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

Proved

OAI.Heisenberg.Selection.field_ordered_cubes

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.Selection.field_ordered_cubes of lean/OAI/MathematicalPhysics/Heisenberg/CubeSelection.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Heisenberg.Selection, with open scoped Classical BigOperators Fin.NatCast Matrix Matrix.Norms.Operator, open Filter Topology, open Heisenberg.Geometry SpatialFrame SlotTensor):

theorem field_ordered_cubes (d ℓ : ℕ) (hd : 2≤d) [NeZero ℓ]
    (τ : State (d+1) ℓ) (hτ : ∀ A B,τ.functional (A*B)=τ.functional (B*A)) :
    ∃ β₀ : ℝ,0<β₀ ∧ ∀ β : ℝ,β₀≤β → ∀ h : ℝ,0<h →
      ∀ᶠ n : ℕ in atTop,
        ((box (d+1) (n+1)).card:ℝ)*ℓ/8≤
          ((gibbsState τ (shiftedField (box (d+1) (n+1)) h) β).functional
            (totalZ (box (d+1) (n+1)))).re

Definitions. The statement is written with OAI.Heisenberg.shiftedField and OAI.Heisenberg.totalZ (definitions bundle OAIW2HM000 of this split); OAI.Heisenberg.gibbsState (definitions bundle OAIW2HM002 of this split); OAI.Heisenberg.Configuration, OAI.Heisenberg.QuasiLocal, OAI.Heisenberg.Site, OAI.Heisenberg.SpinHilbert, OAI.Heisenberg.SpinOperator, OAI.Heisenberg.State and OAI.Heisenberg.box (the published bundle Def_Heisenberg).

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

Preamble

import Mathlib
import Definitions.Def_Heisenberg
import Definitions.Def_OAIW2HM002

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 Fin.NatCast Matrix Matrix.Norms.Operator
open Filter Topology
namespace Heisenberg.Selection
open Heisenberg.Geometry SpatialFrame SlotTensor

theorem field_ordered_cubes (d ℓ : ℕ) (hd : 2≤d) [NeZero ℓ]
    (τ : State (d+1) ℓ) (hτ : ∀ A B,τ.functional (A*B)=τ.functional (B*A)) :
    ∃ β₀ : ℝ,0<β₀ ∧ ∀ β : ℝ,β₀≤β → ∀ h : ℝ,0<h →
      ∀ᶠ n : ℕ in atTop,
        ((box (d+1) (n+1)).card:ℝ)*ℓ/8≤
          ((gibbsState τ (shiftedField (box (d+1) (n+1)) h) β).functional
            (totalZ (box (d+1) (n+1)))).re := by
  sorry

end Heisenberg.Selection
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/CubeSelection.lean (declaration OAI.Heisenberg.Selection.field_ordered_cubes)

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