# ⚛  L1 Principle — Mie Scattering — rigorous spherical-particle scattering

**ID:** `L1-243` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Particle size distribution from Mie pattern*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** particle size distribution
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553989

---

## 🧠 1. Introduction

**Mie Scattering — rigorous spherical-particle scattering** is a **nonlinear inverse problem** whose unknown lives in **particle size distribution** space, within the **Particle size distribution from Mie pattern** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **angular scatter detector** sensing mechanism.

The forward operator applies, in order: L · mie coeff operator; L · series sum operator; L · polydispersity average operator; integration over the solid angle of incidence/emission.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · mie coeff → L · series sum → L · polydispersity average → Angular integration → **y** (detector).

```
y = ∫dΩ `L.polydispersity_average` `L.series_sum` `L.mie_coeff` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.mie_coeff` | L · mie coeff operator |
| `L.series_sum` | L · series sum operator |
| `L.polydispersity_average` | L · polydispersity average operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Particle size distribution from Mie pattern |
| Carrier | em |
| Problem class | nonlinear_inverse |
| Solution space | particle_size_distribution |
| Noise model | gaussian |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `mie` · **Forward operator:** `mie_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| N angles | — | 360 |
| N wavelengths | — | 3 |
| N complex error | — | 0 |
| Concentration error | — | 0 |
| Polydispersity width error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| N angles | — | 18 – 3600 |
| N wavelengths | — | 1 – 50 |
| N complex error | — | 0.0 – 0.05 |
| Concentration error | — | 0.0 – 0.3 |
| Polydispersity width error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xeb1ac9320f533261622240766b5919a639edc4323dd50764cd76131aa2b665c2`
- **Chain tx hash:** `0x466091f9732a9a9834e3540b87e6f0cb4f54be175bc2725f77669214a54c6e6a`
- **Chain block:** `41553989`

---

## File Mapping

This bundle consists of: `L1-243.md`, `L1-243.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-243.md` | Source of truth — edit this | Human or LLM |
| `L1-243.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 `L1-243`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._