# ⚛  L1 Principle — Acoustic Scattering — boundary element method (BEM)

**ID:** `L1-261` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *Exterior BEM for acoustic scattering*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex pressure field 2D
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554036

---

## 🧠 1. Introduction

**Acoustic Scattering — boundary element method (BEM)** is a **linear inverse problem** whose unknown lives in **complex pressure field 2D** space, within the **Exterior BEM for acoustic scattering** sub-domain of **Acoustics**.

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

The forward operator applies, in order: L · source incident operator; L · bem surface integral operator; S · record · farfield 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 incident → L · bem surface integral → S · record · farfield → Frequency integration → **y** (detector).

```
y = ∫dω `S.record.farfield` `L.bem_surface_integral` `L.source_incident` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.source_incident` | L · source incident operator |
| `L.bem_surface_integral` | L · bem surface integral operator |
| `S.record.farfield` | S · record · farfield operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | Exterior BEM for acoustic 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:** `bem_scatter` · **Forward operator:** `bem_scatter_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Ka | — | 5 |
| N mesh | — | 5000 |
| Snr db | dB | 30 |
| N freqs | — | 16 |
| Freq error | — | 0 |
| N receivers | — | 64 |
| Impedance bc error | — | 0 |
| Surface mesh error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Ka | — | 0.1 – 50 |
| N mesh | — | 500 – 200000 |
| Snr db | dB | 0 – 40 |
| N freqs | — | 4 – 128 |
| Freq error | — | 0.0 – 0.02 |
| N receivers | — | 16 – 512 |
| Impedance bc error | — | 0.0 – 0.3 |
| Surface mesh 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 = `700` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x06c41e823d456d08f9c9a98ca67a85be3e601c6570f02d0226e368761520e9d9`
- **Chain tx hash:** `0x2d275464b4d76d233f85566deb12b76e4cdc79ca1a842fa1c382ebb0e40909e7`
- **Chain block:** `41554036`

---

## File Mapping

This bundle consists of: `L1-261.md`, `L1-261.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-261.md` | Source of truth — edit this | Human or LLM |
| `L1-261.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 `L1-261`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._