# ⚛  L1 Principle — Vlasov Equation — collisionless kinetic plasma

**ID:** `L1-344` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Plasma Physics — *6D phase-space Vlasov-Poisson*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** distribution function 6D
> **📡 Carrier:** plasma · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554095

---

## 🧠 1. Introduction

**Vlasov Equation — collisionless kinetic plasma** is a **linear inverse problem** whose unknown lives in **distribution function 6D** space, within the **6D phase-space Vlasov-Poisson** sub-domain of **Plasma Physics**.

Measurements consist of plasma particle and field diagnostics via a **faraday cup lif** sensing mechanism.

The forward operator applies, in order: L · distribution f operator; L · advect phase space operator; L · poisson coupling operator; int · phase space operator.

Observations are corrupted by additive Gaussian noise. 6D phase space + time = enormous compute requirements; filamentation common.

## ⚙ 2. Forward Model

Physical chain: **x** → L · distribution f → L · advect phase space → L · poisson coupling → int · phase space → **y** (detector).

```
y = `int.phase_space` `L.poisson_coupling` `L.advect_phase_space` `L.distribution_f` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.distribution_f` | L · distribution f operator |
| `L.advect_phase_space` | L · advect phase space operator |
| `L.poisson_coupling` | L · poisson coupling operator |
| `int.phase_space` | Int · phase space operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Plasma Physics |
| Sub domain | 6D phase-space Vlasov-Poisson |
| Carrier | plasma |
| Problem class | linear_inverse |
| Solution space | distribution_function_6D |
| Noise model | gaussian |
| Integration axis | phase_space |
| Difficulty delta | 10 |
| L dag | 3.8 |

## 📡 4. Measurement Model

6D phase space + time = enormous compute requirements; filamentation common.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `vlasov` · **Forward operator:** `vlasov_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N v | — | 64 |
| N x | — | 64 |
| Snr db | dB | 25 |
| N times | — | 500 |
| Initial f error | — | 0 |
| Electric field error | — | 0 |
| Grid phase space error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N v | — | 16 – 2048 |
| N x | — | 16 – 2048 |
| Snr db | dB | 0 – 40 |
| N times | — | 50 – 100000 |
| Initial f error | — | 0.0 – 0.3 |
| Electric field error | — | 0.0 – 0.2 |
| Grid phase space error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 17.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `1200` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x30b20f532e5028d4e4138b312a504f25139769f09d61a5e61197cc75b405129b`
- **Chain tx hash:** `0x66515ba6cca2e4be7aa6febcf0b17672879f4926b045904711097201019706c4`
- **Chain block:** `41554095`

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## File Mapping

This bundle consists of: `L1-344.md`, `L1-344.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-344.md` | Source of truth — edit this | Human or LLM |
| `L1-344.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 `L1-344`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._