# ⚛  L1 Principle — Classical Nucleation Theory

**ID:** `L1-332` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Nucleation kinetics*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** J T Dmu surface
> **📡 Carrier:** none · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554128

---

## 🧠 1. Introduction

**Classical Nucleation Theory** is a **nonlinear inverse problem** whose unknown lives in **J T Dmu surface** space, within the **Nucleation kinetics** sub-domain of **Materials Science**.

Measurements consist of none via a **induction time measurement** sensing mechanism.

The forward operator applies, in order: E · cluster thermodynamics operator; E · nucleation barrier operator; K · filter operator; O · J rate operator.

Observations are corrupted by Poisson counting noise. Well-posed forward; J is exponentially sensitive to sigma, making inversion challenging.

## ⚙ 2. Forward Model

Physical chain: **x** → E · cluster thermodynamics → E · nucleation barrier → O · J rate → **y** (detector).

```
y = `O.J_rate` `E.nucleation_barrier` `E.cluster_thermodynamics` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.cluster_thermodynamics` | E · cluster thermodynamics operator |
| `E.nucleation_barrier` | E · nucleation barrier operator |
| `O.J_rate` | O · j rate operator |

**🛰 Estimator components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `K.filter` | K · filter operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Nucleation kinetics |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | J_T_Dmu_surface |
| Noise model | poisson |
| Integration axis | temperature |
| Difficulty delta | 3 |
| L dag | 3.1 |

## 📡 4. Measurement Model

Well-posed forward; J is exponentially sensitive to sigma, making inversion challenging.

| Metric | Value |
|---|---|
| Metric | log_J_relative_error |
| Secondary | tau_induction_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `classical_nucleation` · **Forward operator:** `cnt_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N sites | — | 1000000000000000000000 |
| Dmu kjmol | — | 2 |
| T under k | — | 10 |
| Interface sigma mjm2 | — | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N sites | — | 1000000000000000.0 – 10000000000000000000000000 |
| Dmu kjmol | — | 0.1 – 20 |
| T under k | — | 0.1 – 200 |
| Interface sigma mjm2 | — | 10 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** log10(J) error <= 0.3

| Metric | Range |
|---|---|
| Log j relative error | 0.1 – 3.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **log_J_relative_error**, with κ = `120` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xf8b48cb50eb6adae4d9bd09c2921ffcdd77f5db6e1956136eb7dd7c3d68bb6e3`
- **Chain tx hash:** `0xc11fbb15792fba74fd7d2148570cfbad3f814a325e6c3051f0ee2f28fbe1ac70`
- **Chain block:** `41554128`

---

## File Mapping

This bundle consists of: `L1-332.md`, `L1-332.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-332.md` | Source of truth — edit this | Human or LLM |
| `L1-332.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 `L1-332`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._