# ⚛  L1 Principle — Phase-Field Allen-Cahn Dynamics

**ID:** `L1-327` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Non-conserved order parameter dynamics*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** order parameter field phi
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554127

---

## 🧠 1. Introduction

**Phase-Field Allen-Cahn Dynamics** is a **nonlinear inverse problem** whose unknown lives in **order parameter field phi** space, within the **Non-conserved order parameter dynamics** sub-domain of **Materials Science**.

Measurements consist of none via a **electron or optical micrograph** sensing mechanism.

The forward operator applies, in order: E · free energy operator; E · gradient flow operator; time evolution of the state; O · domain map operator.

Observations are corrupted by additive Gaussian noise. Well-posed parabolic PDE; inversion for (kappa, M, W) is ill-conditioned due to kinematic degeneracy between mobility and barrier.

## ⚙ 2. Forward Model

Physical chain: **x** → E · free energy → E · gradient flow → Time derivative → O · domain map → **y** (detector).

```
y = `O.domain_map` ∂_t `E.gradient_flow` `E.free_energy` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.free_energy` | E · free energy operator |
| `E.gradient_flow` | E · gradient flow operator |
| `D.time` | Time evolution of the state |
| `O.domain_map` | O · domain map operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Non-conserved order parameter dynamics |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | order_parameter_field_phi |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Well-posed parabolic PDE; inversion for (kappa, M, W) is ill-conditioned due to kinematic degeneracy between mobility and barrier.

| Metric | Value |
|---|---|
| Metric | domain_velocity_relative_error |
| Secondary | interface_width_error_nm |

## 📏 5. Operating Range (Ω)

**Center problem class:** `phase_field_allen_cahn` · **Forward operator:** `allen_cahn_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 300 |
| Dt s | s | 0.001 |
| M phi | — | 1 |
| Dx nm | nm | 10 |
| Kappa | — | 1 |
| Grid n | N | 256 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 77 – 1500 |
| M phi | — | 0.1 – 100 |
| Dx nm | nm | 0.5 – 100 |
| Kappa | — | 0.1 – 10 |
| Grid n | N | 64 – 1024 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** velocity rel error <= 0.04

| Metric | Range |
|---|---|
| Domain velocity relative error | 0.01 – 0.4 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **domain_velocity_relative_error**, with κ = `300` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xa7ed16ca675d449b1b4d7c13c443255793c5e53a53f79b8b7b6ad88350d23e13`
- **Chain tx hash:** `0x43384ce95987b9b98eb749d5a726d02a6509a29ec829feb110525c3298ccb60b`
- **Chain block:** `41554127`

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## File Mapping

This bundle consists of: `L1-327.md`, `L1-327.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-327.md` | Source of truth — edit this | Human or LLM |
| `L1-327.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.
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> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-327`._
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