# ⚛  L1 Principle — Bone Fracture Detection from Radiograph (PWDR)

**ID:** `L1-526` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Cortical bone discontinuity recovery from X-ray with fracture categorical readout*
> **🎯 Problem class:** linear inverse with categorical readout · **🧮 Solution space:** 1D fracture class with AO OTA
> **📡 Carrier:** x_ray · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553387

---

## 🧠 1. Introduction

**Bone Fracture Detection from Radiograph (PWDR)** is a **linear inverse with categorical readout** whose unknown lives in **1D fracture class with AO OTA** space, within the **Cortical bone discontinuity recovery from X-ray with fracture categorical readout** sub-domain of **Medical Imaging**.

Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **radiograph with fracture classifier** sensing mechanism.

The forward operator applies, in order: polyenergetic X-ray emission spectrum; line-integral projection through an attenuation map; L · cortical edge detection operator; L · fragment segmentation operator; L · fracture classifier operator; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson counting noise. Existence inherited from L1-031. Uniqueness conditional on adequate views. Stability dominated by overlapping_structures and growth_plate_confounder (pediatric). Joint Hadamard well-posedness established by Müller AO 1996 (foundational classification), Lindsey 2018 (deep learning fracture detection benchmark), Rajpurkar 2017 (CheXNet), Olczak 2017 (Stockholm fracture deep learning).

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → Attenuation projection → L · cortical edge detection → L · fragment segmentation → L · fracture classifier → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.fracture_classifier` `L.fragment_segmentation` `L.cortical_edge_detection` ∫µ dl I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.attenuation_projection` | Line-integral projection through an attenuation map |
| `L.cortical_edge_detection` | L · cortical edge detection operator |
| `L.fragment_segmentation` | L · fragment segmentation operator |
| `L.fracture_classifier` | L · fracture classifier operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Cortical bone discontinuity recovery from X-ray with fracture categorical readout |
| Carrier | x_ray |
| Problem class | linear_inverse_with_categorical_readout |
| Solution space | 1D_fracture_class_with_AO_OTA |
| Noise model | poisson |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 4.6 |

## 📡 4. Measurement Model

Existence inherited from L1-031. Uniqueness conditional on adequate views. Stability dominated by overlapping_structures and growth_plate_confounder (pediatric). Joint Hadamard well-posedness established by Müller AO 1996 (foundational classification), Lindsey 2018 (deep learning fracture detection benchmark), Rajpurkar 2017 (CheXNet), Olczak 2017 (Stockholm fracture deep learning).

| Metric | Value |
|---|---|
| Metric | categorical_accuracy |
| Secondary | sensitivity_for_displaced |

## 📏 5. Operating Range (Ω)

**Center problem class:** `fracture_xray_pwdr` · **Forward operator:** `fracture_pwdr_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 2500 |
| W | px | 3000 |
| Kvp | — | 65 |
| Mas | — | 6 |
| Snr db | dB | 30 |
| View count | — | 2 |
| Anatomic region | — | wrist |
| Rotation artifact | — | 0 |
| Pixel resolution um | µm | 100 |
| Overlapping structures | — | 0 |
| Growth plate confounder | — | 0 |
| Soft tissue swelling obscuring | — | 0 |
| Manual radiologist disagreement | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 1024 – 4096 |
| W | px | 1024 – 4096 |
| View count | — | 1 – 4 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.85_accuracy

| Metric | Range |
|---|---|
| Categorical accuracy | 0.5 – 0.99 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **categorical_accuracy**, with κ = `60` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x06849a8d91dddbb63ca0bd3a1c05c2aec15383b0ef105e58a3a52eae7eb8d4fa`
- **Chain tx hash:** `0x9a7916d79c210232418a539e6ad45df627f016fa3cc4aaccafb2143505e3c4c5`
- **Chain block:** `41553387`

---

## File Mapping

This bundle consists of: `L1-526.md`, `L1-526.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-526.md` | Source of truth — edit this | Human or LLM |
| `L1-526.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 `L1-526`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._