A measurable statistical restart on the closed physical section

Proved

BianchiIX.exists_restartClosedSection_borel_restart

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 budget cap in $(0,1)$ and $0<h_*\le1/8$ . Fix $0<h\le h_*$ , $q>3$ , $D>4$ , $P>6D+2$ , $0<\epsilon\le1/128$ . For sufficiently small section width $0<d\le d_*\le h/8$ partition the open section into angular and signed shear pieces at dyadic radii. Each piece's bad parameters $F_i$ have unselected labels or an actual return outside the typical survivor set. Let $E$ be their physical images together with the dyadic boundary. Write $I_i$ for the integral of the original flux density on the bad fibers, with Jacobian $1/\zeta$ in angular coordinates or $1/(n_2 n_3)$ in shear coordinates. Add the boundary of the small section to $E$ , obtaining a Borel subset $\bar E$ of the closed section. There is $C>0$ before any prescribed budget bound $B_*>0$ ; the threshold $d_*$ may now also depend on $B_*$ . The original flux is $\sum_i I_i$ , is finite, dominates the corrected flux and is at most $Cd(1+|\log d|)^2$ . A measurable map on the whole section coordinate space selects, on the closed section outside $\bar E$ , a positive-time physical restart with budget at most $B_*$ and with the budget and core of one actual successful radial piece. All such points are in the chart domain.

Preamble

import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Literature
import Definitions.Def_BianchiIX_RadialFailureData
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Flux
import Definitions.Def_BianchiIX_Dynamics_AxisMaps
import Definitions.Def_BianchiIX_AxisAndOrbitData
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_SectionRestartData
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_RadialTransportData
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Statistics_EraStatistics
import Definitions.Def_BianchiIX_Dynamics_Events
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_AxisSymmetry
import Definitions.Def_BianchiIX_Dynamics_ProjectiveReflection
import Definitions.Def_BianchiIX_AxisSectionMeasureData
import Definitions.Def_BianchiIX_Dynamics_ConnectionEvents
import Definitions.Def_BianchiIX_Dynamics_CoreEntryDifferential
import Definitions.Def_BianchiIX_Dynamics_ConnectionReturn
import Definitions.Def_BianchiIX_Dynamics_ConnectionPhysicalReturn
import Definitions.Def_BianchiIX_Dynamics_CoreConnectionChart
import Definitions.Def_BianchiIX_EraCodingData
import Definitions.Def_BianchiIX_PhysicalEraData
import Definitions.Def_BianchiIX_GeometricExceptionData
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.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.Topology.Algebra.InfiniteSum.ENNReal
import Mathlib.Algebra.Order.Archimedean.Basic
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.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.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.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.MeasureTheory.Integral.Lebesgue.Markov

noncomputable section
open BianchiIX
open Set MeasureTheory
open scoped ENNReal

Formal statement

theorem BianchiIX.exists_restartClosedSection_borel_restart (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), hstar ≤ 1 / 8 ∧
          ∀ h ∈ Ioc (0 : ℝ) hstar, ∀ q D P : ℝ, 3 < q → 4 < D → 6 * D + 2 < P →
            ∀ ε ∈ Ioc (0 : ℝ) (1 / 128), ∃ C > (0 : ℝ),
              ∀ Bbound : ℝ, 0 < Bbound → ∃ dstar > (0 : ℝ), dstar ≤ h / 8 ∧
                ∀ d ∈ Ioc (0 : ℝ) dstar,
                  let F := restartRadialFailureFamily h q d ε P η J σ Bcap
                  let E := mainWallClosedFailure d (radialFailureSet d ε F)
                  MeasurableSet E ∧ E ⊆ mainWallSmallSection d ∧
                    mainWallOriginalMeasure h E = (∑' i, radialFailureIntegral h d ε F i) ∧
                    mainWallMeasure h E ≤ mainWallOriginalMeasure h E ∧
                    mainWallOriginalMeasure h E ≤ ENNReal.ofReal (C * d * (1 + |Real.log d|) ^ 2) ∧
                    mainWallOriginalMeasure h E ≠ ⊤ ∧
                    ∃ W : SectionCoordinates → CoreRestartData, Measurable W ∧
                      ∀ y ∈ mainWallSmallSection d \ E,
                        y ∈ mainWallChartDomain h ∧ (W y).2.1 ≤ Bbound ∧
                        CoreRestartWitness η J σ Bcap (mainWallChart h y) (W y) ∧
                        ∃ i : RadialPieceIndex, y ∈ physicalRadialPiece d ε i ∧
                          (W y).2 = restartRadialOutput h q d ε i (radialPieceInverse i.1 y) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Measure/RestartClosedSection.lean#L11-L53

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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