Uniform finite-tail coordinate bounds on the compact phase interval

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

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 $r(m,v)=\sqrt{(m-1)^2+v^2}$.

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.

Define the reflected radius and the composed entrance, time and state by

$$R_\eta(Y)=r((\mathcal R P_\eta(Y))_m,(\mathcal R P_\eta(Y))_v), \qquad e_\eta(Y)=\mathcal R P_\eta(Y),$$ $$T_c(\eta,Y)=T_+(e_\eta(Y))+T_\eta(Y),\qquad P_c(\eta,Y)=P_+(e_\eta(Y)).$$

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.

Let $p(y)=P(y_0,y_1,y_3)$ and $\rho(y)=\sqrt{(p(y)-1)^2+y_2^2}$. The finite phase, actual phase and their difference are

$$F_\eta(Y)=\rho(e_\eta(Y)),\quad \widehat F_\eta(Y)=r(P_c(\eta,Y)_0,P_c(\eta,Y)_2),\quad D_\eta=\widehat F_\eta-F_\eta.$$

For every $0<\eta\leq1/10000$ there exist $\varepsilon>0$ and $A\geq1$ such that, for $Y=(k,u,w)$ with $31/8\leq k\leq6$ and $|u|,|w|\leq\varepsilon$, the finite tail is valid on a neighborhood of $Y$, $F_\eta$ is $C^2$ at $Y$, and $\|dF_\eta(Y)\|\leq A$. Each of the two tail multipliers $m_i(\eta,\cdot)$ is $C^2$ at $Y$ and satisfies

$$0<m_i(\eta,Y)\leq A,\qquad\|d_Ym_i(\eta,Y)\|\leq A.$$

import Definitions.Def_BianchiIX_CompactPhaseExpansionData
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_ProjectiveClock
import Definitions.Def_BianchiIX_Dynamics_ReferenceTail
import Definitions.Def_BianchiIX_EntrancePhaseResponseData
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 Filter Metric
open scoped Topology

theorem BianchiIX.exists_compact_connection_coordinate_bounds {η : ℝ} (hη : 0 < η ∧ η ≤ 1 / 10000) :
    ∃ ε > (0 : ℝ), ∃ A : ℝ, 1 ≤ A ∧ ∀ Y : CoreAngleState,
      (31 / 8 ≤ Y 0 ∧ Y 0 ≤ 6) → |Y 1| ≤ ε → |Y 2| ≤ ε →
      (∀ᶠ Z in 𝓝 Y, coreTailValid η Z) ∧ ContDiffAt ℝ 2 (compactConnectionFinitePhase η) Y ∧
        ‖fderiv ℝ (compactConnectionFinitePhase η) Y‖ ≤ A ∧
        ∀ i : Fin 2, ContDiffAt ℝ 2 (fun Z => coreTailMultiplier η Z i) Y ∧
          0 < coreTailMultiplier η Y i ∧ coreTailMultiplier η Y i ≤ A ∧
          ‖fderiv ℝ (fun Z => coreTailMultiplier η Z i) Y‖ ≤ A := by sorry