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Lean statement

Low-temperature spontaneous magnetization for every positive spin

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.Heisenberg.spontaneous_magnetization

Lean statement file: Heisenberg.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 271 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 spontaneous_magnetization (d ℓ : ℕ) (hd : 3≤d) (hℓ : 1≤ℓ) :    DynamicsConverges d ℓ ∧      ∃ β₀ : ℝ,0<β₀ ∧ ∀ β : ℝ,β₀≤β →        ∃ ω : State d ℓ,TranslationInvariant ω ∧ IsKMS β ω ∧          (ω.functional (spinZ d ℓ 0)).im=0 ∧          (ℓ:ℝ)/8≤(ω.functional (spinZ d ℓ 0)).re := 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.Heisenberg.L2 line 14

    abbrev L2 (ι : Type u) := lp (fun _ : ι => ℂ) 2

  • definition OAI.Heisenberg.l2Reindex line 16

    def l2Reindex {ι : Type u} {κ : Type v} (e : ι ≃ κ) : L2 ι ≃ₗᵢ[ℂ] L2 κ where
    Show the remaining 14 lines
      toFun f := ⟨fun j => f (e.symm j), by    apply memℓp_gen    exact e.symm.summable_iff.mpr ((lp.memℓp f).summable (by norm_num))⟩  invFun f := ⟨fun i => f (e i), by    apply memℓp_gen    exact e.summable_iff.mpr ((lp.memℓp f).summable (by norm_num))⟩  left_inv f := by ext i; simp  right_inv f := by ext j; simp  map_add' f g := by ext j; rfl  map_smul' c f := by ext j; rfl  norm_map' f := by    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]    congr 1    exact e.symm.tsum_eq (fun i => ‖f i‖ ^ (2 : ℝ≥0∞).toReal)

  • definition OAI.Heisenberg.l2Projection line 32

    def l2Projection {ι : Type u} (p : ι → Prop) : L2 ι →L[ℂ] L2 ι := by
    Show the remaining 15 lines
      classical  let f : L2 ι →ₗ[ℂ] L2 ι :=    { toFun := fun v => ⟨fun i => if p i then v i else 0,        (lp.memℓp v).mono' (fun i => by split_ifs <;> simp)⟩      map_add' := fun v w => by        ext i        change (if p i then v i + w i else 0) =          (if p i then v i else 0) + (if p i then w i else 0)        by_cases h : p i <;> simp [h]      map_smul' := fun c v => by ext i; by_cases h : p i <;> simp [h] }  exact f.mkContinuous 1 (fun v => by    simpa only [one_mul] using lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      (x := f v) (y := v) (fun i => by        change ‖if p i then v i else 0‖ ≤ ‖v i‖        split_ifs <;> simp))

  • abbreviation OAI.Heisenberg.Site line 49

    abbrev Site (d : ℕ) := Fin d → ℤ

  • abbreviation OAI.Heisenberg.Level line 50

    abbrev Level (ℓ : ℕ) := Fin (ℓ + 1)

  • abbreviation OAI.Heisenberg.Configuration line 51

    abbrev Configuration (d ℓ : ℕ) := Site d →₀ Level ℓ

  • abbreviation OAI.Heisenberg.SpinHilbert line 52

    abbrev SpinHilbert (d ℓ : ℕ) := L2 (Configuration d ℓ)

  • abbreviation OAI.Heisenberg.SpinOperator line 53

    abbrev SpinOperator (d ℓ : ℕ) := SpinHilbert d ℓ →L[ℂ] SpinHilbert d ℓ

  • definition OAI.Heisenberg.configPerm line 55

    def configPerm (d ℓ : ℕ) (x : Site d) (σ : Equiv.Perm (Level ℓ)) :    Equiv.Perm (Configuration d ℓ) where
    Show the remaining 10 lines
      toFun c := c.update x (σ (c x))  invFun c := c.update x (σ.symm (c x))  left_inv c := by    classical    ext y    by_cases h : y = x <;> simp [Finsupp.update_apply, h]  right_inv c := by    classical    ext y    by_cases h : y = x <;> simp [Finsupp.update_apply, h]

  • definition OAI.Heisenberg.matrixUnit line 68

    def matrixUnit (d ℓ : ℕ) (x : Site d) (a b : Level ℓ) : SpinOperator d ℓ :=  (l2Projection (fun c : Configuration d ℓ => c x = a)).comp    (l2Reindex (configPerm d ℓ x (Equiv.swap a b))).toContinuousLinearEquiv.toContinuousLinearMap

  • definition OAI.Heisenberg.operatorRelabel line 72

    def operatorRelabel (d ℓ : ℕ) (e : Equiv.Perm (Site d)) :    SpinOperator d ℓ ≃⋆ₐ[ℂ] SpinOperator d ℓ :=  (l2Reindex (Finsupp.equivCongrLeft e)).conjStarAlgEquiv

  • definition OAI.Heisenberg.quasiLocalAlgebra line 76

    def quasiLocalAlgebra (d ℓ : ℕ) : StarSubalgebra ℂ (SpinOperator d ℓ) :=  (StarAlgebra.adjoin ℂ {A | ∃ x a b, A = matrixUnit d ℓ x a b}).topologicalClosure

  • abbreviation OAI.Heisenberg.QuasiLocal line 79

    abbrev QuasiLocal (d ℓ : ℕ) := quasiLocalAlgebra d ℓ

  • definition OAI.Heisenberg.localUnit line 81

    def localUnit (d ℓ : ℕ) (x : Site d) (a b : Level ℓ) : QuasiLocal d ℓ :=  ⟨matrixUnit d ℓ x a b, (StarSubalgebra.le_topologicalClosure _)    (StarAlgebra.subset_adjoin ℂ _ ⟨x, a, b, rfl⟩)⟩

  • definition OAI.Heisenberg.translation line 85

    def translation (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ →⋆ₐ[ℂ] QuasiLocal d ℓ := by
    Show the remaining 53 lines
      have l2Reindex_apply {ι : Type} {κ : Type} (e : ι ≃ κ) (f : L2 ι) (j : κ) :      l2Reindex e f j = f (e.symm j) := rfl  have l2Projection_apply {ι : Type} (p : ι → Prop) (v : L2 ι) (i : ι) :      l2Projection p v i = if p i then v i else 0 := by classical rfl  have matrixUnit_apply (d ℓ : ℕ) (x : Site d) (a b : Level ℓ)      (v : SpinHilbert d ℓ) (c : Configuration d ℓ) :      matrixUnit d ℓ x a b v c = if c x = a then v (c.update x b) else 0 := by    simp only [matrixUnit, ContinuousLinearMap.comp_apply, l2Projection_apply]    split_ifs with h    · change v (c.update x ((Equiv.swap a b).symm (c x))) = v (c.update x b)      simp [h]    · rfl  have l2Reindex_symm_apply {ι : Type} {κ : Type} (e : ι ≃ κ) (v : L2 κ) (i : ι) :      (l2Reindex e).symm v i = v (e i) := rfl  have relabel_matrixUnit (d ℓ : ℕ) (e : Equiv.Perm (Site d))      (x : Site d) (a b : Level ℓ) :      operatorRelabel d ℓ e (matrixUnit d ℓ x a b) = matrixUnit d ℓ (e x) a b := by    ext v c    simp only [operatorRelabel, LinearIsometryEquiv.conjStarAlgEquiv_apply_apply,      l2Reindex_apply, matrixUnit_apply]    change (if c (e x) = a then      (l2Reindex (Finsupp.equivCongrLeft e)).symm v        (((Finsupp.equivCongrLeft e).symm c).update x b) else 0) = _    rw [l2Reindex_symm_apply]    by_cases h : c (e x) = a    · simp only [h, ite_eq_left]      apply congrArg v      ext y      simp only [Finsupp.equivCongrLeft_apply, Finsupp.equivMapDomain_apply,        Finsupp.update_apply]      by_cases hy : y = e x      · subst y; simp      · have hy' : e.symm y ≠ x := fun hxy => hy (by simpa using congrArg e hxy)        simp [hy, hy']    · simp only [h, ite_false]  have relabel_mem (d ℓ : ℕ) (e : Equiv.Perm (Site d))      {A : SpinOperator d ℓ} (hA : A ∈ quasiLocalAlgebra d ℓ) :      operatorRelabel d ℓ e A ∈ quasiLocalAlgebra d ℓ := by    let f := (operatorRelabel d ℓ e).toStarAlgHom    have h : StarAlgebra.adjoin ℂ {A | ∃ x a b, A = matrixUnit d ℓ x a b} ≤        (quasiLocalAlgebra d ℓ).comap f := by      apply StarAlgebra.adjoin_le      rintro _ ⟨x, a, b, rfl⟩      change operatorRelabel d ℓ e (matrixUnit d ℓ x a b) ∈ quasiLocalAlgebra d ℓ      rw [relabel_matrixUnit]      exact (localUnit d ℓ (e x) a b).property    have hc : IsClosed ((quasiLocalAlgebra d ℓ).comap f : Set (SpinOperator d ℓ)) :=      (StarSubalgebra.isClosed_topologicalClosure _).preimage        (StarAlgEquiv.isometry (operatorRelabel d ℓ e)).continuous    exact StarSubalgebra.topologicalClosure_minimal h hc hA  exact let f := (operatorRelabel d ℓ (Equiv.addLeft x)).toStarAlgHom    (f.comp (quasiLocalAlgebra d ℓ).subtype).codRestrict (quasiLocalAlgebra d ℓ)      (fun A => relabel_mem d ℓ (Equiv.addLeft x) A.property)

  • definition OAI.Heisenberg.magneticLevel line 140

    def magneticLevel (ℓ : ℕ) (k : Level ℓ) : ℝ := (k : ℝ) - (ℓ : ℝ) / 2

  • definition OAI.Heisenberg.spinZ line 142

    def spinZ (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  ∑ k : Level ℓ, (magneticLevel ℓ k : ℂ) • localUnit d ℓ x k k

  • definition OAI.Heisenberg.spinPlus line 145

    def spinPlus (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  ∑ k : Fin ℓ,    (Real.sqrt ((ℓ / 2 : ℝ) * (ℓ / 2 + 1) -      magneticLevel ℓ k.castSucc * (magneticLevel ℓ k.castSucc + 1)) : ℂ) •      localUnit d ℓ x k.succ k.castSucc

  • definition OAI.Heisenberg.spinMinus line 151

    def spinMinus (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ := star (spinPlus d ℓ x)

  • definition OAI.Heisenberg.spinX line 153

    def spinX (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  (1 / 2 : ℂ) • (spinPlus d ℓ x + spinMinus d ℓ x)

  • definition OAI.Heisenberg.spinY line 156

    def spinY (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  (1 / (2 * Complex.I) : ℂ) • (spinPlus d ℓ x - spinMinus d ℓ x)

  • definition OAI.Heisenberg.NearestNeighbor line 159

    def NearestNeighbor {d : ℕ} (x y : Site d) : Prop :=  ∑ i : Fin d, |x i - y i| = (1 : ℤ)

  • definition OAI.Heisenberg.volumeEdges line 162

    def volumeEdges {d : ℕ} (Λ : Finset (Site d)) : Finset (Site d × Site d) :=  (Λ ×ˢ Λ).filter fun p => NearestNeighbor p.1 p.2 ∧ WellOrderingRel p.1 p.2

  • definition OAI.Heisenberg.hamiltonian line 165

    def hamiltonian (d ℓ : ℕ) (Λ : Finset (Site d)) : QuasiLocal d ℓ :=  -∑ e ∈ volumeEdges Λ,    (spinX d ℓ e.1 * spinX d ℓ e.2 + spinY d ℓ e.1 * spinY d ℓ e.2 +      spinZ d ℓ e.1 * spinZ d ℓ e.2)

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

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: dynamics, kms, hamiltonian, translation, invariant, state, integer, 1/2 and two more. Found in the paper’s title, which its main theorem states: spontaneous and magnetization. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 3/2.

Statements in Spontaneous magnetization in the quantum Heisenberg ferromagnet come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Spontaneous magnetization in the quantum Heisenberg ferromagnet

Theorem

For every integer d\geq3 and every S\in\{\frac12,1,\frac32,\ldots\}, there is a finite \beta_0(d,S)>0 such that, for every finite \beta\geq\beta_0(d,S), the dynamics associated with (eq:hamiltonian) has a translation-invariant \beta-KMS state \omega satisfying

\omega(S_0^z)\ge S/4.
Section 1, Introduction Read it in the paper TeX source, line 64
Show the TeX
For every integer $d\geq3$ and every
$S\in\{\frac12,1,\frac32,\ldots\}$, there is a finite
$\beta_0(d,S)>0$ such that, for every finite $\beta\geq\beta_0(d,S)$,
the dynamics associated with \eqref{eq:hamiltonian} has a
translation-invariant $\beta$-KMS state $\omega$ satisfying
\[
                         \omega(S_0^z)\ge S/4.
\]

Other possible paper matches, closest wording first

  • Proposition 8.2 of Spontaneous magnetization in the quantum Heisenberg ferromagnet. Words of the Lean statement, its names split into words, that also appear in the theorem: converges, normalized, translation, invariant, state and 1/2. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: field.
  • Proposition 7.3 of The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet. Words of the Lean statement, its names split into words, that also appear in the theorem: translation, invariant, magnetization and state. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equilibrium.
  • Proposition 6.2 of Spontaneous magnetization in the quantum Heisenberg ferromagnet. Words of the Lean statement, its names split into words, that also appear in the theorem: unit, local, site and eq. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sufficiently.
  • Proposition 7.1 of The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet. Words of the Lean statement, its names split into words, that also appear in the theorem: re, positive, real, integer and eq.
  • Corollary 3.6 of The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet. Words of the Lean statement, its names split into words, that also appear in the theorem: spontaneous, magnetization, positive and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sufficiently.
  • Proposition 5.6 of Spontaneous magnetization in the quantum Heisenberg ferromagnet. Words of the Lean statement, its names split into words, that also appear in the theorem: local, site, one and eq. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least.

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 271’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). Only spontaneous_magnetization (d ℓ) (hd : 3 ≤ d) (hℓ : 1 ≤ ℓ) : DynamicsConverges d ℓ ∧ ∃ β₀ > 0, ∀ β ≥ β₀, ∃ ω, TranslationInvariant ω ∧ IsKMS β ω ∧ (ω.functional (spinZ d ℓ 0)).im = 0 ∧ ℓ/8 ≤ (ω.functional (spinZ d ℓ 0)).re (nearest-neighbor Heisenberg ferromagnet, every spin S = ℓ/2) is stated, i.e. the summary's third claim. The headline, Bloch's T^{3/2} law with its exact coefficient for general finite-range couplings (and the first lattice correction for 3D nearest-neighbor couplings), has no Lean statement (docs: 'The formalization proves spontaneous magnetization ...').

    Definitions the group asks readers to compare with the paper: Entire quantum-spin setup is hand-built: Configuration d ℓ := Site d →₀ Level ℓ (incomplete tensor product over the all-zero reference configuration), QuasiLocal d ℓ := closure of the *-algebra of matrix units, hamiltonian with edges counted via WellOrderingRel, dynamics := Filter.limUnder atTop (finiteDynamics .. (box d n) ..) (with convergence asserted separately by DynamicsConverges) and a hand-written IsKMS (analytic strip function with F(t) = ω(A α_t B), F(t+iβ) = ω(α_t B A)). Plausible, but none is Mathlib's notion of a KMS state or quantum spin system.

  • 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: Bloch's T^{3/2} law with exact coefficient (finite-range couplings) and its first lattice correction (the two October 5 manuscripts, which give the family its title) are not stated; only the September 24 spontaneous-magnetization result is. The spherical magnetization law named in the family title is also absent (it is not a headline clause).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves spontaneous magnetization for the nearest-neighbor isotropic quantum Heisenberg ferromagnet on $\mathbb Z^d$ for every $d\ge3$ and every spin $S\in\{\tfrac12,1,\tfrac32,\ldots\}$. At every sufficiently low positive temperature, it constructs a translation-invariant equilibrium state satisfying the KMS condition for the zero-field dynamics and having magnetization at least $S/4$.”

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

218 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: 139 to its own declarations, 75 to Mathlib’s documentation. 32 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInoncomputable sectionuniverse u vopen scoped ComplexConjugate ENNReal Classicalnamespace Heisenbergabbrev L2 (ι : Type u) := lp (fun _ : ι => ℂ) 2def l2Reindex {ι : Type u} {κ : Type v} (e : ι ≃ κ) : L2 ι ≃ₗᵢ[ℂ] L2 κ where  toFun f := ⟨fun j => f (e.symm j), by    apply memℓp_gen    exact e.symm.summable_iff.mpr ((lp.memℓp f).summable (by norm_num))⟩  invFun f := ⟨fun i => f (e i), by    apply memℓp_gen    exact e.summable_iff.mpr ((lp.memℓp f).summable (by norm_num))⟩  left_inv f := by ext i; simp  right_inv f := by ext j; simp  map_add' f g := by ext j; rfl  map_smul' c f := by ext j; rfl  norm_map' f := by    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]    congr 1    exact e.symm.tsum_eq (fun i => ‖f i‖ ^ (2 : ℝ≥0∞).toReal)def l2Projection {ι : Type u} (p : ι → Prop) : L2 ι →L[ℂ] L2 ι := by  classical  let f : L2 ι →ₗ[ℂ] L2 ι :=    { toFun := fun v => ⟨fun i => if p i then v i else 0,        (lp.memℓp v).mono' (fun i => by split_ifs <;> simp)⟩      map_add' := fun v w => by        ext i        change (if p i then v i + w i else 0) =          (if p i then v i else 0) + (if p i then w i else 0)        by_cases h : p i <;> simp [h]      map_smul' := fun c v => by ext i; by_cases h : p i <;> simp [h] }  exact f.mkContinuous 1 (fun v => by    simpa only [one_mul] using lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)      (x := f v) (y := v) (fun i => by        change ‖if p i then v i else 0‖ ≤ ‖v i‖        split_ifs <;> simp))abbrev Site (d : ℕ) := Fin d → ℤabbrev Level (ℓ : ℕ) := Fin (ℓ + 1)abbrev Configuration (d ℓ : ℕ) := Site d →₀ Level ℓabbrev SpinHilbert (d ℓ : ℕ) := L2 (Configuration d ℓ)abbrev SpinOperator (d ℓ : ℕ) := SpinHilbert d ℓ →L[ℂ] SpinHilbert d ℓdef configPerm (d ℓ : ℕ) (x : Site d) (σ : Equiv.Perm (Level ℓ)) :    Equiv.Perm (Configuration d ℓ) where  toFun c := c.update x (σ (c x))  invFun c := c.update x (σ.symm (c x))  left_inv c := by    classical    ext y    by_cases h : y = x <;> simp [Finsupp.update_apply, h]  right_inv c := by    classical    ext y    by_cases h : y = x <;> simp [Finsupp.update_apply, h]def matrixUnit (d ℓ : ℕ) (x : Site d) (a b : Level ℓ) : SpinOperator d ℓ :=  (l2Projection (fun c : Configuration d ℓ => c x = a)).comp    (l2Reindex (configPerm d ℓ x (Equiv.swap a b))).toContinuousLinearEquiv.toContinuousLinearMapdef operatorRelabel (d ℓ : ℕ) (e : Equiv.Perm (Site d)) :    SpinOperator d ℓ ≃⋆ₐ[ℂ] SpinOperator d ℓ :=  (l2Reindex (Finsupp.equivCongrLeft e)).conjStarAlgEquivdef quasiLocalAlgebra (d ℓ : ℕ) : StarSubalgebra ℂ (SpinOperator d ℓ) :=  (StarAlgebra.adjoin ℂ {A | ∃ x a b, A = matrixUnit d ℓ x a b}).topologicalClosureabbrev QuasiLocal (d ℓ : ℕ) := quasiLocalAlgebra d ℓdef localUnit (d ℓ : ℕ) (x : Site d) (a b : Level ℓ) : QuasiLocal d ℓ :=  ⟨matrixUnit d ℓ x a b, (StarSubalgebra.le_topologicalClosure _)    (StarAlgebra.subset_adjoin ℂ _ ⟨x, a, b, rfl⟩)⟩def translation (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ →⋆ₐ[ℂ] QuasiLocal d ℓ := by  have l2Reindex_apply {ι : Type} {κ : Type} (e : ι ≃ κ) (f : L2 ι) (j : κ) :      l2Reindex e f j = f (e.symm j) := rfl  have l2Projection_apply {ι : Type} (p : ι → Prop) (v : L2 ι) (i : ι) :      l2Projection p v i = if p i then v i else 0 := by classical rfl  have matrixUnit_apply (d ℓ : ℕ) (x : Site d) (a b : Level ℓ)      (v : SpinHilbert d ℓ) (c : Configuration d ℓ) :      matrixUnit d ℓ x a b v c = if c x = a then v (c.update x b) else 0 := by    simp only [matrixUnit, ContinuousLinearMap.comp_apply, l2Projection_apply]    split_ifs with h    · change v (c.update x ((Equiv.swap a b).symm (c x))) = v (c.update x b)      simp [h]    · rfl  have l2Reindex_symm_apply {ι : Type} {κ : Type} (e : ι ≃ κ) (v : L2 κ) (i : ι) :      (l2Reindex e).symm v i = v (e i) := rfl  have relabel_matrixUnit (d ℓ : ℕ) (e : Equiv.Perm (Site d))      (x : Site d) (a b : Level ℓ) :      operatorRelabel d ℓ e (matrixUnit d ℓ x a b) = matrixUnit d ℓ (e x) a b := by    ext v c    simp only [operatorRelabel, LinearIsometryEquiv.conjStarAlgEquiv_apply_apply,      l2Reindex_apply, matrixUnit_apply]    change (if c (e x) = a then      (l2Reindex (Finsupp.equivCongrLeft e)).symm v        (((Finsupp.equivCongrLeft e).symm c).update x b) else 0) = _    rw [l2Reindex_symm_apply]    by_cases h : c (e x) = a    · simp only [h, ite_eq_left]      apply congrArg v      ext y      simp only [Finsupp.equivCongrLeft_apply, Finsupp.equivMapDomain_apply,        Finsupp.update_apply]      by_cases hy : y = e x      · subst y; simp      · have hy' : e.symm y ≠ x := fun hxy => hy (by simpa using congrArg e hxy)        simp [hy, hy']    · simp only [h, ite_false]  have relabel_mem (d ℓ : ℕ) (e : Equiv.Perm (Site d))      {A : SpinOperator d ℓ} (hA : A ∈ quasiLocalAlgebra d ℓ) :      operatorRelabel d ℓ e A ∈ quasiLocalAlgebra d ℓ := by    let f := (operatorRelabel d ℓ e).toStarAlgHom    have h : StarAlgebra.adjoin ℂ {A | ∃ x a b, A = matrixUnit d ℓ x a b} ≤        (quasiLocalAlgebra d ℓ).comap f := by      apply StarAlgebra.adjoin_le      rintro _ ⟨x, a, b, rfl⟩      change operatorRelabel d ℓ e (matrixUnit d ℓ x a b) ∈ quasiLocalAlgebra d ℓ      rw [relabel_matrixUnit]      exact (localUnit d ℓ (e x) a b).property    have hc : IsClosed ((quasiLocalAlgebra d ℓ).comap f : Set (SpinOperator d ℓ)) :=      (StarSubalgebra.isClosed_topologicalClosure _).preimage        (StarAlgEquiv.isometry (operatorRelabel d ℓ e)).continuous    exact StarSubalgebra.topologicalClosure_minimal h hc hA  exact let f := (operatorRelabel d ℓ (Equiv.addLeft x)).toStarAlgHom    (f.comp (quasiLocalAlgebra d ℓ).subtype).codRestrict (quasiLocalAlgebra d ℓ)      (fun A => relabel_mem d ℓ (Equiv.addLeft x) A.property)def magneticLevel (ℓ : ℕ) (k : Level ℓ) : ℝ := (k : ℝ) - (ℓ : ℝ) / 2def spinZ (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  ∑ k : Level ℓ, (magneticLevel ℓ k : ℂ) • localUnit d ℓ x k kdef spinPlus (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  ∑ k : Fin ℓ,    (Real.sqrt ((ℓ / 2 : ℝ) * (ℓ / 2 + 1) -      magneticLevel ℓ k.castSucc * (magneticLevel ℓ k.castSucc + 1)) : ℂ) •      localUnit d ℓ x k.succ k.castSuccdef spinMinus (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ := star (spinPlus d ℓ x)def spinX (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  (1 / 2 : ℂ) • (spinPlus d ℓ x + spinMinus d ℓ x)def spinY (d ℓ : ℕ) (x : Site d) : QuasiLocal d ℓ :=  (1 / (2 * Complex.I) : ℂ) • (spinPlus d ℓ x - spinMinus d ℓ x)def NearestNeighbor {d : ℕ} (x y : Site d) : Prop :=  ∑ i : Fin d, |x i - y i| = (1 : ℤ)def volumeEdges {d : ℕ} (Λ : Finset (Site d)) : Finset (Site d × Site d) :=  (Λ ×ˢ Λ).filter fun p => NearestNeighbor p.1 p.2 ∧ WellOrderingRel p.1 p.2def hamiltonian (d ℓ : ℕ) (Λ : Finset (Site d)) : QuasiLocal d ℓ :=  -∑ e ∈ volumeEdges Λ,    (spinX d ℓ e.1 * spinX d ℓ e.2 + spinY d ℓ e.1 * spinY d ℓ e.2 +      spinZ d ℓ e.1 * spinZ d ℓ e.2)def box (d n : ℕ) : Finset (Site d) :=  Fintype.piFinset fun _ : Fin d => Finset.Icc (-(n : ℤ)) (n : ℤ)def finiteDynamics (d ℓ : ℕ) (Λ : Finset (Site d)) (t : ℝ)    (A : QuasiLocal d ℓ) : QuasiLocal d ℓ :=  NormedSpace.exp ((Complex.I * (t : ℂ)) • hamiltonian d ℓ Λ) * A *    NormedSpace.exp ((-(Complex.I * (t : ℂ))) • hamiltonian d ℓ Λ)def dynamics (d ℓ : ℕ) (t : ℝ) (A : QuasiLocal d ℓ) : QuasiLocal d ℓ :=  Filter.limUnder Filter.atTop (fun n : ℕ => finiteDynamics d ℓ (box d n) t A)def DynamicsConverges (d ℓ : ℕ) : Prop :=  ∀ (t : ℝ) (A : QuasiLocal d ℓ),    Filter.Tendsto (fun n : ℕ => finiteDynamics d ℓ (box d n) t A)      Filter.atTop (nhds (dynamics d ℓ t A))structure State (d ℓ : ℕ) where  functional : QuasiLocal d ℓ →L[ℂ] ℂ  normalized : functional 1 = 1  positive : ∀ A, (functional (star A * A)).im = 0 ∧    0 ≤ (functional (star A * A)).redef TranslationInvariant {d ℓ : ℕ} (ω : State d ℓ) : Prop :=  ∀ (x : Site d) (A : QuasiLocal d ℓ), ω.functional (translation d ℓ x A) = ω.functional Adef IsKMS {d ℓ : ℕ} (β : ℝ) (ω : State d ℓ) : Prop :=  0 < β ∧ ∀ A B : QuasiLocal d ℓ, ∃ F : ℂ → ℂ,    ContinuousOn F {z | 0 ≤ z.im ∧ z.im ≤ β} ∧    DifferentiableOn ℂ F {z | 0 < z.im ∧ z.im < β} ∧    (∃ C : ℝ, ∀ z : ℂ, 0 ≤ z.im → z.im ≤ β → ‖F z‖ ≤ C) ∧    (∀ t : ℝ, F t = ω.functional (A * dynamics d ℓ t B)) ∧    (∀ t : ℝ, F ((t : ℂ) + Complex.I * β) = ω.functional (dynamics d ℓ t B * A))open scoped Classical BigOperators ComplexConjugateopen Filter Topologytheorem spontaneous_magnetization (d ℓ : ℕ) (hd : 3≤d) (hℓ : 1≤ℓ) :    DynamicsConverges d ℓ ∧      ∃ β₀ : ℝ,0<β₀ ∧ ∀ β : ℝ,β₀≤β →        ∃ ω : State d ℓ,TranslationInvariant ω ∧ IsKMS β ω ∧          (ω.functional (spinZ d ℓ 0)).im=0 ∧          (ℓ:ℝ)/8≤(ω.functional (spinZ d ℓ 0)).re := by  sorryend Heisenbergendend OAI
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Spontaneous magnetization in the quantum Heisenberg ferromagnet

Theorem

For every integer d\geq3 and every S\in\{\frac12,1,\frac32,\ldots\}, there is a finite \beta_0(d,S)>0 such that, for every finite \beta\geq\beta_0(d,S), the dynamics associated with (eq:hamiltonian) has a translation-invariant \beta-KMS state \omega satisfying

\omega(S_0^z)\ge S/4.
Section 1, Introduction Read it in the paper TeX source, line 64
Show the TeX
For every integer $d\geq3$ and every
$S\in\{\frac12,1,\frac32,\ldots\}$, there is a finite
$\beta_0(d,S)>0$ such that, for every finite $\beta\geq\beta_0(d,S)$,
the dynamics associated with \eqref{eq:hamiltonian} has a
translation-invariant $\beta$-KMS state $\omega$ satisfying
\[
                         \omega(S_0^z)\ge S/4.
\]
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 271 link it.
Solution module
OAI.MathematicalPhysics.Heisenberg.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
31 local vocabulary declarations
Lines of Lean, without comments or blank lines
169
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

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Plectis Atlas

Low-temperature spontaneous magnetization for every positive spin

Family 271: Bloch’s law, its lattice correction, and the spherical magnetization law

Candidate pairing

Theorem

For every integer d\geq3 and every S\in\{\frac12,1,\frac32,\ldots\}, there is a finite \beta_0(d,S)>0 such that, for every finite \beta\geq\beta_0(d,S), the dynamics associated with (eq:hamiltonian) has a translation-invariant \beta-KMS state \omega satisfying

\omega(S_0^z)\ge S/4.

Spontaneous magnetization in the quantum Heisenberg ferromagnet

TeX label thm:main, build/source/sections/00-introduction.tex, line 64

The Lean statement says

theorem spontaneous_magnetization (d ℓ : ℕ) (hd : 3≤d) (hℓ : 1≤ℓ) :    DynamicsConverges d ℓ ∧      ∃ β₀ : ℝ,0<β₀ ∧ ∀ β : ℝ,β₀≤β →        ∃ ω : State d ℓ,TranslationInvariant ω ∧ IsKMS β ω ∧          (ω.functional (spinZ d ℓ 0)).im=0 ∧          (ℓ:ℝ)/8≤(ω.functional (spinZ d ℓ 0)).re := by  sorry

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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

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

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