Below is an updated **paper blueprint** that cleanly integrates the three gap patches **without breaking your frozen MODEL_SPEC logic** (except the explicit, minimal power-mapping extension for GPS). I’ll show **(i) where each patch lands**, **(ii) what each section must now contain**, and **(iii) what new data/evidence is required** so the added content is rigorous (not “text-only fluff”).

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# Updated Paper Blueprint (with GPS + Monte Carlo UQ + Multi-cycle aging)

## Summary Sheet (1 page)

### Logical progression (updated)

1. Model: continuous-time ODE + CPL closure + **extended power mapping including GPS**.
2. Core outputs: SOC(t), V_term(t), Δ(t), TTE.
3. Key findings:

   * Baseline TTE
   * **Navigation/GPS drain impact**
   * **Uncertainty band** (MC distribution + survival curve)
   * **TTE degradation across cycles** (aging trajectory)
4. Recommendations: user + OS + lifecycle-aware battery management.

### Must include (new evidence)

* A **one-line quantification** of GPS impact on TTE (ΔTTE from turning GPS “on” vs “off” in a navigation segment).
* UQ: mean/CI and at least one survival milestone (e.g., 90% survival time).
* Aging: a mini table/plot of TTE vs cycle index (e.g., cycles 0, 50, 100, 200).

---

## 1) Introduction and framing

### Logical progression (updated)

* “Unpredictability” arises from time-varying usage and environment; **navigation/location services** are a common drain source.
* We address both **short-horizon discharge** and **long-horizon degradation**.
* Outline three analyses:

  1. Mechanistic model with GPS term
  2. Monte Carlo UQ for stochastic usage
  3. Multi-cycle aging forecast for TTE decline

### Must include

* Motivation sentence tying GPS to the real-world “navigation drains phone quickly” phenomenon.
* A roadmap paragraph mapping to sections: baseline → scenario drivers (including GPS) → global sensitivity → UQ → aging forecast → recommendations.

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## 2) Model overview: states/inputs/outputs/assumptions (minor extension)

### What changes

* Add **one new input**: GPS duty variable (G(t)\in[0,1]).
  (This is the minimal extension implied by your patch: add (P_{\text{gps}}(G)) to (P_{\text{tot}}).)

### Must include (new items)

* **Table updates**

  * Inputs now include (G(t)) (unitless, [0,1], “GPS duty / navigation intensity”)
  * Parameters now include (P_{\text{gps},0}), (k_{\text{gps}})
* Assumption: (G(t)) is an externally specified scenario signal (like (L,C,N,\Psi,T_a)), not a new state.

### Evidence required

* A short justification for treating GPS drain as linear in duty cycle (first-order approximation).
* A stated range for (P_{\text{gps},0}), (k_{\text{gps}}) (even if “calibrated / assumed”; must be declared).

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## 3) Governing equations (PATCH P10 + P11)

### 3.1 Power mapping (UPDATED)

#### Logical progression

1. Screen + CPU + Network + background (existing)
2. **GPS term** added additively
3. Total power drives CPL current through quadratic closure

#### Must include (specific equations)

* Replace total power line exactly as patch indicates:
  [
  P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L)+P_{\mathrm{cpu}}(C)+P_{\mathrm{net}}(N,\Psi,w)+P_{\mathrm{gps}}(G).
  ]
* GPS submodel (BLOCK_A):
  [
  P_{\mathrm{gps}}(G) = P_{\mathrm{gps},0}+k_{\mathrm{gps}},G(t).
  ]

#### Evidence/data required to make this rigorous

* Provide either:

  * (Preferred) a citation/value range from a source (your placeholder [REF-GPS-POWER]) **or**
  * (If no citation) a **calibration protocol**: “Set (P_{\text{gps},0},k_{\text{gps}}) so that navigation scenario reproduces observed drain factor X,” and report the chosen values.

### 3.2–3.5 Constitutive + CPL + ODEs (unchanged)

* No new dynamics are needed; GPS affects (P_{\text{tot}}) only.

---

## 4) Time-to-Empty (TTE) and event logic (unchanged structure, stronger interpretation)

### Logical progression (unchanged)

* Event functions (g_V,g_z,g_\Delta)
* earliest crossing via interpolation
* termination reason recorded

### New content to add (one paragraph)

* Explain how GPS affects TTE *indirectly*:

  * (G(t)\uparrow \Rightarrow P_{\text{tot}}\uparrow \Rightarrow I\uparrow) via CPL, accelerating SOC decay and potentially increasing the risk of Δ collapse / voltage cutoff earlier.

### Evidence required

* A navigation/GPS scenario result showing:

  * higher avg (P_{\text{tot}}), higher max (I), and reduced TTE relative to baseline.

---

## 5) Parameterization and data support (must now include GPS + aging-law parameters)

### Logical progression (expanded)

1. Parameter groups: power mapping, battery ECM, thermal, radio tail
2. **GPS parameters** included in power mapping
3. **Aging parameters** (from Section 3.5 SOH law) clearly listed and sourced/assumed
4. Plausibility checks (energy, bounds, monotonicity)

### Must include (new items)

* GPS parameter table entries: (P_{\text{gps},0},k_{\text{gps}})
* Aging-law parameter table entries (whatever Section 3.5 uses; must be explicit)
* Clear labeling:

  * “Measured / literature”
  * “Calibrated”
  * “Assumed for demonstration”

### Evidence required

* For aging: at least one **reference point** like “capacity drops to 80% after N cycles” OR cite your [REF-LIION-AGING].
* If no empirical anchor, you must add a limitation note: aging trajectory is qualitative.

---

## 6) Numerical method and reproducibility (minor add)

### Logical progression

* RK4 nested CPL unchanged.
* Add that (G(t)) is treated identically to other inputs in scenario function.

### Must include

* Updated trajectory column list to include:

  * (G(t)) and (P_{\text{gps}}(t)) (optional but recommended for clarity)
* Reproducibility: seed fixed for MC; dt fixed; step-halving.

---

## 7) Baseline results (update: add one GPS/navigation stress baseline)

### Logical progression (updated)

1. Baseline scenario plots and TTE table (existing)
2. **Navigation with GPS “high duty”** as an extended baseline variant
3. Compare TTE and identify mechanism (P_tot, I, Δ)

### Must include (new evidence)

* A small 2-row comparison:

  * Baseline (G=0 or low)
  * Navigation/GPS-active (G high during navigation segment)
* Plot overlay or table:

  * ΔTTE, avg (P_{\text{tot}}), avg (P_{\text{gps}})

---

## 8) Scenario analysis: drivers of rapid drain (expand the matrix to include GPS)

### Logical progression (updated)

* The scenario matrix should now include a GPS-focused scenario explicitly.

### Must include

* Add scenario like:

  * **S8: “Navigation + GPS high duty”** (or fold into your existing navigation_poor_signal segment by setting G(t)=1 there)
* Keep the ranking output but ensure GPS is represented in driver comparisons.

### Evidence required

* Quantified ΔTTE for GPS scenario.
* Mechanistic signature entries include avg (P_{\text{gps}}) and show how it shifts current draw.

---

## 9) Sensitivity analysis (optional: include GPS parameters)

### Logical progression

* Your current Sobol set is fine; but the blueprint should specify a choice:

  * Either keep the 6-parameter set unchanged **or**
  * Replace the weakest contributor with (k_{\text{gps}}) to test GPS importance.

### Must include (if you include GPS)

* Ranges for (k_{\text{gps}}) and/or (P_{\text{gps},0}) (±20% around baseline).
* Updated ranking interpretation: whether GPS is a primary driver *in navigation-dominant regimes*.

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## 10) Uncertainty Quantification (PATCH P12: MC is now required, not optional)

### Logical progression (updated)

10.1 Define uncertainty source (usage variability)
10.2 Deterministic solver stability/step-halving (existing)
10.3 **Monte Carlo UQ** (BLOCK_B)
10.4 Survival curve and uncertainty reporting

### Must include (new “hard” components)

* MC method statement:

  * number of paths (M=300)
  * perturbation model (OU on L,C,N; optionally also N/Ψ/G if you want)
  * fixed seed
* Outputs:

  * mean TTE, CI, p10/p50/p90, survival curve (P(\text{TTE}>t))

### Evidence required

* UQ summary table + survival curve plot/table.
* A brief comparison: deterministic baseline TTE vs MC mean vs percentile spread (to interpret “unpredictable”).

---

## 11) Multi-cycle aging and lifespan TTE forecasting (PATCH P13)

### Logical progression

1. Explain time-scale separation: discharge seconds vs aging days.
2. Define outer-loop over cycles (j).
3. At each cycle: run discharge simulation → compute throughput → update SOH → update (R_0,Q_{\text{eff}}) → next cycle.
4. Produce TTE degradation trajectory.

### Must include (new evidence)

* A formal algorithm box for the outer loop (BLOCK_C).
* Define (Q_{\text{thr},j}=\int |I(t)|,dt) and how it drives your SOH update (must reference Section 3.5 law).
* A plot/table:

  * cycle index (j) vs (S_j) and TTE(_j)
* Interpretation:

  * explain why TTE declines (capacity loss + resistance increase).

### Evidence required

* Explicit SOH update equation (from your Section 3.5).
* At least one aging reference anchor (or clearly marked as “illustrative”).

---

## 12) Recommendations (updated: add GPS + lifecycle-aware policy)

### Logical progression

* Convert scenario rankings + Sobol + UQ + aging forecast into actions.

### Must include (new recommendation types)

* **GPS/location service policy**:

  * adaptive duty-cycling, batching location updates, “navigation mode” warnings
  * quantify expected gain using your GPS scenario ΔTTE
* **Lifecycle-aware** recommendations:

  * as S declines, OS should lower peak power demands to avoid V_cut/Δ collapse earlier
  * user guidance: avoid high-drain use in cold/poor signal when battery aged

### Evidence required

* Each recommendation must cite a model result:

  * “This action targets parameter/driver X and yields ΔTTE ≈ Y in scenario tests.”

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## 13) Validation, limitations, and extensions (expanded)

### Must include (new limitation + validation points)

* GPS model limitation: linear duty approximation; could refine with acquisition bursts.
* Aging limitation: if no calibrated dataset, trajectory is qualitative.
* UQ limitation: OU is a stylized model; could use empirical traces.

### Validation evidence (additions)

* Show GPS inclusion doesn’t break:

  * unit checks, Δ feasibility checks, step-halving convergence.

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# What you should update in your appendix/tables (minimum edits)

1. **Variable table**: add (G(t)).
2. **Parameter table**: add (P_{\text{gps},0},k_{\text{gps}}) + aging-law parameters.
3. **Scenario matrix**: add one GPS-heavy scenario (navigation).
4. **Results**:

   * Baseline + GPS variant TTE comparison
   * MC summary + survival curve
   * Multi-cycle TTE vs cycle plot/table

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If you paste your current section headings (or your LaTeX/Word outline), I can produce a **“diff-style” outline**: exact headings to add/renumber, and exactly which existing paragraphs need one new sentence vs a full new subsection.