# ⚛  L1 Principle — Helmholtz Equation — frequency-domain acoustic scattering / interior field

**ID:** `L1-258` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *Frequency-domain acoustic inverse scattering*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex pressure field 2D
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554035

---

## 🧠 1. Introduction

**Helmholtz Equation — frequency-domain acoustic scattering / interior field** is a **linear inverse problem** whose unknown lives in **complex pressure field 2D** space, within the **Frequency-domain acoustic inverse scattering** sub-domain of **Acoustics**.

Measurements consist of acoustic pressure waves recorded by transducers via a **pressure receivers** sensing mechanism.

The forward operator applies, in order: L · source injection operator; L · helmholtz solve operator; S · record · receivers 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 · source injection → L · helmholtz solve → S · record · receivers → Frequency integration → **y** (detector).

```
y = ∫dω `S.record.receivers` `L.helmholtz_solve` `L.source_injection` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.source_injection` | L · source injection operator |
| `L.helmholtz_solve` | L · helmholtz solve operator |
| `S.record.receivers` | S · record · receivers operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | Frequency-domain acoustic inverse scattering |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | complex_pressure_field_2D |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 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:** `helmholtz` · **Forward operator:** `helmholtz_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Snr db | dB | 30 |
| K mean | — | 20 |
| N freqs | — | 16 |
| K error | — | 0 |
| N receivers | — | 32 |
| Geometry error | — | 0 |
| Impedance bc error | — | 0 |
| Receiver pos error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 – 1024 |
| W | px | 64 – 1024 |
| Snr db | dB | 0 – 40 |
| K mean | — | 1 – 200 |
| N freqs | — | 4 – 128 |
| K error | — | 0.0 – 0.05 |
| N receivers | — | 8 – 256 |
| Geometry error | — | 0.0 – 0.05 |
| Impedance bc error | — | 0.0 – 0.3 |
| Receiver pos error | — | 0.0 – 0.02 |

## 🎯 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 = `1000` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xf0c96d119c0fb83452aab765fecad67b9eab4a4edd1df308232ed2a41a19bea7`
- **Chain tx hash:** `0x494a4b6d46a007355770f52f90848ff6989453012689871ee1552dd9530b91e9`
- **Chain block:** `41554035`

---

## File Mapping

This bundle consists of: `L1-258.md`, `L1-258.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-258.md` | Source of truth — edit this | Human or LLM |
| `L1-258.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 `L1-258`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._