Joint regularity for the same shear return in both small-wall orderings

Proved

BianchiIX.exists_restartShearReturn_common_regular

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

bianchi-ix-era-statisticsdynamical-systems

Write $\Phi_t x$ for the normalized vacuum Bianchi IX flow and $x_h(Z)$ for the constraint-one, negative-first-shear chart with primary wall $h$ and center coordinates $Z=(v,b,c)$ (second shear and the other two walls). At time $t$ , denote its primary wall by $n_t$ and center by $Z_t$ . Set $r(Z)=\sqrt{v^2+(b-c)^2}$ . For $u=\sqrt{1-h^2}$ define $A=3/(2+u)$ , $B=3\sqrt3 h/(2+u)^2$ , $H=9(1+u)/(2(2+u)^2)$ , $Q_h(v,w)=(Av+Bw)^2+H^2w^2$ and $F_h(Z)=\sqrt{Q_h(v,b-c)}/\sqrt3$ . On a shear annulus take $0<a$ , $\rho\in[a,2a]$ , $Z=(\rho,n_2,n_3)$ , $s=(q/2)\log(1/a)$ , $Y=\Pi\Phi_s x_h(Z)$ , $k=\sqrt{(Y_0-1)^2+Y_2^2}$ and $F=F_h(Z)$ . A signed annulus uses $Z=(\omega\rho,n_2,n_3)$ , $|\omega|=1$ . Here $h>0$ is sufficiently small and $a>0$ is sufficiently small thereafter. Primes are ordinary radial derivatives with labels and cutoff fixed. Take $q>3$ , $D>4$ , $P>6D+2$ , $0\le\epsilon\le1/128$ , $0<n_2,n_3\le2\epsilon a$ and $\min(n_2,n_3)\le a^P$ . The connection parameters, reflected return state, reciprocal phase and original arrival time are jointly $C^2$ at the given pair of labels and radius.

Preamble

import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_SectionRestartData
import Definitions.Def_BianchiIX_ShearTransportData
import Definitions.Def_BianchiIX_Dynamics_ConnectionExistence
import Definitions.Def_BianchiIX_Dynamics_ConnectionReturn
import Definitions.Def_BianchiIX_Dynamics_EntranceReturn
import Definitions.Def_BianchiIX_EraReturnData
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
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.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.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.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.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
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.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.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.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_common_regular :
    ∃ hstar > (0 : ℝ), ∀ h ∈ Ioc (0 : ℝ) hstar,
      ∀ q D P : ℝ, 3 < q → 4 < D → 6 * D + 2 < P →
        ∃ astar > (0 : ℝ), ∀ a ∈ Ioc (0 : ℝ) astar, ∀ ε n₂ n₃ : ℝ,
          (0 ≤ ε ∧ ε ≤ 1 / 128) → (0 < n₂ ∧ n₂ ≤ 2 * ε * a) →
          (0 < n₃ ∧ n₃ ≤ 2 * ε * a) → min n₂ 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/ShearRestartCommonReturn.lean#L11-L35

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