From survivor blocks to a physical sequence of complete eras

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 $\lfloor X_0/2\rfloor=m+1$.

For a budget cap $b$ define $A_m(b)$ by the conditions on $(B,X)$

$$0<B\le b,\quad m<\lfloor B^{-\sigma}\rfloor,\quad X_1,X_2>0,\quad X_1+X_2\le B,\quad m+1+B^\sigma\le X_0/2\le m+2-B^\sigma.$$

The selected domain $D_m(b)$ additionally requires $2B^\sigma<P_m(X)<1-2B^\sigma$; the budget return is $\mathcal F_m(B,X)=(B^{4/3},N_m(X))$. In a small-budget assertion there exists $0<\delta\le1/40000$ such that for every $0<\eta\le\delta$, $J\in\mathbb N$ with $J\ge20$, $0<\sigma<1$ and $2J\sigma<1$, the stated constants can be chosen.

Let $p$ be a budget and core state surviving all selected returns at cap $b$. Its successive digits are $a_n$, its transported states are $X_n$, its phases are $y_n=2/(X_n)_0$, and $m_n=a_n-1$. Put

$$d_n=d_{m_n}(X_n),\qquad s_n=\sum_{j<n}d_j,\qquad A_0=1,\quad A_{n+1}=L_{m_n}A_n.$$

Frame indices use the chosen composition on the six labels. For restart data $w=((i,s),(B,X))$, let $t_n=s+s_n$ and let $P_n$ denote the frame with index chosen for $A_n Q_i$, where $Q_i$ is the initial frame of label $i$. A restart witness for $x$ requires $s>0$, $s\in I(x,0)$ and a survivor with initial data $(B,X)$, full era statistics, and entry $e_+(X)=\Pi(Q_i\Phi^0_s(x))$. Let $\lambda$ be Lebesgue measure on $\mathbb R$, and let $\lambda_0$ be its restriction to $(0,1)$. The Gauss measure is

$$d\mu_G(x)=\frac{1}{\log2\,(1+x)}\,d\lambda_0(x).$$

Write $w(x)=1/(\log2\,(1+x))$ and $P(x)=\log(1+x)/\log2$ for $x\geq0$.

Let $G(x)=1/x-\lfloor1/x\rfloor$ for $x\ne0$, with $G(0)=0$. For a finite word $w=(a_0,\ldots,a_{k-1})$ of natural numbers, set

$$C_w=(g_{a_0}\circ\cdots\circ g_{a_{k-1}})((0,1)),\qquad g_a(t)=1/(a+t),$$

where the empty composition is the identity and $0^{-1}=0$. Define

$$f_w=\mathbf1_{C_w},\qquad m_w=\mu_G(C_w),\qquad \mathcal B_G=\left\{x\in\mathbb R:\ \forall w,\ \frac1n\sum_{j=0}^{n-1}f_w(G^jx)\longrightarrow m_w\right\}.$$

For a positive-digit sequence $a$, write $\pi(a)$ for its continued-fraction code, $\sigma^j a$ for its tail beginning at $j$, and $a|_k$ for its first $k$ digits.

Define the natural-valued digit, tail observables, and truncations by

$$D(x)=\lfloor1/x\rfloor_{\mathbb N},\qquad H_k(x)=\mathbf1_{(0,1/k)}(x)\quad(k\geq1),\qquad D_R(x)=\min(D(x),R)\quad(R\in\mathbb N).$$

Here the natural floor is zero for negative arguments, and $0^{-1}=0$. Write

$$p_k=\frac{\log((k+1)^2/(k(k+2)))}{\log2},\qquad q_k=\frac{\log(1+1/k)}{\log2},\qquad t_R=\frac{\log(R+1)}{\log2}.$$

For a real sequence $u$, define

$$A_n(u)=\frac1n\sum_{j=0}^{n-1}u_j\quad(n\geq1),\qquad A_0(u)=0.$$

For a positive-digit sequence $a$ and a real sequence $y$, let $\mathcal E(a,y)$ mean all of the following statements:

$$A_N(f\circ y)\to\int f\,d\mu_G \quad\text{for every }f:\mathbb R\to\mathbb R \text{ continuous on }[0,1],$$ $$A_N(j\mapsto\mathbf1_{\{(\sigma^ja)|_{|w|}=w\}})\to m_w \quad\text{for every finite positive-digit word }w,$$ $$A_N(j\mapsto\mathbf1_{\{a_j=k\}})\to p_k,\qquad A_N(j\mapsto\mathbf1_{\{k\leq a_j\}})\to q_k\quad(k\geq1),$$ $$A_N(j\mapsto\min(a_j,R))\to t_R\quad(R\in\mathbb N),\qquad A_N(j\mapsto a_j)\to+\infty.$$

Let $w$ be restart data, $p$ a survivor and $x$ a constraint-one state. Suppose the survivor digits and phases satisfy $\mathcal E$, every $\mathcal I_{m_n}(X_n)$ holds with $e_+(X_{n+1})=S_{m_n}(X_n)$, the $t_n$ increase strictly to $+\infty$, and for all $n$ one has $t_n>0$ and $\Xi(e_+(X_n))=P_n\Phi^0_{t_n}(x)$. Then

$$(a_{n+1},X_{n+1},t_{n+1},P_{n+1})_{n\ge0}$$

is a physical era sequence on the original trajectory: it retains the statistics, entrance identities and event blocks, the next state, time and frame agree with the actual return, and its first entrance is the terminal arrival of an earlier complete block.

import Definitions.Def_BianchiIX_EraCodingData
import Definitions.Def_BianchiIX_PhysicalEraData
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.Normed.Module.FiniteDimension
import Mathlib.LinearAlgebra.Determinant
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.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.Data.Real.Basic
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
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.Data.Matrix.Mul
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Analysis.Calculus.ImplicitContDiff
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.Topology.Instances.Real.Lemmas
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.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.Topology.Order.Compact
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Tactic.Positivity
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.Normed.Operator.Basic
import Mathlib.LinearAlgebra.Matrix.Notation
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Data.List.Induction
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Data.PNat.Basic
import Mathlib.Data.Rat.Floor
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Order.OrderIsoNat
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.JacobianOneDim
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.Dynamics.BirkhoffSum.Average
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Topology.UniformSpace.HeineCantor
import Mathlib.Dynamics.BirkhoffSum.NormedSpace
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Analysis.Normed.Group.Tannery
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.Order.MonotoneConvergence
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics

noncomputable section
open BianchiIX
open Set Filter
open scoped Topology
open BianchiIX.EraSurvivor
variable {η σ Bcap : ℝ} {J : ℕ}

theorem BianchiIX.EraSurvivor.physical_era_sequence {x : NormalizedState} (w : CoreRestartData)
    (p : eraSurvivorDomain η J σ Bcap) (hc : stateConstraint x = 1)
    (Hstat : EraStatistics (digits p) (phase p))
    (Hblocks : ∀ n, EraEventItinerary η J (digits p n).natPred (state p n) ∧
      coreRadiusEntry (state p (n + 1)) = terminalNaturalState η J (eraCoreExit (digits p n).natPred (state p n)))
    (Hmono : StrictMono (physicalTime w p))
    (Htop : Tendsto (physicalTime w p) atTop atTop)
    (Hactual : ∀ n, 0 < physicalTime w p n ∧
      fromProjectiveState (coreRadiusEntry (state p n)) =
        (axisFrame (physicalFrame w p n)).map (normalizedMaxCurve x 0 (physicalTime w p n))) :
    PhysicalEraSequence η J x (fun n => digits p (n + 1)) (fun n => state p (n + 1))
      (fun n => physicalTime w p (n + 1)) (fun n => physicalFrame w p (n + 1)) := by sorry