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

Proved

OAI.Mahler.source_punctured_scaled_extDeriv_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.Mahler.source_punctured_scaled_extDeriv of lean/OAI/Analysis/Mahler/Mahler/SourceBoundaryIntegral.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem source_punctured_scaled_extDeriv_eq (h : MassHypotheses n N m U f G) (k : ℕ)
    (Ψ : (Fin ((2*k+1)+1) → ℝ) ≃L[ℝ] ComplexEuclidean n)
    {R r : ℝ} (hR : 0 < R) (hR1 : R < 1)
    (hreg : ∀ x ∈ U, tau f x = R → fderiv ℝ (tau f) x ≠ 0)
    (hr : 0 < r)
    (hB : coordClosedBall ((2*k+1)+1) r ⊆ regularSublevel (Ψ ⁻¹' U) (fun x => tau f (Ψ x)) R) :
    (R⁻¹)^(k+1) *
      (∫ x in regularSublevel (Ψ ⁻¹' U) (fun x => tau f (Ψ x)) R,
        extDeriv (sourceCoordinateForm Ψ (energy f) k) x (coordinateBasis ((2*k+1)+1))) -
      _root_.OAI.MahlerStokes.sphereFlux r (sourceCoordinateForm Ψ (logTau f) k) =
      ∫ x in regularSublevel (Ψ ⁻¹' U) (fun x => tau f (Ψ x)) R \ coordClosedBall ((2*k+1)+1) r,
        extDeriv (sourceCoordinateForm Ψ (logTau f) k) x (coordinateBasis ((2*k+1)+1))

Definitions. The statement is written with OAI.Mahler.ComplexEuclidean, OAI.Mahler.MassHypotheses, OAI.Mahler.energy, OAI.Mahler.logTau, OAI.Mahler.sourceCoordinateForm, OAI.Mahler.tau, OAI.MahlerStokes.coordClosedBall, OAI.MahlerStokes.coordinateBasis, OAI.MahlerStokes.regularSublevel and OAI.MahlerStokes.sphereFlux (definitions bundle OAIQ2Ma000 of this split).

Used by. The proof of OAI.Mahler.source_regular_massBelow_of_flux_limit_eq of this split imports it.

Preamble

import Mathlib
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

open Set Filter MeasureTheory ContinuousAlternatingMap MahlerStokes
open scoped Topology Manifold

namespace Mahler
noncomputable section
variable {n N m : ℕ} {U : Set (ComplexEuclidean n)}
  {f : Fin N → ComplexEuclidean n → ℂ} {G : Fin N → MvPolynomial (Fin n) ℂ}

theorem source_punctured_scaled_extDeriv_eq (h : MassHypotheses n N m U f G) (k : ℕ)
    (Ψ : (Fin ((2*k+1)+1) → ℝ) ≃L[ℝ] ComplexEuclidean n)
    {R r : ℝ} (hR : 0 < R) (hR1 : R < 1)
    (hreg : ∀ x ∈ U, tau f x = R → fderiv ℝ (tau f) x ≠ 0)
    (hr : 0 < r)
    (hB : coordClosedBall ((2*k+1)+1) r ⊆ regularSublevel (Ψ ⁻¹' U) (fun x => tau f (Ψ x)) R) :
    (R⁻¹)^(k+1) *
      (∫ x in regularSublevel (Ψ ⁻¹' U) (fun x => tau f (Ψ x)) R,
        extDeriv (sourceCoordinateForm Ψ (energy f) k) x (coordinateBasis ((2*k+1)+1))) -
      _root_.OAI.MahlerStokes.sphereFlux r (sourceCoordinateForm Ψ (logTau f) k) =
      ∫ x in regularSublevel (Ψ ⁻¹' U) (fun x => tau f (Ψ x)) R \ coordClosedBall ((2*k+1)+1) r,
        extDeriv (sourceCoordinateForm Ψ (logTau f) k) x (coordinateBasis ((2*k+1)+1)) := 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/SourceBoundaryIntegral.lean (declaration OAI.Mahler.source_punctured_scaled_extDeriv)

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