Phase and decreasing radius at the first upward entrance after a long tail

Proved

BianchiIX.exists_restartShearTail_entrance_phase

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. The long cutoff adds $S=(D/a)\log(1/a)$ to $s$ . Write $Y_L=\Pi\Phi_{s+S}x_h(Z)$ , and write $(\zeta,\nu)=(n_2,n_3)$ . Take $0\le\epsilon\le1/128$ , $0<\zeta,\nu\le2\epsilon a$ , $\zeta\le a^P$ , $P>6D+2$ . For $q,D>2$ there is $C>0$ before $P$ . The first upward return from $Y_L$ is valid. Its radius $R$ is differentiable, $1/(48a)\le R\le128/a$ and $1/(36864a^2)\le-R'\le24576/a^2$ . For its phase $f=2/R$ : The phase $f$ is differentiable, with $|f-F|\le C a^2$ , $|f'-F'|\le Ca$ , and $1/(4\sqrt3)\le f'\le2/\sqrt3+1$ .

Preamble

import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_EntranceRegion
import Definitions.Def_BianchiIX_RadialTransportData
import Definitions.Def_BianchiIX_ShearTransportData
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.Topology.Order.Compact
import Mathlib.Tactic.NormNum
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.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.SpecialFunctions.Arsinh
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.Analysis.Calculus.FDeriv.Partial
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Data.Matrix.Mul
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
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
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Log.Basic

noncomputable section
open BianchiIX
open Set

Formal statement

theorem BianchiIX.exists_restartShearTail_entrance_phase :
    ∃ hstar > (0 : ℝ), ∀ h ∈ Ioc (0 : ℝ) hstar,
      ∀ q D : ℝ, 2 < q → 2 < D → ∃ C > (0 : ℝ),
        ∀ P : ℝ, 6 * D + 2 < P → ∃ astar > (0 : ℝ),
          ∀ a ∈ Ioc (0 : ℝ) astar, ∀ ε ζ ν : ℝ,
            (0 ≤ ε ∧ ε ≤ 1 / 128) →
            (0 < ζ ∧ ζ ≤ a ^ P ∧ ζ ≤ 2 * ε * a) →
            (0 < ν ∧ ν ≤ 2 * ε * a) → ∀ ρ ∈ Icc a (2 * a),
            let Φ := restartShearTailEntrancePhase h a q D ζ ν
            let F := restartShearTailEntranceRadius h a q D ζ ν
            let f₀ := fun r => restartReferenceRadialPhase h (restartShearCurve 1 ζ ν r)
            entranceReturnValid (restartShearTailProjectiveEntry h a q ζ ν (shearTailTime a D) ρ) ∧
            DifferentiableAt ℝ Φ ρ ∧ |Φ ρ - f₀ ρ| ≤ C * a ^ 2 ∧
            |deriv Φ ρ - deriv f₀ ρ| ≤ C * a ∧
            1 / (4 * Real.sqrt 3) ≤ deriv Φ ρ ∧ deriv Φ ρ ≤ 2 / Real.sqrt 3 + 1 ∧
            DifferentiableAt ℝ F ρ ∧ 1 / (48 * a) ≤ F ρ ∧ F ρ ≤ 128 / a ∧
            1 / (36864 * a ^ 2) ≤ -deriv F ρ ∧ -deriv F ρ ≤ 24576 / a ^ 2 := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ShearRestartTailEntrancePhase.lean#L18-L122

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