Lean statement

Asymptotic outer-electron radii of neutral atoms

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.NeutralAtom.generalized_outer_radii

Lean statement file: CoulombRadii.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 263 link it.

See the candidate theorem Who ran the proof-verification program Report 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 generalized_outer_radii    (Ψ : ∀ N : ℕ, Wavefunction (N + 1))    (hΨ : ∀ N : ℕ, IsNormalizedGroundState (N + 1) (Ψ N)) :    Tendsto (fun m : ℕ => (((m : ℝ) ^ (1 / 3 : ℝ) : ℝ) : EReal) * upperRadius Ψ m)      atTop (𝓝 (bTF : EReal)) ∧    Tendsto (fun m : ℕ => (((m : ℝ) ^ (1 / 3 : ℝ) : ℝ) : EReal) * lowerRadius Ψ m)      atTop (𝓝 (bTF : EReal)) := 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.NeutralAtom.Position line 12

abbrev Position := EuclideanSpace ℝ (Fin 3)

abbreviation OAI.NeutralAtom.Configuration line 13

abbrev Configuration (n : ℕ) := Fin n → Position

abbreviation OAI.NeutralAtom.Spins line 14

abbrev Spins (n : ℕ) := Fin n → Fin 2

abbreviation OAI.NeutralAtom.Wavefunction line 15

abbrev Wavefunction (n : ℕ) := Spins n → Configuration n → ℂ

abbreviation OAI.NeutralAtom.Gradient line 16

abbrev Gradient (n : ℕ) := Spins n → Fin n → Fin 3 → Configuration n → ℂ

definition OAI.NeutralAtom.coordinateDirection line 18

def coordinateDirection {n : ℕ} (i : Fin n) (a : Fin 3) : Configuration n :=  Pi.single i (EuclideanSpace.single a 1)

definition OAI.NeutralAtom.HasWeakGradient line 21

def HasWeakGradient {n : ℕ} (ψ : Wavefunction n) (g : Gradient n) : Prop :=  ∀ σ i a (φ : Configuration n → ℝ), ContDiff ℝ ∞ φ → HasCompactSupport φ →    (∫ x, (fderiv ℝ φ x (coordinateDirection i a) : ℂ) * ψ σ x) =      -(∫ x, (φ x : ℂ) * g σ i a x)

definition OAI.NeutralAtom.IsAntisymmetric line 26

def IsAntisymmetric {n : ℕ} (ψ : Wavefunction n) : Prop :=  ∀ (p : Equiv.Perm (Fin n)) (σ : Spins n),    (fun x : Configuration n => ψ (σ ∘ p) (x ∘ p)) =ᵐ[volume]      (fun x => (((Equiv.Perm.sign p : ℤ) : ℂ)) * ψ σ x)

definition OAI.NeutralAtom.FormDomain line 31

def FormDomain {n : ℕ} (ψ : Wavefunction n) (g : Gradient n) : Prop :=  IsAntisymmetric ψ ∧ HasWeakGradient ψ g ∧  (∀ σ, MemLp (ψ σ) 2 volume) ∧  (∀ σ i a, MemLp (g σ i a) 2 volume) ∧  (∀ σ i, Integrable (fun x : Configuration n => ‖x i‖⁻¹ * ‖ψ σ x‖ ^ 2)) ∧  (∀ σ i j, i < j →    Integrable (fun x : Configuration n => ‖x i - x j‖⁻¹ * ‖ψ σ x‖ ^ 2))

definition OAI.NeutralAtom.normSquared line 39

def normSquared {n : ℕ} (ψ : Wavefunction n) : ℝ :=  ∑ σ : Spins n, ∫ x : Configuration n, ‖ψ σ x‖ ^ 2

definition OAI.NeutralAtom.coulombPotential line 42

def coulombPotential {n : ℕ} (Z : ℕ) (x : Configuration n) : ℝ :=  -(Z : ℝ) * (∑ i : Fin n, ‖x i‖⁻¹) +    ∑ i : Fin n, ∑ j ∈ Finset.univ.filter (fun j : Fin n => i < j), ‖x i - x j‖⁻¹

definition OAI.NeutralAtom.energy line 46

def energy {n : ℕ} (Z : ℕ) (ψ : Wavefunction n) (g : Gradient n) : ℝ :=  (1 / 2 : ℝ) * (∑ σ : Spins n, ∑ i : Fin n, ∑ a : Fin 3,    ∫ x : Configuration n, ‖g σ i a x‖ ^ 2) +  ∑ σ : Spins n, ∫ x : Configuration n, coulombPotential Z x * ‖ψ σ x‖ ^ 2

definition OAI.NeutralAtom.IsNormalizedGroundState line 51

def IsNormalizedGroundState (Z : ℕ) (ψ : Wavefunction Z) : Prop :=  ∃ g : Gradient Z, FormDomain ψ g ∧ normSquared ψ = 1 ∧    ∀ (χ : Wavefunction Z) (h : Gradient Z),      FormDomain χ h → normSquared χ = 1 → energy Z ψ g ≤ energy Z χ h

definition OAI.NeutralAtom.density line 56

def density {N : ℕ} (ψ : Wavefunction (N + 1)) (x : Position) : ℝ :=  (N + 1 : ℝ) * ∑ σ : Spins (N + 1),    ∫ y : Configuration N, ‖ψ σ (Fin.cons x y)‖ ^ 2

definition OAI.NeutralAtom.exteriorMass line 60

def exteriorMass {N : ℕ} (ψ : Wavefunction (N + 1)) (r : ℝ) : ℝ :=  ∫ x in {x : Position | r < ‖x‖}, density ψ x

definition OAI.NeutralAtom.radius line 63

def radius {N : ℕ} (ψ : Wavefunction (N + 1)) (m : ℕ) : ℝ :=  sInf {r : ℝ | 0 ≤ r ∧ exteriorMass ψ r ≤ (m : ℝ)}

definition OAI.NeutralAtom.bTF line 66

def bTF : ℝ := (81 * Real.pi ^ 2 / 2) ^ (1 / 3 : ℝ)

definition OAI.NeutralAtom.largeCharge line 68

def largeCharge (m : ℕ) : Filter ℕ :=  atTop ⊓ Filter.principal {N : ℕ | m < N + 1}

definition OAI.NeutralAtom.upperRadius line 71

def upperRadius (Ψ : ∀ N : ℕ, Wavefunction (N + 1)) (m : ℕ) : EReal :=  Filter.limsup (fun N : ℕ => (radius (Ψ N) m : EReal)) (largeCharge m)

definition OAI.NeutralAtom.lowerRadius line 74

def lowerRadius (Ψ : ∀ N : ℕ, Wavefunction (N + 1)) (m : ℕ) : EReal :=  Filter.liminf (fun N : ℕ => (radius (Ψ N) m : EReal)) (largeCharge m)

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: 2/2, 1/3, generalized, radii, tf, pi, normalized, ground and two more. Found in the paper’s title, which its main theorem states: outer and coulomb. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: limits, choice and neutral.

Statements in Generalized outer-electron radii of neutral Coulomb atoms come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Generalized outer-electron radii of neutral Coulomb atoms

Theorem (Generalized ionization conjecture: radii)

Set b_{\mathrm{TF}}=(81\pi^2/2)^{1/3}. For every choice of normalized neutral ground states (\Psi_Z)_{Z\ge1} of (intro:hamiltonian),

\begin{aligned}\lim_{m\to\infty}m^{1/3} \left[\limsup_{\substack{Z\to\infty\\ Z>m}}R_m(\Psi_Z)\right] &=b_{\mathrm{TF}},\\ \lim_{m\to\infty}m^{1/3} \left[\liminf_{\substack{Z\to\infty\\ Z>m}}R_m(\Psi_Z)\right] &=b_{\mathrm{TF}}.\end{aligned}

All limits are over integers. The inner limits hold m fixed; no convergence in Z at a fixed m is assumed.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

Set $b_{\mathrm{TF}}=(81\pi^2/2)^{1/3}$.  For every choice of normalized
neutral ground states $(\Psi_Z)_{Z\ge1}$ of \eqref{intro:hamiltonian},
\begin{align}
 \lim_{m\to\infty}m^{1/3}
       \left[\limsup_{\substack{Z\to\infty\\ Z>m}}R_m(\Psi_Z)\right]
       &=b_{\mathrm{TF}},\\
 \lim_{m\to\infty}m^{1/3}
       \left[\liminf_{\substack{Z\to\infty\\ Z>m}}R_m(\Psi_Z)\right]
       &=b_{\mathrm{TF}}.
\end{align}
All limits are over integers.  The inner limits hold $m$ fixed; no
convergence in $Z$ at a fixed $m$ is assumed.

Other possible paper matches, closest wording first

Theorem 1.1 of Generalized ionization energies for full Coulomb atoms. Words of the Lean statement, its names split into words, that also appear in the theorem: generalized, energy, spin, tf, coulomb, integers and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: asserted, taken, two, limits, atoms, electron and one more.
Theorem 5.4 of Generalized outer-electron radii of neutral Coulomb atoms. Words of the Lean statement, its names split into words, that also appear in the theorem: exterior, tf, normalized, ground and states. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: asymptotic, expected, neutral and longrightarrow.
Theorem 1.1 of Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms. Words of the Lean statement, its names split into words, that also appear in the theorem: positions, real, charge and integer. Found in the paper’s title, which its main theorem states: radius, outer and coulomb. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nuclear and longrightarrow.
Corollary 1.3 of Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms. Words of the Lean statement, its names split into words, that also appear in the theorem: energy, spin, charge, coulomb, integer and states. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: first, nuclear, two, atom, electron and neutral.
Proposition 2.1 of Generalized outer-electron radii of neutral Coulomb atoms. Words of the Lean statement, its names split into words, that also appear in the theorem: configuration, normalized, ground and state. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: electron and neutral.
Proposition 6.2 of Generalized ionization energies for full Coulomb atoms. Words of the Lean statement, its names split into words, that also appear in the theorem: real, spin, charges and tf. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two and limit.

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 263’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: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: CoulombIonization.generalized_ionization : ∃ a > 0, TFCharacterization a ∧ JointLimit a ∧ IteratedLimits a (I_m(Z)/m^{7/3} → a_TF as m → ∞ and Z/m → ∞, and the Z → ∞ first iterated limsup/liminf; energies are infima of the full antisymmetric two-spin form with FormAdmissible) and CoulombRadii.generalized_outer_radii (m^{1/3} R_m → (81π²/2)^{1/3} for both limsup and liminf over any sequence of normalized ground states). Not stated: the summary's first claim, a molecule with M nuclei and total charge Z strictly binds at most Z + CM electrons, and the universal positive upper and lower bounds on first ionization energies/half-electron radii (docs: only 'large-ionization asymptotics').
Definitions the group asks readers to compare with the paper: Custom 'energy': energy Z N := if N = 0 then 0 else sInf {e | ∃ ψ : FormVector N, FormAdmissible ψ ∧ formEnergy Z ψ = e} (a real sInf, junk 0 if unbounded below, which atoms are not) and IteratedLimits uses real-valued limsup/liminf of ionization m (junk 0 if unbounded); CoulombRadii instead uses EReal-valued limsup/liminf and takes the ground states Ψ N as hypotheses (IsNormalizedGroundState), so it is vacuous if such minimizers did not exist (they do for neutral atoms). TF functional constant tfKinetic = (3/10)(3π²)^(2/3) written by hand.
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: Not stated: the uniform excess-charge bound Z + CM for molecules, and the universal two-sided bounds on neutral-atom first ionization energies and half-electron radii. These are the family's first manuscript, which contains the ionization conjecture of the family title.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For neutral Coulomb atoms with two electron spin states, the formalization proves the asymptotic law for generalized outer-electron radii defined by an expected exterior electron mass $m$. For every choice of normalized ground states, both the upper and lower large-nuclear-charge radius limits satisfy $m^{1/3}R_m\longrightarrow (81\pi^2/2)^{1/3}$ as $m\to\infty$. The large-nuclear-charge limit is taken first. Ground states are inputs to the statement; their existence is not asserted separately.”
Quoted from OpenAI’s Lean note for this family

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

89 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: 61 to its own declarations, 50 to Mathlib’s documentation. 21 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves generalized_outer_radii exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 20 other declarations of the file; with the theorem itself they span 57 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.NeutralAtom are listed without it.

abbrev Positionline 12library
abbrev Configurationline 13library
abbrev Spinsline 14library
abbrev Wavefunctionline 15in the statementlibrary
abbrev Gradientline 16library
def coordinateDirectionline 18library
def HasWeakGradientline 21library
def IsAntisymmetricline 26library
def FormDomainline 31library
def normSquaredline 39library
def coulombPotentialline 42library
def energyline 46library
def IsNormalizedGroundStateline 51in the statementlibrary
def densityline 56library
def exteriorMassline 60library
def radiusline 63library
def bTFline 66in the statementlibrary
def largeChargeline 68library
def upperRadiusline 71in the statementlibrary
def lowerRadiusline 74in the statementlibrary
theorem generalized_outer_radiiline 77selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheory Filteropen scoped BigOperators Topology ContDiffnamespace NeutralAtomabbrev Position := EuclideanSpace ℝ (Fin 3)abbrev Configuration (n : ℕ) := Fin n → Positionabbrev Spins (n : ℕ) := Fin n → Fin 2abbrev Wavefunction (n : ℕ) := Spins n → Configuration n → ℂabbrev Gradient (n : ℕ) := Spins n → Fin n → Fin 3 → Configuration n → ℂdef coordinateDirection {n : ℕ} (i : Fin n) (a : Fin 3) : Configuration n :=  Pi.single i (EuclideanSpace.single a 1)def HasWeakGradient {n : ℕ} (ψ : Wavefunction n) (g : Gradient n) : Prop :=  ∀ σ i a (φ : Configuration n → ℝ), ContDiff ℝ ∞ φ → HasCompactSupport φ →    (∫ x, (fderiv ℝ φ x (coordinateDirection i a) : ℂ) * ψ σ x) =      -(∫ x, (φ x : ℂ) * g σ i a x)def IsAntisymmetric {n : ℕ} (ψ : Wavefunction n) : Prop :=  ∀ (p : Equiv.Perm (Fin n)) (σ : Spins n),    (fun x : Configuration n => ψ (σ ∘ p) (x ∘ p)) =ᵐ[volume]      (fun x => (((Equiv.Perm.sign p : ℤ) : ℂ)) * ψ σ x)def FormDomain {n : ℕ} (ψ : Wavefunction n) (g : Gradient n) : Prop :=  IsAntisymmetric ψ ∧ HasWeakGradient ψ g ∧  (∀ σ, MemLp (ψ σ) 2 volume) ∧  (∀ σ i a, MemLp (g σ i a) 2 volume) ∧  (∀ σ i, Integrable (fun x : Configuration n => ‖x i‖⁻¹ * ‖ψ σ x‖ ^ 2)) ∧  (∀ σ i j, i < j →    Integrable (fun x : Configuration n => ‖x i - x j‖⁻¹ * ‖ψ σ x‖ ^ 2))def normSquared {n : ℕ} (ψ : Wavefunction n) : ℝ :=  ∑ σ : Spins n, ∫ x : Configuration n, ‖ψ σ x‖ ^ 2def coulombPotential {n : ℕ} (Z : ℕ) (x : Configuration n) : ℝ :=  -(Z : ℝ) * (∑ i : Fin n, ‖x i‖⁻¹) +    ∑ i : Fin n, ∑ j ∈ Finset.univ.filter (fun j : Fin n => i < j), ‖x i - x j‖⁻¹def energy {n : ℕ} (Z : ℕ) (ψ : Wavefunction n) (g : Gradient n) : ℝ :=  (1 / 2 : ℝ) * (∑ σ : Spins n, ∑ i : Fin n, ∑ a : Fin 3,    ∫ x : Configuration n, ‖g σ i a x‖ ^ 2) +  ∑ σ : Spins n, ∫ x : Configuration n, coulombPotential Z x * ‖ψ σ x‖ ^ 2def IsNormalizedGroundState (Z : ℕ) (ψ : Wavefunction Z) : Prop :=  ∃ g : Gradient Z, FormDomain ψ g ∧ normSquared ψ = 1 ∧    ∀ (χ : Wavefunction Z) (h : Gradient Z),      FormDomain χ h → normSquared χ = 1 → energy Z ψ g ≤ energy Z χ hdef density {N : ℕ} (ψ : Wavefunction (N + 1)) (x : Position) : ℝ :=  (N + 1 : ℝ) * ∑ σ : Spins (N + 1),    ∫ y : Configuration N, ‖ψ σ (Fin.cons x y)‖ ^ 2def exteriorMass {N : ℕ} (ψ : Wavefunction (N + 1)) (r : ℝ) : ℝ :=  ∫ x in {x : Position | r < ‖x‖}, density ψ xdef radius {N : ℕ} (ψ : Wavefunction (N + 1)) (m : ℕ) : ℝ :=  sInf {r : ℝ | 0 ≤ r ∧ exteriorMass ψ r ≤ (m : ℝ)}def bTF : ℝ := (81 * Real.pi ^ 2 / 2) ^ (1 / 3 : ℝ)def largeCharge (m : ℕ) : Filter ℕ :=  atTop ⊓ Filter.principal {N : ℕ | m < N + 1}def upperRadius (Ψ : ∀ N : ℕ, Wavefunction (N + 1)) (m : ℕ) : EReal :=  Filter.limsup (fun N : ℕ => (radius (Ψ N) m : EReal)) (largeCharge m)def lowerRadius (Ψ : ∀ N : ℕ, Wavefunction (N + 1)) (m : ℕ) : EReal :=  Filter.liminf (fun N : ℕ => (radius (Ψ N) m : EReal)) (largeCharge m)theorem generalized_outer_radii    (Ψ : ∀ N : ℕ, Wavefunction (N + 1))    (hΨ : ∀ N : ℕ, IsNormalizedGroundState (N + 1) (Ψ N)) :    Tendsto (fun m : ℕ => (((m : ℝ) ^ (1 / 3 : ℝ) : ℝ) : EReal) * upperRadius Ψ m)      atTop (𝓝 (bTF : EReal)) ∧    Tendsto (fun m : ℕ => (((m : ℝ) ^ (1 / 3 : ℝ) : ℝ) : EReal) * lowerRadius Ψ m)      atTop (𝓝 (bTF : EReal)) := by  sorryend NeutralAtomendend 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.

Generalized ionization energies for full Coulomb atoms

Theorem (Generalized ionization energy)

For full nonrelativistic Coulomb atoms with two electron spin states,

\frac{I_m(Z)}{m^{7/3}}\longrightarrow a_{\TF}>0 \quad\text{whenever}\quad m\longrightarrow\infty, \qquad \frac Zm\longrightarrow\infty,

where Z>m\ge1 are integers. In particular,

\begin{aligned}\lim_{m\to\infty} \frac{\limsup_{Z\to\infty} I_m(Z)}{m^{7/3}} &=a_{\TF},\\ \lim_{m\to\infty} \frac{\liminf_{Z\to\infty} I_m(Z)}{m^{7/3}} &=a_{\TF}.\end{aligned}

The inner limits are taken over integer Z with m fixed. No convergence of I_m(Z) at a fixed m is asserted or assumed.

Section 1, Introduction Read it in the paper TeX source, line 78

Show the TeX

For full nonrelativistic Coulomb atoms with two electron spin states,
\begin{equation}
 \frac{I_m(Z)}{m^{7/3}}\longrightarrow a_{\TF}>0
 \quad\text{whenever}\quad m\longrightarrow\infty,
 \qquad \frac Zm\longrightarrow\infty,
\end{equation}
where $Z>m\ge1$ are integers. In particular,
\begin{align}
 \lim_{m\to\infty}
  \frac{\limsup_{Z\to\infty} I_m(Z)}{m^{7/3}}
   &=a_{\TF},\\
 \lim_{m\to\infty}
  \frac{\liminf_{Z\to\infty} I_m(Z)}{m^{7/3}}
   &=a_{\TF}.
\end{align}
The inner limits are taken over integer $Z$ with $m$ fixed. No convergence
of $I_m(Z)$ at a fixed $m$ is asserted or assumed.

Theorem (Deficit at a prescribed price)

There is a universal number q>0 such that, for every sequence satisfying

\lambda=s^{-4},\qquad s\longrightarrow0, \qquad Zs^3\longrightarrow\infty,

and every choice of n minimizing E_n+\lambda n, one has

s^3(Z-n)\longrightarrow q.

The number q is the exterior Coulomb charge of the unique classical solution, tending uniformly to zero at infinity, of

\Delta F=4\pi k\bigl((F-1)_+\bigr)^{3/2} \quad\text{on }\R^3\setminus\{0\}, \qquad k=(5T/3)^{-3/2},

that is bounded above and below by positive multiples of |x|^{-4} near zero. The density k((F-1)_+)^{3/2} has bounded support, and F(x)=q/|x| outside that support. Moreover,

a_{\TF}=\frac37q^{-4/3}.

Section 1, Introduction Read it in the paper TeX source, line 129

Show the TeX

There is a universal number $q>0$ such that, for every sequence satisfying
\[
 \lambda=s^{-4},\qquad s\longrightarrow0,
 \qquad Zs^3\longrightarrow\infty,
\]
and every choice of $n$ minimizing $E_n+\lambda n$, one has
\[
 s^3(Z-n)\longrightarrow q.
\]
The number $q$ is the exterior Coulomb charge of the unique classical solution, tending uniformly to zero at infinity,
of
\[
 \Delta F=4\pi k\bigl((F-1)_+\bigr)^{3/2}
 \quad\text{on }\R^3\setminus\{0\},
 \qquad k=(5T/3)^{-3/2},
\]
that is bounded above and below by positive multiples of $|x|^{-4}$ near
zero. The density $k((F-1)_+)^{3/2}$ has bounded support, and
$F(x)=q/|x|$ outside that support.
Moreover,
\[
 a_{\TF}=\frac37q^{-4/3}.
\]

Generalized outer-electron radii of neutral Coulomb atoms

Theorem (Generalized ionization conjecture: radii)

Set b_{\mathrm{TF}}=(81\pi^2/2)^{1/3}. For every choice of normalized neutral ground states (\Psi_Z)_{Z\ge1} of (intro:hamiltonian),

All limits are over integers. The inner limits hold m fixed; no convergence in Z at a fixed m is assumed.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

Set $b_{\mathrm{TF}}=(81\pi^2/2)^{1/3}$.  For every choice of normalized
neutral ground states $(\Psi_Z)_{Z\ge1}$ of \eqref{intro:hamiltonian},
\begin{align}
 \lim_{m\to\infty}m^{1/3}
       \left[\limsup_{\substack{Z\to\infty\\ Z>m}}R_m(\Psi_Z)\right]
       &=b_{\mathrm{TF}},\\
 \lim_{m\to\infty}m^{1/3}
       \left[\liminf_{\substack{Z\to\infty\\ Z>m}}R_m(\Psi_Z)\right]
       &=b_{\mathrm{TF}}.
\end{align}
All limits are over integers.  The inner limits hold $m$ fixed; no
convergence in $Z$ at a fixed $m$ is assumed.

Configuration and file facts

File: lean/ComparatorChallenges/CoulombRadii.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 263 link it.
Solution module: OAI.Analysis.CoulombRadii.Main
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: 20 local vocabulary declarations
Lines of Lean, without comments or blank lines: 57
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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