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

Proved

OAI.Heisenberg.Flow.indexed_prefix_leakage

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.Flow.indexed_prefix_leakage of lean/OAI/MathematicalPhysics/Heisenberg/PeriodicLeakage.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem indexed_prefix_leakage (hd : 2 ≤ d) (L : ℝ) (hL : 0 ≤ L) :
    ∃ M : ℕ,∃ C ≥ 0,∀ (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)
      (ε : ℝ),0 < ε →
      ∀ᶠ δ in 𝓝 (0:ℝ),0 < δ → δ ≤ 2 →
      ∀ (T : Type) [Fintype T] (m Q : ℕ) [NeZero m]
        (s : Equiv.Perm T) (ι : Fin m ≃ T)
        (_hι : ∀ j,s (ι j)=ι (finRotate m j))
        (_hQ : 0 < Q) (_hqQ : q ≤ Q) (_hdiv : Q∣m)
        (a b : T → Slots (B:=B) Λ)
        (_hab : ∀ j : Fin m,∀ _hj : j.val%Q < q,
          a (ι j)=a₀ (j.val%Q) ∧ b (ι j)=b₀ (j.val%Q))
        (_hsite : ∀ j : Fin m,q ≤ j.val%Q → (a (ι j)).1.val=(b (ι j)).1.val)
        (p : T → ℝ) (_hp : ∀ i,0 ≤ p i ∧ p i ≤ 1)
        (_hrate : ∀ j : Fin m,j.val%Q < q → p (ι j)=δ/2)
        (P : Set (T×Slots (B:=B) Λ)) (t : Fin m)
        (J : Finset (Slots (B:=B) Λ))
        (f : (T → Bool) → EuclideanSpace ℝ (Slots (B:=B) Λ))
        (n R : ℕ) (ρ : ℝ),
        0 < n → Heisenberg.Schedule.cut t Q n ≤ m →
        F⊆Λ → (∀ z∈F,∀ w,Step z w → w∈Λ) →
        (∀ base j,j∈J → j∈holeCut s (fun i=>Equiv.swap (a i) (b i)) base P (ι t) ∧ j.1.val∈F) →
        (∀ base j,j∈J → (f base).ofLp j=1) →
        (∀ base j,0 ≤ (f base).ofLp j ∧ (f base).ofLp j ≤ 1) →
        (∀ base j,j∉holeCut s (fun i=>Equiv.swap (a i) (b i)) base P (ι t)∩{j | j.1.val∈F} → (f base).ofLp j=0) →
        (∀ 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)) →
        (∀ (i : Fin n) (j : J), (∑ base,Exposure.layerMass s a b P p base *
          (if ¬Regular (blocked Λ (holeCut s (fun l=>Equiv.swap (a l) (b l)) base P (ι (run t (Heisenberg.Schedule.cut t Q i)))))
            j.val.1.val R M (1/(100*(d+1:ℝ))) then (1:ℝ) else 0)) ≤ ρ) →
        (J.card:ℝ)-2*((ρ+((n+2+R).choose R:ℝ)*(2*(4*(d+1)*(Fintype.card B)^2:ℕ)*δ)^R)*J.card)-
          2*(C*(1+(R:ℝ))^((1:ℝ)/100)*J.card)/((δ/2)*(n:ℝ))-
          (n:ℝ)*(ε*δ*Fintype.card (Slots (B:=B) Λ)+δ^2*q^2) ≤
        ∑ base,Exposure.layerMass s a b P p base *
          (‖f base‖^2-‖exposedSweep s a b base P {v | v.1.val∈F} p (fun l=>ι (run t l)) (Heisenberg.Schedule.cut t Q n) (f base)‖^2)

Definitions. The statement is written with OAI.Heisenberg.Exposure.exposedSweep, OAI.Heisenberg.Exposure.holeCut, OAI.Heisenberg.Exposure.layerMass, OAI.Heisenberg.Flow.periodicDecEq, OAI.Heisenberg.Geometry.Site, OAI.Heisenberg.Geometry.Step, OAI.Heisenberg.LatticeRegular.Regular, OAI.Heisenberg.Schedule.cut, OAI.Heisenberg.Schedule.run, 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.Bootstrap.periodic_return_estimate 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

noncomputable section
open scoped Classical BigOperators Fin.NatCast RealInnerProductSpace
open Filter Topology
namespace Heisenberg.Flow
open Heisenberg.Geometry Heisenberg.Schedule Heisenberg.SpatialFrame Heisenberg.MeshPair Heisenberg.Exposure Heisenberg.LatticeRegular
variable {d : ℕ}

attribute [local instance] OAI.Heisenberg.Flow.periodicDecEq
theorem indexed_prefix_leakage (hd : 2 ≤ d) (L : ℝ) (hL : 0 ≤ L) :
    ∃ M : ℕ,∃ C ≥ 0,∀ (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)
      (ε : ℝ),0 < ε →
      ∀ᶠ δ in 𝓝 (0:ℝ),0 < δ → δ ≤ 2 →
      ∀ (T : Type) [Fintype T] (m Q : ℕ) [NeZero m]
        (s : Equiv.Perm T) (ι : Fin m ≃ T)
        (_hι : ∀ j,s (ι j)=ι (finRotate m j))
        (_hQ : 0 < Q) (_hqQ : q ≤ Q) (_hdiv : Q∣m)
        (a b : T → Slots (B:=B) Λ)
        (_hab : ∀ j : Fin m,∀ _hj : j.val%Q < q,
          a (ι j)=a₀ (j.val%Q) ∧ b (ι j)=b₀ (j.val%Q))
        (_hsite : ∀ j : Fin m,q ≤ j.val%Q → (a (ι j)).1.val=(b (ι j)).1.val)
        (p : T → ℝ) (_hp : ∀ i,0 ≤ p i ∧ p i ≤ 1)
        (_hrate : ∀ j : Fin m,j.val%Q < q → p (ι j)=δ/2)
        (P : Set (T×Slots (B:=B) Λ)) (t : Fin m)
        (J : Finset (Slots (B:=B) Λ))
        (f : (T → Bool) → EuclideanSpace ℝ (Slots (B:=B) Λ))
        (n R : ℕ) (ρ : ℝ),
        0 < n → Heisenberg.Schedule.cut t Q n ≤ m →
        F⊆Λ → (∀ z∈F,∀ w,Step z w → w∈Λ) →
        (∀ base j,j∈J → j∈holeCut s (fun i=>Equiv.swap (a i) (b i)) base P (ι t) ∧ j.1.val∈F) →
        (∀ base j,j∈J → (f base).ofLp j=1) →
        (∀ base j,0 ≤ (f base).ofLp j ∧ (f base).ofLp j ≤ 1) →
        (∀ base j,j∉holeCut s (fun i=>Equiv.swap (a i) (b i)) base P (ι t)∩{j | j.1.val∈F} → (f base).ofLp j=0) →
        (∀ 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)) →
        (∀ (i : Fin n) (j : J), (∑ base,Exposure.layerMass s a b P p base *
          (if ¬Regular (blocked Λ (holeCut s (fun l=>Equiv.swap (a l) (b l)) base P (ι (run t (Heisenberg.Schedule.cut t Q i)))))
            j.val.1.val R M (1/(100*(d+1:ℝ))) then (1:ℝ) else 0)) ≤ ρ) →
        (J.card:ℝ)-2*((ρ+((n+2+R).choose R:ℝ)*(2*(4*(d+1)*(Fintype.card B)^2:ℕ)*δ)^R)*J.card)-
          2*(C*(1+(R:ℝ))^((1:ℝ)/100)*J.card)/((δ/2)*(n:ℝ))-
          (n:ℝ)*(ε*δ*Fintype.card (Slots (B:=B) Λ)+δ^2*q^2) ≤
        ∑ base,Exposure.layerMass s a b P p base *
          (‖f base‖^2-‖exposedSweep s a b base P {v | v.1.val∈F} p (fun l=>ι (run t l)) (Heisenberg.Schedule.cut t Q n) (f base)‖^2) := by
  sorry

end Heisenberg.Flow

end
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/PeriodicLeakage.lean (declaration OAI.Heisenberg.Flow.indexed_prefix_leakage)

View graph