MCM/A题/数值分析检验/Response/补丁1.md

1. GAP_CLASSIFICATION_v1

{
  "GPS_power": {
    "requires_equation_change": true,
    "requires_simulation_logic_change": false,
    "text_only_addition": false,
    "one_sentence_rationale": "Adding a GPS power term requires modifying the primary total power mapping equation."
  },
  "UQ_monte_carlo": {
    "requires_equation_change": false,
    "requires_simulation_logic_change": false,
    "text_only_addition": true,
    "one_sentence_rationale": "Uncertainty quantification is a statistical wrapper around the existing model using stochastic input paths."
  },
  "Aging_dynamic_TTE": {
    "requires_equation_change": false,
    "requires_simulation_logic_change": true,
    "text_only_addition": false,
    "one_sentence_rationale": "Forecasting TTE across the battery lifespan requires an outer-loop logic to update state-of-health between discharge cycles."
  }
}

2. PATCH_SET_v1

- patch_id: "P10-GPS-EQ"
  patch_type: "REPLACE_EQUATION_LINE"
  anchor_heading_verbatim: "### 4. Multiphysics Power Mapping: (L,C,N,\Psi\rightarrow P_{\mathrm{tot}}(t))"
  target_snippet_verbatim: "P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}\big(L(t)\big)+P_{\mathrm{cpu}}\big(C(t)\big)+P_{\mathrm{net}}\big(N(t),\Psi(t),w(t)\big)."
  replacement_snippet: "P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}\big(L(t)\big)+P_{\mathrm{cpu}}\big(C(t)\big)+P_{\mathrm{net}}\big(N(t),\Psi(t),w(t)\big)+P_{\mathrm{gps}}\big(G(t)\big)."

- patch_id: "P11-GPS-TEXT"
  patch_type: "INSERT_AFTER_HEADING"
  anchor_heading_verbatim: "#### 4.3 Network power with signal-quality penalty and radio tail"
  insertion_block_id: "BLOCK_A"

- patch_id: "P12-UQ-TEXT"
  patch_type: "INSERT_AFTER_HEADING"
  anchor_heading_verbatim: "#### 10.2 Step size, stability, and convergence criterion"
  insertion_block_id: "BLOCK_B"

- patch_id: "P13-AGING-TEXT"
  patch_type: "INSERT_AFTER_HEADING"
  anchor_heading_verbatim: "#### 3.5 SOH dynamics: explicit long-horizon mechanism (SEI-inspired)"
  insertion_block_id: "BLOCK_C"

3. INSERT_TEXT_BLOCKS_v1

-----BEGIN BLOCK_A-----

4.4 GPS power and location services

Location-based services introduce a distinct power profile characterized by periodic satellite signal acquisition and processing. We define a GPS duty variable G(t) \in [0,1], which acts as a proxy for navigation-intensive usage segments. The GPS power contribution is modeled as: [ \boxed{ P_{\mathrm{gps}}(G) = P_{\mathrm{gps},0} + k_{\mathrm{gps}} G(t) } ] where P_{\mathrm{gps},0} is the baseline receiver standby power and k_{\mathrm{gps}} is the active tracking coefficient [REF-GPS-POWER]. -----END BLOCK_A-----

-----BEGIN BLOCK_B-----

10.3 Uncertainty Quantification via Monte Carlo Simulation

To quantify the impact of "unpredictable" user behavior on TTE, we employ a Monte Carlo (MC) framework. We generate M=300 stochastic usage paths by perturbing the baseline inputs (L, C, N) with Ornstein-Uhlenbeck processes to simulate realistic fluctuations [REF-MONTE-CARLO]. For a fixed seed, we compute the distribution of TTE across these paths. The primary outputs include the mean TTE, the 95% confidence interval, and the empirical survival curve P(\mathrm{TTE} > t), which represents the probability that the device remains operational at time t. -----END BLOCK_B-----

-----BEGIN BLOCK_C-----

3.6 Multi-cycle Aging and Time-to-Empty Forecasting

While a single discharge reveals immediate performance, the long-term TTE is a function of the cycle index j. We implement an outer-loop procedure to bridge the time-scale separation between discharge (seconds) and aging (days):

1. Initialize S_0 = 1 and battery parameters.
2. For each cycle j, execute the single-discharge simulation until the cutoff condition V_{\mathrm{term}} \le V_{\mathrm{cut}}.
3. Record \mathrm{TTE}_j and calculate the total charge throughput Q_{\mathrm{thr},j} = \int |I(t)| dt.
4. Update the state of health S_{j+1} using the dynamical equation in Section 3.5.
5. Update R_0 and Q_{\mathrm{eff}} for the subsequent cycle based on the new S_{j+1} [REF-LIION-AGING]. This sequence generates a TTE degradation trajectory, capturing how the "remaining life" of the phone contracts as the battery chemically matures. -----END BLOCK_C-----

4. MODIFICATION_AUDIT_v1

{
  "edited_existing_text": false,
  "changed_headings_or_numbering": false,
  "patch_ids_emitted": [
    "P10-GPS-EQ",
    "P11-GPS-TEXT",
    "P12-UQ-TEXT",
    "P13-AGING-TEXT"
  ],
  "notes": "Only additive blocks + minimal equation line replace (if any)."
}