A twice differentiable first downward wall-crossing branch

Proved

BianchiIX.exists_activeWall_first_downcrossing_two

by trppppp · Nov 6, 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).

For real $k$ , put $\varepsilon_k=1/(10000k)$ and define

\mathcal K_k=\{(m,u,v,w): |E(m,v)-k^2|\leq1/8,\quad v\geq1, \quad0\leq u\leq\varepsilon_k,\quad0\leq vw\leq\varepsilon_k\}.

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 $G(y)=6F(y)/D(y)$ , $\lambda(y)=6r_yd_y/D(y)$ , and $S_k(R,U,P)=(R,kU,P)$ . Define

\mathcal P_y\xi=d\eta_y\xi-d\theta_y\xi\,\mathcal V(\theta_y,\eta(y)), \qquad \mathcal W_{y,t_0,w}(t)= \mathcal P_{\Psi_{t_0}(y,t)}(\mathcal J_{y,t_0}(t)w).

The entrance parametrization and radial exit factor are

e(k,u,w)=(1+\sqrt{k^2-1},u,1,w),

f_y=\mathcal V(\theta_y,\eta(y))_0,\qquad c_y=-\tanh\theta_y, \qquad h_y=c_y+f_y/r_y,\qquad \rho_y=c_y/h_y.

For projective initial data $y$ , put

A_{y,t_0}(t)=1+(\alpha_{\Xi(y),t_0}(t))_0.

Coordinates of three-vectors are indexed by $0,1,2$ ; projective coordinates are indexed by $m,u,v,w$ .

For normalized initial data $x$ , let $X=\alpha_{x,t_0}(t)$ and $g(X)=(\Pi(X))_v$ . Define the event differential by

\ell_{x,t_0,t}(v)=-[dg_XN(X)]^{-1}dg_XJ_{x,t_0}(t)v, \qquad \mathscr H_{x,t_0,t}(v)=J_{x,t_0}(t)v+\ell_{x,t_0,t}(v)N(X).

Its expression in projective initial and final coordinates is

L_{y,t_0,t}=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ\mathscr H_{\Xi(y),t_0,t}\circ d\Xi_y.

For a time function $\tau$ , write $M_\tau(z)=\Psi_{t_0}(z,\tau(z))$ . Let $s<t$ and assume the normalized orbit from $(x,t_0)$ exists on $[s,t]$ with positive chart clock. At every $u\in[s,t)$ , suppose either $g(\alpha_{x,t_0}(u))\ne h$ or the event rate is positive. At $t$ , assume $g(\alpha_{x,t_0}(t))=h$ and the event rate is negative. There is a time branch $\tau$ , $C^2$ at $x$ , with $\tau(x)=t$ and $d\tau(x)=\ell_{x,t_0,t}$ . Its projective endpoint is $C^2$ , with derivative $d\Pi\circ\mathscr H_{x,t_0,t}$ . For every initial state near $x$ , $\tau$ is the least negative-rate crossing after $s$ in the maximal domain, and the whole interval from $s$ to that crossing remains in the positive-clock chart.

Preamble

import Definitions.Def_BianchiIX_Dynamics_Normalized
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 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

noncomputable section
open BianchiIX
open Set Filter
open scoped Topology

Formal statement

theorem BianchiIX.exists_activeWall_first_downcrossing_two {x : NormalizedState} {t₀ s t η : ℝ}
    (hst : s < t)
    (hreg : ∀ u ∈ Icc s t, u ∈ normalizedMaxDomain x t₀ ∧ 0 < 1 + normalizedMaxCurve x t₀ u 0)
    (hbefore : ∀ u ∈ Ico s t,
      projectiveActiveWall (normalizedMaxCurve x t₀ u) ≠ η ∨
        0 < normalizedEventRate projectiveActiveWall x t₀ u)
    (hrate : normalizedEventRate projectiveActiveWall x t₀ t < 0)
    (hhit : projectiveActiveWall (normalizedMaxCurve x t₀ t) = η) :
    ∃ τ : NormalizedState → ℝ, τ x = t ∧ ContDiffAt ℝ 2 τ x ∧
      ContDiffAt ℝ 2 (fun y => toProjectiveState (normalizedMaxCurve y t₀ (τ y))) x ∧
      HasFDerivAt τ (normalizedEventTimeDifferential projectiveActiveWall x t₀ t) x ∧
      HasFDerivAt (fun y => toProjectiveState (normalizedMaxCurve y t₀ (τ y)))
        ((fderiv ℝ toProjectiveState (normalizedMaxCurve x t₀ t)).comp
          (normalizedEventDifferential projectiveActiveWall x t₀ t)) x ∧
      (∀ᶠ y in 𝓝 x,
        IsLeast {u | s < u ∧ u ∈ normalizedMaxDomain y t₀ ∧
          projectiveActiveWall (normalizedMaxCurve y t₀ u) = η ∧
          normalizedEventRate projectiveActiveWall y t₀ u < 0} (τ y) ∧
        (∀ u ∈ Icc s (τ y), u ∈ normalizedMaxDomain y t₀ ∧ 0 < 1 + normalizedMaxCurve y t₀ u 0)) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ActiveWallDowncrossing.lean#L72-L124

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