A uniform second-order expansion of the compact finite matching phase

Proved

BianchiIX.exists_compact_connection_phase_expansion

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 $i=0,1$ put $j(0)=1$ , $j(1)=3$ , and define

m_i(\eta,Y)=\exp\left(\log(1+(x_Y)_0) -\log(1+(\alpha_{x_Y,0}(T_\eta(Y)))_0) +\int_0^{T_\eta(Y)}r_{j(i)}(\alpha_{x_Y,0}(t))\,dt\right).

Let $\mathcal R(m,u,v,w)=(-m,u,w,v)$ .

On the invariant plane with zero active wall, use $y=(m,u,w)\in\mathbb R^3$ , the inclusion $\iota(m,u,w)=(m,u,0,w)$ and the restriction $\pi_s(m,u,v,w)=(m,u,w)$ . Define

f(y)=\left(mu^2-(m+1)uw+w^2, \frac{1-m^2-w^2+u^2}{2}u,-(m+1+uw-u^2)w\right),

\Delta(y)=3+m^2+(u-w)^2,\qquad f_o(y)=6f(y)/\Delta(y).

Write $J=df$ , $H=d^2f$ , $J_o=df_o$ , $H_o=d^2f_o$ for their explicit Jacobian and Hessian expressions. The denominator derivatives are $d\Delta=(2m,2(u-w),2(w-u))$ and $d^2\Delta(v,z)=2v_0z_0+2(v_1-v_2)(z_1-z_2)$ . The exact original-time boundary flow and its projected variation are

\Phi(y,t)=\pi_s\Psi_0(\iota y,t),\qquad L(y,t)=\pi_s\mathcal J_{\iota y,0}(t)\iota.

For a fixed scale $K$ , set $\Phi_K(y,s)=\Phi(y,Ks)$ , $L_K(y,s)=L(y,Ks)$ and $Q_K(y,s)=d_yL_K(y,s)$ . For directions $v,z$ , define

S_K(y,s;v,z)=K\left[H_o(\Phi_K)(L_Kv,L_Kz)+J_o(\Phi_K)Q_K(v,z)\right].

All vector norms are sup norms, and operator norms are induced by them. Write $b(y)=|u|+|w|$ , $E(y)=e^{512b(y)}$ , $v_s=(v_1,v_2)$ , and

\mathcal H(y;v,z)=b(y)\|v\|\|z\|+\|v_s\|\|z\|+\|v\|\|z_s\|,

B_2(y)=192b(y)(40000E(y)^3)+10000E(y)^2.

The closed boundary region is $\mathcal S_c=\{(m,u,0,w):|m-c|\leq1/10,\ |u|,|w|\leq1/100\}$ . The boundary projection $P(m,u,w)$ is the limit of $\Phi(m,u,w;t)_0$ as $t\to\infty$ , using the total limit convention of the formal definitions.

For $y\in\mathbb R^4$ , put

P_e(y)=P(y_0,y_1,y_3),\quad R_e(y)=\sqrt{(P_e(y)-1)^2+y_2^2}.

The reflected finite-tail entry is $e_\eta(Y)=\mathcal R P_\eta(Y)$ , and its finite matching phase is $F_\eta(Y)=R_e(e_\eta(Y))$ . For $A_Y(t)=1+(\alpha_{x_Y,0}(t))_0$ , define the weak-wall multiplier

M_w(Y,t)=\exp\left(\log(1+(x_Y)_0)-\log A_Y(t) +\int_0^t r_3(\alpha_{x_Y,0}(q))\,dq\right),

where $r_3$ is the third normalized wall rate. Set

B_\eta(Y)=\int_0^{T_\eta(Y)}2A_Y(t)(\beta_Y(t)_0-1)M_w(Y,t)^2\,dt.

For every $0<\eta\leq1/10000$ there exist $\varepsilon,C>0$ such that, whenever $31/8\leq Y_0\leq6$ and $|Y_1|,|Y_2|\leq\varepsilon$ , the tail is valid, $F_\eta$ is $C^2$ at $Y$ ,

\left|F_\eta(Y)-\left(Y_0-2-\frac{2Y_1}{Y_0}\right)\right| \leq C(|Y_1|+|Y_2|)^2,

and for each $i\in\{0,1,2\}$ ,

\left|dF_\eta(Y)e_i-(1,-2/Y_0,0)_i\right| \leq C(|Y_1|+|Y_2|).

Preamble

import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_CompactPhaseExpansionData
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_ProjectiveRegularity
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_ProjectiveClock
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_Events
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_ReferenceTail
import Definitions.Def_BianchiIX_Dynamics_ProjectiveReflection
import Definitions.Def_BianchiIX_Dynamics_ProjectiveWallMultipliers
import Definitions.Def_BianchiIX_Dynamics_CoreTail
import Definitions.Def_BianchiIX_Dynamics_CoreTailMultipliers
import Definitions.Def_BianchiIX_Dynamics_CompactConnectionReturn
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
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Data.Matrix.Mul
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.LinearAlgebra.Determinant
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.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Order.ProjIcc

noncomputable section
open BianchiIX
open Set

Formal statement

theorem BianchiIX.exists_compact_connection_phase_expansion {η : ℝ} (hη : 0 < η ∧ η ≤ 1 / 10000) :
    ∃ ε > (0 : ℝ), ∃ C > (0 : ℝ), ∀ Y : CoreAngleState,
      (31 / 8 ≤ Y 0 ∧ Y 0 ≤ 6) → |Y 1| ≤ ε → |Y 2| ≤ ε →
      coreTailValid η Y ∧ ContDiffAt ℝ 2 (compactConnectionFinitePhase η) Y ∧
        |compactConnectionFinitePhase η Y - (Y 0 - 2 - 2 * Y 1 / Y 0)| ≤
          C * (|Y 1| + |Y 2|) ^ 2 ∧
        ∀ i : Fin 3, |fderiv ℝ (compactConnectionFinitePhase η) Y (Pi.single i 1) -
          (![1, -2 / Y 0, 0] : Fin 3 → ℝ) i| ≤ C * (|Y 1| + |Y 2|) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/CompactConnectionPhaseExpansion.lean#L10-L105

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