\section{Assumptions}\label{sec:assumptions}

To balance physical fidelity, interpretability, and computational tractability, we adopt the following modeling assumptions. These assumptions are consistent with the continuous-time, mechanism-driven framework developed in Sections~\ref{sec:model_formulation}--\ref{sec:numerics} and are intended to match typical smartphone operating conditions.

\subsection{Structural Assumptions}\label{sec:assumptions_structural}
\begin{enumerate}
  \item \textbf{Single-cell lumped equivalent.}
  The battery pack is represented by an equivalent single cell with lumped electrical and thermal states. The terminal behavior is captured by a first-order equivalent circuit model (ECM) comprising open-circuit voltage (OCV), an ohmic resistance, and one polarization (RC) branch.

  \item \textbf{Additive component power mapping.}
  The device power demand is decomposed into additive contributions from background processes, display, CPU, and network subsystems:
  \(
  P_{\mathrm{tot}} = P_{\mathrm{bg}} + P_{\mathrm{scr}}(L) + P_{\mathrm{cpu}}(C) + P_{\mathrm{net}}(N,\Psi,w).
  \)
  Cross-couplings among subsystems (e.g., CPU--network interactions) are treated as second-order effects and are absorbed into the calibrated parameters of the component maps.

  \item \textbf{Normalized inputs and bounded states.}
  Usage inputs are normalized to dimensionless intensities \(L,C,N\in[0,1]\), and the radio-tail state satisfies \(w\in[0,1]\). State variables are interpreted physically and are constrained to admissible ranges (e.g., \(z\in[0,1]\), \(S\in[0,1]\)) up to numerical tolerances.
\end{enumerate}

\subsection{Load-Side Assumptions}\label{sec:assumptions_load}
\begin{enumerate}
  \item \textbf{Constant-power load (CPL) closure.}
  Over the modeling time scale, the smartphone power management system is approximated as imposing an instantaneous power demand \(P_{\mathrm{tot}}(t)\) at the battery terminals, i.e.,
  \(
  P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t)\,I(t).
  \)
  This CPL closure is used to capture the key nonlinear feedback whereby decreasing terminal voltage can induce increasing current draw under fixed power demand.

  \item \textbf{Feasibility interpretation via discriminant.}
  The quadratic CPL relation yields a discriminant \(\Delta(t)\). When \(\Delta(t)<0\), sustaining the requested power with the current electrical state is infeasible, indicating a voltage-collapse risk. This provides a mechanistic explanation for ``sudden shutdown'' events observed in practice.

  \item \textbf{Optional derating (current/power limiting).}
  Smartphones typically derate performance (e.g., frequency throttling or PMIC current limiting) under low-voltage or high-temperature conditions. We therefore allow an optional saturation policy, e.g.,
  \(
  I(t)=\min\{I_{\mathrm{CPL}}(t),\,I_{\max}(T_b(t))\},
  \)
  which preserves the original CPL behavior when \(I_{\mathrm{CPL}}\le I_{\max}\) while enabling safe operation (at reduced delivered power) when the requested power would otherwise drive excessive current.
\end{enumerate}

\subsection{Thermal Assumptions}\label{sec:assumptions_thermal}
\begin{enumerate}
  \item \textbf{Lumped thermal capacitance.}
  The battery temperature is modeled by a single lumped node \(T_b(t)\) with effective thermal capacitance \(C_{\mathrm{th}}\). Spatial gradients within the cell or across the device chassis are neglected.

  \item \textbf{Dominant heat sources and linear heat rejection.}
  Heat generation is attributed to ohmic loss and polarization-branch dissipation, while heat rejection to the environment is modeled by linear convection/conduction:
  \[
  \dot T_b=\frac{1}{C_{\mathrm{th}}}\Big(I^2R_0+\frac{v_p^2}{R_1}-hA\,(T_b-T_a)\Big).
  \]
  Radiative effects and temperature dependence of \(hA\) are neglected over normal operating ranges.

  \item \textbf{Ambient temperature as an exogenous input.}
  The ambient temperature \(T_a(t)\) is treated as an external forcing. In typical usage scenarios, \(T_a\) varies slowly compared to the electrical dynamics.
\end{enumerate}

\subsection{Aging Assumptions}\label{sec:assumptions_aging}
\begin{enumerate}
  \item \textbf{Slow-time-scale degradation.}
  The state-of-health \(S(t)\) evolves on a slower time scale than \(z(t)\), \(v_p(t)\), and \(T_b(t)\). Over short horizons (single discharge), \(S\) may be approximated as quasi-static; over longer horizons, cumulative degradation is captured by the aging ODE.

  \item \textbf{SEI-dominated capacity fade surrogate.}
  Capacity fade is represented by a compact SEI-driven rate law:
  \[
  \dot S=-\lambda_{\mathrm{sei}}|I|^{m}\exp\!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right),
  \qquad 0\le m\le 1,
  \]
  which captures acceleration with higher current magnitude and higher temperature. More detailed mechanistic extensions (e.g., explicit SEI thickness) are outside the present scope.

  \item \textbf{Aging impacts through resistance and effective capacity.}
  The influence of \(S\) on instantaneous discharge behavior is mediated through (i) an SOH correction in the ohmic resistance \(R_0(T_b,S)\) and (ii) a proportional scaling of effective capacity \(Q_{\mathrm{eff}}(T_b,S)\). Other aging pathways (e.g., lithium plating, impedance spectra changes beyond a single RC branch) are neglected.
\end{enumerate}

\subsection{Boundaries and Applicability}\label{sec:assumptions_scope}
The proposed model is intended for \emph{discharge-dominated} smartphone operation under typical environmental conditions and is not designed to capture the following regimes without further extensions:
\begin{enumerate}
  \item \textbf{Fast charging or charging--discharging transients.}
  Charging dynamics, CC--CV charging protocols, and charger-induced thermal effects are not modeled.

  \item \textbf{Extreme temperatures and protection-layer behavior.}
  Very low-temperature operation (where diffusion limitations, severe capacity loss, or protection circuitry dominates) and very high temperatures beyond normal thermal management limits are outside scope.

  \item \textbf{Severe voltage-sag and hardware protection events.}
  Hard cutoffs triggered by hardware protection (e.g., overcurrent, undervoltage lockout, or thermal shutdown) are approximated by the terminal-voltage cutoff \(V_{\mathrm{cut}}\) and the CPL feasibility indicator; detailed PMIC internal logic is not explicitly modeled.

  \item \textbf{Detailed multi-physics and spatial effects.}
  Spatially distributed thermal fields, electrode-level electrochemical PDE models, and multi-cell balancing are not included; the goal is a compact mechanism-driven model suitable for scenario simulation and sensitivity analysis.

  \item \textbf{Application-specific internal scheduling.}
  Fine-grained OS scheduling, DVFS at sub-second resolution, and app-level state machines are abstracted into the exogenous inputs \(L(t),C(t),N(t),\Psi(t)\) and (optionally) the derating function \(I_{\max}(T_b)\).
\end{enumerate}

In summary, these assumptions yield a compact continuous-time model that remains physically interpretable, numerically stable, and sufficiently expressive to study runtime prediction, voltage-collapse risk, and the impact of temperature and aging under representative smartphone usage patterns.