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

Proved

OAI.Heisenberg.cubeBoundary_lower

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.cubeBoundary_lower of lean/OAI/MathematicalPhysics/Heisenberg/SpinAveragedKMS.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 cubeBoundary_lower (Λ : Finset (Site d)) (ω : State d ℓ) (h : ℝ)
    (A B : QuasiLocal d ℓ) (t : ℝ) :
    cubeBoundary Λ ω h (ContinuousLinearMap.mul ℂ (QuasiLocal d ℓ) A) B t=
      (Λ.card:ℂ)⁻¹*∑ x∈Λ,ω.functional (translation d ℓ x A*
        GibbsStrip.evolve (shiftedField Λ h) t (translation d ℓ x B))

Definitions. The statement is written with OAI.GibbsStrip.evolve, OAI.Heisenberg.cubeBoundary, OAI.Heisenberg.quasiLocalClosed 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, OAI.Heisenberg.State and OAI.Heisenberg.translation (the published bundle Def_Heisenberg).

Used by. The proof of OAI.Heisenberg.finite_cube_gibbs_strip 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 cubeBoundary_lower (Λ : Finset (Site d)) (ω : State d ℓ) (h : ℝ)
    (A B : QuasiLocal d ℓ) (t : ℝ) :
    cubeBoundary Λ ω h (ContinuousLinearMap.mul ℂ (QuasiLocal d ℓ) A) B t=
      (Λ.card:ℂ)⁻¹*∑ x∈Λ,ω.functional (translation d ℓ x A*
        GibbsStrip.evolve (shiftedField Λ h) t (translation d ℓ x B)) := 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/SpinAveragedKMS.lean (declaration OAI.Heisenberg.cubeBoundary_lower)

View graph