# ⚛  L1 Principle — Level-Set Method — implicit interface evolution

**ID:** `L1-184` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Level-set interface representation for multiphase*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** signed distance field
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553967

---

## 🧠 1. Introduction

**Level-Set Method — implicit interface evolution** is a **nonlinear inverse problem** whose unknown lives in **signed distance field** space, within the **Level-set interface representation for multiphase** sub-domain of **Fluid Dynamics**.

Measurements consist of mechanical vibrations or strain signals via a **imaging piv** sensing mechanism.

The forward operator applies, in order: L · level set advect operator; L · reinitialization operator; L · curvature operator; detector accumulates flux over the exposure window.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · level set advect → L · reinitialization → L · curvature → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.curvature` `L.reinitialization` `L.level_set_advect` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.level_set_advect` | L · level set advect operator |
| `L.reinitialization` | L · reinitialization operator |
| `L.curvature` | L · curvature operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Level-set interface representation for multiphase |
| Carrier | mechanical |
| Problem class | nonlinear_inverse |
| Solution space | signed_distance_field |
| Noise model | gaussian |
| Integration axis | temporal |
| 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:** `level_set` · **Forward operator:** `level_set_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Snr db | dB | 30 |
| N times | — | 1000 |
| Reinit error | — | 0 |
| Curvature error | — | 0 |
| Mass loss error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 8192 |
| W | px | 32 – 8192 |
| Snr db | dB | 0 – 40 |
| N times | — | 100 – 1000000 |
| Reinit error | — | 0.0 – 0.3 |
| Curvature error | — | 0.0 – 0.3 |
| Mass loss error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| 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:** `0x4658c27c835d77d9a67e7589abb8b1bd6a97e86224ed313bb8805e4829bc3c1b`
- **Chain tx hash:** `0xc12d9cc6e03fef38a7137c7d4b4b118360fa847218848140d85623bbcbc950da`
- **Chain block:** `41553967`

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## File Mapping

This bundle consists of: `L1-184.md`, `L1-184.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-184.md` | Source of truth — edit this | Human or LLM |
| `L1-184.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 `L1-184`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._