Coverage of each open radial shell by four physical pieces

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 $(P,Q,r_1a,r_2b,r_3c)$.

Let $X=\mathbb R^5$ with coordinates $x=(p,s,a,b,c)$ and the sup norm. Write

$$F(x)=(P,Q,r_1a,r_2b,r_3c),\qquad \mathcal C(x)=C(p,s,a,b,c),$$

and let $W=\{(p,s,a,b,c)\in X:a>0,\ b>0,\ c>0\}$.

Write $N_1(x)=a$, $N_2(x)=b$, and $N_3(x)=c$. For initial data $(x_0,t_0)$, let $I_{\max}(x_0,t_0)$ be the union of all open convex domains of solutions with these initial data, and let $\alpha_{x_0,t_0}$ be their compatible curve. A solution $S$ has such a domain $I_S$, a curve $\alpha_S$, and $\alpha_S(t_0)=x_0$.

Let $\mathcal L(X,X)$ be the continuous linear operators on $X$ and set

$$A(x)=dF_x,\qquad \widetilde F(x,J)=(F(x),A(x)\circ J).$$

For a fixed initial time $t_0$, put

$$\mathcal D_{t_0}=\{(x,t):t\in I_{\max}(x,t_0)\},\qquad \Phi_{t_0}(x,t)=\alpha_{x,t_0}(t),\qquad J_{x,t_0}(t)=d_x\Phi_{t_0}(x,t).$$

Write $\lambda_x=d\mathcal C_x:X\to\mathbb R$ for the constraint differential.

For $x=(p,s,a,b,c)$, define

$$H(x)=(abc)^{2/3},\quad w(x)=abc(1+4H(x)),\quad \delta(x)=2K-3S,\quad \bar\delta(x)=\frac{2(K+4H(x))}{1+4H(x)},$$ $$V(x,t_0,t)=\frac{w(x)\det J_{x,t_0}(t)}{w(\alpha_{x,t_0}(t))}.$$

Write $[L]$ for the matrix of a continuous linear operator $L:X\to X$ in the standard basis. Set $\Omega(v_0,\ldots,v_4)=\det(v_0,\ldots,v_4)$,

$$\omega_x(v_0,\ldots,v_3)=\frac{\Omega(x/2,v_0,\ldots,v_3)}{w(x)},\qquad \varphi_x(v_0,v_1,v_2)=\omega_x(F(x),v_0,v_1,v_2).$$

A normalized symmetry is a continuous linear equivalence $L:X\to X$ satisfying

$$\mathcal C(Lx)=\mathcal C(x),\quad Lx\in W\ \Longleftrightarrow\ x\in W, \quad F(Lx)=LF(x),\quad w(Lx)=w(x),\quad\det L=1.$$

These symmetries include the identity, composition, inverse, and the two maps

$$R(p,s,a,b,c)=((-p+\sqrt3s)/2,(-\sqrt3p-s)/2,b,c,a),\qquad S_0(p,s,a,b,c)=(p,-s,a,c,b).$$

Transforming a solution replaces its curve $\alpha$ by $L\alpha$ and keeps its time domain.

Write $\Phi^s_t(x)=\alpha_{x,s}(t)$, with domain $I(x,s)$, and $\mathcal X=\{x:\mathcal C(x)=1,\ N_i(x)>0\ (i=1,2,3)\}$. For $y=(v,l,r)\in\mathbb R^3$ and a wall height $h$, put

$$b=e^{-l},\quad c=e^{-r},\quad d=1-v^2-K(h,b,c),\quad U_h=\{y:d>0\},\quad c_h(y)=(-\sqrt d,v,h,b,c).$$

For a normalized axis symmetry $A$, the chart is $c_{A,h}=A^{-1}c_h$, with coordinates $\pi_A(x)=((Ax)_1,-\log (Ax)_3,-\log (Ax)_4)$. At $p=-\sqrt d$ its flux density and measure are

$$\rho_h(y)=\frac{p^2+v^2-2p}{2|p|(1+4(hbc)^{2/3})}, \qquad d\mu_h={\bf1}_{U_h}\rho_h^+\,dy.$$

Write $\varphi$ for the signed section flux and $\bar\delta$ for the corrected divergence. The three primary frames interchange axis zero with the chosen dominant axis.

In wall coordinates $y=(v,l,r)$ put $u=e^{-l}$, $w=e^{-r}$, $M=\max(u,w)$, $S_d^\circ=\{|v|,M<d\}$, and let $S_d$ use weak inequalities. Put $R=|v|$ when $M\le\epsilon|v|$ and $R=M$ otherwise. For $a=a_j=d2^{-j-1}$ and $a<\rho<2a$, $\rho<d$, there are four charts $L_k$ with domains $D_{kj}$. The two shear charts send $(\rho,u,w)$ to $(\pm\rho,-\log u,-\log w)$ for $u,w>0$, $M\le\epsilon\rho$. The angular chart sends $(\rho,\theta,\zeta)$ to $(\theta\rho,-\log\rho,-\log\zeta)$, with $|\theta|<L=1/\epsilon$, $|\theta|\rho<d$ and $0<\zeta\le\rho$. Its reflected chart composes with $(v,l,r)\mapsto(-v,r,l)$ and has $\zeta<\rho$.

If $\epsilon>0$, $y\in S_d^\circ$ and $a_j<R(y)<2a_j$, then $y\in L_k(D_{kj})$ for some $k$.

import Definitions.Def_BianchiIX_RadialFailureData
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.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
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.LinearAlgebra.Determinant
import Mathlib.Data.Real.Basic
import Mathlib.LinearAlgebra.Alternating.Curry
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.Topology.Algebra.InfiniteSum.ENNReal
import Mathlib.Algebra.Order.Archimedean.Basic

noncomputable section
open BianchiIX
open Set MeasureTheory

theorem BianchiIX.exists_mem_physicalRadialPiece {d ε : ℝ} {z : SectionCoordinates} {j : ℕ}
    (hε : 0 < ε) (hz : z ∈ mainWallOpenSection d)
    (hr : mainWallRadialRadius ε z ∈ Ioo (dyadicScale d j) (2 * dyadicScale d j)) :
    ∃ k : Bool ⊕ Bool, z ∈ physicalRadialPiece d ε (k, j) := by sorry