# ⚛  L1 Principle — Underwater Normal Modes — modal propagation (KRAKEN)

**ID:** `L1-263` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *Shallow-water modal expansion*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** modal eigenfunction set
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554049

---

## 🧠 1. Introduction

**Underwater Normal Modes — modal propagation (KRAKEN)** is a **linear inverse problem** whose unknown lives in **modal eigenfunction set** space, within the **Shallow-water modal expansion** 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 · eigenvalue solve operator; L · mode sum operator; L · range propagate 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 · eigenvalue solve → L · mode sum → L · range propagate → Frequency integration → **y** (detector).

```
y = ∫dω `L.range_propagate` `L.mode_sum` `L.eigenvalue_solve` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.eigenvalue_solve` | L · eigenvalue solve operator |
| `L.mode_sum` | L · mode sum operator |
| `L.range_propagate` | L · range propagate operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | Shallow-water modal expansion |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | modal_eigenfunction_set |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 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:** `kraken_normal_modes` · **Forward operator:** `kraken_normal_modes_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| N depth | — | 256 |
| N modes | — | 20 |
| Freq hz | Hz | 500 |
| Ssp error | — | 0 |
| Waveguide depth error | — | 0 |
| Bottom sound speed error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | -5 – 40 |
| N depth | — | 32 – 1024 |
| N modes | — | 3 – 100 |
| Freq hz | Hz | 20 – 5000 |
| Ssp error | — | 0.0 – 0.02 |
| Waveguide depth error | — | 0.0 – 5.0 |
| Bottom sound speed error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 23.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xc90ab53ff8d85c26a9e9b23713ab46c1fedd15142992c9cdb21816ea3b7ffa6a`
- **Chain tx hash:** `0x4553b253d9b17884b506948ea280f081d4b0e3588d1e0fba2bf2e3a2c78706cd`
- **Chain block:** `41554049`

---

## File Mapping

This bundle consists of: `L1-263.md`, `L1-263.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-263.md` | Source of truth — edit this | Human or LLM |
| `L1-263.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 `L1-263`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._