# ⚛  L1 Principle — X-Ray Fluorescence (XRF) Mapping

**ID:** `L1-118` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Industrial Inspection — *Elemental mapping via characteristic X-ray fluorescence*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** elemental concentration map
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554200

---

## 🧠 1. Introduction

**X-Ray Fluorescence (XRF) Mapping** is a **linear inverse problem** whose unknown lives in **elemental concentration map** space, within the **Elemental mapping via characteristic X-ray fluorescence** sub-domain of **Industrial Inspection**.

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

The forward operator applies, in order: L · xray excitation operator; L · fluorescence emission operator; D · energy dispersive operator; detector sums all spectral bands.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered elemental concentration map 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 moderately conditioned (kappa_eff ~= 10); matrix_effect dominates the stability cliff; detector_dead_time 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** → L · xray excitation → L · fluorescence emission → D · energy dispersive → Spectral integration → **y** (detector).

```
y = Σ_λ `D.energy_dispersive` `L.fluorescence_emission` `L.xray_excitation` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_excitation` | L · xray excitation operator |
| `L.fluorescence_emission` | L · fluorescence emission operator |
| `D.energy_dispersive` | D · energy dispersive operator |
| `int.spectral` | Detector sums all spectral bands |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Industrial Inspection |
| Sub domain | Elemental mapping via characteristic X-ray fluorescence |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | elemental_concentration_map |
| Noise model | shot_poisson |
| Integration axis | spectral |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered elemental concentration map 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 moderately conditioned (kappa_eff ~= 10); matrix_effect dominates the stability cliff; detector_dead_time 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:** `xrf_mapping` · **Forward operator:** `xrf_mapping_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Kv | — | 30 |
| Dwell ms | ms | 100 |
| N elements | — | 10 |
| Peak counts | — | 1000 |
| Peak overlap | — | 0 |
| Matrix effect | — | 0 |
| Detector dead time | — | 0.05 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 – 4096 |
| W | px | 64 – 4096 |
| Kv | — | 10 – 50 |
| Dwell ms | ms | 1 – 10000 |
| N elements | — | 2 – 30 |
| Peak counts | — | 20 – 100000 |
| Peak overlap | — | 0.0 – 0.5 |
| Matrix effect | — | 0.0 – 0.3 |
| Self absorption | — | 0.0 – 0.5 |
| Detector dead time | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 26.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xcf09b2dc819f67c811fd9056d3087ec4dbd8966173db2fa3f7e6b1669f849d9d`
- **Chain tx hash:** `0x07d27827c55f2a2ada2a79815290bca084c190e32e3fc9f542ff8dbf35146971`
- **Chain block:** `41554200`

---

## File Mapping

This bundle consists of: `L1-118.md`, `L1-118.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-118.md` | Source of truth — edit this | Human or LLM |
| `L1-118.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 `L1-118`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._