# ⚛  L1 Principle — Solid-State Diffusion (Fick + Onsager)

**ID:** `L1-333` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Multi-component diffusion*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** concentration field c i
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554128

---

## 🧠 1. Introduction

**Solid-State Diffusion (Fick + Onsager)** is a **linear inverse problem** whose unknown lives in **concentration field c i** space, within the **Multi-component diffusion** sub-domain of **Materials Science**.

Measurements consist of none via a **concentration profile measurement** sensing mechanism.

The forward operator applies, in order: E · onsager flux operator; time evolution of the state; O · concentration profile operator.

Observations are corrupted by additive Gaussian noise. Parabolic; well-posed; inversion of off-diagonal D_ij ill-conditioned for small gradients.

## ⚙ 2. Forward Model

Physical chain: **x** → E · onsager flux → Time derivative → O · concentration profile → **y** (detector).

```
y = `O.concentration_profile` ∂_t `E.onsager_flux` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.onsager_flux` | E · onsager flux operator |
| `D.time` | Time evolution of the state |
| `O.concentration_profile` | O · concentration profile operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Multi-component diffusion |
| Carrier | none |
| Problem class | linear_inverse |
| Solution space | concentration_field_c_i |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Parabolic; well-posed; inversion of off-diagonal D_ij ill-conditioned for small gradients.

| Metric | Value |
|---|---|
| Metric | concentration_profile_L2_error |
| Secondary | D_ij_relative_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `solid_state_diffusion` · **Forward operator:** `diffusion_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 1000 |
| T s | s | 3600 |
| Dx um | µm | 1 |
| Grid n | N | 256 |
| N species | — | 3 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 300 – 2500 |
| T s | s | 1 – 10000000.0 |
| Dx um | µm | 0.01 – 100 |
| Grid n | N | 64 – 2048 |
| N species | — | 2 – 10 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** profile L2 <= 0.02

| Metric | Range |
|---|---|
| Concentration profile l2 error | 0.005 – 0.2 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **concentration_profile_L2_error**, with κ = `100` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x85a9cab7dc4e5b17b598593ad14c0755bf9c524f0dcf5145ff8eb2779fef0198`
- **Chain tx hash:** `0x816a06882aa267d452e0e2c8a58ee216411dcb58e24fbd453b10d956484ee16e`
- **Chain block:** `41554128`

---

## File Mapping

This bundle consists of: `L1-333.md`, `L1-333.json`.

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|------|------|-------------------|
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| `L1-333.json` | Structured metadata for the registry | LLM regenerates from the sections above |

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