The exact Einstein trajectory admits the logarithmic connection clock

Proved

BianchiIX.exists_normalized_connectionLog_reparametrization

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 C_\mu=\{(m,u,v,w):|-m-\mu|\leq1/2, \ 0\leq u\leq1/10000,\ 0\leq v,w\leq1\},

\omega_-(y)=s(-y_m,y_u,y_v,y_w),\qquad H(y)=F(y)/\omega_-(y).

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).

Let $J\subset I$ be open and convex and $s\in J$ . Assume $\mathcal C(x)=1$ , $A>0$ , $\beta_w>0$ and $\omega_-(\beta)>0$ on $J$ . There is a strictly increasing open partial homeomorphism $e$ with source $J$ and convex target containing zero, such that

e(q)=\log\beta(q)_w-\log\beta(s)_w,\qquad e^{-1}(0)=s.

For every $z$ in its target,

(e^{-1})'(z)=\frac1{A(e^{-1}z)\omega_-(\beta(e^{-1}z))},\quad \beta(e^{-1}z)_w=\beta(s)_we^z,\quad(\beta\circ e^{-1})'(z)=H(\beta(e^{-1}z)).

Preamble

import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_ConnectionClock
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.Deriv.Inv
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.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.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_connectionLog_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)
    (hw : ∀ t ∈ I, 0 < toProjectiveState (normalizedMaxCurve x t₀ t) 3)
    (hD : ∀ t ∈ I, 0 < connectionClockRate (toProjectiveState (normalizedMaxCurve x t₀ t))) :
    let β := fun t => toProjectiveState (normalizedMaxCurve x t₀ t)
    ∃ e : OpenPartialHomeomorph ℝ ℝ,
      e.source = I ∧
      (∀ t ∈ I, e t = Real.log (β t 3) - Real.log (β s 3)) ∧
      Convex ℝ e.target ∧ 0 ∈ e.target ∧ e.symm 0 = s ∧ StrictMonoOn e I ∧
      (∀ z ∈ e.target,
        HasDerivAt e.symm (1 / ((1 + normalizedMaxCurve x t₀ (e.symm z) 0) *
          connectionClockRate (β (e.symm z)))) z ∧
        β (e.symm z) 3 = β s 3 * Real.exp z ∧
        HasDerivAt (fun u => β (e.symm u)) (connectionLogField (β (e.symm z))) z) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ConnectionClock.lean#L109-L132

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