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

Proved

OAI.Mahler.source_regular_massBelow_of_flux_limit

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.SymmetricMahler.symmetric_mahler, 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.Mahler.source_regular_massBelow_of_flux_limit of lean/OAI/Analysis/Mahler/Mahler/SourceRegularMass.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Mahler, with open Set MeasureTheory Filter MahlerStokes, open scoped Topology ENNReal):

theorem source_regular_massBelow_of_flux_limit (h : MassHypotheses (k+1) N m U f G)
    {R L : ℝ} (hR : 0 < R) (hR1 : R < 1)
    (hreg : ∀ z ∈ U, tau f z = R → fderiv ℝ (tau f) z ≠ 0)
    (hlimit : Tendsto
      (fun r => _root_.OAI.MahlerStokes.sphereFlux r (sourceCoordinateForm (sourceCoordinates k) (logTau f) k))
      (𝓝[>] (0 : ℝ)) (𝓝 L)) :
    ENNReal.ofReal (R^(k+1) * L) ≤ massBelow U f R

Definitions. The statement is written with OAI.Mahler.ComplexEuclidean, OAI.Mahler.MassHypotheses, OAI.Mahler.logTau, OAI.Mahler.massBelow, OAI.Mahler.sourceCoordinateForm, OAI.Mahler.sourceCoordinates, OAI.Mahler.tau and OAI.MahlerStokes.sphereFlux (definitions bundle OAISMh000 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAISMh000

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

open Set MeasureTheory Filter MahlerStokes
open scoped Topology ENNReal
namespace Mahler
noncomputable section
variable {k N m : ℕ} {U : Set (ComplexEuclidean (k+1))}
  {f : Fin N → ComplexEuclidean (k+1) → ℂ} {G : Fin N → MvPolynomial (Fin (k+1)) ℂ}

theorem source_regular_massBelow_of_flux_limit (h : MassHypotheses (k+1) N m U f G)
    {R L : ℝ} (hR : 0 < R) (hR1 : R < 1)
    (hreg : ∀ z ∈ U, tau f z = R → fderiv ℝ (tau f) z ≠ 0)
    (hlimit : Tendsto
      (fun r => _root_.OAI.MahlerStokes.sphereFlux r (sourceCoordinateForm (sourceCoordinates k) (logTau f) k))
      (𝓝[>] (0 : ℝ)) (𝓝 L)) :
    ENNReal.ofReal (R^(k+1) * L) ≤ massBelow U f R := by
  sorry

end
end Mahler

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/SourceRegularMass.lean (declaration OAI.Mahler.source_regular_massBelow_of_flux_limit)

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