# ⚛  L1 Principle — Electrostatics — Poisson equation inversion

**ID:** `L1-250` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Poisson equation for electrostatic potential*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D potential field
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554000

---

## 🧠 1. Introduction

**Electrostatics — Poisson equation inversion** is a **linear inverse problem** whose unknown lives in **3D potential field** space, within the **Poisson equation for electrostatic potential** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **e field probes** sensing mechanism.

The forward operator applies, in order: L · charge source operator; L · poisson operator; L · boundary match operator; pixel-level spatial averaging on the detector.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · charge source → L · poisson → L · boundary match → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.boundary_match` `L.poisson` `L.charge_source` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.charge_source` | L · charge source operator |
| `L.poisson` | L · poisson operator |
| `L.boundary_match` | L · boundary match operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Poisson equation for electrostatic potential |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | 3D_potential_field |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 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:** `electrostatic` · **Forward operator:** `electrostatic_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 32 |
| Snr db | dB | 30 |
| N probes | — | 64 |
| Eps r error | — | 0 |
| Boundary error | — | 0 |
| Sensor pos error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Z | — | 8 – 512 |
| Snr db | dB | 0 – 40 |
| N probes | — | 4 – 4096 |
| Eps r error | — | 0.0 – 0.1 |
| Boundary error | — | 0.0 – 0.2 |
| Sensor pos error | — | 0.0 – 0.02 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 25.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x75b95302856b368ca1e67824c6a406fbe89ecbea8909af56b297c44c374168d3`
- **Chain tx hash:** `0x50c0de0159c2e99287fef4f63edc1a88888475a337b851bb91b1d8206f088a69`
- **Chain block:** `41554000`

---

## File Mapping

This bundle consists of: `L1-250.md`, `L1-250.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-250.md` | Source of truth — edit this | Human or LLM |
| `L1-250.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 `L1-250`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._