OAI.Heisenberg.field_local_small: intermediate lemma from OpenAI, Spontaneous magnetization in the quantum Heisenberg ferromagnet

Proved

OAI.Heisenberg.field_local_small

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Heisenberg.spontaneous_magnetization, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Spontaneous magnetization in the quantum Heisenberg ferromagnet, OpenAI Math Release, September 24, 2026, result family 271, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Heisenberg.field_local_small of lean/OAI/MathematicalPhysics/Heisenberg/SpinFieldUniform.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Heisenberg, with open scoped Classical BigOperators ComplexConjugate, open Filter Topology):

lemma field_local_small (F : Finset (Site d)) (B : QuasiLocal d ℓ)
    (hB : B∈localAlgebra d ℓ (F : Set (Site d))) (K : Set ℝ) (hK : IsCompact K)
    {ε : ℝ} (hε : 0<ε) : ∃ δ : ℝ,0<δ ∧ ∀ Γ : Finset (Site d),F⊆Γ →
      ∀ h : ℝ,|h|<δ → ∀ t∈K,
        dist (GibbsStrip.evolve (shiftedField Γ h) t B) (finiteDynamics d ℓ Γ t B)<ε

Definitions. The statement is written with OAI.GibbsStrip.evolve, OAI.Heisenberg.localAlgebra, OAI.Heisenberg.quasiLocalClosed, OAI.Heisenberg.quasiLocalStarRing and OAI.Heisenberg.shiftedField (definitions bundle OAIW2HM000 of this split); OAI.Heisenberg.Configuration, OAI.Heisenberg.QuasiLocal, OAI.Heisenberg.Site, OAI.Heisenberg.SpinHilbert, OAI.Heisenberg.SpinOperator and OAI.Heisenberg.finiteDynamics (the published bundle Def_Heisenberg).

Used by. The proof of OAI.Heisenberg.cubeAverage_limit_KMS of this split imports it.

Preamble

import Mathlib
import Definitions.Def_Heisenberg
import Definitions.Def_OAIW2HM000

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section
noncomputable section
open scoped Classical BigOperators ComplexConjugate
open Filter Topology
namespace Heisenberg
variable {d ℓ : ℕ}

lemma field_local_small (F : Finset (Site d)) (B : QuasiLocal d ℓ)
    (hB : B∈localAlgebra d ℓ (F : Set (Site d))) (K : Set ℝ) (hK : IsCompact K)
    {ε : ℝ} (hε : 0<ε) : ∃ δ : ℝ,0<δ ∧ ∀ Γ : Finset (Site d),F⊆Γ →
      ∀ h : ℝ,|h|<δ → ∀ t∈K,
        dist (GibbsStrip.evolve (shiftedField Γ h) t B) (finiteDynamics d ℓ Γ t B)<ε := by
  sorry

end Heisenberg
end
end

end OAI
end

Source

OpenAI, Spontaneous magnetization in the quantum Heisenberg ferromagnet, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 271; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Heisenberg/SpinFieldUniform.lean (declaration OAI.Heisenberg.field_local_small)

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