# ⚛  L1 Principle — Phase Unwrapping (2D modulo-2pi inversion)

**ID:** `L1-390` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Signal Processing — *Wrapped-to-absolute phase reconstruction*
> **🎯 Problem class:** integer ambiguity inverse · **🧮 Solution space:** 2D scalar phase
> **📡 Carrier:** electromagnetic · **🌫 Noise:** wrapped gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555198

---

## 🧠 1. Introduction

**Phase Unwrapping (2D modulo-2pi inversion)** is a **integer ambiguity inverse** whose unknown lives in **2D scalar phase** space, within the **Wrapped-to-absolute phase reconstruction** sub-domain of **Signal Processing**.

Measurements consist of electromagnetic field measurements via a **phase wrapping** sensing mechanism.

The forward operator applies, in order: L · modulo · 2pi operator; pixel-level spatial averaging on the detector.

Observations are corrupted by wrapped gaussian. Well-posed when Itoh condition holds everywhere; ill-posed around phase residues (where gradient circulation is non-zero). LS unwrap smooths but biases; branch-cut correct when cuts routed through low-coherence regions.

## ⚙ 2. Forward Model

Physical chain: **x** → L · modulo · 2pi → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.modulo.2pi` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.modulo.2pi` | L · modulo · 2pi operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Signal Processing |
| Sub domain | Wrapped-to-absolute phase reconstruction |
| Carrier | electromagnetic |
| Problem class | integer_ambiguity_inverse |
| Solution space | 2D_scalar_phase |
| Noise model | wrapped_gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 2 |

## 📡 4. Measurement Model

Well-posed when Itoh condition holds everywhere; ill-posed around phase residues (where gradient circulation is non-zero). LS unwrap smooths but biases; branch-cut correct when cuts routed through low-coherence regions.

| Metric | Value |
|---|---|
| Metric | RMSE_rad |
| Secondary | cycle_slip_count |

## 📏 5. Operating Range (Ω)

**Center problem class:** `2d_phase_unwrap` · **Forward operator:** `modulo_2pi`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Sigma phi | — | 0.3 |
| Coherence floor | — | 0.8 |
| Undersampling ratio | — | 1 |
| Discontinuity fraction | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 128 – 4096 |
| W | px | 128 – 4096 |
| Sigma phi | — | 0.05 – 2.0 |
| Coherence floor | — | 0.1 – 1.0 |
| Undersampling ratio | — | 0.5 – 1.0 |
| Local fringe gradient | — | 0.1 – 3.0 |
| Discontinuity fraction | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.1 rad RMSE

| Metric | Range |
|---|---|
| Rmse rad | 0.01 – 2.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **RMSE_rad**, with κ = `1000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x5e389706d22b79eedc948cfe29ad82ea682ec67a463242fe55e02f2c12bfc889`
- **Chain tx hash:** `0x41ae3fa547a068a3780388a3e2a4c3b0c1033c41ffea15b6576d12766d209a6e`
- **Chain block:** `41555198`

---

## File Mapping

This bundle consists of: `L1-390.md`, `L1-390.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-390.md` | Source of truth — edit this | Human or LLM |
| `L1-390.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 `L1-390`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._