OAI.SymmetricMahler.equality_strip_images_dense_eq: intermediate lemma from OpenAI, The symmetric Mahler conjecture and its equality cases

Proved

OAI.SymmetricMahler.equality_strip_images_dense_eq

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.SymmetricMahler.symmetric_mahler_equality, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The symmetric Mahler conjecture and its equality cases, OpenAI Math Release, September 22, 2026, result family 87, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SymmetricMahler.equality_strip_images_dense of lean/OAI/Analysis/Mahler/Mahler/Lens.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SymmetricMahler, with open Set Filter Metric MeasureTheory, open scoped Topology ENNReal, open Set Filter Metric MeasureTheory, open scoped Topology ENNReal Pointwise):

theorem equality_strip_images_dense_eq {n : ℕ} (hn : 1 ≤ n)
    {B : Set (Fin n → ℝ)} (hB : IsCompact B) (hconv : Convex ℝ B)
    (hsym : ∀ x ∈ B, -x ∈ B) (hint : (interior B).Nonempty)
    (heq : (volume B).toReal * (volume (coordinatePolar B)).toReal =
      (4:ℝ)^n/(Nat.factorial n:ℝ))
    (a : ℕ → coordinatePolar B)
    (ha : ∀ j, Function.Injective (measurement (fun i : Fin (n+j) => (a i).val)))
    (hcompact : ∀ j, IsCompact (stripBody (fun i : Fin (n+j) => (a i).val)))
    (hsub : ∀ j, B ⊆ stripBody (fun i : Fin (n+j) => (a i).val))
    (hvol : Tendsto (fun j => (volume (stripBody (fun i : Fin (n+j) => (a i).val))).toReal)
      atTop (𝓝 (volume B).toReal))
    (m : ℕ → ℕ) (hm : ∀ j, 2 ≤ m j)
    (herr : Tendsto (fun j => ((n+j:ℕ):ℝ)*planarError (m j)) atTop (𝓝 0))
    {x e : Fin n → ℝ} (hx : x ∈ B) (he : e ∈ coordinatePolar B) :
    ∃ j : ℕ → ℕ, ∃ z : ℕ → ((Fin n → ℝ) × (Fin n → ℝ)),
      Tendsto j atTop atTop ∧
      (∀ k, z k ∈ stripSublevel (fun i : Fin (n+j k) => (a i).val) (m (j k))) ∧
      Tendsto (fun k => stripMap (fun i : Fin (n+j k) => (a i).val) (m (j k)) (z k))
        atTop (𝓝 (x,e))

Definitions. The statement is written with OAI.SymmetricMahler.measurement, OAI.SymmetricMahler.planarError, OAI.SymmetricMahler.stripBody, OAI.SymmetricMahler.stripMap and OAI.SymmetricMahler.stripSublevel (definitions bundle OAIQ2Ma000 of this split); OAI.SymmetricMahler.coordinatePolar (the published bundle Def_SymmetricMahlerEquality).

Used by. The proof of OAI.SymmetricMahler.equality_polar_gauge_medians_eq of this split imports it.

Preamble

import Mathlib
import Definitions.Def_SymmetricMahlerEquality
import Definitions.Def_OAIQ2Ma000

Formal statement

section

attribute [-instance] _root_.Convexity.StdSimplex.instInhabited _root_.Convexity.StdSimplex.instNonempty _root_.Convexity.StdSimplex.instSubsingleton _root_.Convexity.StdSimplex.instUnique _root_.Convexity.StdSimplex.topologicalSpace
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

open Set Filter Metric MeasureTheory
open scoped Topology ENNReal
namespace SymmetricMahler

open Set Filter Metric MeasureTheory
open scoped Topology ENNReal Pointwise

theorem equality_strip_images_dense_eq {n : ℕ} (hn : 1 ≤ n)
    {B : Set (Fin n → ℝ)} (hB : IsCompact B) (hconv : Convex ℝ B)
    (hsym : ∀ x ∈ B, -x ∈ B) (hint : (interior B).Nonempty)
    (heq : (volume B).toReal * (volume (coordinatePolar B)).toReal =
      (4:ℝ)^n/(Nat.factorial n:ℝ))
    (a : ℕ → coordinatePolar B)
    (ha : ∀ j, Function.Injective (measurement (fun i : Fin (n+j) => (a i).val)))
    (hcompact : ∀ j, IsCompact (stripBody (fun i : Fin (n+j) => (a i).val)))
    (hsub : ∀ j, B ⊆ stripBody (fun i : Fin (n+j) => (a i).val))
    (hvol : Tendsto (fun j => (volume (stripBody (fun i : Fin (n+j) => (a i).val))).toReal)
      atTop (𝓝 (volume B).toReal))
    (m : ℕ → ℕ) (hm : ∀ j, 2 ≤ m j)
    (herr : Tendsto (fun j => ((n+j:ℕ):ℝ)*planarError (m j)) atTop (𝓝 0))
    {x e : Fin n → ℝ} (hx : x ∈ B) (he : e ∈ coordinatePolar B) :
    ∃ j : ℕ → ℕ, ∃ z : ℕ → ((Fin n → ℝ) × (Fin n → ℝ)),
      Tendsto j atTop atTop ∧
      (∀ k, z k ∈ stripSublevel (fun i : Fin (n+j k) => (a i).val) (m (j k))) ∧
      Tendsto (fun k => stripMap (fun i : Fin (n+j k) => (a i).val) (m (j k)) (z k))
        atTop (𝓝 (x,e)) := by
  sorry

end SymmetricMahler

open Set Filter Metric MeasureTheory Complex
open scoped Topology

open Set Filter
open scoped Topology

end

end OAI
end

Source

OpenAI, The symmetric Mahler conjecture and its equality cases, OpenAI Math Release, September 22, 2026, https://github.com/openai/math (result family 87; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Mahler/Mahler/Lens.lean (declaration OAI.SymmetricMahler.equality_strip_images_dense)

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