The canonical entrance output obeys explicit logarithmic wall bounds

Proved

BianchiIX.entranceReturn_decay

by trppppp · Nov 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

For real variables $p,s,a,b,c$ , set

K=a^2+b^2+c^2-2(ab+bc+ca),\qquad S=p^2+s^2,\qquad C=S+K.

The shear components and wall rates are

P=Kp-2a^2+b^2+c^2+ab-2bc+ac,

Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac),

r_1=-S+2p,\qquad r_2=-S-p+\sqrt{3}s,\qquad r_3=-S-p-\sqrt{3}s.

These functions define the normalized vacuum Bianchi IX vector field $(P,Q,r_1a,r_2b,r_3c)$ . For $y=(m,u,v,w)\in\mathbb R^4$ , define

K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u),

F_m=-v^2+w^2+u(v-w)-m\kappa,\qquad F_u=\tfrac12(1-m^2-(v-w)^2+u^2)u,

F_v=(m-1-\kappa)v,\qquad F_w=(-m-1-\kappa)w,\qquad F=(F_m,F_u,F_v,F_w),

and $E(m,v)=(m-1)^2+v^2$ .

For $x=(p,s,a,b,c)$ , put

\psi(p,z)=\frac{\sqrt3\,z}{1+p},\qquad \Pi(x)=(\psi(p,s),\psi(p,a),\psi(p,b),\psi(p,c)).

For $y=(m,u,v,w)$ , set $D(y)=3+m^2+K(u,v,w)$ and

\Xi_A(y)=\left(A-1,\frac{Am}{\sqrt3},\frac{Au}{\sqrt3}, \frac{Av}{\sqrt3},\frac{Aw}{\sqrt3}\right),\qquad \Xi(y)=\Xi_{6/D(y)}(y).

Write $N(x)=(P,Q,r_1a,r_2b,r_3c)$ for the normalized vector field and $\mathcal C(x)=C(p,s,a,b,c)$ for its constraint.

The normalized and projective state spaces are $\mathbb R^5$ and $\mathbb R^4$ with their finite-product norms. The maps $\Pi,\Xi,D,F$ act on these state spaces.

The projective vector field in normalized time is

G(y)=\frac6{D(y)}F(y).

Write $X=\mathbb R^5$ , $Y=\mathbb R^4$ , and let $\alpha_{x,t_0}$ be the maximal solution of $\alpha'=N(\alpha)$ with initial value $x$ at $t_0$ and domain $I_{\max}(x,t_0)$ . Put

J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad \Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)),

\mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ J_{\Xi(y),t_0}(t)\circ d\Xi_y.

Put $c=u(v+w-u)$ , $s(m,u,v,w)=m-1-c$ , $b(m,u,v,w)=(1-m^2-(v-w)^2+u^2)/2$ , $D(m,u,v,w)=3+m^2+K(u,v,w)$ , and $r(m,v)=\sqrt{(m-1)^2+v^2}$ .

Define

\mathcal E_k=\{(m,u,v,w):m\geq1,\ |r(m,v)-k|\leq k/100, \ 0\leq u,w\leq1/10000,\ 0\leq v\leq1\},

and $\omega_+(m,u,v,w)=s(m,u,v,w)$ . For a normalized trajectory write

\beta(t)=\Pi(\alpha_{x,t_0}(t)),\quad A(t)=1+(\alpha_{x,t_0}(t))_0, \quad I=I_{\max}(x,t_0).

For $y\in\mathbb R^4$ , put $x_y=\Xi(y)$ ,

T_+(y)=\inf\{t>0:t\in I_{\max}(x_y,0),\ (\Psi_0(y,t))_v=1\}, \qquad P_+(y)=\Psi_0(y,T_+(y)).

Let $V_+(y)$ mean that this infimum is a least element, the full interval $[0,T_+(y)]$ lies in the maximal domain with $1+(\alpha_{x_y,0}(t))_0>0$ , the active wall is below one before the endpoint, and its normalized-time derivative is strictly positive throughout the interval. The infimum has its total real-order meaning.

Let $y=(m,u,v,w)$ satisfy $r(m,v)\geq7/4$ , $m\geq1$ , $0\leq u,w\leq1/10000$ , and $0<v<1$ . Write $k=r(m,v)$ . Put $L=-\log v$ . Then

(P_+(y))_u\leq uv e^{-kL/3},\qquad (P_+(y))_w\leq wv e^{-L/k}.

Preamble

import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Dynamics_ProjectiveRegularity
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_Events
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_EntranceReturn
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

noncomputable section
open BianchiIX
open Set

Formal statement

theorem BianchiIX.entranceReturn_decay {y : ProjectiveState}
    (hk : 7 / 4 ≤ coreRadius (y 0) (y 2)) (hm : 1 ≤ y 0)
    (hu : 0 ≤ y 1 ∧ y 1 ≤ 1 / 10000) (hv : 0 < y 2 ∧ y 2 < 1)
    (hw : 0 ≤ y 3 ∧ y 3 ≤ 1 / 10000) :
    let k := coreRadius (y 0) (y 2)
    let L := -Real.log (y 2)
    entranceReturnState y 1 ≤ y 1 * y 2 * Real.exp (-k * L / 3) ∧
      entranceReturnState y 3 ≤ y 3 * y 2 * Real.exp (-L / k) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/EntranceReturnDecay.lean#L10-L41

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