Partial-core returns, complete era maps and successor curves

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 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

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

At $(z,Y)$ use the abbreviations

$$s=1/\cosh z,\quad c=-\tanh z,\quad \delta=u-w,$$ $$C=u(v+w-u),\quad a=-r/2+\tanh z+w/\cosh z+(u^2-w^2)/(2r), \quad g=-C+c\delta^2/r.$$

Let $\mathcal Q(z,Y)=(C,d,a,g)$. Define the $4\times3$ matrix $H(z,Y)$ by its rows, in the order $(C,d,a,g)$ and columns $(r,u,p)$:

$$H_C=(u(s-w/r),\ v-2u+w,\ u/v),$$ $$H_d=(2\delta(u-2w)/r^3,\ -2\delta/r^2,\ 2\delta/(vr^2)),$$ $$H_a=(-1/2-sw/r+(3w^2-u^2)/(2r^2),\ u/r,\ (1-w/v)/r),$$ $$H_g=(-u(s-w/r)+c\delta(3w-u)/r^2,\ -(v-2u+w)+2c\delta/r,\ -u/v-2c\delta/(rv)).$$

Define the two rates and the $2\times3$ matrix $J$ by

$$L_0(z,Y)=a/d,\qquad L_1(z,Y)=-2(1+C)/(rd),$$ $$J_{0j}=H_{aj}/d-aH_{dj}/d^2,$$ $$J_{1j}=\delta_{j0}\frac{2(1+C)}{r^2d} -\frac{2H_{Cj}}{rd}+\frac{2(1+C)H_{dj}}{rd^2},\qquad j=0,1,2.$$

Here $\delta_{j0}$ is the Kronecker symbol. The total formulas are defined also when a wall amplitude vanishes, and use $x/0=0$.

Define the rates in normalized time and the positive flow factors by

$$R(y)=\lambda(y)L(\theta_y,\eta(y)),\qquad M_i(y,t_0,a,b)=\exp\left(\int_a^b R_i(\Psi_{t_0}(y,t))\,dt\right), \quad i=0,1.$$

The interval integral is oriented and uses its total Bochner-integral definition for every choice of endpoints and initial data.

For real $k,z$, define

$$\Gamma_k(z)=2\arctan(\sinh z)+\frac{24}{10000k}z, \qquad C_* = \exp\left(2\pi+\frac{48}{10000}\right).$$

Division at zero has its total real-field meaning.

Write $\mathfrak r(y)=r(y_0,y_2)$. For a time function $\tau:\mathbb R^4\to\mathbb R$ put $m_i^\tau(y)=M_i(y,0,0,\tau(y))$. Let $\tau_\downarrow(y)$ be the infimum of positive times in the normalized maximal domain at which $(\Psi_0(y,t))_v=1$ with negative event rate, and set $F_\downarrow(y)=\eta(\Psi_0(y,\tau_\downarrow(y)))$. The incoming condition means $y_v>1$, or $y_v=1$ and $y_m\geq1$. Define

$$Q_k(u,p;v,dz)=|v_0|+k|v_1|+|v_2|+(k^2u+p/k)|dz|.$$

All vector norms are supremum norms and operator norms are induced by them. 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 $\mathcal H_N=\bigcup_{m<N}(h_m(1-2\beta),h_m(2\beta))$.

A curve $\Gamma=(U,W)$ is admissible at budget $B$ if, for every $r>2$, $U(r),W(r)>0$, $U(r)+W(r)\leq B$, both functions are $C^1$ at $r$, and $|U'(r)|\leq\sqrt{U(r)}$, $|W'(r)|\leq\sqrt{W(r)}$. For an arbitrary graph, define its successor walls by

$$U_m^+=\mathfrak E_{k_m,u_m}^{a,b;\,2/(1-2\beta),\,1/\beta},\qquad W_m^+=\mathfrak E_{k_m,w_m}^{a,b;\,2/(1-2\beta),\,1/\beta},$$

where $\mathfrak E$ is the canonical amplitude extension after composing with the choice inverse of $k_m$, with slope $u_m'/k_m'$ or $w_m'/k_m'$. The one-step property at $(\eta,J,\sigma,B,C)$ consists of the following. Put $\beta=B^\sigma$, $N=\lfloor1/\beta\rfloor_{\mathbb N}$, $K_\beta=[2\beta,1-2\beta]$. One has $0<\beta<1/160$. For every $m<N$, $h_m$ is strictly decreasing on $K_\beta$, and for $y\in K_\beta$,

$$\frac1{m+2}<h_m(y)<\frac1{m+1},\quad \phi_m(2/h_m(y))=y, \quad h_m\text{ is }C^1\text{ at }y,\quad h_m'(y)<0,$$ $$|h_m(y)-\gamma_{m+1}(y)|\leq CB/(m+1)^2,\quad \left|\log\frac{|h_m'(y)|}{|\gamma_{m+1}'(y)|}\right|\leq5C\sqrt B.$$

The successor $(U_m^+,W_m^+)$ is admissible at budget $B^{4/3}$ and, with $r=2/h_m(y)$, satisfies $k_m(r)=2/y$, $U_m^+(2/y)=u_m(r)$, $W_m^+(2/y)=w_m(r)$. The retained branch intervals are pairwise disjoint and $\lambda((0,1)\setminus\mathcal H_N)\leq10B^\sigma$. Define

$$\chi(t)=\max(1-t,0),\qquad P(t)=\int_0^t\chi(u)\,du,\qquad \mathcal R_{s,d,e}(x)=s+edsP(ex/s).$$

The integral is oriented, and division uses the total reciprocal $0^{-1}=0$.

For real functions $f,d_f$ and real endpoints $l,r$, define $F$ and $V$ by

$$F(x)=\begin{cases} \mathcal R_{f(l),d_f(l),-1}(x-l),&x<l,\\ \mathcal R_{f(r),d_f(r),1}(x-r),&x\geq l,\ r<x,\\ f(x),&\text{otherwise}, \end{cases}$$ $$V(x)=\begin{cases} d_f(l)\chi(-(x-l)/f(l)),&x<l,\\ d_f(r)\chi((x-r)/f(r)),&x\geq l,\ r<x,\\ d_f(x),&\text{otherwise}. \end{cases}$$

Write $F_f,V_f$ when comparing several pairs of input functions.

For real functions $U,d_U$, set

$$r_U(x)=\sqrt{U(x)},\qquad b_U(x)=\frac{d_U(x)}{2\sqrt{U(x)}}.$$

Let $F_U,V_U$ be the root-curve construction applied to $(r_U,b_U)$, and set

$$\widetilde U(x)=F_U(x)^2.$$

For another pair $W,d_W$, use the analogous notation.

Let $g=\operatorname{invFunOn}k\,[a,b]$, $A(t)=U(g(t))$, and $V(t)=U'(g(t))/k'(g(t))$. Denote by $\mathfrak E_{k,U}^{a,b;l,r}$ the canonical amplitude extension of $A,V$ from $[l,r]$, and by $\mathfrak E_{k,U}^{a,b}$ the extension from $[k(b),k(a)]$.

The data includes the scalar local-inverse predicate: a $C^1$ inverse at $f(a)$ taking $f(a)$ to $a$, with derivative $1/f'(a)$ and two-sided inverse germs. It also includes the diagonal map $S_k(R,U,P)=(R,kU,P)$, the graph restrictions, the retained phase unions, and the complete one-step property of the era curve.