# ⚛  L1 Principle — X-ray Crystallography (single-crystal diffraction)

**ID:** `L1-125` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Scientific Instrumentation — *Atomic-resolution crystal structure determination*
> **🎯 Problem class:** phase retrieval inverse · **🧮 Solution space:** electron density
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554213

---

## 🧠 1. Introduction

**X-ray Crystallography (single-crystal diffraction)** is a **phase retrieval inverse** whose unknown lives in **electron density** space, within the **Atomic-resolution crystal structure determination** sub-domain of **Scientific Instrumentation**.

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

The forward operator applies, in order: L · diffract · bragg operator; L · project · rescipocal operator; integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Phase problem fundamentally ill-posed from intensities alone; experimental phasing (MAD/SAD) adds constraints. Well-posed at atomic resolution (<1.2 Å) via direct methods.

## ⚙ 2. Forward Model

Physical chain: **x** → L · diffract · bragg → L · project · rescipocal → Angular integration → **y** (detector).

```
y = ∫dΩ `L.project.rescipocal` `L.diffract.bragg` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.diffract.bragg` | L · diffract · bragg operator |
| `L.project.rescipocal` | L · project · rescipocal operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Scientific Instrumentation |
| Sub domain | Atomic-resolution crystal structure determination |
| Carrier | photon |
| Problem class | phase_retrieval_inverse |
| Solution space | electron_density |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3 |

## 📡 4. Measurement Model

Phase problem fundamentally ill-posed from intensities alone; experimental phasing (MAD/SAD) adds constraints. Well-posed at atomic resolution (<1.2 Å) via direct methods.

| Metric | Value |
|---|---|
| Metric | R_free |
| Secondary | atomic_coord_RMSD_angstrom |

## 📏 5. Operating Range (Ω)

**Center problem class:** `sad_phasing_refinement` · **Forward operator:** `xrd_bragg_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N refl | — | 50000 |
| N atoms | — | 1000 |
| Resolution a | — | 2 |
| Mosaicity deg | deg | 0.3 |
| Radiation damage | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N refl | — | 100 – 1000000 |
| N atoms | — | 10 – 100000 |
| Resolution a | — | 0.8 – 4.0 |
| Mosaicity deg | deg | 0.05 – 2.0 |
| Radiation damage | — | 0.0 – 1.0 |
| Twinning fraction | — | 0.0 – 0.5 |
| Absorption anisotropy | — | 0.0 – 0.3 |
| Crystal size variability | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** R_free 0.22

| Metric | Range |
|---|---|
| R free | 0.15 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **R_free**, with κ = `1000` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x4578dd83185a94e3d1c8032fcdb46c12dd20d6bda35d7a107ac5bbbe60ae7049`
- **Chain tx hash:** `0xd56d1d748d8253073d6526feeb18173c43487ebf92bc6094a2965abfa4dec1fc`
- **Chain block:** `41554213`

---

## File Mapping

This bundle consists of: `L1-125.md`, `L1-125.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-125.md` | Source of truth — edit this | Human or LLM |
| `L1-125.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 `L1-125`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._