# 1) 必须文档 ①：Project Memory（核心模型备忘录）

> **用途**：下个对话里快速恢复我们已完成的“假设 + 模型建立 + 求解框架”。
> **你要做的**：原样粘贴到新对话开头（Prompt A 会包含它）。

## A. Problem & Scope

* Contest: **2026 MCM Problem A (continuous-time smartphone battery drain)**
* Completed sections: **Assumptions + Model Formulation and Solution (Q1 core)**
* Constraints: **mechanism-driven, no black-box regression**, continuous-time ODE/DAE, include numerical method + stability/convergence statements.

## B. State, Inputs, Outputs

* **State**: (\mathbf{x}(t)=[z(t),v_p(t),T_b(t),S(t),w(t)]^\top)

  * (z): SOC, (v_p): polarization voltage, (T_b): battery temperature, (S): SOH (capacity fraction), (w): radio tail state
* **Inputs**: (\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top)

  * (L): brightness, (C): CPU load, (N): network activity, (\Psi): signal quality (higher better), (T_a): ambient temp
* **Outputs**: (V_{\text{term}}(t)), SOC (z(t)), **TTE**

## C. Power mapping (component-level, explicit (\Psi) effect)

[
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L)+P_{\mathrm{cpu}}(C)+P_{\mathrm{net}}(N,\Psi,w)
]
[
P_{\mathrm{scr}}(L)=P_{\mathrm{scr},0}+k_L L^\gamma,;\gamma>1
]
[
P_{\mathrm{cpu}}(C)=P_{\mathrm{cpu},0}+k_C C^\eta,;\eta>1
]
[
P_{\mathrm{net}}(N,\Psi,w)=P_{\mathrm{net},0}+k_N\frac{N}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w,;\kappa>0
]
Tail dynamics (continuous, avoids discrete FSM):
[
\dot w=\frac{\sigma(N)-w}{\tau(N)},\quad
\tau(N)=\begin{cases}\tau_\uparrow,&\sigma(N)\ge w\ \tau_\downarrow,&\sigma(N)<w\end{cases},;
\tau_\uparrow\ll\tau_\downarrow,;
\sigma(N)=\min(1,N)
]

## D. ECM + CPL current closure (nonlinear feedback source)

Terminal voltage:
[
V_{\mathrm{term}}=V_{\mathrm{oc}}(z)-v_p-I R_0(T_b,S)
]
CPL constraint:
[
P_{\mathrm{tot}}=V_{\mathrm{term}}I=\big(V_{\mathrm{oc}}-v_p-IR_0\big)I
]
Quadratic current:
[
I=\frac{V_{\mathrm{oc}}-v_p-\sqrt{\Delta}}{2R_0},\quad
\Delta=(V_{\mathrm{oc}}-v_p)^2-4R_0P_{\mathrm{tot}}
]
Shutdown/feasibility:

* Require (\Delta\ge0); if (\Delta\le0) ⇒ power infeasible ⇒ voltage collapse/shutdown.

## E. Coupled ODEs (SOC–polarization–thermal–SOH)

[
\dot z=-\frac{I}{3600,Q_{\mathrm{eff}}(T_b,S)}
]
[
\dot v_p=\frac{I}{C_1}-\frac{v_p}{R_1C_1}
]
[
\dot T_b=\frac{1}{C_{\mathrm{th}}}\Big(I^2R_0+Iv_p-hA(T_b-T_a)\Big)
]
SOH (Option A compact, used for Q1):
[
\dot S=-\lambda_{\mathrm{sei}}|I|^{m}\exp!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right),;0\le m\le1
]
(Option B SEI thickness (\delta) exists as upgrade path if needed.)

## F. Constitutive relations

Modified Shepherd OCV:
[
V_{\mathrm{oc}}(z)=E_0-K\Big(\frac{1}{z}-1\Big)+A e^{-B(1-z)}
]
Arrhenius resistance + SOH correction:
[
R_0(T_b,S)=R_{\mathrm{ref}}\exp!\Big[\frac{E_a}{R_g}\Big(\frac{1}{T_b}-\frac{1}{T_{\mathrm{ref}}}\Big)\Big],(1+\eta_R(1-S))
]
Effective capacity:
[
Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}}S\Big[1-\alpha_Q(T_{\mathrm{ref}}-T_b)\Big]_+
]

## G. Initial conditions & TTE

[
z(0)=z_0,;v_p(0)=0,;T_b(0)=T_a(0),;S(0)=S_0,;w(0)=0
]
[
\mathrm{TTE}=\inf{t>0:;V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le0\ \text{or}\ \Delta(t)\le0}
]

## H. Numerical solution standard

* Use RK4 (or ode45) with **nested algebraic solve** for (I) at each substep.
* Step size: (\Delta t\le0.05,\tau_p) where (\tau_p=R_1C_1).
* Convergence: step-halving until (|z_{\Delta t}-z_{\Delta t/2}|_\infty<10^{-4}); TTE change <1%.

## I. Parameter estimation (hybrid, reproducible)

* OCV params ((E_0,K,A,B)): least squares to OCV–SOC curve.
* (R_0): pulse instantaneous drop (\Delta V(0^+)/\Delta I).
* (R_1,C_1): pulse relaxation exponential fit.
* (\kappa): fit (\ln P_{\mathrm{net}}) vs (-\ln(\Psi)) at fixed throughput.

## J. References (BibTeX you already used)

* Shepherd (1965), Tremblay & Dessaint (2009), Plett (2004) + smartphone energy paper as needed.

---

# 2) 必须文档 ②：“不可预测机制叙事”一句话模板

> **用途**：下次写 Introduction/Modeling/Results 时保持口径一致

> Battery-life variability arises from (i) time-varying usage inputs ((L,C,N,\Psi,T_a)), (ii) nonlinear CPL closure (P=VI) that amplifies current when voltage drops, and (iii) state memory through polarization (v_p) and thermal inertia (T_b), producing history-dependent discharge trajectories.

---

# 3) 必须文档 ③：你下次对话开场的 Prompt（复制即用）

## Prompt A（必用：恢复上下文 + 锁定写作风格与约束）

把下面整段复制到新对话的第一条消息：

```markdown
You are my MCM/ICM continuous-modeling O-award mentor and paper lead writer. 
We have already completed Assumptions + full Model Formulation and Solution (Q1 core). 
Do NOT reinvent the model; strictly continue from the finalized framework below, keeping all symbols consistent and mechanism-driven (no black-box regression). 
Write in academic English (SIAM/IEEE), equations in LaTeX, and ensure solution logic matches paper narrative.

## Project Memory (do not alter)
[PASTE THE ENTIRE "Project Memory" SECTION HERE]
```

> 你只需要把上面那个 `[PASTE ... HERE]` 换成我给你的 **Project Memory** 全文即可。

---

## Prompt B（如果你下一步要做 Q2/Q3：不确定性、策略、灵敏度）

```markdown
Continue with the same model. Now do: (1) uncertainty modeling for future usage inputs using a continuous-time stochastic process (e.g., OU / regime switching), (2) Monte Carlo to obtain a TTE distribution, (3) global sensitivity (Sobol or variance-based) on key parameters (k_L, gamma, k_N, kappa, T_a, etc.), and (4) produce figure descriptions that match the simulations. Keep all derivations and algorithmic steps explicit.
```

---

## Prompt C（如果你下一步要做“Parameter Estimation”章节写作）

```markdown
Write a complete "Parameter Estimation" section for the existing model: 
- specify which parameters come from literature/datasheets vs which are fitted;
- provide objective functions and constraints for fitting (OCV curve, pulse response for R0/R1/C1, signal exponent kappa);
- include identifiability discussion and practical calibration workflow.
No new model components unless strictly necessary.
```