Physical axis charts, core flight times and the Brehm attractor input

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.

Let $\mathcal T$ be the union, over primary frames, of the subspaces where the transverse shear and the difference of the other two walls both vanish. Set $\Delta_{ij}(x)=2\sqrt{|N_i(x)N_j(x)|}$. The Brehm attractor input used here is the following proposition: for each $x\in\mathcal X\setminus\mathcal T$ and $i\ne j$,

$$\Delta_{ij}(\Phi^0_t(x))\longrightarrow0\qquad(t\to+\infty).$$

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.

Define the core radius, angle, and clock factor by

$$r(m,v)=\sqrt{(m-1)^2+v^2},\qquad \theta(m,v)=\operatorname{arsinh}\frac{1-m}{v},\qquad d(r,u,w)=1-\frac{(u-w)^2}{r^2}.$$

For $y=(m,u,v,w)$, write $r_y=r(m,v)$, $\theta_y=\theta(m,v)$, and $d_y=d(r_y,u,w)$. All quotients use the total convention $x/0=0$.

The core state space is $\mathbb R^3$ with coordinates $Y=(r,u,p)$. Define

$$\eta(m,u,v,w)=(r(m,v),u,vw),\qquad \iota_z(r,u,p)=\left(1-r\tanh z,u,\frac r{\cosh z}, \frac p{r/\cosh z}\right).$$

For fixed $(z,Y)$ put $v=r/\cosh z$, $w=p/v$, $d=d(r,u,w)$, and define

$$\mathcal V(z,Y)=\left( \frac{-\kappa(u,v,w)-\tanh z\,(u-w)^2/r}{d},\quad \frac{u[-r/2+\tanh z+w/\cosh z+(u^2-w^2)/(2r)]}{d},\quad \frac{-2p(1+\kappa(u,v,w))}{rd}\right).$$

For real $k$, put $\varepsilon_k=1/(10000k)$ and define

$$\mathcal K_k=\{(m,u,v,w): |E(m,v)-k^2|\leq1/8,\quad v\geq1, \quad0\leq u\leq\varepsilon_k,\quad0\leq vw\leq\varepsilon_k\}.$$

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

$$J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad \Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)),$$ $$\mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ J_{\Xi(y),t_0}(t)\circ d\Xi_y.$$

Write $G(y)=6F(y)/D(y)$, $\lambda(y)=6r_yd_y/D(y)$, and $S_k(R,U,P)=(R,kU,P)$. Define

$$\mathcal P_y\xi=d\eta_y\xi-d\theta_y\xi\,\mathcal V(\theta_y,\eta(y)), \qquad \mathcal W_{y,t_0,w}(t)= \mathcal P_{\Psi_{t_0}(y,t)}(\mathcal J_{y,t_0}(t)w).$$

The entrance parametrization and radial exit factor are

$$e(k,u,w)=(1+\sqrt{k^2-1},u,1,w),$$ $$f_y=\mathcal V(\theta_y,\eta(y))_0,\qquad c_y=-\tanh\theta_y, \qquad h_y=c_y+f_y/r_y,\qquad \rho_y=c_y/h_y.$$

For projective initial data $y$, put

$$A_{y,t_0}(t)=1+(\alpha_{\Xi(y),t_0}(t))_0.$$

Coordinates of three-vectors are indexed by $0,1,2$; projective coordinates are indexed by $m,u,v,w$.

For $Y\in\mathbb R^3$, put $x_Y=\Xi(e(Y))$, $\beta_Y(t)=\Psi_0(e(Y),t)$, and

$$H_Y=\{t>0:t\in I_{\max}(x_Y,0),\ (\beta_Y(t))_v=1\}, \qquad T(Y)=\inf H_Y,\qquad F(Y)=\eta(\beta_Y(T(Y))).$$

The infimum uses the total real-order convention. Write $\operatorname{Valid}(Y)$ when $T(Y)$ is the least element of $H_Y$, every $t\in[0,T(Y)]$ lies in $I_{\max}(x_Y,0)$ with $A_{e(Y),0}(t)>0$, the active wall is strictly above one for $0<t<T(Y)$, and

$$dg_{\alpha_{x_Y,0}(T(Y))}N(\alpha_{x_Y,0}(T(Y)))<0, \qquad g(x)=(\Pi(x))_v.$$

All vector and operator norms below use the finite-product supremum norm. 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.

Write $e_+(R,u,w)=(1+\sqrt{R^2-1},u,1,w)$, $e_-(R,u,w)=(1-\sqrt{R^2-1},u,1,w)$, $p_-(R,u,w)=(\sqrt{R^2-1}-1,u,w)$, and $\mathcal R(m,u,v,w)=(-m,u,w,v)$. The physical connection coordinates are

$$\mathscr C(p)=\left(\sqrt{(-P_-(p)_0-1)^2+1},P_-(p)_1,P_-(p)_2\right).$$

Below $T$ denotes the canonical core return time from $e_+$ and $F(Y)=\eta(\Psi_0(e_+(Y),T(Y)))$ its radial and wall coordinates. Let $\mathscr R(Y)=\mathscr C(p_-(Y))$, $t_r(Y)=T_-(p_-(Y))$, $G=\mathscr R\circ F$, and $t_G(Y)=T(Y)+t_r(F(Y))$. A cone direction at $Y$ has $d_0=1$, $|d_1|\leq\sqrt{Y_1}$, $|d_2|\leq\sqrt{Y_2}$. For a map $H$ with positive output walls, its positive cone property means $dH(Y)d$ has positive radial component and $|(dH(Y)d)_i|\leq\sqrt{H(Y)_i}(dH(Y)d)_0$ for $i=1,2$. 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 $y\in\mathbb R^4$, 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.

For $y=(m,u,v,w)$ define the cyclic chart by

$$H(y)=(m+1)^2+K(u,v,w),\qquad \mathcal A(y)=(1-4(m+1)/H,4v/H,4w/H,4u/H).$$

Its normalized symmetry is

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

Put $z_h(Y)=\mathcal A(P_h(Y))$ and

$$\mathcal T(h,Y)=T_h(Y)+T_+(z_h(Y)),\qquad \mathcal P(h,Y)=P_+(z_h(Y)),\qquad x_Y^c=A\Xi(e_-(Y)).$$

Write $r(m,v)=\sqrt{(m-1)^2+v^2}$ and $g_*(x)=(\Pi(x))_v$.

In the era formulas below, $F$ denotes the core return map and $G=\mathscr R\circ F$ its composition with reflection. For $Z=(R,u,w)$ use the natural height $h_*(Z)=\eta((R-2)/2)^J$ and define $S_*(Z)=\mathcal P(h_*(Z),Z)$, $R_*(Z)=r(S_*(Z)_0,S_*(Z)_2)$, $Q_*(Z)=2/R_*(Z)$ and $t_*(Z)=\mathcal T(h_*(Z),Z)$. The complete era maps are

$$Z_m=F\circ G^m,\qquad S_m=S_*\circ Z_m,\qquad R_m=R_*\circ Z_m,\qquad P_m=2/R_m.$$

For a graph $\Gamma=(U,W)$ write $\Gamma[r]=(r,U(r),W(r))$, $\phi_m(r)=P_m(\Gamma[r])$, $k_m(r)=R_m(\Gamma[r])$, $u_m(r)=S_m(\Gamma[r])_1$, $w_m(r)=S_m(\Gamma[r])_3$. Let $\operatorname{Inv}_{a,b}(f)$ denote Lean's total choice inverse $\operatorname{invFunOn}f\,[a,b]$. For $a=2(m+1+\beta)$, $b=2(m+2-\beta)$, define

$$g_m=\operatorname{Inv}_{a,b}(\phi_m),\qquad h_m(y)=2/g_m(y).$$

Write $\gamma_n(y)=1/(n+y)$ for the Gauss inverse branch. The retained phase set is $\mathcal H_N=\bigcup_{m<N}(h_m(1-2\beta),h_m(2\beta))$.

Write $x(Y)=\Xi(e_+(Y))$, $\Phi^s_t(x)=\alpha_{x,s}(t)$ and $I(x,s)$ for its maximal domain. Set $Y_j=G^jX$, $Z=F(Y_m)$, $N_m(X)=(R_m(X),S_m(X)_1,S_m(X)_3)$, and

$$d_m(X)=\sum_{j<m}t_G(Y_j)+T(Y_m)+t_*(Z).$$

The six axis frames realize all permutations of the normalized walls and diagonal shears. Write $L_m$ for the frame obtained by $m$ swaps followed by the cyclic permutation. The event itinerary $\mathcal I_m(X)$ has the following meaning. For $j\le m$, the radius of $Y_j$ exceeds two, the entry is an upward unit-wall crossing, its core first return is valid, and its endpoint is $e_-(F(Y_j))$. For $j<m$ the exit radius exceeds four, and the first positive unit-wall hit in the swapped chart reaches $e_+(Y_{j+1})$, the reflection of the endpoint at time $t_G(Y_j)$. The final exit has radius strictly between two and four. In the cycled exit chart its first positive unit-wall hit occurs at $t_*(Z)$, has state $S_*(Z)$ and positive event rate. Finally $\lfloor X_0/2\rfloor=m+1$.

These data include the positive constraint phase space, the Euclidean distances to the three Taub points, their invariant subspaces, and the pairwise wall products. In the primary frame the squared transverse radius is $s^2+(b-c)^2$; its weighted form multiplies this by $e^{-8a}$. Orbit saturation is the union of the existing flow images at all times. The frame changes, wall charts, actual arrival branches and core flight times specify the events of each finite era.