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

Proved

OAI.Heisenberg.Selection.physical_field_mean

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.physical_field_mean of lean/OAI/MathematicalPhysics/Heisenberg/PhysicalTilt.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 Matrix Matrix.Norms.Operator, open SlotTensor):

lemma physical_field_mean {d ℓ : ℕ} (Λ : Finset (Site d))
    (τ : State d ℓ) (hτ : ∀ A B,τ.functional (A*B)=τ.functional (B*A))
    (β h L : ℝ) (hb : 0≤β*h) (hL : 0<L)
    (htail : L*Matrix.trace (NormedSpace.exp ((β/2) • spinGenerator (fun x y : Λ=>NearestNeighbor x.val y.val) ℓ))≤
      Matrix.trace (NormedSpace.exp ((β/2) • spinGenerator (fun x y : Λ=>NearestNeighbor x.val y.val) ℓ)*
        Matrix.diagonal (fun k : Λ → SpinSlots.Level ℓ=>if (Λ.card:ℝ)*ℓ/4≤physicalMagnetization ℓ k then (1:ℝ) else 0)))
    (hsmall : L⁻¹*Real.exp (-((β*h)*((Λ.card:ℝ)*ℓ)/16))≤1/11) :
    (Λ.card:ℝ)*ℓ/8≤((gibbsState τ (shiftedField Λ h) β).functional (totalZ Λ)).re

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

Used by. The proof of OAI.Heisenberg.Selection.field_ordered_cubes 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 Matrix Matrix.Norms.Operator
namespace Heisenberg.Selection
open SlotTensor

lemma physical_field_mean {d ℓ : ℕ} (Λ : Finset (Site d))
    (τ : State d ℓ) (hτ : ∀ A B,τ.functional (A*B)=τ.functional (B*A))
    (β h L : ℝ) (hb : 0≤β*h) (hL : 0<L)
    (htail : L*Matrix.trace (NormedSpace.exp ((β/2) • spinGenerator (fun x y : Λ=>NearestNeighbor x.val y.val) ℓ))≤
      Matrix.trace (NormedSpace.exp ((β/2) • spinGenerator (fun x y : Λ=>NearestNeighbor x.val y.val) ℓ)*
        Matrix.diagonal (fun k : Λ → SpinSlots.Level ℓ=>if (Λ.card:ℝ)*ℓ/4≤physicalMagnetization ℓ k then (1:ℝ) else 0)))
    (hsmall : L⁻¹*Real.exp (-((β*h)*((Λ.card:ℝ)*ℓ)/16))≤1/11) :
    (Λ.card:ℝ)*ℓ/8≤((gibbsState τ (shiftedField Λ h) β).functional (totalZ Λ)).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/PhysicalTilt.lean (declaration OAI.Heisenberg.Selection.physical_field_mean)

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