Prove2Me

Joint smoothness of the positive shear return across the active-wall threshold

Proved
BianchiIX.exists_restartShearReturn_all_active_joint_regular

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

bianchi-ix-era-statisticsdynamical-systems

Write Φtx\Phi_t x for the normalized vacuum Bianchi IX flow and xh(Z)x_h(Z) for the constraint-one, negative-first-shear chart with primary wall hh and center coordinates Z=(v,b,c)Z=(v,b,c) (second shear and the other two walls). At time tt, denote its primary wall by ntn_t and center by ZtZ_t. Set r(Z)=v2+(b−c)2r(Z)=\sqrt{v^2+(b-c)^2}. For u=1−h2u=\sqrt{1-h^2} define A=3/(2+u)A=3/(2+u), B=33h/(2+u)2B=3\sqrt3 h/(2+u)^2, H=9(1+u)/(2(2+u)2)H=9(1+u)/(2(2+u)^2), Qh(v,w)=(Av+Bw)2+H2w2Q_h(v,w)=(Av+Bw)^2+H^2w^2 and Fh(Z)=Qh(v,b−c)/3F_h(Z)=\sqrt{Q_h(v,b-c)}/\sqrt3. On a shear annulus take 0<a0<a, ρ∈[a,2a]\rho\in[a,2a], Z=(ρ,n2,n3)Z=(\rho,n_2,n_3), s=(q/2)log⁡(1/a)s=(q/2)\log(1/a), Y=ΠΦsxh(Z)Y=\Pi\Phi_s x_h(Z), k=(Y0−1)2+Y22k=\sqrt{(Y_0-1)^2+Y_2^2} and F=Fh(Z)F=F_h(Z). A signed annulus uses Z=(ωρ,n2,n3)Z=(\omega\rho,n_2,n_3), ∣ω∣=1|\omega|=1. Here h>0h>0 is sufficiently small and a>0a>0 is sufficiently small thereafter. Primes are ordinary radial derivatives with labels and cutoff fixed. Take q>3q>3, P>4P>4, 0≤ϵ≤1/640\le\epsilon\le1/64, 0≤n2,n3≤2ϵa0\le n_2,n_3\le2\epsilon a and n3≤aPn_3\le a^P. Assume both weak walls are positive. The connection parameters, reflected return state, reciprocal phase and original arrival time are jointly C2C^2 at the given pair of labels and radius.

Preamble
import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Analysis_CoreGrowthBudget
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_SectionRestartData
import Definitions.Def_BianchiIX_Dynamics_WallEvolution
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_RadialTransportData
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_ProjectiveRegularity
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_CoreCoordinateDifferential
import Definitions.Def_BianchiIX_Dynamics_FlowRegularity
import Definitions.Def_BianchiIX_Dynamics_FlowDifferential
import Definitions.Def_BianchiIX_Dynamics_Events
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_CoreEntry
import Definitions.Def_BianchiIX_Dynamics_ConnectionExistence
import Definitions.Def_BianchiIX_Dynamics_ConnectionEvents
import Definitions.Def_BianchiIX_Dynamics_ConnectionReturn
import Definitions.Def_BianchiIX_Dynamics_EntranceReturn
import Definitions.Def_BianchiIX_EraReturnData
import Definitions.Def_BianchiIX_Dynamics_ConnectionPhysicalReturn
import Definitions.Def_BianchiIX_Dynamics_CoreConnectionChart
import Definitions.Def_BianchiIX_ShearTransportData
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.Calculus.ImplicitContDiff
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arcosh
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Tactic.Positivity
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.LinearAlgebra.Determinant
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.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
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.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

noncomputable section
open BianchiIX
open Set
Formal statement
theorem BianchiIX.exists_restartShearReturn_all_active_joint_regular :
    ∃ hstar > (0 : ℝ), ∀ h ∈ Ioc (0 : ℝ) hstar,
      ∀ q P : ℝ, 3 < q → 4 < P → ∃ astar > (0 : ℝ), ∀ a ∈ Ioc (0 : ℝ) astar,
        ∀ ε n₂ n₃ : ℝ, (0 ≤ ε ∧ ε ≤ 1 / 64) →
          (0 < n₂ ∧ n₂ ≤ 2 * ε * a) → (0 < n₃ ∧ n₃ ≤ 2 * ε * a) →
          n₃ ≤ a ^ P → ∀ ρ ∈ Icc a (2 * a),
            let s := q / 2 * Real.log (1 / a)
            ContDiffAt ℝ 2 (fun v : (ℝ × ℝ) × ℝ => restartShearReturnParameters h s v.1.1 v.1.2 v.2) ((n₂, n₃), ρ) ∧
            ContDiffAt ℝ 2 (fun v : (ℝ × ℝ) × ℝ => restartShearReturnState h s v.1.1 v.1.2 v.2) ((n₂, n₃), ρ) ∧
            ContDiffAt ℝ 2 (fun v : (ℝ × ℝ) × ℝ => restartShearReturnPhase h s v.1.1 v.1.2 v.2) ((n₂, n₃), ρ) ∧
            ContDiffAt ℝ 2 (fun v : (ℝ × ℝ) × ℝ => restartShearReturnTime h s v.1.1 v.1.2 v.2) ((n₂, n₃), ρ) := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ShearRestartAllActiveReturn.lean#L10-L37

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