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% Section 3: Nomenclature (Symbols and Variables)
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\section{Nomenclature: Symbols and Variables}\label{sec:nomenclature}

To ensure clarity and reproducibility, this section summarizes the state variables, exogenous inputs, model outputs, derived quantities, and the parameter set used throughout the paper. All symbols are consistent with the mechanistic continuous-time framework defined in Sections~\ref{sec:model}--\ref{sec:numerics}.

\subsection{State Vector $\mathbf{x}(t)$}\label{subsec:state}
We define the state vector
\begin{equation}
\mathbf{x}(t)=\big[z(t),\,v_p(t),\,T_b(t),\,S(t),\,w(t)\big]^\top.
\end{equation}
\begin{table}[h!]
\centering
\caption{State variables in $\mathbf{x}(t)$.}\label{tab:states}
\begin{tabular}{lll}
\hline
Symbol & Meaning & Typical range / unit \\
\hline
$z(t)$ & State of charge (SOC) & $[0,1]$ \\
$v_p(t)$ & Polarization voltage (RC memory state) & V \\
$T_b(t)$ & Battery (cell) temperature & K \\
$S(t)$ & State of health (SOH), capacity fraction & $[0,1]$ \\
$w(t)$ & Radio tail state (continuous tail activity) & $[0,1]$ \\
\hline
\end{tabular}
\end{table}

\subsection{Input Vector $\mathbf{u}(t)$}\label{subsec:input}
The exogenous usage/environment inputs are
\begin{equation}
\mathbf{u}(t)=\big[L(t),\,C(t),\,N(t),\,\Psi(t),\,T_a(t)\big]^\top.
\end{equation}
\begin{table}[h!]
\centering
\caption{Inputs in $\mathbf{u}(t)$.}\label{tab:inputs}
\begin{tabular}{lll}
\hline
Symbol & Meaning & Typical range / unit \\
\hline
$L(t)$ & Screen brightness (normalized) & $[0,1]$ \\
$C(t)$ & CPU load (normalized) & $[0,1]$ \\
$N(t)$ & Network activity / throughput proxy (normalized) & $[0,1]$ \\
$\Psi(t)$ & Signal quality (larger is better) & dimensionless or normalized \\
$T_a(t)$ & Ambient temperature & K \\
\hline
\end{tabular}
\end{table}

\subsection{Outputs and Derived Quantities}\label{subsec:outputs-derived}
The primary outputs are the terminal voltage $V_{\mathrm{term}}(t)$, the SOC $z(t)$, and the time-to-empty (TTE). In addition, several derived quantities are used to couple the load-side power demand to the battery-side electro-thermal dynamics.

\paragraph{(i) Total power demand.}
The total power consumption is decomposed into background, screen, CPU, and networking components:
\begin{equation}
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L(t))+P_{\mathrm{cpu}}(C(t))+P_{\mathrm{net}}(N(t),\Psi(t),w(t)),
\label{eq:Ptot_def}
\end{equation}
where
\begin{align}
P_{\mathrm{scr}}(L)&=P_{\mathrm{scr},0}+k_L L^\gamma,\qquad \gamma>1, \label{eq:Pscr_def}\\
P_{\mathrm{cpu}}(C)&=P_{\mathrm{cpu},0}+k_C C^\eta,\qquad \eta>1, \label{eq:Pcpu_def}\\
P_{\mathrm{net}}(N,\Psi,w)&=P_{\mathrm{net},0}+k_N\frac{N}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w,\qquad \kappa>0. \label{eq:Pnet_def}
\end{align}

\paragraph{(ii) Terminal voltage.}
The battery terminal voltage is given by the first-order equivalent circuit model (ECM):
\begin{equation}
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z(t)) - v_p(t) - I(t)\,R_0(T_b(t),S(t)).
\label{eq:Vterm_def}
\end{equation}

\paragraph{(iii) CPL feasibility discriminant.}
Under the constant-power-load (CPL) closure $P_{\mathrm{tot}}=V_{\mathrm{term}}I$, the algebraic current solve yields the discriminant
\begin{equation}
\Delta(t)=\big(V_{\mathrm{oc}}(z(t))-v_p(t)\big)^2-4\,R_0(T_b(t),S(t))\,P_{\mathrm{tot}}(t).
\label{eq:Delta_def}
\end{equation}
The CPL current is real-valued only if $\Delta(t)\ge 0$. When $\Delta(t)<0$, the requested power is infeasible given the instantaneous electrochemical state, which corresponds to a voltage-collapse risk event in the model.

\paragraph{(iv) Time-to-empty (TTE).}
The operational end time is defined by the earliest occurrence among voltage cutoff and SOC depletion:
\begin{equation}
\mathrm{TTE}=\inf\left\{t>0:\;V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le 0\right\}.
\label{eq:TTE_def}
\end{equation}

\begin{table}[h!]
\centering
\caption{Key outputs and derived quantities.}\label{tab:derived}
\begin{tabular}{lll}
\hline
Symbol & Meaning & Unit \\
\hline
$V_{\mathrm{term}}(t)$ & Terminal voltage & V \\
$I(t)$ & Battery current (discharge positive) & A \\
$P_{\mathrm{tot}}(t)$ & Total power demand & W \\
$V_{\mathrm{oc}}(z)$ & Open-circuit voltage (OCV) & V \\
$\Delta(t)$ & CPL feasibility discriminant & V$^2$ \\
$\mathrm{TTE}$ & Time-to-empty (operation end time) & s (or min, h) \\
$V_{\mathrm{cut}}$ & Voltage cutoff threshold & V \\
\hline
\end{tabular}
\end{table}

\subsection{Parameter Set and Units}\label{subsec:params}
Let $\Theta$ denote the full parameter set. For transparency, we group parameters by subsystem: load-side power mapping, ECM, thermal, and aging. Parameters may be identified from pulse tests, OCV--SOC curves, and device-level power measurements as described in Section~\ref{sec:numerics}.

\paragraph{(a) Power mapping parameters.}
\begin{table}[h!]
\centering
\caption{Power mapping parameters (load-side).}\label{tab:params_power}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$P_{\mathrm{bg}}$ & Background power & W & idle measurement \\
$P_{\mathrm{scr},0}$ & Screen baseline power & W & brightness sweep \\
$k_L$ & Screen power coefficient & W & brightness sweep \\
$\gamma$ & Screen superlinearity exponent & -- & brightness sweep \\
$P_{\mathrm{cpu},0}$ & CPU baseline power & W & CPU micro-benchmark \\
$k_C$ & CPU power coefficient & W & CPU micro-benchmark \\
$\eta$ & CPU superlinearity exponent & -- & CPU micro-benchmark \\
$P_{\mathrm{net},0}$ & Network baseline power & W & network idle \\
$k_N$ & Network activity coefficient & W & fixed-throughput tests \\
$\kappa$ & Signal-quality penalty exponent & -- & $\log$--$\log$ fit vs $\Psi$ \\
$\varepsilon$ & Signal-quality regularizer & same as $\Psi$ & chosen small, prevents singularity \\
$k_{\mathrm{tail}}$ & Tail power coefficient & W & tail decay fit \\
$\tau_\uparrow,\tau_\downarrow$ & Tail rise/decay time constants & s & tail transient fit \\
\hline
\end{tabular}
\end{table}

\paragraph{(b) ECM and electrochemical parameters.}
\begin{table}[h!]
\centering
\caption{ECM/electrochemical parameters.}\label{tab:params_ecm}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$E_0,K,A,B$ & Modified Shepherd OCV parameters & (V, V, V, --) & OCV--SOC curve fit \\
$R_{\mathrm{ref}}$ & Reference ohmic resistance & $\Omega$ & pulse $\Delta V(0^+)/\Delta I$ \\
$E_a$ & Activation energy for $R_0(T)$ & J/mol & multi-$T$ resistance fit \\
$T_{\mathrm{ref}}$ & Reference temperature & K & fixed (e.g., 298 K) \\
$\eta_R$ & SOH-to-resistance coefficient & -- & multi-SOH resistance fit \\
$R_1$ & Polarization resistance & $\Omega$ & pulse relaxation \\
$C_1$ & Polarization capacitance & F & pulse relaxation \\
\hline
\end{tabular}
\end{table}

\paragraph{(c) Capacity and thermal parameters.}
\begin{table}[h!]
\centering
\caption{Capacity and thermal parameters.}\label{tab:params_thermal}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$Q_{\mathrm{nom}}$ & Nominal capacity & Ah & datasheet / capacity test \\
$\alpha_Q$ & Temperature-capacity coefficient & 1/K & multi-$T$ capacity test \\
$C_{\mathrm{th}}$ & Lumped thermal capacitance & J/K & heating transient fit \\
$hA$ & Effective heat transfer coefficient & W/K & cooling transient fit \\
\hline
\end{tabular}
\end{table}

\paragraph{(d) Aging (SOH) parameters.}
\begin{table}[h!]
\centering
\caption{SOH degradation parameters (SEI-driven compact model).}\label{tab:params_aging}
\begin{tabular}{llll}
\hline
Parameter & Meaning & Unit & Source / identification \\
\hline
$\lambda_{\mathrm{sei}}$ & SEI degradation rate prefactor & s$^{-1}$A$^{-m}$ & aging dataset fit \\
$m$ & Current-stress exponent & -- & aging dataset fit \\
$E_{\mathrm{sei}}$ & SEI activation energy & J/mol & aging dataset fit \\
$R_g$ & Universal gas constant & J/(mol$\cdot$K) & constant \\
\hline
\end{tabular}
\end{table}

\paragraph{(e) Robustness/control micro-adjustments.}
The following quantities support numerical robustness and device-level throttling without altering the core mechanism:
\begin{equation}
z_{\min}\in(0,1)\ \text{(low-SOC guard for OCV evaluation)},\qquad
V_{\mathrm{cut}}\ \text{(shutdown voltage)},\qquad
I_{\max,0},\rho_T\ \text{(current limit parameters)}.
\end{equation}
Their calibration and usage are detailed in Section~\ref{sec:numerics}.

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