Research-Stack/6-Documentation/docs/specs/KernelEquationSheet.tex

\documentclass[11pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{geometry}
\geometry{margin=2.5cm}

\title{Revised Kernel Equation Sheet}
\author{Sovereign Stack}
\date{\today}

\begin{document}

\maketitle

\appendix
\section{Revised Kernel Equation Sheet}

\subsection{Symbols}

Let the global system state at time $t$ be
\begin{equation}
\mathcal{K}_t = \bigl(G_t,\mathcal{L}_t,\mathcal{T}_t,\mathcal{I}_t,\mathcal{R}_t,\mathcal{A}_t,\mathcal{S}_t\bigr)
\label{eq:global_state}
\end{equation}
where $G_t$ is the weighted N-DAG, $\mathcal{L}_t$ the set of explicit lanes, $\mathcal{T}_t$ the set of throats, $\mathcal{I}_t$ the set of throat islands, $\mathcal{R}_t$ the hierarchical routing objects, $\mathcal{A}_t$ the AVMR summaries/log, and $\mathcal{S}_t$ the canal sections.

\subsection{DIAT integer geometry}

For $n\in\mathbb{N}$ define
\begin{equation}
k = \lfloor \sqrt{n} \rfloor
\label{eq:k_floor}
\end{equation}
\begin{equation}
\underline{s} = k^2,
\qquad
\overline{s} = (k+1)^2
\label{eq:squares}
\end{equation}
\begin{equation}
a(n)=n-\underline{s},
\qquad
b(n)=\overline{s}-n
\label{eq:a_b}
\end{equation}
so that
\begin{equation}
a(n)+b(n)=2k+1.
\label{eq:a_plus_b}
\end{equation}

The DIAT encoding is
\begin{equation}
\mathrm{DIAT}(n)=\bigl(a,b,ab,a-b\bigr)
\label{eq:diat}
\end{equation}
with shell index
\begin{equation}
\sigma(n)=k
\label{eq:shell_index}
\end{equation}
and shell partition
\begin{equation}
\mathcal{S}_k=\{n\mid k^2\le n<(k+1)^2\}.
\label{eq:shell_partition}
\end{equation}

Optional normalized form:
\begin{equation}
\widehat{\mathrm{DIAT}}(n)=
\left(
\frac{a}{2k+1},
\frac{b}{2k+1},
\frac{ab}{(2k+1)^2},
\frac{a-b}{2k+1}
\right).
\label{eq:diat_normalized}
\end{equation}

\subsection{Event lifting and lane state}

Each encoded event is lifted to
\begin{equation}
z_i=
\bigl(
d_i,\phi_i,\tau_i,\iota_i,\chi_i,\omega_i,m_i,e_i,r_i,u_i
\bigr)
\label{eq:event_lift}
\end{equation}
where $d_i=\mathrm{DIAT}(n_i)$.

A lane state is
\begin{equation}
\ell=
\bigl(
vtx,p,q,\phi,s,P,\lambda_{\mathrm{eff}},K,E,m,r,\eta
\bigr).
\label{eq:lane_state}
\end{equation}

\subsection{AVMR aggregation}

Given leaves $z_1,\dots,z_q$, define the AVMR merge
\begin{equation}
R=\bigoplus_{\mathrm{avmr}}(z_1,\dots,z_q).
\label{eq:avmr_merge}
\end{equation}

Aggregate fields:
\begin{equation}
c_R=\sum_{i=1}^{q}1,
\qquad
m_R=\sum_{i=1}^{q}m_i,
\qquad
E_R=\sum_{i=1}^{q}e_i,
\qquad
\iota_R=\sum_{i=1}^{q}\iota_i
\label{eq:avmr_basic_aggregates}
\end{equation}
\begin{equation}
\omega_R=\min_i \omega_i,
\qquad
m_{R,\max}=\max_i \|m_i\|.
\label{eq:avmr_extrema}
\end{equation}

Phase is merged by phasor accumulation:
\begin{equation}
z_{\phi}=\sum_{i=1}^{q}w_i
\begin{pmatrix}
\cos\phi_i\\
\sin\phi_i
\end{pmatrix}
\label{eq:phasor_sum}
\end{equation}
\begin{equation}
\phi_R=\operatorname{atan2}(z_{\phi,y},z_{\phi,x}),
\qquad
\kappa_R=\|z_\phi\|.
\label{eq:phase_merge}
\end{equation}

Associativity requirement:
\begin{equation}
(A\oplus B)\oplus C=A\oplus(B\oplus C).
\label{eq:avmr_assoc}
\end{equation}

\subsection{Dynamic Canal law}

Pressure evolves as
\begin{equation}
P_{t+1}=\gamma P_t+\mathrm{stress}_t
\label{eq:pressure_update}
\end{equation}
with optional external coupling
\begin{equation}
P_{t+1}^{\mathrm{cal}}=P_{t+1}+w_{\mathrm{bio}}\Pi_{\mathrm{bio},t}.
\label{eq:pressure_cal}
\end{equation}

Effective canal resistance:
\begin{equation}
\lambda_{\mathrm{eff}}(P)=
\lambda_0\left[\sigma+(1-\sigma)e^{-\xi P}\right].
\label{eq:lambda_eff}
\end{equation}
Bounds:
\begin{equation}
\lambda_0\sigma\le\lambda_{\mathrm{eff}}(P)\le\lambda_0
\label{eq:lambda_bounds}
\end{equation}
Derivative:
\begin{equation}
\frac{d\lambda_{\mathrm{eff}}}{dP}
=
-\lambda_0(1-\sigma)\xi e^{-\xi P}<0.
\label{eq:lambda_derivative}
\end{equation}

Compliance:
\begin{equation}
K(P)=\frac{1}{\lambda_{\mathrm{eff}}(P)+\varepsilon}
\label{eq:compliance}
\end{equation}
and canal width:
\begin{equation}
W_c(P)=W_{c,0}\frac{\lambda_0}{\lambda_{\mathrm{eff}}(P)}.
\label{eq:canal_width}
\end{equation}

\subsection{Stress law}

\begin{equation}
\mathrm{stress}_t=
\alpha\,\mathrm{surprise}_t+\beta\,\mathrm{regret}_t
\label{eq:stress}
\end{equation}
with
\begin{equation}
\mathrm{surprise}_t=-\log p_{\mathrm{actual},t}
\label{eq:surprise}
\end{equation}
\begin{equation}
\mathrm{regret}_t=
\max\bigl(0,\log p_{\mathrm{best},t}-\log p_{\mathrm{actual},t}\bigr).
\label{eq:regret}
\end{equation}

For graph-based continuation:
\begin{equation}
p_e=
\frac{\exp(\kappa\,\mathrm{score}_e)}
{\sum_{e'\in\mathrm{Out}(v)}\exp(\kappa\,\mathrm{score}_{e'})}
\label{eq:softmax_edge}
\end{equation}
and
\begin{equation}
p_{\mathrm{actual}}=p_{e^\star},
\qquad
p_{\mathrm{best}}=\max_e p_e.
\label{eq:p_actual_best}
\end{equation}

\subsection{N-DAG and Laplacian}

\begin{equation}
G_t=(V,E,W_t)
\label{eq:graph}
\end{equation}
\begin{equation}
W_{ij}(t)=\sum_{e:i\to j}w_e(t)
\label{eq:adjacency}
\end{equation}
\begin{equation}
D_{ii}(t)=\sum_j W_{ij}(t)
\label{eq:degree}
\end{equation}
\begin{equation}
L(t)=D(t)-W(t).
\label{eq:laplacian}
\end{equation}

Symmetrized form:
\begin{equation}
W^{(s)}(t)=\frac{W(t)+W(t)^\top}{2},
\qquad
L^{(s)}(t)=D^{(s)}(t)-W^{(s)}(t).
\label{eq:sym_laplacian}
\end{equation}

\subsection{Edge scoring}

\begin{equation}
\mathrm{score}_e(\ell,t)=
\alpha_w w_e(t)
-\alpha_d d_e(p)
-\alpha_\phi |\operatorname{wrap}(\phi-\phi_e^\star)|
-\lambda_{\mathrm{eff}}(P_\ell)\,\mathrm{stressProxy}_e
-\alpha_m m_\ell
+\alpha_E \Delta E_e
\label{eq:edge_score}
\end{equation}
with
\begin{equation}
\mathrm{stressProxy}_e=
c_\tau\tau_e+c_\delta\|\delta_e(t)\|+c_q\|q_\perp\|.
\label{eq:stress_proxy}
\end{equation}
The selected edge is
\begin{equation}
e^\star=\arg\max_{e\in\mathrm{Out}(vtx)}\mathrm{score}_e(\ell,t).
\label{eq:edge_choice}
\end{equation}

\subsection{Regime law}

\begin{equation}
r_{t+1}=
\begin{cases}
C, & m_{t+1}\le\theta_C \land s_{t+1}\le s_C\\
S, & \theta_C<m_{t+1}<\theta_T \text{ or } s_{t+1}>s_C\\
T, & m_{t+1}\ge\theta_T \land \text{throat admissible}\\
S, & m_{t+1}\ge\theta_T \land \text{no throat admissible}
\end{cases}
\label{eq:regime_update}
\end{equation}

\subsection{Lane dynamics}

\paragraph{Coherent regime}
\begin{align}
p_{t+1}&=p_t+q_t+\Pi_e+F_{\mathrm{island}}(p_t)
\label{eq:coh_pos}\\
q_{t+1}&=q_t+\beta_x\Delta x_e-\beta_f\nabla U(v)+F_{\mathrm{island}}(p_t)
\label{eq:coh_vel}\\
\phi_{t+1}&=\phi_t+\Delta\phi_e
\label{eq:coh_phase}\\
s_{t+1}&=\max\bigl(0,s_t+\beta_\tau\tau_e-\beta_r K(P_t)\bigr)
\label{eq:coh_stress}\\
E_{t+1}&=E_t+\Delta E_e-\beta_\gamma\gamma_e
\label{eq:coh_energy}\\
m_{t+1}&=\max\bigl(0,m_t+\beta_\mu\|\delta_e\|-\beta_h h_t\bigr).
\label{eq:coh_mismatch}
\end{align}

\paragraph{Stressed regime}
\begin{align}
p_{t+1}&=p_t+q_t+\Pi_e+\Xi_t+F_{\mathrm{island}}(p_t)
\label{eq:str_pos}\\
q_{t+1}&=\rho_q q_t+\beta_x\Delta x_e-\beta_f\nabla U(v)-\beta_\tau T_e(q_t)+F_{\mathrm{island}}(p_t)
\label{eq:str_vel}\\
\phi_{t+1}&=\phi_t+\Delta\phi_e+\epsilon_\phi
\label{eq:str_phase}\\
s_{t+1}&=s_t+\beta_\tau\tau_e+\beta_m m_t-\beta_r K(P_t)
\label{eq:str_stress}\\
E_{t+1}&=E_t+\Delta E_e-\beta_\gamma\gamma_e-\beta_s s_t
\label{eq:str_energy}\\
m_{t+1}&=m_t+\beta_\mu\|\delta_e\|+\beta_x\|\Xi_t\|-\beta_h h_t.
\label{eq:str_mismatch}
\end{align}

\paragraph{Throat regime}
\begin{equation}
\widetilde{x}=A_e(x_t),\qquad \widehat{x}=B_e(t)\widetilde{x},\qquad x_{t+1}=H_e(\widehat{x})
\label{eq:throat_transfer}
\end{equation}
\begin{align}
p_{t+1}&=H_p(B_p(A_p(p_t)))
\label{eq:thr_pos}\\
q_{t+1}&=\rho_T q_t+\xi_q
\label{eq:thr_vel}\\
\phi_{t+1}&=\phi_t+\Delta\phi_e+\epsilon_T
\label{eq:thr_phase}\\
s_{t+1}&=s_t+\beta_{T,s}\|\delta_t\|
\label{eq:thr_stress}\\
E_{t+1}&=\rho_E E_t-\Lambda_T(\delta_t)-\Lambda_P(P_t)
\label{eq:thr_energy}\\
m_{t+1}&=\rho_m m_t+\beta_{T,m}\|\delta_t\|-\beta_h h_t
\label{eq:thr_mismatch}
\end{align}
with
\begin{equation}
\Lambda_P(P_t)=c_P\bigl(K(P_t)-K(0)\bigr).
\label{eq:pressure_penalty}
\end{equation}

\subsection{Dual geometry and throat generation}

Intrinsic merge:
\begin{equation}
R_{ij}^{\mathrm{geom}}=R_i\oplus R_j
\label{eq:intrinsic_merge}
\end{equation}
Projection embedding:
\begin{equation}
h_i=H(R_i)
\label{eq:hash_embed}
\end{equation}
Projection composition:
\begin{equation}
v_{ij}^{\mathrm{proj}}=\Psi(h_i,h_j),
\qquad
\Psi(h_i,h_j)=H(h_i\|h_j)
\label{eq:projection_comp}
\end{equation}
Projection decode:
\begin{equation}
\widetilde{R}_{ij}^{\mathrm{proj}}=D(v_{ij}^{\mathrm{proj}})
\label{eq:projection_decode}
\end{equation}
Mismatch:
\begin{equation}
\delta_{ij}=
\mathrm{Dist}\bigl(R_{ij}^{\mathrm{geom}},\widetilde{R}_{ij}^{\mathrm{proj}}\bigr).
\label{eq:projection_mismatch}
\end{equation}

A throat is generated if
\begin{equation}
\delta_{ij}\ge \theta_{\mathrm{throat-gen}}.
\label{eq:throat_gen}
\end{equation}

Dynamic throat weight:
\begin{equation}
w_{ij}(t)=
\max\Bigl(
0,\,
w_{0,ij}
-\lambda_\delta \delta_{ij}(t)
+\lambda_P(K(P_t)-K(0))
-\lambda_s s_t
\Bigr).
\label{eq:throat_weight}
\end{equation}

\subsection{Throat superposition}

\begin{equation}
\mathrm{Interf}(T_1,T_2)=\mathrm{Dist}_{hash}(v_1,v_2)
\label{eq:interference}
\end{equation}
\begin{equation}
v_\oplus=\mathrm{Fuse}(v_1,v_2)
\label{eq:fused_vec}
\end{equation}
\begin{equation}
\delta_\oplus=\delta_1+\delta_2+\eta\,\mathrm{Interf}(T_1,T_2)
\label{eq:superposed_mismatch}
\end{equation}
\begin{equation}
w_\oplus=\max(0,w_1+w_2-\lambda_\delta\delta_\oplus)
\label{eq:superposed_weight}
\end{equation}
\begin{equation}
\kappa_\oplus=\delta_\oplus.
\label{eq:superposed_curvature}
\end{equation}

\subsection{Throat islands}

For a cluster $\mathcal{T}=\{T_1,\dots,T_k\}$,
\begin{equation}
I(T_i,T_j)=\mathrm{Dist}_{hash}(v_i,v_j)
\label{eq:pairwise_interference}
\end{equation}
\begin{equation}
\mathrm{Coh}(\mathcal{T})=
\frac{1}{k^2}\sum_{i,j}\exp\bigl(-\alpha I(T_i,T_j)\bigr)
\label{eq:island_coherence}
\end{equation}
\begin{equation}
W_{\mathcal{T}}=
\sum_i w_i-\lambda_{\mathrm{int}}\sum_{i<j}I(T_i,T_j)
\label{eq:island_weight}
\end{equation}
Island criterion:
\begin{equation}
\mathbf{1}_{\mathrm{island}}(\mathcal{T})=
\mathbf{1}\left[
\mathrm{Coh}(\mathcal{T})>\theta_{\mathrm{coh}}
\land
W_{\mathcal{T}}>\theta_{\mathrm{stable}}
\right].
\label{eq:island_indicator}
\end{equation}

\subsection{Island potential}

\begin{equation}
U_{\mathcal{I}}(x)=
-\beta_{\mathcal{I}}W_{\mathcal{I}}
\exp\bigl(-\gamma_{\mathcal{I}}\mathrm{dist}(x,c_{\mathcal{I}})\bigr)
\label{eq:island_potential}
\end{equation}
\begin{equation}
F_{\mathcal{I}}(x)=-\nabla U_{\mathcal{I}}(x)
\label{eq:island_force}
\end{equation}
\begin{equation}
F_{\mathrm{island}}(x)=\sum_{\mathcal{I}}F_{\mathcal{I}}(x).
\label{eq:total_island_force}
\end{equation}

\subsection{Canal section equations}

\begin{equation}
\rho_i^{t+1}
=
\rho_i^t-(F_{i+1/2}^t-F_{i-1/2}^t)-S_i^t+I_i^t
\label{eq:density}
\end{equation}
\begin{equation}
P_i^{t+1}=\gamma P_i^t+\bar{\mathrm{stress}}_i^t
\label{eq:section_pressure}
\end{equation}
\begin{equation}
\lambda_{\mathrm{eff},i}^{t+1}
=
\lambda_0\left[\sigma+(1-\sigma)e^{-\xi P_i^{t+1}}\right]
\label{eq:section_lambda}
\end{equation}
\begin{equation}
K_i^{t+1}=\frac{1}{\lambda_{\mathrm{eff},i}^{t+1}+\varepsilon}
\label{eq:section_compliance}
\end{equation}
\begin{equation}
W_{c,i}^{t+1}=W_{c,0,i}\frac{\lambda_0}{\lambda_{\mathrm{eff},i}^{t+1}}
\label{eq:section_width}
\end{equation}
\begin{equation}
C_i^{t+1}=
C_{0,i}\frac{\lambda_0}{\lambda_{\mathrm{eff},i}^{t+1}}
e^{-a_R R_i^t}e^{-a_m\bar m_i^t}
\label{eq:section_capacity_exp}
\end{equation}
or linearized
\begin{equation}
C_i^{t+1}=
\operatorname{sat}(C_{0,i}+c_P P_i^{t+1}-c_R R_i^t-c_m\bar m_i^t)
\label{eq:section_capacity_lin}
\end{equation}
\begin{equation}
v_{\mathrm{eff},i}^{t+1}
=
v_{0,i}+g_GG_i-g_D\rho_i-g_RR_i-g_m\bar m_i+g_KK_i
\label{eq:veff}
\end{equation}
\begin{equation}
F_i^{t+1}=\rho_i^{t+1}v_{\mathrm{eff},i}^{t+1}
\label{eq:flux}
\end{equation}
\begin{equation}
S_i^{t+1}=\sum_{e\in\mathcal{T}(i)}\pi_{e,i}^{t+1}F_i^{t+1}
\label{eq:siphon}
\end{equation}
with
\begin{equation}
\pi_{e,i}^{t+1}\propto \mathrm{throatBias}_e\,g(\delta_e)\,K_i.
\label{eq:siphon_fraction}
\end{equation}

\subsection{Pack/unpack rule}

\begin{equation}
\Theta_i=
\theta_\rho \rho_i+\theta_m\bar m_i+\theta_s\bar s_i+\theta_I I_i+\theta_{PT}K_iT_i
\label{eq:unpack_score}
\end{equation}
\begin{equation}
U_i=\mathbf{1}[\Theta_i\ge \Theta_{\mathrm{crit}}].
\label{eq:mode_indicator}
\end{equation}

\subsection{Hierarchical routing}

For each level $k\in[n_{\min},n_{\max}]$ define
\begin{equation}
X_i^{(k)}=
\bigl(
E_i^{(k)},P_i^{(k)},M_i^{(k)},\kappa_i^{(k)},C_i^{(k)},W_i^{(k)},H_i^{(k)}
\bigr).
\label{eq:route_obj}
\end{equation}

Pressure law:
\begin{equation}
P_i^{(k)}(t+1)=
\gamma_k P_i^{(k)}(t)
+\alpha_k \mathrm{stress}_i^{(k)}(t)
+\beta_k \mathrm{cong}_i^{(k)}(t)
+\chi_k \mathrm{interf}_i^{(k)}(t)
\label{eq:level_pressure}
\end{equation}
\begin{equation}
\lambda_i^{(k)}(t+1)=
\lambda_{0,k}\left[\sigma_k+(1-\sigma_k)e^{-\xi_k P_i^{(k)}(t+1)}\right]
\label{eq:level_lambda}
\end{equation}
\begin{equation}
K_i^{(k)}(t+1)=\frac{1}{\lambda_i^{(k)}(t+1)+\varepsilon}
\label{eq:level_compliance}
\end{equation}

Capacity:
\begin{equation}
C_i^{(k)}(t+1)=
C_{0,k}\mu_C^k
\frac{1+\beta_P K_i^{(k)}}{1+\beta_M M_i^{(k)}}
(1+\beta_H H_i^{(k)})
\label{eq:level_capacity}
\end{equation}

Lateral flow:
\begin{equation}
F_{ij}^{(k)}(t)=
w_{ij}^{(k)}(t)\min(C_i^{(k)},C_j^{(k)})
\frac{E_i^{(k)}(t)}{\sum_m w_{im}^{(k)}(t)+\varepsilon}
\label{eq:level_flow}
\end{equation}

Loss:
\begin{equation}
\Lambda_i^{(k)}(t)=
\lambda_k^{\mathrm{base}}+a_kM_i^{(k)}+b_k\kappa_i^{(k)}-c_kH_i^{(k)}
\label{eq:level_loss}
\end{equation}

Energy:
\begin{equation}
E_i^{(k)}(t+1)=
E_i^{(k)}(t)
-\sum_j F_{ij}^{(k)}(t)
+\sum_j F_{ji}^{(k)}(t)
-\Lambda_i^{(k)}(t)
+\Pi_i^{(k)}(t)
-\Delta_i^{(k)}(t).
\label{eq:level_energy}
\end{equation}

Promotion:
\begin{equation}
E_i^{(k)} \ge E_k^\uparrow,\qquad
H_i^{(k)}\ge \theta_k^\uparrow,\qquad
k+1\le n_{\max}
\label{eq:promotion_cond}
\end{equation}
\begin{equation}
\Pi_i^{(k+1)}=
\eta_k^\uparrow E_i^{(k)}-a_kM_i^{(k)}-b_k\kappa_i^{(k)}
\label{eq:promotion_energy}
\end{equation}

Demotion:
\begin{equation}
H_i^{(k)}<\theta_k^\downarrow
\;\lor\;
M_i^{(k)}>M_k^{\max}
\label{eq:demotion_cond}
\end{equation}
\begin{equation}
\Delta_i^{(k)}=
(1-\eta_k^\downarrow)E_i^{(k)}+d_kM_i^{(k)}.
\label{eq:demotion_energy}
\end{equation}

\subsection{Spectral quantities}

\begin{equation}
R_{\mathrm{eff}}(i,j)=(e_i-e_j)^\top L^+(e_i-e_j)
\label{eq:effective_resistance}
\end{equation}
\begin{equation}
L\psi_k=\lambda_k\psi_k
\label{eq:eigenmodes}
\end{equation}
\begin{equation}
I_R(t)=
\frac{\sum_{k\in K_{\mathrm{hi}}}\|\psi_k|_R\|^2}
{\sum_{k\in K_{\mathrm{lo}}}\|\psi_k|_R\|^2+\varepsilon}.
\label{eq:modal_instability}
\end{equation}

\subsection{Poincar\'e return map}

\begin{equation}
Y_k(s)=
(\rho_k,C_k,F_k,S_k,\bar E_k,\bar m_k,\bar s_k,P_k,\lambda_k,U_k)
\label{eq:poincare_state}
\end{equation}
\begin{equation}
Y_{k+1}=P_\Sigma(Y_k)
\label{eq:poincare_map}
\end{equation}
with components
\begin{align}
P_{k+1}&=\gamma P_k+\bar{\mathrm{stress}}_k
\label{eq:pmap_pressure}\\
\lambda_{k+1}&=\lambda_0\left[\sigma+(1-\sigma)e^{-\xi P_{k+1}}\right]
\label{eq:pmap_lambda}\\
C_{k+1}&=\operatorname{sat}(C_0+c_P P_{k+1}-c_RR_k-c_m\bar m_k)
\label{eq:pmap_capacity}\\
F_{k+1}&=\rho_{k+1}\operatorname{sat}(v_0+g_GG_k-g_PP_k-g_RR_k-g_m\bar m_k+g_KK_{k+1})
\label{eq:pmap_flux}\\
S_{k+1}&=\sum_{e\in\mathcal{T}}\pi_e(P_{k+1},\delta_{e,k+1})F_{k+1}
\label{eq:pmap_siphon}\\
U_{k+1}&=\mathbf{1}[\Theta_{k+1}\ge \Theta_{\mathrm{crit}}].
\label{eq:pmap_unpack}
\end{align}

\subsection{Unified kernel map}

\begin{equation}
\mathcal{K}_{t+1}=\mathrm{stepKernel}(\mathcal{K}_t)
\label{eq:kernel_map}
\end{equation}
with substeps
\begin{align}
\mathcal{S}_{t+1}&=\mathrm{stepSection}(\mathcal{S}_t)
\label{eq:kernel_sections}\\
B_i&=\mathrm{sectionBundleState}(i)
\label{eq:kernel_bundle}\\
\mathcal{T}_t&=\mathrm{synthesizeAllThroats}(B_1,\dots,B_n)
\label{eq:kernel_throats}\\
\mathcal{T}_t^\oplus&=\mathrm{superposeOverlappingThroats}(\mathcal{T}_t)
\label{eq:kernel_superposed}\\
\mathcal{I}_t&=\mathrm{buildIslands}(\mathcal{T}_t^\oplus)
\label{eq:kernel_islands}\\
G_t'&=G_t\cup\mathcal{T}_t^\oplus
\label{eq:kernel_injected}\\
\mathcal{L}_{t+1}&=\mathrm{stepLaneWithIslands}(\mathcal{L}_t,G_t',\mathcal{I}_t)
\label{eq:kernel_lanes}\\
\mathcal{R}_{t+1}&=\mathrm{iteratePromotions}(\mathcal{L}_{t+1},\mathcal{T}_t^\oplus,\mathcal{I}_t)
\label{eq:kernel_routes}\\
\mathcal{A}_{t+1}&=\mathcal{A}_t\cup\mathrm{emitLeaves}(\mathcal{L}_{t+1}).
\label{eq:kernel_avmr}
\end{align}

\subsection{Master coupled system}

\begin{equation}
\boxed{
\begin{aligned}
&\mathrm{DIAT}(n)=(a,b,ab,a-b)\\
&R=\bigoplus_{\mathrm{avmr}}z_i\\
&P_{t+1}=\gamma P_t+\mathrm{stress}_t\\
&\lambda_{\mathrm{eff}}(P)=\lambda_0[\sigma+(1-\sigma)e^{-\xi P}]\\
&K(P)=\frac{1}{\lambda_{\mathrm{eff}}(P)+\varepsilon}\\
&\ell_{t+1}=\mathcal{T}_{r_t}(\ell_t,G_t,\delta_t,F_{\mathrm{island}})\\
&\delta_{ij}=\mathrm{Dist}(R_i\oplus R_j,D(\Psi(H(R_i),H(R_j))))\\
&w_{ij}(t)=\max(0,w_0-\lambda_\delta\delta_{ij}+\lambda_P(K-K_0)-\lambda_s s_t)\\
&\mathrm{Coh}(\mathcal{T})=\frac{1}{k^2}\sum_{i,j}\exp(-\alpha I(T_i,T_j))\\
&E_i^{(k)}(t+1)=E_i^{(k)}-\sum_jF_{ij}^{(k)}+\sum_jF_{ji}^{(k)}-\Lambda_i^{(k)}+\Pi_i^{(k)}-\Delta_i^{(k)}\\
&\mathcal{K}_{t+1}=\mathrm{stepKernel}(\mathcal{K}_t)
\end{aligned}
}
\label{eq:master_system}
\end{equation}

\end{document}