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

Proved

OAI.SymmetricMahler.equality_polar_gauge_medians_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_polar_gauge_medians of lean/OAI/Analysis/Mahler/Mahler/EqualityMedians.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem equality_polar_gauge_medians_eq {n : ℕ} (hn : 1 ≤ n)
    {K : Set (Fin n → ℝ)} (hK : IsCompact K) (hc : Convex ℝ K)
    (hs : ∀ x ∈ K, -x ∈ K) (hi : (interior K).Nonempty)
    (heq : (volume K).toReal * (volume (coordinatePolar K)).toReal = (4:ℝ)^n/Nat.factorial n) :
    ∃ g h : RealNormData (Fin n → ℝ), g.unit = K ∧ h.unit = coordinatePolar K ∧
      g.IsDual h ∧ HasMetricMedians h.Space ∧ HasMetricMedians g.Space

Definitions. The statement is written with OAI.SymmetricMahler.HasMetricMedians, OAI.SymmetricMahler.RealNormData, OAI.SymmetricMahler.RealNormData.IsDual, OAI.SymmetricMahler.RealNormData.Space, OAI.SymmetricMahler.RealNormData.unit and OAI.SymmetricMahler.RealNormData.Space.instNormedAddCommGroup (definitions bundle OAIQ2Ma000 of this split); OAI.SymmetricMahler.coordinatePolar (the published bundle Def_SymmetricMahlerEquality).

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

Preamble

import Mathlib
import Definitions.Def_SymmetricMahlerEquality
import Definitions.Def_OAIQ2Ma000

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

noncomputable section

namespace SymmetricMahler

open Set Filter Finset Metric
open scoped Topology

open Set Filter Finset Metric Complex
open scoped Topology
open MahlerConformal

open Set Filter Finset Metric Complex MeasureTheory
open scoped Topology
open MahlerConformal

open Set Finset MeasureTheory
open MahlerConformal

open Set MeasureTheory Metric

theorem equality_polar_gauge_medians_eq {n : ℕ} (hn : 1 ≤ n)
    {K : Set (Fin n → ℝ)} (hK : IsCompact K) (hc : Convex ℝ K)
    (hs : ∀ x ∈ K, -x ∈ K) (hi : (interior K).Nonempty)
    (heq : (volume K).toReal * (volume (coordinatePolar K)).toReal = (4:ℝ)^n/Nat.factorial n) :
    ∃ g h : RealNormData (Fin n → ℝ), g.unit = K ∧ h.unit = coordinatePolar K ∧
      g.IsDual h ∧ HasMetricMedians h.Space ∧ HasMetricMedians g.Space := by
  sorry

end SymmetricMahler

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/EqualityMedians.lean (declaration OAI.SymmetricMahler.equality_polar_gauge_medians)

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