Prove2Me

The closed connection phase and wall region

Definition
BianchiIX_Dynamics_ConnectionRegion

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

bianchi-ix-era-statisticsdynamical-systems

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

K=a2+b2+c2−2(ab+bc+ca),S=p2+s2,C=S+K.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−2a2+b2+c2+ab−2bc+ac,P=Kp-2a^2+b^2+c^2+ab-2bc+ac, Q=Ks+3(−b2+c2+ab−ac),Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac), r1=−S+2p,r2=−S−p+3s,r3=−S−p−3s.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,r1a,r2b,r3c)(P,Q,r_1a,r_2b,r_3c). For y=(m,u,v,w)∈R4y=(m,u,v,w)\in\mathbb R^4, define

K(u,v,w)=u2+v2+w2−2(uv+vw+wu),κ=u(v+w−u),K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u), Fm=−v2+w2+u(v−w)−mκ,Fu=12(1−m2−(v−w)2+u2)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, Fv=(m−1−κ)v,Fw=(−m−1−κ)w,F=(Fm,Fu,Fv,Fw),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+v2E(m,v)=(m-1)^2+v^2.

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

ψ(p,z)=3 z1+p,Π(x)=(ψ(p,s),ψ(p,a),ψ(p,b),ψ(p,c)).\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)y=(m,u,v,w), set D(y)=3+m2+K(u,v,w)D(y)=3+m^2+K(u,v,w) and

ΞA(y)=(A−1,Am3,Au3,Av3,Aw3),Ξ(y)=Ξ6/D(y)(y).\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,r1a,r2b,r3c)N(x)=(P,Q,r_1a,r_2b,r_3c) for the normalized vector field and C(x)=C(p,s,a,b,c)\mathcal C(x)=C(p,s,a,b,c) for its constraint.

The normalized and projective state spaces are R5\mathbb R^5 and R4\mathbb R^4 with their finite-product norms. The maps Π,Ξ,D,F\Pi,\Xi,D,F act on these state spaces.

The projective vector field in normalized time is

G(y)=6D(y)F(y).G(y)=\frac6{D(y)}F(y).

Write X=R5X=\mathbb R^5, Y=R4Y=\mathbb R^4, and let αx,t0\alpha_{x,t_0} be the maximal solution of α′=N(α)\alpha'=N(\alpha) with initial value xx at t0t_0 and domain Imax⁡(x,t0)I_{\max}(x,t_0). Put

Jx,t0(t)=dxαx,t0(t),Ψt0(y,t)=Π(αΞ(y),t0(t)),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)), Jy,t0(t)=dΠαΞ(y),t0(t)∘JΞ(y),t0(t)∘dΞy.\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)c=u(v+w-u), s(m,u,v,w)=m−1−cs(m,u,v,w)=m-1-c, b(m,u,v,w)=(1−m2−(v−w)2+u2)/2b(m,u,v,w)=(1-m^2-(v-w)^2+u^2)/2, D(m,u,v,w)=3+m2+K(u,v,w)D(m,u,v,w)=3+m^2+K(u,v,w), and r(m,v)=(m−1)2+v2r(m,v)=\sqrt{(m-1)^2+v^2}.

Define

Cμ={(m,u,v,w):∣−m−μ∣≤1/2, 0≤u≤1/10000, 0≤v,w≤1},\mathcal C_\mu=\{(m,u,v,w):|-m-\mu|\leq1/2, \ 0\leq u\leq1/10000,\ 0\leq v,w\leq1\}, ω−(y)=s(−ym,yu,yv,yw),H(y)=F(y)/ω−(y).\omega_-(y)=s(-y_m,y_u,y_v,y_w),\qquad H(y)=F(y)/\omega_-(y).

For a normalized trajectory write

β(t)=Π(αx,t0(t)),A(t)=1+(αx,t0(t))0,I=Imax⁡(x,t0).\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).
Definition code
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
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.Topology.Instances.Real.Lemmas
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
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed

noncomputable section

namespace BianchiIX

open Set

def connectionRegion (μ : ℝ) (y : ProjectiveState) : Prop :=
  |-y 0 - μ| ≤ 1 / 2 ∧ (0 ≤ y 1 ∧ y 1 ≤ 1 / 10000) ∧
    (0 ≤ y 2 ∧ y 2 ≤ 1) ∧ (0 ≤ y 3 ∧ y 3 ≤ 1)







end BianchiIX
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ConnectionRegion.lean

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