Statistical core futures outside a finite-flux angular exception

Proved

BianchiIX.exists_angularRegion_statistics

by trppppp · Nov 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

Assume the pointwise Birkhoff theorem on the real line for every ergodic probability-preserving map and integrable real observable. There exists $0<\delta\le1/40000$ ; for each $0<\eta\le\delta$ , integer $J\ge20$ , $0<\sigma<1$ with $2J\sigma<1$ , the following choices can be made. Choose a cap in $(0,1)$ and $h_*>0$ . For $0<h\le h_*$ , $L\ge0$ , $q>3$ , $P>4$ and all sufficiently small $d>0$ let $M$ be the larger weak wall. In $M\le d$ , $|v|\le LM$ there is a Borel failure set of finite corrected flux at most $C d(1+|\log d|+2\log2)$ with $C$ depending on $L,\sigma,P$ . Every remaining point is in the section chart and at some positive existing time its orbit enters, in an axis frame, a surviving core with the Gauss era statistics.

Preamble

import Definitions.Def_BianchiIX_Literature
import Definitions.Def_BianchiIX_RestartFluxData
import Definitions.Def_BianchiIX_Algebra_Flux
import Definitions.Def_BianchiIX_Dynamics_AxisMaps
import Definitions.Def_BianchiIX_AxisAndOrbitData
import Definitions.Def_BianchiIX_RadialTransportData
import Definitions.Def_BianchiIX_Statistics_EraStatistics
import Definitions.Def_BianchiIX_Dynamics_AxisSymmetry
import Definitions.Def_BianchiIX_Dynamics_ProjectiveReflection
import Definitions.Def_BianchiIX_AxisSectionMeasureData
import Definitions.Def_BianchiIX_EraCodingData
import Definitions.Def_BianchiIX_PhysicalFutureData
import Definitions.Def_BianchiIX_RadialFailureData
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Topology.Instances.Matrix
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.LinearAlgebra.Determinant
import Mathlib.Data.Real.Basic
import Mathlib.LinearAlgebra.Alternating.Curry
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.Topology.Order.Compact
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Tactic.NormNum
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arcosh
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Data.Matrix.Mul
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Tactic.Positivity
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.Normed.Operator.Basic
import Mathlib.LinearAlgebra.Matrix.Notation
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Data.List.Induction
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Data.PNat.Basic
import Mathlib.Data.Rat.Floor
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Order.OrderIsoNat
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.JacobianOneDim
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.Dynamics.BirkhoffSum.Average
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Topology.UniformSpace.HeineCantor
import Mathlib.Dynamics.BirkhoffSum.NormedSpace
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Analysis.Normed.Group.Tannery
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.Topology.Algebra.InfiniteSum.ENNReal
import Mathlib.MeasureTheory.Integral.Lebesgue.Markov
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Algebra.Order.Archimedean.Basic

noncomputable section
open BianchiIX
open Set MeasureTheory
open scoped ENNReal

Formal statement

theorem BianchiIX.exists_angularRegion_statistics (hBirkhoff : Literature.BirkhoffErgodicInput ℝ) :
    ∃ δ > (0 : ℝ), δ ≤ 1 / 40000 ∧ ∀ η : ℝ, 0 < η → η ≤ δ →
      ∀ J : ℕ, 20 ≤ J → ∀ σ : ℝ, 0 < σ → σ < 1 → σ * (2 * J : ℕ) < 1 →
        ∃ Bcap > (0 : ℝ), Bcap < 1 ∧ ∃ hstar > (0 : ℝ),
          ∀ h ∈ Ioc (0 : ℝ) hstar, ∀ L : ℝ, 0 ≤ L →
            ∀ q P : ℝ, 3 < q → 4 < P → ∃ dstar > (0 : ℝ),
              ∀ d ∈ Ioc (0 : ℝ) dstar,
                let F := angularFailureRegion h q d L P η J σ Bcap
                let C := 12 * L * logarithmicFluxConstantOne (angularFailureFluxConstant L σ) σ +
                  24 * L * P * Real.log 2
                MeasurableSet F ∧ mainWallMeasure h F ≤ ENNReal.ofReal (C * d * (1 + |Real.log d| + 2 * Real.log 2)) ∧
                  mainWallMeasure h F ≠ ⊤ ∧
                  ∀ z ∈ angularSection d L \ F,
                    z ∈ mainWallChartDomain h ∧ StatisticalCoreFuture η J σ Bcap (mainWallChart h z) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Measure/AngularRegionStatistics.lean#L18-L94

Human review

Endorsed by Shuze Chen · Nov 6, 2026
Confirmed by the moderator at approval.
Endorsed by trppppp · Nov 6, 2026
Confirmed by the mission captain (proposal self-audit).

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