# ⚛  L1 Principle — Helmholtz Scattering — frequency-domain EM scattering

**ID:** `L1-233` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Frequency-domain vector Helmholtz scattering*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex E field freq
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553987

---

## 🧠 1. Introduction

**Helmholtz Scattering — frequency-domain EM scattering** is a **linear inverse problem** whose unknown lives in **complex E field freq** space, within the **Frequency-domain vector Helmholtz scattering** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **far field antenna** sensing mechanism.

The forward operator applies, in order: L · incident wave operator; L · scatter boundary operator; L · green function 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 · incident wave → L · scatter boundary → L · green function → Frequency integration → **y** (detector).

```
y = ∫dω `L.green_function` `L.scatter_boundary` `L.incident_wave` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.incident_wave` | L · incident wave operator |
| `L.scatter_boundary` | L · scatter boundary operator |
| `L.green_function` | L · green function operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Frequency-domain vector Helmholtz scattering |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | complex_E_field_freq |
| 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:** `helmholtz_em` · **Forward operator:** `helmholtz_em_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 64 |
| Snr db | dB | 30 |
| N freqs | — | 16 |
| K error | — | 0 |
| Bc error | — | 0 |
| N angles | — | 256 |
| Surface error | — | 0 |
| Material error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 2048 |
| W | px | 32 – 2048 |
| Z | — | 32 – 1024 |
| Snr db | dB | 0 – 40 |
| N freqs | — | 1 – 256 |
| K error | — | 0.0 – 0.05 |
| Bc error | — | 0.0 – 0.2 |
| N angles | — | 16 – 4096 |
| Surface error | — | 0.0 – 0.1 |
| Material error | — | 0.0 – 0.1 |

## 🎯 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 = `800` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xf0218231898026cd11536b7b79fe90409ca7abbef8bd2f81d3acad7b479e021d`
- **Chain tx hash:** `0x3348d75b5c4133103dea3a01afb716e100de7e200388754b90860b9614fb20e7`
- **Chain block:** `41553987`

---

## File Mapping

This bundle consists of: `L1-233.md`, `L1-233.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-233.md` | Source of truth — edit this | Human or LLM |
| `L1-233.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 `L1-233`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._