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

Proved

OAI.Heisenberg.Surface.mesh_matrix_surface

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.Surface.mesh_matrix_surface of lean/OAI/MathematicalPhysics/Heisenberg/MeshMatrixSurface.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Heisenberg.Surface, with open scoped Classical BigOperators Fin.NatCast, open Filter Topology, open Heisenberg.Cycle Pin Return LayeredExchange Exposure LatticeRegular SpatialFrame Heisenberg.Geometry):

theorem mesh_matrix_surface (d : ℕ) (hd : 2≤d) (B : Type) [Fintype B] [Nonempty B] :
    ∃ s : ℝ,0<s ∧
      ∀ (Λ F : Finset (Site (d+1))) (_hΛ : Λ.Nonempty) (q : ℕ) [NeZero q]
        (edge : Fin q ≃ Edges (B:=B) Step Λ) (a₀ b₀ : ℕ → Slots (B:=B) Λ)
        (_ha₀ : ∀ j : Fin q,a₀ j=(edge j).val.1)
        (_hb₀ : ∀ j : Fin q,b₀ j=(edge j).val.2),
        ∀ᶠ δ in 𝓝 (0:ℝ),0<δ →
        ∀ (L Q : ℕ) (_hL : 0<L) (_hQ : 0<Q) (_hqQ : q≤Q) (_hdiv : Q∣L)
          (a b : Fin L → Slots (B:=B) Λ)
          (_hab : ∀ j : Fin L,j.val%Q<q → a j=a₀ (j.val%Q) ∧ b j=b₀ (j.val%Q))
          (_hsite : ∀ j : Fin L,q≤j.val%Q → (a j).1.val=(b j).1.val)
          (rates : Fin L → ℝ) (_hp : ∀ j,0≤rates j ∧ rates j≤1)
          (_hrate : ∀ j : Fin L,j.val%Q<q → rates j=δ/2),
          F⊆Λ → (∀ z∈F,∀ w,Step z w → w∈Λ) →
          2*(⌊s/δ⌋₊+1)*Q ≤ L →
          let K:=Finset.univ.filter (fun j : Fin L=>j.val%Q<q ∧ ((a j).1.val∉F ∨ (b j).1.val∉F))
          let T:=Serial.product L (fun j=>colorCoin (a j) (b j) (rates j))
          Real.exp (-(Real.log 2)*(((Finset.univ.filter (fun v : Slots (B:=B) Λ=>v.1.val∉F)).card:ℝ)+2*δ*K.card))/2 *
            Matrix.trace T ≤ Matrix.trace (T*tailProjection (Slots (B:=B) Λ))

Definitions. The statement is written with OAI.Heisenberg.Geometry.Site, OAI.Heisenberg.Geometry.Step, OAI.Heisenberg.Serial.product, OAI.Heisenberg.SpatialFrame.Edges, OAI.Heisenberg.SpatialFrame.Slots, OAI.Heisenberg.Surface.colorCoin and OAI.Heisenberg.Surface.tailProjection (definitions bundle OAIW2HM000 of this split); OAI.matrixSurfaceDecEq (definitions bundle OAIW2HM001 of this split); OAI.Heisenberg.Site (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_OAIW2HM000
import Definitions.Def_OAIW2HM001

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
open Filter Topology
attribute [-instance] instDecidableEqFin instDecidableEqProd
namespace Heisenberg.Surface
open Heisenberg.Cycle Pin Return LayeredExchange Exposure LatticeRegular SpatialFrame Heisenberg.Geometry

attribute [local instance] OAI.matrixSurfaceDecEq
theorem mesh_matrix_surface (d : ℕ) (hd : 2≤d) (B : Type) [Fintype B] [Nonempty B] :
    ∃ s : ℝ,0<s ∧
      ∀ (Λ F : Finset (Site (d+1))) (_hΛ : Λ.Nonempty) (q : ℕ) [NeZero q]
        (edge : Fin q ≃ Edges (B:=B) Step Λ) (a₀ b₀ : ℕ → Slots (B:=B) Λ)
        (_ha₀ : ∀ j : Fin q,a₀ j=(edge j).val.1)
        (_hb₀ : ∀ j : Fin q,b₀ j=(edge j).val.2),
        ∀ᶠ δ in 𝓝 (0:ℝ),0<δ →
        ∀ (L Q : ℕ) (_hL : 0<L) (_hQ : 0<Q) (_hqQ : q≤Q) (_hdiv : Q∣L)
          (a b : Fin L → Slots (B:=B) Λ)
          (_hab : ∀ j : Fin L,j.val%Q<q → a j=a₀ (j.val%Q) ∧ b j=b₀ (j.val%Q))
          (_hsite : ∀ j : Fin L,q≤j.val%Q → (a j).1.val=(b j).1.val)
          (rates : Fin L → ℝ) (_hp : ∀ j,0≤rates j ∧ rates j≤1)
          (_hrate : ∀ j : Fin L,j.val%Q<q → rates j=δ/2),
          F⊆Λ → (∀ z∈F,∀ w,Step z w → w∈Λ) →
          2*(⌊s/δ⌋₊+1)*Q ≤ L →
          let K:=Finset.univ.filter (fun j : Fin L=>j.val%Q<q ∧ ((a j).1.val∉F ∨ (b j).1.val∉F))
          let T:=Serial.product L (fun j=>colorCoin (a j) (b j) (rates j))
          Real.exp (-(Real.log 2)*(((Finset.univ.filter (fun v : Slots (B:=B) Λ=>v.1.val∉F)).card:ℝ)+2*δ*K.card))/2 *
            Matrix.trace T ≤ Matrix.trace (T*tailProjection (Slots (B:=B) Λ)) := by
  sorry

end Heisenberg.Surface
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/MeshMatrixSurface.lean (declaration OAI.Heisenberg.Surface.mesh_matrix_surface)

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