# ⚛  L1 Principle — Cahn-Hilliard Phase Separation

**ID:** `L1-328` · **Status:** ⊙ Testnet (genesis catalog)

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

---

## 🧠 1. Introduction

**Cahn-Hilliard Phase Separation** is a **nonlinear inverse problem** whose unknown lives in **composition field phi** space, within the **Conserved order parameter dynamics** sub-domain of **Materials Science**.

Measurements consist of none via a **scattering morphology** sensing mechanism.

The forward operator applies, in order: E · chemical potential operator; E · conservative flux operator; time evolution of the state; O · morphology operator.

Observations are corrupted by additive Gaussian noise. Fourth-order PDE; well-posed; inversion ill-conditioned (M-chi degeneracy).

## ⚙ 2. Forward Model

Physical chain: **x** → E · chemical potential → E · conservative flux → Time derivative → O · morphology → **y** (detector).

```
y = `O.morphology` ∂_t `E.conservative_flux` `E.chemical_potential` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.chemical_potential` | E · chemical potential operator |
| `E.conservative_flux` | E · conservative flux operator |
| `D.time` | Time evolution of the state |
| `O.morphology` | O · morphology operator |

## 🔬 3. Physics Fingerprint

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

## 📡 4. Measurement Model

Fourth-order PDE; well-posed; inversion ill-conditioned (M-chi degeneracy).

| Metric | Value |
|---|---|
| Metric | S_q_time_L2_error |
| Secondary | domain_size_error_nm |

## 📏 5. Operating Range (Ω)

**Center problem class:** `phase_field_cahn_hilliard` · **Forward operator:** `cahn_hilliard_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| M | — | 1 |
| T k | — | 300 |
| Dx nm | nm | 5 |
| Phi 0 | — | 0.5 |
| Chi ab | — | 3 |
| Grid n | N | 256 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| M | — | 0.01 – 100 |
| T k | — | 77 – 1500 |
| Dx nm | nm | 0.5 – 100 |
| Phi 0 | — | 0.1 – 0.9 |
| Chi ab | — | 1.5 – 20 |
| Grid n | N | 64 – 2048 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** S(q,t) L2 <= 0.04

| Metric | Range |
|---|---|
| S q time l2 error | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **S_q_time_L2_error**, with κ = `350` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x85f7bf568c22adc5dff5da9d39e3048af5b622de9fdb4ff57917723c89b5f063`
- **Chain tx hash:** `0x5d78a235c117f3465ea34a83f617e04f3c129c0647b062ab0a9d9885bbb8dde6`
- **Chain block:** `41554127`

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## File Mapping

This bundle consists of: `L1-328.md`, `L1-328.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-328.md` | Source of truth — edit this | Human or LLM |
| `L1-328.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 `L1-328`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._