# ⚛  L1 Principle — SAXS — Small-Angle X-ray Scattering

**ID:** `L1-126` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Scientific Instrumentation — *Nanoscale structural scattering (1-100 nm)*
> **🎯 Problem class:** inverse fourier scatter · **🧮 Solution space:** pair distribution or envelope
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554213

---

## 🧠 1. Introduction

**SAXS — Small-Angle X-ray Scattering** is a **inverse fourier scatter** whose unknown lives in **pair distribution or envelope** space, within the **Nanoscale structural scattering (1-100 nm)** sub-domain of **Scientific Instrumentation**.

Measurements consist of photons collected by an optical detector via a **xray small 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. Inverse Fourier transform from I(q) to P(r) is ill-posed (ATSAS-GNOM uses regularization); shape reconstruction requires ab-initio bead models. Mismatch: aggregation, polydispersity, buffer mismatch.

## ⚙ 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 | Nanoscale structural scattering (1-100 nm) |
| Carrier | photon |
| Problem class | inverse_fourier_scatter |
| Solution space | pair_distribution_or_envelope |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 3 |
| L dag | 2.7 |

## 📡 4. Measurement Model

Inverse Fourier transform from I(q) to P(r) is ill-posed (ATSAS-GNOM uses regularization); shape reconstruction requires ab-initio bead models. Mismatch: aggregation, polydispersity, buffer mismatch.

| Metric | Value |
|---|---|
| Metric | Rg_RMSE_nm |
| Secondary | chi2_reduced |

## 📏 5. Operating Range (Ω)

**Center problem class:** `biosaxs_globular_protein` · **Forward operator:** `saxs_iq_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N q | — | 400 |
| Rg nm | nm | 2 |
| Q max | — | 0.5 |
| Q min | — | 0.01 |
| Concentration mg ml | — | 1 |
| Aggregation fraction | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N q | — | 50 – 2000 |
| Rg nm | nm | 0.5 – 50 |
| Q max | — | 0.1 – 1.0 |
| Q min | — | 0.001 – 0.1 |
| Beam damage | — | 0.0 – 0.5 |
| Polydispersity | — | 0.0 – 0.5 |
| Buffer mismatch | — | 0.0 – 0.3 |
| Q range truncation | — | 0.0 – 0.5 |
| Concentration mg ml | — | 0.1 – 20 |
| Aggregation fraction | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.05 nm Rg RMSE

| Metric | Range |
|---|---|
| Rg rmse nm | 0.01 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **Rg_RMSE_nm**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x6ea68f7f1986db6a5ed685b31657281b924c5f5c3a38a47cae6b5e03d6b986ae`
- **Chain tx hash:** `0x671a3fa51d16ae2af60593a60dbc756a1ec43d71c82ba0962cef38eb9f046716`
- **Chain block:** `41554213`

---

## File Mapping

This bundle consists of: `L1-126.md`, `L1-126.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-126.md` | Source of truth — edit this | Human or LLM |
| `L1-126.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 `L1-126`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._