Angular reparametrization of the exact normalized Einstein flow

Proved

BianchiIX.exists_normalized_coreAngle_reparametrization

by trppppp · Nov 4, 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.

Define the core radius, angle, and clock factor by

r(m,v)=\sqrt{(m-1)^2+v^2},\qquad \theta(m,v)=\operatorname{arsinh}\frac{1-m}{v},\qquad d(r,u,w)=1-\frac{(u-w)^2}{r^2}.

For $y=(m,u,v,w)$ , write $r_y=r(m,v)$ , $\theta_y=\theta(m,v)$ , and $d_y=d(r_y,u,w)$ . All quotients use the total convention $x/0=0$ .

The core state space is $\mathbb R^3$ with coordinates $Y=(r,u,p)$ . Define

\eta(m,u,v,w)=(r(m,v),u,vw),\qquad \iota_z(r,u,p)=\left(1-r\tanh z,u,\frac r{\cosh z}, \frac p{r/\cosh z}\right).

For fixed $(z,Y)$ put $v=r/\cosh z$ , $w=p/v$ , $d=d(r,u,w)$ , and define

\mathcal V(z,Y)=\left( \frac{-\kappa(u,v,w)-\tanh z\,(u-w)^2/r}{d},\quad \frac{u[-r/2+\tanh z+w/\cosh z+(u^2-w^2)/(2r)]}{d},\quad \frac{-2p(1+\kappa(u,v,w))}{rd}\right).

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.

Write $G(y)=6F(y)/D(y)$ for the projective field in normalized time and

\lambda(y)=\frac6{D(y)}r_y d_y

for its angular clock rate. The projective angle observable is $y\mapsto\theta_y$ . Let $I\subset I_{\max}(x,t_0)$ be open and convex, with $s\in I$ , and suppose $\mathcal C(x)=1$ . Put $\beta(t)=\Pi(\alpha_{x,t_0}(t))$ . Assume, for every $t\in I$ ,

1+(\alpha_{x,t_0}(t))_0>0,\qquad (\beta(t))_v>0,\qquad d_{\beta(t)}>0.

There is an open partial homeomorphism $e$ with source $I$ and open convex target $J$ , with $0\in J$ , $e^{-1}(0)=s$ , and $e$ strictly increasing on $I$ , such that $e(t)=\theta_{\beta(t)}-\theta_{\beta(s)}$ on $I$ . For $Y(z)=\eta(\beta(e^{-1}(z)))$ and $z\in J$ ,

\theta_{\beta(e^{-1}(z))}=z+\theta_{\beta(s)},\qquad Y'(z)=\mathcal V(z+\theta_{\beta(s)},Y(z)),

\iota_{z+\theta_{\beta(s)}}(Y(z))=\beta(e^{-1}(z)).

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 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.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.FieldSimp
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.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.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial

noncomputable section
open BianchiIX
open Set

Formal statement

theorem BianchiIX.exists_normalized_coreAngle_reparametrization {x : NormalizedState} {t₀ s : ℝ}
    {I : Set ℝ} (hI : Convex ℝ I) (ho : IsOpen I) (hs : s ∈ I)
    (hdom : I ⊆ normalizedMaxDomain x t₀) (hc : stateConstraint x = 1)
    (hA : ∀ t ∈ I, 0 < 1 + normalizedMaxCurve x t₀ t 0)
    (hv : ∀ t ∈ I, 0 < toProjectiveState (normalizedMaxCurve x t₀ t) 2)
    (hd : ∀ t ∈ I, let y := toProjectiveState (normalizedMaxCurve x t₀ t)
      0 < coreClockFactor (coreRadius (y 0) (y 2)) (y 1) (y 3)) :
    let β := fun t => toProjectiveState (normalizedMaxCurve x t₀ t)
    ∃ e : OpenPartialHomeomorph ℝ ℝ,
      e.source = I ∧
      (∀ t ∈ I, e t = coreAngle (β t 0) (β t 2) - coreAngle (β s 0) (β s 2)) ∧
      Convex ℝ e.target ∧ 0 ∈ e.target ∧ e.symm 0 = s ∧ StrictMonoOn e I ∧
      (∀ z ∈ e.target,
        coreAngle (β (e.symm z) 0) (β (e.symm z) 2) = z + coreAngle (β s 0) (β s 2) ∧
        HasDerivAt (fun u => toCoreAngleState (β (e.symm u)))
          (coreAngleField (z + coreAngle (β s 0) (β s 2)) (toCoreAngleState (β (e.symm z)))) z ∧
        fromCoreAngleState (z + coreAngle (β s 0) (β s 2)) (toCoreAngleState (β (e.symm z))) = β (e.symm z)) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/CoreAngleClock.lean#L78-L103

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