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

Proved

OAI.Mahler.MassHypotheses.homogeneousResidualFlux_eq_zero_of_transgression

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.MassHypotheses.homogeneousResidualFlux_eq_zero_of_transgression of lean/OAI/Analysis/Mahler/Mahler/HomogeneousTransgressionFTC.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Mahler, with open Complex Set Metric, open scoped Topology):

theorem MassHypotheses.homogeneousResidualFlux_eq_zero_of_transgression {k N m : ℕ}
    {U : Set (ComplexEuclidean (k+2))} {f : Fin N → ComplexEuclidean (k+2) → ℂ}
    {G : Fin N → MvPolynomial (Fin (k+2)) ℂ} (h : MassHypotheses (k+2) N m U f G)
    (t : ℝ)
    (ht : ∀ x : ComplexEuclidean (k+2), ‖x‖ = 1 →
      ∀ v : Fin (2*(k+1)+1) → ComplexEuclidean (k+2),
      (∀ i, inner ℝ x (v i) = 0) →
      boundaryResidualFin (alphaPath (polynomialMap G) (coordinateMap (k+2)) m t)
        (betaLinear (polynomialMap G) (coordinateMap (k+2)) m) x v =
        -((k+1 : ℕ) : ℂ) * extDeriv (sourceHomogeneousPrimitive (m := m) G t) x v) :
    sphereFlux (k+1) (boundaryResidualFin
      (alphaPath (polynomialMap G) (coordinateMap (k+2)) m t)
      (betaLinear (polynomialMap G) (coordinateMap (k+2)) m)) = 0

Definitions. The statement is written with OAI.Mahler.ComplexEuclidean, OAI.Mahler.MassHypotheses, OAI.Mahler.alphaPath, OAI.Mahler.betaLinear, OAI.Mahler.boundaryResidualFin, OAI.Mahler.coordinateMap, OAI.Mahler.polynomialMap, OAI.Mahler.sourceHomogeneousPrimitive and OAI.Mahler.sphereFlux (definitions bundle OAISMh000 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAISMh000

Formal statement

section

attribute [-instance] _root_.OAI.Mahler.finiteDimensional_continuousAlternatingMap
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

open Complex Set Metric
open scoped Topology

noncomputable section
namespace Mahler

theorem MassHypotheses.homogeneousResidualFlux_eq_zero_of_transgression {k N m : ℕ}
    {U : Set (ComplexEuclidean (k+2))} {f : Fin N → ComplexEuclidean (k+2) → ℂ}
    {G : Fin N → MvPolynomial (Fin (k+2)) ℂ} (h : MassHypotheses (k+2) N m U f G)
    (t : ℝ)
    (ht : ∀ x : ComplexEuclidean (k+2), ‖x‖ = 1 →
      ∀ v : Fin (2*(k+1)+1) → ComplexEuclidean (k+2),
      (∀ i, inner ℝ x (v i) = 0) →
      boundaryResidualFin (alphaPath (polynomialMap G) (coordinateMap (k+2)) m t)
        (betaLinear (polynomialMap G) (coordinateMap (k+2)) m) x v =
        -((k+1 : ℕ) : ℂ) * extDeriv (sourceHomogeneousPrimitive (m := m) G t) x v) :
    sphereFlux (k+1) (boundaryResidualFin
      (alphaPath (polynomialMap G) (coordinateMap (k+2)) m t)
      (betaLinear (polynomialMap G) (coordinateMap (k+2)) m)) = 0 := by
  sorry

end Mahler

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/HomogeneousTransgressionFTC.lean (declaration OAI.Mahler.MassHypotheses.homogeneousResidualFlux_eq_zero_of_transgression)

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