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

Proved

OAI.Heisenberg.Bootstrap.periodic_return_estimate

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.Bootstrap.periodic_return_estimate of lean/OAI/MathematicalPhysics/Heisenberg/PeriodicReturnEstimate.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Heisenberg.Bootstrap, with open scoped Classical BigOperators Fin.NatCast, open Filter Topology, open Heisenberg.Geometry Heisenberg.Schedule Heisenberg.SpatialFrame Heisenberg.MeshPair Heisenberg.Exposure Heisenberg.LatticeRegular Heisenberg.Cycle):

theorem periodic_return_estimate (d : ℕ) (hd : 2 ≤ d) (L : ℝ) (hL : 0 ≤ L) :
    ∃ M : ℕ,∃ C≥0,∀ (p s : ℝ) (R : ℕ) (ρ : ℝ),0<p → 0<s → 1≤R →
      (2*(ρ+1/(1+(R:ℝ)))+8*C*(1+(R:ℝ))^((1:ℝ)/100)/s < p/4) →
      ∀ (B : Type) [Fintype B] [Nonempty B]
        (Λ F : Finset (Site (d+1))) (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),
        8*(4*(d+1)*(Fintype.card B)^2:ℕ)*Real.exp 1*s ≤ R →
        ∀ᶠ δ in 𝓝 (0:ℝ),0<δ →
        ∀ (N Q : ℕ) (_hQ : 0<Q) (_hqQ : q≤Q) (_hdiv : Q∣N+1)
          (a b : Fin (N+1) → Slots (B:=B) Λ)
          (_hab : ∀ j : Fin (N+1),j.val%Q<q →
            a j=a₀ (j.val%Q) ∧ b j=b₀ (j.val%Q))
          (_hsite : ∀ j : Fin (N+1),q≤j.val%Q → (a j).1.val=(b j).1.val)
          (rates : Fin (N+1) → ℝ) (_hp : ∀ j,0≤rates j ∧ rates j≤1)
          (_hrate : ∀ j : Fin (N+1),j.val%Q<q → rates j=δ/2)
          (P : Set (Fin (N+1)×Slots (B:=B) Λ))
          (_hP : ∀ z,z.2.1.val∉F → z∈P)
          (t : Fin (N+1)) (T : Finset P) (J : Finset (Slots (B:=B) Λ)),
          T.image Subtype.val=Heisenberg.Serial.targetsAt N t J → J.Nonempty →
          (∀ j∈J,j.1.val∈F) → (∀ j∈J,(t,j)∈P) →
          F⊆Λ → (∀ z∈F,∀ w,Step z w → w∈Λ) →
          (∀ z u,0≤u → ((Finset.univ.filter (fun j : J=>dist j.val.1.val z≤u)).card:ℝ)≤L*(1+u)^((1:ℝ)/100)) →
          2*(⌊s/δ⌋₊+1)*Q ≤ N+1 →
          (∀ u x,(∑ c,layerMass (finRotate (N+1)) a b P rates c *
            (if ¬Regular (blocked Λ (holeCut (finRotate (N+1)) (fun i=>Equiv.swap (a i) (b i)) c P u))
              x R M (1/(100*(d+1:ℝ))) then (1:ℝ) else 0)) ≤ ρ) →
          (∑ c,layerMass (finRotate (N+1)) a b P rates c *
            returnCount (layerHistory (finRotate (N+1)) (fun i=>Equiv.swap (a i) (b i)) c) P T) ≤ p/2*J.card

Definitions. The statement is written with OAI.Heisenberg.Bootstrap.retDecEq, OAI.Heisenberg.Exposure.holeCut, OAI.Heisenberg.Exposure.layerHistory, OAI.Heisenberg.Exposure.layerMass, OAI.Heisenberg.Exposure.returnCount, OAI.Heisenberg.Geometry.Site, OAI.Heisenberg.Geometry.Step, OAI.Heisenberg.LatticeRegular.Regular, OAI.Heisenberg.Serial.targetsAt, OAI.Heisenberg.SpatialFrame.Edges, OAI.Heisenberg.SpatialFrame.Slots and OAI.Heisenberg.SpatialFrame.blocked (definitions bundle OAIW2HM000 of this split); OAI.Heisenberg.Site (the published bundle Def_Heisenberg).

Used by. The proof of OAI.Heisenberg.Surface.mesh_matrix_surface 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 Fin.NatCast
open Filter Topology
namespace Heisenberg.Bootstrap
attribute [-instance] instDecidableEqFin
attribute [local irreducible] Pin.pinDensity LayeredExchange.layerWeight LayeredExchange.coinWiring Heisenberg.Exposure.layerMass Heisenberg.Exposure.layerHistory Heisenberg.Cycle.MissingSet Heisenberg.Return.HasHit Heisenberg.Exposure.exposedSweep
open Heisenberg.Geometry Heisenberg.Schedule Heisenberg.SpatialFrame Heisenberg.MeshPair Heisenberg.Exposure Heisenberg.LatticeRegular Heisenberg.Cycle

attribute [local instance] OAI.Heisenberg.Bootstrap.retDecEq
theorem periodic_return_estimate (d : ℕ) (hd : 2 ≤ d) (L : ℝ) (hL : 0 ≤ L) :
    ∃ M : ℕ,∃ C≥0,∀ (p s : ℝ) (R : ℕ) (ρ : ℝ),0<p → 0<s → 1≤R →
      (2*(ρ+1/(1+(R:ℝ)))+8*C*(1+(R:ℝ))^((1:ℝ)/100)/s < p/4) →
      ∀ (B : Type) [Fintype B] [Nonempty B]
        (Λ F : Finset (Site (d+1))) (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),
        8*(4*(d+1)*(Fintype.card B)^2:ℕ)*Real.exp 1*s ≤ R →
        ∀ᶠ δ in 𝓝 (0:ℝ),0<δ →
        ∀ (N Q : ℕ) (_hQ : 0<Q) (_hqQ : q≤Q) (_hdiv : Q∣N+1)
          (a b : Fin (N+1) → Slots (B:=B) Λ)
          (_hab : ∀ j : Fin (N+1),j.val%Q<q →
            a j=a₀ (j.val%Q) ∧ b j=b₀ (j.val%Q))
          (_hsite : ∀ j : Fin (N+1),q≤j.val%Q → (a j).1.val=(b j).1.val)
          (rates : Fin (N+1) → ℝ) (_hp : ∀ j,0≤rates j ∧ rates j≤1)
          (_hrate : ∀ j : Fin (N+1),j.val%Q<q → rates j=δ/2)
          (P : Set (Fin (N+1)×Slots (B:=B) Λ))
          (_hP : ∀ z,z.2.1.val∉F → z∈P)
          (t : Fin (N+1)) (T : Finset P) (J : Finset (Slots (B:=B) Λ)),
          T.image Subtype.val=Heisenberg.Serial.targetsAt N t J → J.Nonempty →
          (∀ j∈J,j.1.val∈F) → (∀ j∈J,(t,j)∈P) →
          F⊆Λ → (∀ z∈F,∀ w,Step z w → w∈Λ) →
          (∀ z u,0≤u → ((Finset.univ.filter (fun j : J=>dist j.val.1.val z≤u)).card:ℝ)≤L*(1+u)^((1:ℝ)/100)) →
          2*(⌊s/δ⌋₊+1)*Q ≤ N+1 →
          (∀ u x,(∑ c,layerMass (finRotate (N+1)) a b P rates c *
            (if ¬Regular (blocked Λ (holeCut (finRotate (N+1)) (fun i=>Equiv.swap (a i) (b i)) c P u))
              x R M (1/(100*(d+1:ℝ))) then (1:ℝ) else 0)) ≤ ρ) →
          (∑ c,layerMass (finRotate (N+1)) a b P rates c *
            returnCount (layerHistory (finRotate (N+1)) (fun i=>Equiv.swap (a i) (b i)) c) P T) ≤ p/2*J.card := by
  sorry

end Heisenberg.Bootstrap
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/PeriodicReturnEstimate.lean (declaration OAI.Heisenberg.Bootstrap.periodic_return_estimate)

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