# ⚛  L1 Principle — MHD Stability — linear eigenmode growth rates

**ID:** `L1-349` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Plasma Physics — *Tokamak/stellarator linear stability*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** mode growth rates
> **📡 Carrier:** plasma · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554096

---

## 🧠 1. Introduction

**MHD Stability — linear eigenmode growth rates** is a **linear inverse problem** whose unknown lives in **mode growth rates** space, within the **Tokamak/stellarator linear stability** sub-domain of **Plasma Physics**.

Measurements consist of plasma particle and field diagnostics via a **magnetics fluctuations** sensing mechanism.

The forward operator applies, in order: L · linearize mhd operator; L · eigenvalue operator; L · mode classify operator; integration over a band of frequencies.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · linearize mhd → L · eigenvalue → L · mode classify → Frequency integration → **y** (detector).

```
y = ∫dω `L.mode_classify` `L.eigenvalue` `L.linearize_mhd` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.linearize_mhd` | L · linearize mhd operator |
| `L.eigenvalue` | L · eigenvalue operator |
| `L.mode_classify` | L · mode classify operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Plasma Physics |
| Sub domain | Tokamak/stellarator linear stability |
| Carrier | plasma |
| Problem class | linear_inverse |
| Solution space | mode_growth_rates |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `mhd_stability` · **Forward operator:** `mhd_stability_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 25 |
| N modes | — | 20 |
| N radial | — | 64 |
| Rotation error | — | 0 |
| Equilibrium error | — | 0 |
| Pressure profile error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| N modes | — | 1 – 200 |
| N radial | — | 16 – 1024 |
| Rotation error | — | 0.0 – 0.3 |
| Equilibrium error | — | 0.0 – 0.1 |
| Pressure profile error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 24.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xdc12de2a17fb3c52f900c1b3dcb95e0478b4e223d0ef25418fbd5b55ed9f43a3`
- **Chain tx hash:** `0x648e8509678c982db94b0b419d18ed974feb98f15215bf249230ddfa95acd87b`
- **Chain block:** `41554096`

---

## File Mapping

This bundle consists of: `L1-349.md`, `L1-349.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-349.md` | Source of truth — edit this | Human or LLM |
| `L1-349.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 `L1-349`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._