# ⚛  L1 Principle — X-ray Non-Destructive Testing (2D radiography)

**ID:** `L1-112` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Industrial Inspection — *2D projection radiography for defect detection*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D attenuation projection
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554199

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## 🧠 1. Introduction

**X-ray Non-Destructive Testing (2D radiography)** is a **linear inverse problem** whose unknown lives in **2D attenuation projection** space, within the **2D projection radiography for defect detection** sub-domain of **Industrial Inspection**.

Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **xray projection** sensing mechanism.

The forward operator applies, in order: polyenergetic X-ray emission spectrum; exponential attenuation along the propagation path; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D attenuation projection is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is well-conditioned (kappa_eff ~= 8); beam_hardening dominates the stability cliff; scatter and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → Beer-Lambert attenuation → Spatial integration → **y** (detector).

```
y = ∫_A dA exp(-∫µ dl) I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.beer_lambert` | Exponential attenuation along the propagation path |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Industrial Inspection |
| Sub domain | 2D projection radiography for defect detection |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 2D_attenuation_projection |
| Noise model | shot_poisson |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered 2D attenuation projection is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is well-conditioned (kappa_eff ~= 8); beam_hardening dominates the stability cliff; scatter and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `xray_ndt` · **Forward operator:** `xray_ndt_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 2048 |
| W | px | 2048 |
| Kv | — | 150 |
| Scatter | — | 0 |
| Pixel um | µm | 100 |
| Photon count | — | 5000 |
| Beam hardening | — | 0 |
| Thickness variation | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 512 – 8192 |
| W | px | 512 – 8192 |
| Kv | — | 20 – 450 |
| Scatter | — | 0.0 – 0.4 |
| Pixel um | µm | 1 – 500 |
| Photon count | — | 100 – 100000 |
| Beam hardening | — | 0.0 – 0.3 |
| Thickness variation | — | 0.0 – 0.5 |
| Source intensity drift | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 30.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `160` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x45ae0a0745a91fb54ed4ea22c7385cd1c7c8987636f0b49d00f54529f86ede3b`
- **Chain tx hash:** `0xefd2535572644fe7ea98926b807253d2982b005655e87797e178e56372fc1791`
- **Chain block:** `41554199`

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## File Mapping

This bundle consists of: `L1-112.md`, `L1-112.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-112.md` | Source of truth — edit this | Human or LLM |
| `L1-112.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 `L1-112`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._