# ⚛  L1 Principle — X-ray Phase Contrast — propagation / grating / analyzer

**ID:** `L1-254` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *X-ray phase and attenuation imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D refractive index delta beta
> **📡 Carrier:** em · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554001

---

## 🧠 1. Introduction

**X-ray Phase Contrast — propagation / grating / analyzer** is a **linear inverse problem** whose unknown lives in **3D refractive index delta beta** space, within the **X-ray phase and attenuation imaging** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **xray ccd** sensing mechanism.

The forward operator applies, in order: polyenergetic X-ray emission spectrum; L · refract sample operator; L · propagate operator; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → L · refract sample → L · propagate → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.propagate` `L.refract_sample` I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.refract_sample` | L · refract sample operator |
| `L.propagate` | L · propagate operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | X-ray phase and attenuation imaging |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | 3D_refractive_index_delta_beta |
| Noise model | shot_poisson |
| Integration axis | spatial |
| 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:** `xray_pc` · **Forward operator:** `xray_pc_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 2048 |
| W | px | 2048 |
| Z | — | 128 |
| Snr db | dB | 30 |
| Z error | — | 0 |
| N distances | — | 3 |
| Coherence error | — | 0 |
| Beam harmonics error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 8192 |
| W | px | 256 – 8192 |
| Z | — | 32 – 1024 |
| Snr db | dB | 0 – 40 |
| Z error | — | 0.0 – 0.1 |
| N distances | — | 1 – 50 |
| Coherence error | — | 0.0 – 0.3 |
| Beam harmonics error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 25.0

| 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:** `0xf3df13574937e5b2987af88ac19d4520895b2819e64c29616282a70955a7fde8`
- **Chain tx hash:** `0x630fccc1c8e7fdbbe95916e6199752865d47f2449ed24e02ac4bde7fd134dab0`
- **Chain block:** `41554001`

---

## File Mapping

This bundle consists of: `L1-254.md`, `L1-254.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-254.md` | Source of truth — edit this | Human or LLM |
| `L1-254.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 `L1-254`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._