# ⚛  L1 Principle — Underwater Acoustics — Parabolic Equation (RAM)

**ID:** `L1-264` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *Pade-expansion split-step PE*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex pressure range depth 2D
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554049

---

## 🧠 1. Introduction

**Underwater Acoustics — Parabolic Equation (RAM)** is a **linear inverse problem** whose unknown lives in **complex pressure range depth 2D** space, within the **Pade-expansion split-step PE** sub-domain of **Acoustics**.

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

The forward operator applies, in order: L · split step pe operator; L · marching · range operator; L · absorb bottom operator; int · range operator.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · split step pe → L · marching · range → L · absorb bottom → int · range → **y** (detector).

```
y = `int.range` `L.absorb_bottom` `L.marching.range` `L.split_step_pe` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.split_step_pe` | L · split step pe operator |
| `L.marching.range` | L · marching · range operator |
| `L.absorb_bottom` | L · absorb bottom operator |
| `int.range` | Int · range operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | Pade-expansion split-step PE |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | complex_pressure_range_depth_2D |
| Noise model | gaussian |
| Integration axis | range |
| 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:** `ram_pe` · **Forward operator:** `ram_pe_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| N depth | — | 256 |
| N range | — | 1024 |
| Freq hz | Hz | 100 |
| Ssp error | — | 0 |
| Pade order | — | 8 |
| Range step dr | — | 0 |
| Bottom atten error | — | 0 |
| Pade order truncation | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | -5 – 40 |
| N depth | — | 64 – 2048 |
| N range | — | 128 – 8192 |
| Freq hz | Hz | 10 – 2000 |
| Ssp error | — | 0.0 – 0.02 |
| Pade order | — | 1 – 10 |
| Range step dr | — | 0.0 – 1.0 |
| Bottom atten error | — | 0.0 – 0.2 |
| Pade order truncation | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 23.5

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `440` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xa31398de1ec9a1e678488d83603531bb330038b2aa81a409634b6116a831baf7`
- **Chain tx hash:** `0xd74677a1202c180d7d19e8589407a3ee6f134759ed4eae112ae58e2e43ecd450`
- **Chain block:** `41554049`

---

## File Mapping

This bundle consists of: `L1-264.md`, `L1-264.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-264.md` | Source of truth — edit this | Human or LLM |
| `L1-264.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 `L1-264`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._