# ⚛  L1 Principle — Battery Electrode Electrochemistry (Newman-Doyle-Fuller)

**ID:** `L1-340` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Porous-electrode Li-ion transport*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** p2d battery state
> **📡 Carrier:** ion · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554129

---

## 🧠 1. Introduction

**Battery Electrode Electrochemistry (Newman-Doyle-Fuller)** is a **nonlinear inverse problem** whose unknown lives in **p2d battery state** space, within the **Porous-electrode Li-ion transport** sub-domain of **Materials Science**.

Measurements consist of ions detected after acceleration or extraction via a **battery cycling impedance** sensing mechanism.

The forward operator applies, in order: E · electrolyte transport operator; E · solid diffusion operator; E · butler volmer operator; a fixed-point or gradient iteration on the unknown; O · V cell operator.

Observations are corrupted by additive Gaussian noise. Stiff coupled PDE; well-posed in forward; inversion of (D_s, i_0) from V(t,I) is moderately ill-conditioned.

## ⚙ 2. Forward Model

Physical chain: **x** → E · electrolyte transport → E · solid diffusion → E · butler volmer → O · V cell → **y** (detector).

```
y = `O.V_cell` `E.butler_volmer` `E.solid_diffusion` `E.electrolyte_transport` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.electrolyte_transport` | E · electrolyte transport operator |
| `E.solid_diffusion` | E · solid diffusion operator |
| `E.butler_volmer` | E · butler volmer operator |
| `O.V_cell` | O · v cell operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Porous-electrode Li-ion transport |
| Carrier | ion |
| Problem class | nonlinear_inverse |
| Solution space | p2d_battery_state |
| Noise model | gaussian |
| Integration axis | time_state_of_charge |
| Difficulty delta | 5 |
| L dag | 4.1 |

## 📡 4. Measurement Model

Stiff coupled PDE; well-posed in forward; inversion of (D_s, i_0) from V(t,I) is moderately ill-conditioned.

| Metric | Value |
|---|---|
| Metric | V_cell_L2_mV |
| Secondary | capacity_fade_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `battery_electrode` · **Forward operator:** `p2d_battery_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 298 |
| C rate | — | 1 |
| N mesh | — | 100 |
| Dod range | — | 0 – 1 |
| Thickness um | µm | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 253 – 353 |
| C rate | — | 0.01 – 100 |
| N mesh | — | 20 – 1000 |
| Cycles | — | 1 – 10000 |
| Thickness um | µm | 10 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** V_cell L2 <= 10 mV

| Metric | Range |
|---|---|
| V cell l2 mv | 2.0 – 200.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **V_cell_L2_mV**, with κ = `800` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x7dd4698a1356ff03b4222b69c4310c1c0854941c4d07a01bc2cb3a23f917be5c`
- **Chain tx hash:** `0xf35d5c75d4707471da1f38b986bd3ccc5b60c496eba71816e02b048298e5352d`
- **Chain block:** `41554129`

---

## File Mapping

This bundle consists of: `L1-340.md`, `L1-340.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-340.md` | Source of truth — edit this | Human or LLM |
| `L1-340.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-340`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._