The canonical connection return is twice differentiable in all entry parameters

Proved

BianchiIX.connectionReturn_regular

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).

For $p=(\mu,u,w)\in\mathbb R^3$ , set $e(p)=(-\mu,u,1,w)$ , $x_p=\Xi(e(p))$ , and

T_-(p)=\inf\{t>0:t\in I_{\max}(x_p,0),\ (\Psi_0(e(p),t))_w=1\}, \qquad P_-(p)=\Psi_0(e(p),T_-(p)).

Let $V_-(p)$ mean that this infimum is a least element, the full interval $[0,T_-(p)]$ lies in the maximal domain with positive $A$ , the growing wall is below one before the endpoint, and its normalized-time derivative is strictly positive at the endpoint. The infimum has its total real-order meaning.

Assume $p=(\mu,u,w)$ with $\mu\geq4$ , $0\leq u\leq1/(10000\mu)$ , and $0<w\leq1/10000$ .

For an observable $g$ , put $q=\alpha_{x,t_0}(T)$ and define

\ell_{g,x,t_0,T}(h)=-\frac{dg_q(J_{x,t_0}(T)h)}{dg_qN(q)}, \qquad E_{g,x,t_0,T}(h)=J_{x,t_0}(T)h+\ell_{g,x,t_0,T}(h)N(q).

Then $V_-(p)$ holds, $T_-$ and $P_-$ are $C^2$ at $p$ , and $V_-$ holds throughout a neighborhood of $p$ . With $g=\Pi_w$ , $T=T_-(p)$ , $q=\alpha_{x_p,0}(T)$ and $i(p)=x_p$ ,

dT_-|_p=\ell_{g,x_p,0,T}\circ di_p,\qquad dP_-|_p=d\Pi_q\circ E_{g,x_p,0,T}\circ di_p.

Preamble

import Definitions.Def_BianchiIX_Dynamics_ConnectionClock
import Definitions.Def_BianchiIX_Dynamics_ConnectionRegion
import Definitions.Def_BianchiIX_Dynamics_ConnectionReturn
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.Topology.Instances.Real.Lemmas
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
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.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Order.Compact
import Mathlib.Tactic.NormNum

noncomputable section
open BianchiIX
open Set Filter
open scoped Topology

Formal statement

theorem BianchiIX.connectionReturn_regular {p : ConnectionParameters} (hμ : 4 ≤ p 0)
    (hu : 0 ≤ p 1 ∧ p 1 ≤ (1 / 10000) / p 0) (hw : 0 < p 2 ∧ p 2 ≤ 1 / 10000) :
    let x := connectionInitialState p
    let T := connectionReturnTime p
    connectionReturnValid p ∧ ContDiffAt ℝ 2 connectionReturnTime p ∧
      ContDiffAt ℝ 2 connectionReturnState p ∧
      HasFDerivAt connectionReturnTime
        ((normalizedEventTimeDifferential projectiveGrowingWall x 0 T).comp
          (fderiv ℝ connectionInitialState p)) p ∧
      HasFDerivAt connectionReturnState
        (((fderiv ℝ toProjectiveState (normalizedMaxCurve x 0 T)).comp
          (normalizedEventDifferential projectiveGrowingWall x 0 T)).comp
          (fderiv ℝ connectionInitialState p)) p ∧
      (∀ᶠ q in 𝓝 p, connectionReturnValid q) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ConnectionReturn.lean#L46-L104

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