# ⚛  L1 Principle — Grain Growth and Recrystallization

**ID:** `L1-331` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Microstructure evolution*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** grain orientation map
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554128

---

## 🧠 1. Introduction

**Grain Growth and Recrystallization** is a **nonlinear inverse problem** whose unknown lives in **grain orientation map** space, within the **Microstructure evolution** sub-domain of **Materials Science**.

Measurements consist of none via a **ebsd micrograph** sensing mechanism.

The forward operator applies, in order: E · curvature flow operator; adds a prior term that biases the solution toward smoothness/sparsity; O · grain statistics operator.

Observations are corrupted by additive Gaussian noise. Well-posed; parabolic in mean-field; statistical fluctuations bounded.

## ⚙ 2. Forward Model

Physical chain: **x** → E · curvature flow → O · grain statistics → **y** (detector).

```
y = `O.grain_statistics` `E.curvature_flow` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.curvature_flow` | E · curvature flow operator |
| `O.grain_statistics` | O · grain statistics operator |

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

| Primitive | What it does |
|---|---|
| `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Microstructure evolution |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | grain_orientation_map |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 3.1 |

## 📡 4. Measurement Model

Well-posed; parabolic in mean-field; statistical fluctuations bounded.

| Metric | Value |
|---|---|
| Metric | grain_size_relative_error |
| Secondary | texture_KL_divergence |

## 📏 5. Operating Range (Ω)

**Center problem class:** `grain_growth` · **Forward operator:** `grain_growth_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 1000 |
| Grid n | N | 256 |
| F pinning | — | 0 |
| T anneal s | s | 100 |
| Initial grains | — | 10000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 300 – 2000 |
| Grid n | N | 64 – 2048 |
| F pinning | — | 0 – 0.2 |
| T anneal s | s | 1 – 100000 |
| Initial grains | — | 100 – 1000000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** <r> rel error <= 0.04

| Metric | Range |
|---|---|
| Grain size relative error | 0.01 – 0.4 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **grain_size_relative_error**, with κ = `150` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xe452da87a79ee73252faff58d86558b28a575cd76ffd77c983cae44f0700e925`
- **Chain tx hash:** `0x86bdfac8e9db203afe5c6644a7f12906b5ede8a14f3772155cd71c889acd5183`
- **Chain block:** `41554128`

---

## File Mapping

This bundle consists of: `L1-331.md`, `L1-331.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-331.md` | Source of truth — edit this | Human or LLM |
| `L1-331.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 `L1-331`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._