Continuous auxiliary height choices preserve the physical connection germ

Proved

BianchiIX.exists_terminal_connection_height_freezing

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.

For $Y=(k,u,w)\in\mathbb R^3$ , define

e_-(Y)=(1-\sqrt{k^2-1},u,1,w),\quad x_Y=\Xi(e_-(Y)),\quad \beta_Y(t)=\Psi_0(e_-(Y),t),\quad g(x)=(\Pi(x))_v.

For any real height $\eta$ , put

H_{\eta,Y}=\{t>0:t\in I_{\max}(x_Y,0),\ g(\alpha_{x_Y,0}(t))=\eta\},

T_\eta(Y)=\inf H_{\eta,Y},\qquad P_\eta(Y)=\beta_Y(T_\eta(Y)).

The infimum uses the total real-order convention. Let $\operatorname{TailValid}_\eta(Y)$ mean that $T_\eta(Y)$ is the least element of $H_{\eta,Y}$ , and, writing $T=T_\eta(Y)$ , that

t\in I_{\max}(x_Y,0),\quad 1+(\alpha_{x_Y,0}(t))_0>0 \quad(0\leq t\leq T),

g(\alpha_{x_Y,0}(t))>\eta\quad(0\leq t<T),

dg_{\alpha_{x_Y,0}(t)}N(\alpha_{x_Y,0}(t))<0,\quad |(\Pi(\alpha_{x_Y,0}(t)))_w|<1/2\quad(0\leq t\leq T).

All vector and operator norms use the finite-product supremum norms.

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.

For $y=(m,u,v,w)$ define the cyclic chart by

H(y)=(m+1)^2+K(u,v,w),\qquad \mathcal A(y)=(1-4(m+1)/H,4v/H,4w/H,4u/H).

Its normalized symmetry is

A(p,s,a,b,c)=((-p+\sqrt3s)/2,(-\sqrt3p-s)/2,b,c,a).

Put $z_h(Y)=\mathcal A(P_h(Y))$ and

\mathcal T(h,Y)=T_h(Y)+T_+(z_h(Y)),\qquad \mathcal P(h,Y)=P_+(z_h(Y)),\qquad x_Y^c=A\Xi(e_-(Y)).

Write $r(m,v)=\sqrt{(m-1)^2+v^2}$ and $g_*(x)=(\Pi(x))_v$ .

Fix $0<\varepsilon_{\rm in}\leq1/10000$ , $0<\eta\leq1/100$ , $q,J\in\mathbb N$ with $q+2\leq J$ , and $\eta\leq\varepsilon_{\rm in}/[4(11/5)^q]$ . For $0<\alpha\leq1$ and $u,w>0$ , set $Y=(2+2\alpha,u,w)$ and $h=\eta\alpha^J$ .

There exists $\varepsilon>0$ such that, if $0<b\leq\varepsilon\alpha^{2J}$ and $u+w\leq b$ , the derivatives of $v\mapsto\mathcal T(v,Y)$ and $v\mapsto\mathcal P(v,Y)$ at $h$ are zero. For every height function $a$ continuous at $Y$ with $a(Y)=h$ ,

\mathcal T(a(Z),Z)=\mathcal T(h,Z),\qquad \mathcal P(a(Z),Z)=\mathcal P(h,Z)

near $Y$ , and both maps on the left are $C^2$ at $Y$ .

Preamble

import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_TerminalConnectionData
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.Normed.Module.FiniteDimension
import Mathlib.LinearAlgebra.Determinant
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.Topology.Instances.Matrix
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Data.Real.Basic
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.Log.Deriv
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Tactic.Positivity
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Topology.Order.Compact
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.Module
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence

noncomputable section
open BianchiIX
open Set Filter
open scoped Topology

Formal statement

theorem BianchiIX.exists_terminal_connection_height_freezing {εin η : ℝ}
    (hεin : 0 < εin ∧ εin ≤ 1 / 10000) (hη : 0 < η ∧ η ≤ 1 / 100)
    {q J : ℕ} (hJ : q + 2 ≤ J) (hηin : η ≤ εin / (4 * (11 / 5) ^ q)) :
    ∃ ε > (0 : ℝ), ∀ α b u w : ℝ,
      (0 < α ∧ α ≤ 1) → 0 < b → b ≤ ε * α ^ (2 * J) →
      |u| + |w| ≤ b → 0 < u → 0 < w →
      let Y : CoreAngleState := ![2 + 2 * α, u, w]
      let h := η * α ^ J
      HasDerivAt (fun v => terminalConnectionTime v Y) 0 h ∧
        HasDerivAt (fun v => terminalConnectionState v Y) 0 h ∧
        ∀ a : CoreAngleState → ℝ, ContinuousAt a Y → a Y = h →
          (∀ᶠ Z in 𝓝 Y, terminalConnectionTime (a Z) Z = terminalConnectionTime h Z ∧
            terminalConnectionState (a Z) Z = terminalConnectionState h Z) ∧
          ContDiffAt ℝ 2 (fun Z => terminalConnectionTime (a Z) Z) Y ∧
          ContDiffAt ℝ 2 (fun Z => terminalConnectionState (a Z) Z) Y := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/TerminalHeightFreezing.lean#L95-L116

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