# ⚛  L1 Principle — Thin Film Growth Kinetics

**ID:** `L1-337` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Nucleation-growth-coalescence*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** film morphology island distribution
> **📡 Carrier:** none · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554129

---

## 🧠 1. Introduction

**Thin Film Growth Kinetics** is a **nonlinear inverse problem** whose unknown lives in **film morphology island distribution** space, within the **Nucleation-growth-coalescence** sub-domain of **Materials Science**.

Measurements consist of none via a **rheed afm insitu** sensing mechanism.

The forward operator applies, in order: E · deposition flux operator; E · surface diffusion operator; applies a smooth nonlinear function element-wise; O · film morphology operator.

Observations are corrupted by Poisson counting noise. Well-posed forward; inversion of (D, E_mig) from N(s,t) moderately ill-conditioned (Arrhenius vs pre-factor).

## ⚙ 2. Forward Model

Physical chain: **x** → E · deposition flux → E · surface diffusion → Pointwise nonlinearity → O · film morphology → **y** (detector).

```
y = `O.film_morphology` f(·) `E.surface_diffusion` `E.deposition_flux` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.deposition_flux` | E · deposition flux operator |
| `E.surface_diffusion` | E · surface diffusion operator |
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.film_morphology` | O · film morphology operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Nucleation-growth-coalescence |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | film_morphology_island_distribution |
| Noise model | poisson |
| Integration axis | time |
| Difficulty delta | 5 |
| L dag | 3.7 |

## 📡 4. Measurement Model

Well-posed forward; inversion of (D, E_mig) from N(s,t) moderately ill-conditioned (Arrhenius vs pre-factor).

| Metric | Value |
|---|---|
| Metric | island_density_relative_error |
| Secondary | roughness_rms_error_nm |

## 📏 5. Operating Range (Ω)

**Center problem class:** `thin_film_growth` · **Forward operator:** `thinfilm_kmc_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 500 |
| T s | s | 10 |
| F ml s | s | 0.1 |
| E mig ev | — | 0.8 |
| Lattice l | — | 1024 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 200 – 1500 |
| T s | s | 0.1 – 3600 |
| F ml s | s | 0.0001 – 10 |
| E mig ev | — | 0.1 – 2.5 |
| Lattice l | — | 128 – 8192 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** island density rel error <= 0.05

| Metric | Range |
|---|---|
| Island density relative error | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **island_density_relative_error**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xd081498a3276ff5939256a9779e12660f541ce2186d7cbdce273e5fcb121cb1e`
- **Chain tx hash:** `0xeb4df93a151b6acab71b5a4cbb88cedeb85ab6ea5e7fcef7d8b147bbefb5746d`
- **Chain block:** `41554129`

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## File Mapping

This bundle consists of: `L1-337.md`, `L1-337.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-337.md` | Source of truth — edit this | Human or LLM |
| `L1-337.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 `L1-337`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._