# ⚛  L1 Principle — WAXS — Wide-Angle X-ray Scattering

**ID:** `L1-133` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Scientific Instrumentation — *X-ray scattering probing Å-scale structure*
> **🎯 Problem class:** inverse fourier scatter · **🧮 Solution space:** radial structure function
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554214

---

## 🧠 1. Introduction

**WAXS — Wide-Angle X-ray Scattering** is a **inverse fourier scatter** whose unknown lives in **radial structure function** space, within the **X-ray scattering probing Å-scale structure** sub-domain of **Scientific Instrumentation**.

Measurements consist of photons collected by an optical detector via a **xray wide angle scatter** sensing mechanism.

The forward operator applies, in order: L · scatter · elastic operator; L · project · q operator; integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Fourier inversion is ill-conditioned for truncated q-range; regularization (max-entropy, oversampling) stabilizes. Mismatch: background subtraction, termination ripples.

## ⚙ 2. Forward Model

Physical chain: **x** → L · scatter · elastic → L · project · q → Angular integration → **y** (detector).

```
y = ∫dΩ `L.project.q` `L.scatter.elastic` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.scatter.elastic` | L · scatter · elastic operator |
| `L.project.q` | L · project · q operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Scientific Instrumentation |
| Sub domain | X-ray scattering probing Å-scale structure |
| Carrier | photon |
| Problem class | inverse_fourier_scatter |
| Solution space | radial_structure_function |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 3 |
| L dag | 2.7 |

## 📡 4. Measurement Model

Fourier inversion is ill-conditioned for truncated q-range; regularization (max-entropy, oversampling) stabilizes. Mismatch: background subtraction, termination ripples.

| Metric | Value |
|---|---|
| Metric | g_r_NMSE |
| Secondary | Rwp_rietveld |

## 📏 5. Operating Range (Ω)

**Center problem class:** `waxs_polymer_analysis` · **Forward operator:** `waxs_iq_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N q | — | 800 |
| Q max | — | 30 |
| Q min | — | 5 |
| Sample thickness um | µm | 100 |
| Background subtraction | — | 0.05 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N q | — | 100 – 5000 |
| Q max | — | 15 – 50 |
| Q min | — | 1 – 10 |
| Q truncation | — | 0.0 – 0.5 |
| Sample thickness um | µm | 1 – 1000 |
| Termination ripples | — | 0.0 – 0.3 |
| Beam profile mismatch | — | 0.0 – 0.2 |
| Preferred orientation | — | 0.0 – 0.5 |
| Background subtraction | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** g(r) NMSE 5e-3

| Metric | Range |
|---|---|
| G r nmse | 0.0001 – 0.3 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x03efadead8ee044135c98acd73d0a6aead576d7b75d7141b0c8fbd6c32b5f664`
- **Chain tx hash:** `0x7416be9e99e0ab6284a27b574c8a532a8786a68ce691caae4f03c1553b76afd8`
- **Chain block:** `41554214`

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## File Mapping

This bundle consists of: `L1-133.md`, `L1-133.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-133.md` | Source of truth — edit this | Human or LLM |
| `L1-133.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 `L1-133`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._