# ⚛  L1 Principle — Fokker-Planck — Coulomb-collision kinetic

**ID:** `L1-347` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Plasma Physics — *Plasma Coulomb collision operator*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** distribution f with collisions
> **📡 Carrier:** plasma · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554095

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## 🧠 1. Introduction

**Fokker-Planck — Coulomb-collision kinetic** is a **linear inverse problem** whose unknown lives in **distribution f with collisions** space, within the **Plasma Coulomb collision operator** sub-domain of **Plasma Physics**.

Measurements consist of plasma particle and field diagnostics via a **thomson scattering fast diagnostics** sensing mechanism.

The forward operator applies, in order: L · collision operator operator; L · drift diffusion v operator; L · relaxation operator; int · phase space operator.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · collision operator → L · drift diffusion v → L · relaxation → int · phase space → **y** (detector).

```
y = `int.phase_space` `L.relaxation` `L.drift_diffusion_v` `L.collision_operator` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.collision_operator` | L · collision operator operator |
| `L.drift_diffusion_v` | L · drift diffusion v operator |
| `L.relaxation` | L · relaxation operator |
| `int.phase_space` | Int · phase space operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Plasma Physics |
| Sub domain | Plasma Coulomb collision operator |
| Carrier | plasma |
| Problem class | linear_inverse |
| Solution space | distribution_f_with_collisions |
| Noise model | gaussian |
| Integration axis | phase_space |
| 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:** `fokker_planck` · **Forward operator:** `fokker_planck_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N v | — | 128 |
| Snr db | dB | 25 |
| N times | — | 1000 |
| Log lambda error | — | 0 |
| Mass ratio error | — | 0 |
| Collision freq error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N v | — | 16 – 2048 |
| Snr db | dB | 0 – 40 |
| N times | — | 50 – 100000 |
| Log lambda error | — | 0.0 – 0.1 |
| Mass ratio error | — | 0.0 – 0.1 |
| Collision freq error | — | 0.0 – 0.2 |

## 🎯 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 = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x2766624865e2ccec9ad01f38ebf0def6fc2dd97e1b78779a9e4024ba1b1a5b33`
- **Chain tx hash:** `0x59d14c6dae81c0f5bb37ea9c0f7813af156903b231b4b87faf613b85d7533627`
- **Chain block:** `41554095`

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## File Mapping

This bundle consists of: `L1-347.md`, `L1-347.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-347.md` | Source of truth — edit this | Human or LLM |
| `L1-347.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 `L1-347`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._