# ⚛  L1 Principle — X-ray Diffraction Crystal Space-Group Classification (PWDR)

**ID:** `L1-519` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Crystal symmetry classification from XRD lattice + structure-factor recovery*
> **🎯 Problem class:** linear inverse with categorical readout · **🧮 Solution space:** 1D space group label
> **📡 Carrier:** x_ray · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554142

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## 🧠 1. Introduction

**X-ray Diffraction Crystal Space-Group Classification (PWDR)** is a **linear inverse with categorical readout** whose unknown lives in **1D space group label** space, within the **Crystal symmetry classification from XRD lattice + structure-factor recovery** sub-domain of **Materials Science**.

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

The forward operator applies, in order: polyenergetic X-ray emission spectrum; L · bragg diffraction operator; L · structure factor operator; L · lattice parameter recovery operator; L · systematic absence test operator; L · space group decision tree operator; integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson counting noise. Existence and uniqueness inherited from L1-518 with additional constraint: pseudosymmetry can cause space-group ambiguity (e.g., P2_1/n vs Pn — distinguishing requires additional statistics or chemistry constraint). Stability inherits L1-518's high condition number plus additive systematic_absence_threshold_uncertainty. Joint Hadamard well-posedness for the space-group classification established by Hahn 2005 (International Tables Vol A), Cromer-Liberman 1970 (anomalous scattering for centrosymmetry tests), Le Bail 2005 (powder indexing), Kabsch 2010 (XDS for space-group determination), Spek 2003 (PLATON ADDSYM).

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → L · bragg diffraction → L · structure factor → L · lattice parameter recovery → L · systematic absence test → L · space group decision tree → Angular integration → **y** (detector).

```
y = ∫dΩ `L.space_group_decision_tree` `L.systematic_absence_test` `L.lattice_parameter_recovery` `L.structure_factor` `L.bragg_diffraction` I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.bragg_diffraction` | L · bragg diffraction operator |
| `L.structure_factor` | L · structure factor operator |
| `L.lattice_parameter_recovery` | L · lattice parameter recovery operator |
| `L.systematic_absence_test` | L · systematic absence test operator |
| `L.space_group_decision_tree` | L · space group decision tree operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Crystal symmetry classification from XRD lattice + structure-factor recovery |
| Carrier | x_ray |
| Problem class | linear_inverse_with_categorical_readout |
| Solution space | 1D_space_group_label |
| Noise model | poisson |
| Integration axis | angular_spatial |
| Difficulty delta | 5 |
| L dag | 6.8 |

## 📡 4. Measurement Model

Existence and uniqueness inherited from L1-518 with additional constraint: pseudosymmetry can cause space-group ambiguity (e.g., P2_1/n vs Pn — distinguishing requires additional statistics or chemistry constraint). Stability inherits L1-518's high condition number plus additive systematic_absence_threshold_uncertainty. Joint Hadamard well-posedness for the space-group classification established by Hahn 2005 (International Tables Vol A), Cromer-Liberman 1970 (anomalous scattering for centrosymmetry tests), Le Bail 2005 (powder indexing), Kabsch 2010 (XDS for space-group determination), Spek 2003 (PLATON ADDSYM).

| Metric | Value |
|---|---|
| Metric | categorical_accuracy |
| Secondary | top_3_accuracy |

## 📏 5. Operating Range (Ω)

**Center problem class:** `xrd_space_group_pwdr` · **Forward operator:** `xrd_space_group_pwdr_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Two theta max | — | 120 |
| Two theta min | — | 5 |
| Two theta step | — | 0.02 |
| Lambda angstrom | — | 1.5418 |
| Twinning artifact | — | 0 |
| Instrumental broadening | — | 0.05 |
| Pseudosymmetry ambiguity | — | 0 |
| Modulation or superlattice | — | 0 |
| Lattice parameter uncertainty | — | 0 |
| Systematic absence threshold uncertainty | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Two theta max | — | 60 – 160 |
| Two theta min | — | 1 – 30 |
| Lambda angstrom | — | 0.4 – 2.5 |
| Instrumental broadening | — | 0.01 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.85_accuracy

| Metric | Range |
|---|---|
| Categorical accuracy | 0.3 – 0.99 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **categorical_accuracy**, with κ = `600` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xeecb4fc500389a8009bfab9a6b2e0ddb83b0b17ed2ade9f3f25506d88e0d71a7`
- **Chain tx hash:** `0xecefcf2feafe0b852896a9ab692100199979b48b3c591987bb3aa9d43b794dda`
- **Chain block:** `41554142`

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## File Mapping

This bundle consists of: `L1-519.md`, `L1-519.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-519.md` | Source of truth — edit this | Human or LLM |
| `L1-519.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 `L1-519`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._