The canonical first-return differential includes the event-time variation

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.

For the positive reflected phase variable $m$, put

$$n(a,b)=\sqrt{a^2+b^2},\quad j(a,b)=\frac1{n(a,b)(n(a,b)+a)},$$ $$\phi=n(m+1,v)+n(m-1,w)-m,\quad h=\frac{m+1}{n(m+1,v)}+\frac{m-1}{n(m-1,w)}-1,$$ $$Q=mh+\frac{v^2}{n(m+1,v)}+\frac{w^2}{n(m-1,w)},\quad k=j(m+1,v)-j(m-1,w).$$

Write $\kappa=u(v+w-u)$, $s=m-1-\kappa$, $b=(1-m^2-(v-w)^2+u^2)/2$, and

$$M_v=-\frac{v^2}{n(m+1,v)h},\qquad M_w=-\frac{w^2}{n(m-1,w)h}.$$

The four coordinate differential operators, in the order $(\phi,\log u,\log v,\log w)$, are

$$L_0=h^{-1}\partial_m,\quad L_1=u\partial_u,\quad L_2=M_v\partial_m+v\partial_v,\quad L_3=M_w\partial_m+w\partial_w.$$

Subscripts $i$ below denote the rational expressions obtained by these rules, with total real-field division. On the regular positive-wall region they are the corresponding coordinate partial derivatives. Set

$$N=h(v-w)+(v+w-u)Q,\quad R=h(v-w)+(v+w-u)(Q-s),$$ $$A=-N/s,\quad B=k/s,\quad G=uA+(vw)^2B,\quad \alpha=b/s,\quad \beta=-1-2(1+\kappa)/s.$$

The coefficient matrix $C$ has rows $(G_i)_i$, $(\alpha_i)_i$ and $(\beta_i)_i$.

For a projective state $y$, evaluate the reflected formulas at $(m,u,v,w)=(-y_0,y_1,y_2,y_3)$. Define

$$\chi(y)=(\phi,\log y_1,\log y_2,\log y_3),$$ $$\mathcal T_y\xi=\left(-h\xi_0+ \frac{y_2\xi_2}{n(-y_0+1,y_2)}+ \frac{y_3\xi_3}{n(-y_0-1,y_3)}, \frac{\xi_1}{y_1},\frac{\xi_2}{y_2},\frac{\xi_3}{y_3}\right).$$

Let $\mathcal A_y d=(\sum_i C_{0i}d_i,\sum_i C_{1i}d_i, \sum_i C_{2i}d_i,0)$ and $W_\mu d=(d_1,\mu^2d_2)$, with the sup norm. The original-time clock rate is $\Omega(y)=6\omega_-(y)/D(y)$. For $Y=\Psi_{t_0}(y,t)$ and $\xi=d_y\Psi_{t_0}(y,t)v$, set

$$\mathcal P_{y,t_0,t}v=\xi-(\xi_3/Y_3-v_3/y_3)H(Y),\qquad \mathcal D_{y,t_0,t}v=\mathcal T_Y(\mathcal P_{y,t_0,t}v).$$

Thus $\mathcal D$ is the logarithmic variation at fixed relative growing-wall clock.

For entry amplitudes $u,p$, define the feedback budget

$$\Lambda_\mu(u,p)=64\left(\frac{2u}{\mu^3}+\frac{p^2}{4\mu^3}\right) +8192\left(\frac{4u}{\mu^2}+\frac{p^2}{16\mu^3}\right).$$

Write $p=(\mu,u,w)$, $L=-\log w$, and define the physical return and phase error by

$$P^{\rm phys}(p)=(n(-P_-(p)_0-1,1),P_-(p)_1,P_-(p)_2),$$ $$E(p)=n(-P_-(p)_0-1,1)-n(\mu+1,1)+2.$$

Assume $p=(\mu,u,w)$ with $\mu\geq4$, $0\leq u\leq1/(10000\mu)$, and $0<w\leq1/10000$. For a direction $v\in\mathbb R^3$, let $q=\alpha_{x_p,0}(T_-(p))$ and $\xi=d\Pi_q[J_{x_p,0}(T_-(p))\,d_px_p(v)]$. With $g=\Pi_w$ and $\rho=dg_qN(q)$,

$$dT_-(p)v=-\xi_3/\rho,\qquad dP_-(p)v=\xi-\frac{\xi_3}{F(P_-(p))_3}F(P_-(p)).$$

import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_Linearization
import Definitions.Def_BianchiIX_Dynamics_FlowDifferential
import Definitions.Def_BianchiIX_Dynamics_Events
import Definitions.Def_BianchiIX_Dynamics_ConnectionExistence
import Definitions.Def_BianchiIX_Dynamics_ConnectionEvents
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

theorem BianchiIX.connectionReturn_differential {p : ConnectionParameters} (hμ : 4 ≤ p 0)
    (hu : 0 ≤ p 1 ∧ p 1 ≤ (1 / 10000) / p 0) (hw : 0 < p 2 ∧ p 2 ≤ 1 / 10000)
    (v : ConnectionParameters) :
    let x := connectionInitialState p
    let T := connectionReturnTime p
    let y := normalizedMaxCurve x 0 T
    let ξ := fderiv ℝ toProjectiveState y (normalizedFlowDifferential x 0 T (fderiv ℝ connectionInitialState p v))
    fderiv ℝ connectionReturnTime p v = -(normalizedEventRate projectiveGrowingWall x 0 T)⁻¹ * ξ 3 ∧
      fderiv ℝ connectionReturnState p v =
        ξ - (ξ 3 / projectiveField (connectionReturnState p) 3) • projectiveField (connectionReturnState p) := by sorry