# ⚛  L1 Principle — Proton Radiography & CT (pRad, pCT)

**ID:** `L1-131` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Scientific Instrumentation — *Proton transmission imaging for materials and medicine*
> **🎯 Problem class:** linear tomographic inverse · **🧮 Solution space:** rsp map
> **📡 Carrier:** proton · **🌫 Noise:** shot poisson plus MCS
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554214

---

## 🧠 1. Introduction

**Proton Radiography & CT (pRad, pCT)** is a **linear tomographic reconstruction** whose unknown lives in **rsp map** space, within the **Proton transmission imaging for materials and medicine** sub-domain of **Scientific Instrumentation**.

Measurements consist of protons in a particle imaging system via a **energy loss scatter** sensing mechanism.

The forward operator applies, in order: L · scatter · MCS operator; L · eloss · BetheBloch operator; L · project · MLP operator; pixel-level spatial averaging on the detector.

Observations are corrupted by shot poisson plus MCS. CT reconstruction via FBP with MLP curvilinear path (not straight-line like X-ray). Spatial resolution ~0.5-1 mm limited by MCS.

## ⚙ 2. Forward Model

Physical chain: **x** → L · scatter · MCS → L · eloss · BetheBloch → L · project · MLP → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.project.MLP` `L.eloss.BetheBloch` `L.scatter.MCS` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.scatter.MCS` | L · scatter · mcs operator |
| `L.eloss.BetheBloch` | L · eloss · bethebloch operator |
| `L.project.MLP` | L · project · mlp operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Scientific Instrumentation |
| Sub domain | Proton transmission imaging for materials and medicine |
| Carrier | proton |
| Problem class | linear_tomographic_inverse |
| Solution space | rsp_map |
| Noise model | shot_poisson_plus_MCS |
| Integration axis | projection |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

CT reconstruction via FBP with MLP curvilinear path (not straight-line like X-ray). Spatial resolution ~0.5-1 mm limited by MCS.

| Metric | Value |
|---|---|
| Metric | RSP_NMSE |
| Secondary | spatial_resolution_mm |

## 📏 5. Operating Range (Ω)

**Center problem class:** `pct_reconstruction` · **Forward operator:** `pct_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N view | — | 360 |
| Pixel mm | mm | 1 |
| E beam mev | — | 200 |
| N proton per view | — | 1e+08 |
| Mls path model error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N view | — | 10 – 720 |
| Pixel mm | mm | 0.2 – 5 |
| E beam mev | — | 70 – 330 |
| Motion blur | — | 0.0 – 5 |
| Dose limited snr | — | 0.5 – 100 |
| N proton per view | — | 100000.0 – 10000000000.0 |
| Mls path model error | — | 0.0 – 0.3 |
| Calorimeter calibration | — | 0.0 – 0.05 |
| Nuclear interaction fraction | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** RSP NMSE 1e-3

| Metric | Range |
|---|---|
| Rsp nmse | 0.0001 – 0.1 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **RSP_NMSE**, with κ = `300` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x20549f20256d3bd8b64bd29b1c0b4f8bbd707395468485815f7e8f378d7e761d`
- **Chain tx hash:** `0xd28f26b34f52dd063fca19e10e3ee7f49e9a8a75aa5aca36f30c36e18439e6c6`
- **Chain block:** `41554214`

---

## File Mapping

This bundle consists of: `L1-131.md`, `L1-131.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-131.md` | Source of truth — edit this | Human or LLM |
| `L1-131.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 `L1-131`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._