# ⚛  L1 Principle — Eddy Currents — conductor inversion from induced fields

**ID:** `L1-238` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Quasi-static eddy-current inspection*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** conductivity 3D map
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553988

---

## 🧠 1. Introduction

**Eddy Currents — conductor inversion from induced fields** is a **linear inverse problem** whose unknown lives in **conductivity 3D map** space, within the **Quasi-static eddy-current inspection** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **induction coil** sensing mechanism.

The forward operator applies, in order: L · induced current operator; L · quasi static operator; L · diffuse through conductor 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 · induced current → L · quasi static → L · diffuse through conductor → Frequency integration → **y** (detector).

```
y = ∫dω `L.diffuse_through_conductor` `L.quasi_static` `L.induced_current` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.induced_current` | L · induced current operator |
| `L.quasi_static` | L · quasi static operator |
| `L.diffuse_through_conductor` | L · diffuse through conductor operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Quasi-static eddy-current inspection |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | conductivity_3D_map |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 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:** `eddy_current` · **Forward operator:** `eddy_current_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 32 |
| N pos | — | 64 |
| Snr db | dB | 30 |
| N freqs | — | 16 |
| Mu r error | — | 0 |
| Sigma error | — | 0 |
| Coil pos error | — | 0 |
| Coil lift off error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Z | — | 8 – 512 |
| N pos | — | 4 – 1024 |
| Snr db | dB | 0 – 40 |
| N freqs | — | 1 – 256 |
| Mu r error | — | 0.0 – 0.1 |
| Sigma error | — | 0.0 – 0.1 |
| Coil pos error | — | 0.0 – 0.02 |
| Coil lift off error | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

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

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xb42e8313c53bfab8498afd7c391ceb25e2c6cd9b6b7bdbbfaf33b3c408a3c5fd`
- **Chain tx hash:** `0x0f46e71c1b23d7f577aad33fbfa2c5f675ad7a646fdd10054790d6ed4ef3c4fc`
- **Chain block:** `41553988`

---

## File Mapping

This bundle consists of: `L1-238.md`, `L1-238.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-238.md` | Source of truth — edit this | Human or LLM |
| `L1-238.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 `L1-238`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._