# ⚛  L1 Principle — Dendritic Solidification

**ID:** `L1-334` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Phase-field solidification*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** dendrite morphology and temperature
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554128

---

## 🧠 1. Introduction

**Dendritic Solidification** is a **nonlinear inverse problem** whose unknown lives in **dendrite morphology and temperature** space, within the **Phase-field solidification** sub-domain of **Materials Science**.

Measurements consist of none via a **insitu imaging X ray synchrotron** sensing mechanism.

The forward operator applies, in order: E · phi evolution operator; E · thermal diffusion operator; a fixed-point or gradient iteration on the unknown; O · dendrite morphology operator.

Observations are corrupted by additive Gaussian noise. Stiff reaction-diffusion; well-posed under Karma-Rappel anti-trapping; inversion from morphology is moderate-ill-conditioned.

## ⚙ 2. Forward Model

Physical chain: **x** → E · phi evolution → E · thermal diffusion → O · dendrite morphology → **y** (detector).

```
y = `O.dendrite_morphology` `E.thermal_diffusion` `E.phi_evolution` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.phi_evolution` | E · phi evolution operator |
| `E.thermal_diffusion` | E · thermal diffusion operator |
| `O.dendrite_morphology` | O · dendrite morphology operator |

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

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Phase-field solidification |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | dendrite_morphology_and_temperature |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Stiff reaction-diffusion; well-posed under Karma-Rappel anti-trapping; inversion from morphology is moderate-ill-conditioned.

| Metric | Value |
|---|---|
| Metric | dendrite_tip_velocity_relative_error |
| Secondary | arm_spacing_error_um |

## 📏 5. Operating Range (Ω)

**Center problem class:** `solidification` · **Forward operator:** `solidification_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Dt k | — | 5 |
| Grid n | N | 512 |
| V pull um s | s | 100 |
| K partition | — | 0.15 |
| Lambda 1 um | µm | 20 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Dt k | — | 0.1 – 100 |
| Grid n | N | 128 – 4096 |
| V pull um s | s | 1 – 10000 |
| K partition | — | 0.01 – 0.99 |
| Lambda 1 um | µm | 1 – 500 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** tip velocity rel error <= 0.05

| Metric | Range |
|---|---|
| Dendrite tip velocity relative error | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **dendrite_tip_velocity_relative_error**, with κ = `600` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x9deaf9dd67d72f543ca6acdbff2197e1517826064235e3deca58f47255a5e55e`
- **Chain tx hash:** `0x5807e5796a1d240c02c89c781f4146a64ce7d6dfd314106b7415b388b6c21489`
- **Chain block:** `41554128`

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## File Mapping

This bundle consists of: `L1-334.md`, `L1-334.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-334.md` | Source of truth — edit this | Human or LLM |
| `L1-334.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 `L1-334`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._