomial (a : Fin 15) : Polynomial ℂ :=  C (4 * (kappa : ℂ)^2) * ∏ r ∈ Finset.univ.erase a, atomFactor rdef dAtomPolynomial (a : Fin 15) : Polynomial ℂ :=  C (Complex.I * (kappa : ℂ)) *    (C (phase (residue a)) - C (phase (-residue a)) * X^2) *      ∏ r ∈ Finset.univ.erase a, atomFactor rdef fourierCoeff (j : Fin 31) : ℂ := atomPolynomial.coeff j.valdef frequency (j : Fin 31) : ℝ := ((j.val : ℝ)-15)/18def waveDerivative (k : ℕ) (s : ℂ) : ℂ :=  ∑ j : Fin 31, fourierCoeff j * (I*Real.pi*frequency j)^k *    exp (I*Real.pi*frequency j*s)def Q (n : ℤ) : ℝ := (waveDerivative 2 n).re / 2def D (n : ℤ) : ℝ := (waveDerivative 3 n).re / (6 * Q n)end AtomicTriangular.Constructionnamespace AtomicTriangular.FiniteCertificateopen Constructiondef cShiftPolynomial (a : Fin 15) : Polynomial ℂ := X * cAtomPolynomial adef cCoefficient (a : Fin 15) (j : Fin 31) : ℂ := (cShiftPolynomial a).coeff j.valdef dCoefficient (a : Fin 15) (j : Fin 31) : ℂ := (dAtomPolynomial a).coeff j.valend AtomicTriangular.FiniteCertificatenamespace AtomicTriangular.Constructionabbrev Plane := EuclideanSpace ℝ (Fin 2)def radial (x : Plane) : ℝ := b*‖x‖^2def waveDen (k t : ℝ) : ℝ := k^2+t^2def Xwave (k t : ℝ) : ℝ := (4/3)/waveDen k tdef lambda (k t : ℝ) : ℂ := ((k:ℂ)+I*t)/(b*waveDen k t : ℝ)def zeta (k v t : ℝ) : ℂ := -(t:ℂ)*(Xwave k t:ℂ)+I*(k*Xwave k t-v:ℝ)def transformedWave (k v t : ℝ) (s : ℂ) : ℂ := lambda k t*exp (I*Real.pi*zeta k v t*s)def finiteColumn (col : Fin 23) (j : Fin 31) : ℂ :=  if hc : col.val<22 then    if col.val%2=0 then FiniteCertificate.cCoefficient ⟨col.val/2,by omega⟩ j      else FiniteCertificate.dCoefficient ⟨col.val/2,by omega⟩ j  else fourierCoeff jdef finiteNode (n : Fin 10) : ℤ := residue ⟨n.val+1,by omega⟩def jetMultiplier (k v : ℝ) (n : ℤ) (d : Fin 2) (t : ℝ) : ℂ :=  (Q n:ℂ)⁻¹*transformedWave k v t n *    (if d.val=0 then -1 else (D n:ℂ)-I*Real.pi*zeta k v t)def jet (k v : ℝ) (n : ℤ) (d : Fin 2) (S : Fin 31 → ℂ) : ℝ :=  (∑ j : Fin 31, jetMultiplier k v n d (frequency j)*S j).redef fullJetMatrix : Matrix (Fin 20) (Fin 23) ℝ := fun i col =>  jet (if col.val=22 then h else H) H    (finiteNode ⟨i.val/2,by omega⟩) ⟨i.val%2,by omega⟩ (finiteColumn col)def R : Matrix (Fin 20) (Fin 20) ℝ := fun i j => fullJetMatrix i ⟨j.val+2,by omega⟩def R0 : Matrix (Fin 20) (Fin 3) ℝ := fun i j =>  fullJetMatrix i (if j.val<2 then ⟨j.val,by omega⟩ else 22)abbrev TailNode := {n : Node // n.val ∉ FiniteNodes}structure InputCoefficients where  C : ℝ  finiteC : Fin 11 → ℝ  finiteD : Fin 11 → ℝ  tailC : TailNode → ℝ  tailD : TailNode → ℝ  summable : Summable (fun n => |tailC n|+|tailD n|)def tailInput (a : InputCoefficients) (s : ℂ) : ℂ :=  (a.C:ℂ)*P s + ∑' n : TailNode, ((a.tailC n:ℂ)*cBasis n.val s+(a.tailD n:ℂ)*dBasis n.val s)def finiteInput (a : InputCoefficients) (s : ℂ) : ℂ :=  ∑ j : Fin 11, ((a.finiteC j:ℂ)*finiteC ⟨j.val,by omega⟩ s +    (a.finiteD j:ℂ)*finiteD ⟨j.val,by omega⟩ s)def inputComplex (a : InputCoefficients) (x : Plane) : ℂ :=  exp (-(Real.pi:ℂ)*h*(radial x:ℂ))*tailInput a (radial x) +    exp (-(Real.pi:ℂ)*H*(radial x:ℂ))*finiteInput a (radial x)def targetCoefficient (n : Fin 10) : ℝ := Q 1 / Q (finiteNode n) * Real.exp (Real.pi*eta*(finiteNode n))def Yf : Matrix (Fin 20) (Fin 8) ℝ := fun i j =>  if i.val<8 then    targetCoefficient ⟨i.val/2,by omega⟩ *      (if i.val%2=0 then (if j.val=i.val then 1 else 0)       else (if j.val=i.val then -1 else if j.val=i.val-1 then Real.pi*eta-D (finiteNode ⟨i.val/2,by omega⟩) else 0))  else 0def gamma (side : Fin 2) : Matrix (Fin 3) (Fin 8) ℝ := fun i j =>  if j.val=1 then (if i.val=0 then 11/25 else if i.val=1 then 0 else if side.val=0 then -(3/500) else 3/500) else 0def Uplus : Matrix (Fin 20) (Fin 8) ℝ :=  (1-R)⁻¹*(Yf+R0*(gamma 1+gamma 0))def Uminus : Matrix (Fin 20) (Fin 8) ℝ :=  (1+R)⁻¹*(Yf+R0*(gamma 1-gamma 0))def finiteSolution (side : Fin 2) : Matrix (Fin 20) (Fin 8) ℝ :=  if side.val=0 then (1/2:ℝ) • (Uplus+Uminus) else (1/2:ℝ) • (Uplus-Uminus)def Acoefficient (side : Fin 2) : Matrix (Fin 23) (Fin 8) ℝ := fun i j =>  if hi : i.val<2 then gamma side ⟨i.val,by omega⟩ j  else if h22 : i.val=22 then gamma side 2 j  else finiteSolution side ⟨i.val-2,by omega⟩ jdef gaussianZ (α : ℝ) : ℝ := 1/(Real.pi*(α/b-h))def actualParameter (z : ℝ) : Fin 8 → ℝ :=  ![z,1,z*Real.exp (-2/z),Real.exp (-2/z),z*Real.exp (-3/z),Real.exp (-3/z),    z*Real.exp (-6/z),Real.exp (-6/z)]def normalizingDen (z : ℝ) := Q 1*z*Real.exp (1/z)end AtomicTriangular.Constructionendnoncomputable sectionopen scoped BigOperators FourierTransformnamespace AtomicTriangular.Constructiondef correctedFinite (side : Fin 2) (z : ℝ) (δ : Fin 20 → ℝ) (j : Fin 22) : ℝ :=  (∑ r : Fin 8, Acoefficient side ⟨j.val, by omega⟩ r * actualParameter z r) +    if hj : j.val < 2 then 0 else δ ⟨j.val - 2, by omega⟩end AtomicTriangular.Constructionnamespace AtomicTriangularopen Constructionlocal instance finite_pair_neZero : NeZero (2 : ℕ) := inferInstancelocal instance three_atLeastTwo : Nat.AtLeastTwo 3 := inferInstancelocal instance ten_atLeastTwo : Nat.AtLeastTwo 10 := inferInstancelocal instance five_hundred_atLeastTwo : Nat.AtLeastTwo 500 := inferInstancelocal instance plane_borelSpace : BorelSpace Plane := inferInstancelocal instance plane_finite : Module.Finite ℝ Plane := inferInstancedef AtomicConstruction (α : ℝ) (f : Plane → ℝ) : Prop :=  IsUnit (1 - R) ∧ IsUnit (1 + R) ∧  (1 - R) * Uplus = Yf + R0 * (gamma 1 + gamma 0) ∧  (1 + R) * Uminus = Yf + R0 * (gamma 1 - gamma 0) ∧  ∃ (δ : Fin 2 → Fin 20 → ℝ) (a : Fin 2 → InputCoefficients),    (∀ side, (∑ j : Fin 20, |δ side j|) < (1 / 10^5 : ℝ)) ∧    (∀ side, (a side).C = if side.val = 0 then -(3/500) else 3/500) ∧    (∀ side (j : Fin 11),      (a side).finiteC j =        correctedFinite side (gaussianZ α) (δ side) ⟨2*j.val, by omega⟩ ∧      (a side).finiteD j =        correctedFinite side (gaussianZ α) (δ side) ⟨2*j.val+1, by omega⟩) ∧    (∀ side, Summable (fun n => |(a side).tailC n| + |(a side).tailD n|) ∧      (∑' n, (|(a side).tailC n| + |(a side).tailD n|)) < (3 / 10^9 : ℝ)) ∧    (∀ x, (f x : ℂ) =      (inputComplex (a 0) x + 𝓕 (inputComplex (a 1)) x) /        (normalizingDen (gaussianZ α) : ℂ))end AtomicTriangularendnoncomputable sectionopen scoped BigOperators FourierTransformnamespace AtomicTriangularopen Constructiontheorem sharp_gaussian_minorants_with_construction (α : ℝ) (hα : 0 < α) :    ∃ f : SchwartzMap Plane ℝ,      Radial f ∧ (∀ x, f x ≤ gaussian α x) ∧      (∀ w, (realFourier f w).im = 0 ∧ 0 ≤ (realFourier f w).re) ∧      (∀ a ∈ A, a ≠ 0 → f a = gaussian α a) ∧      (∀ w ∈ dualA, w ≠ 0 → realFourier f w = 0) ∧      (1 ≤ α → AtomicConstruction α f) := by  sorryend AtomicTriangularendend 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.

An atomic certificate for triangular-lattice universal optimality

Theorem (Universal energy minimum)

For every smooth nonnegative completely monotone function g on (0,\infty) and every locally finite planar set \mathcal C of centered disk density one,

E_g(\mathcal C)\ge \sum_{a\in A\setminus\{0\}}g(|a|^2)=E_g(A).

The comparison and the equality are in the extended nonnegative reals.

Section 1, Introduction Read it in the paper TeX source, line 68
Show the TeX
For every smooth nonnegative completely monotone function $g$ on
$(0,\infty)$ and every locally finite planar set $\mathcal C$ of centered
disk density one,
\[
 E_g(\mathcal C)\ge
 \sum_{a\in A\setminus\{0\}}g(|a|^2)=E_g(A).
\]
The comparison and the equality are in the extended nonnegative reals.
Theorem (Sharp Gaussian minorants)

For every \alpha>0 there is a real radial Schwartz function f_\alpha:\mathbb R^2\to\mathbb R such that

f_\alpha(x)\le k_\alpha(x),\qquad \widehat f_\alpha(\xi)\ge0 \quad(x,\xi\in\mathbb R^2),

and

f_\alpha(a)=k_\alpha(a)\quad(a\in A\setminus\{0\}),\qquad \widehat f_\alpha(w)=0\quad(w\in A^*\setminus\{0\}).

For \alpha\ge1, the construction uses two finite 20-by-20 interpolation blocks and an absolutely summable infinite correction.

Section 1, Introduction Read it in the paper TeX source, line 95
Show the TeX
For every $\alpha>0$ there is a real radial Schwartz function
$f_\alpha:\mathbb R^2\to\mathbb R$ such that
\[
 f_\alpha(x)\le k_\alpha(x),\qquad
 \widehat f_\alpha(\xi)\ge0
 \quad(x,\xi\in\mathbb R^2),
\]
and
\[
 f_\alpha(a)=k_\alpha(a)\quad(a\in A\setminus\{0\}),\qquad
 \widehat f_\alpha(w)=0\quad(w\in A^*\setminus\{0\}).
\]
For $\alpha\ge1$, the construction uses two finite $20$-by-$20$ interpolation
blocks and an absolutely summable infinite correction.

A sharp Fourier certificate for planar circle packing

Theorem (Sharp Fourier certificate)

There is a real radial Schwartz function f:\RR^2\to\RR such that

\widehat f(0)=1,\qquad f(0)=\frac{2}{\sqrt3},\qquad \widehat f(\xi)\ge0\quad(\xi\in\RR^2),\qquad f(x)\le0\quad(|x|\ge1).
Section 1, Introduction Read it in the paper TeX source, line 34
Show the TeX
There is a real radial Schwartz function $f:\RR^2\to\RR$ such that
\begin{equation}
  \widehat f(0)=1,\qquad f(0)=\frac{2}{\sqrt3},\qquad
  \widehat f(\xi)\ge0\quad(\xi\in\RR^2),\qquad
  f(x)\le0\quad(|x|\ge1).
\end{equation}
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 090 link it.
Solution module
OAI.Analysis.Triangular.GaussianMinorants
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
73 local vocabulary declarations
Lines of Lean, without comments or blank lines
149
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Gaussian Fourier minorants with their construction

Family 090: Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy

Candidate pairing

Theorem (Sharp Gaussian minorants)

For every \alpha>0 there is a real radial Schwartz function f_\alpha:\mathbb R^2\to\mathbb R such that

f_\alpha(x)\le k_\alpha(x),\qquad \widehat f_\alpha(\xi)\ge0 \quad(x,\xi\in\mathbb R^2),

and

f_\alpha(a)=k_\alpha(a)\quad(a\in A\setminus\{0\}),\qquad \widehat f_\alpha(w)=0\quad(w\in A^*\setminus\{0\}).

For \alpha\ge1, the construction uses two finite 20-by-20 interpolation blocks and an absolutely summable infinite correction.

An atomic certificate for triangular-lattice universal optimality

TeX label thm:gaussian, build/main.tex, line 95

The Lean statement says

theorem sharp_gaussian_minorants_with_construction (α : ℝ) (hα : 0 < α) :    ∃ f : SchwartzMap Plane ℝ,      Radial f ∧ (∀ x, f x ≤ gaussian α x) ∧      (∀ w, (realFourier f w).im = 0 ∧ 0 ≤ (realFourier f w).re) ∧      (∀ a ∈ A, a ≠ 0 → f a = gaussian α a) ∧      (∀ w ∈ dualA, w ≠ 0 → realFourier f w = 0) ∧      (1 ≤ α → AtomicConstruction α f) := 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.

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

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

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