# ⚛  L1 Principle — Maxwell Full-Wave — time-domain 3D Maxwell equations

**ID:** `L1-232` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *3D vector Maxwell time-domain simulation*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D E H field spacetime
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41553987

---

## 🧠 1. Introduction

**Maxwell Full-Wave — time-domain 3D Maxwell equations** is a **linear inverse problem** whose unknown lives in **3D E H field spacetime** space, within the **3D vector Maxwell time-domain simulation** sub-domain of **Electromagnetics**.

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

The forward operator applies, in order: L · source injection operator; L · curl curl operator; L · constitutive operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. CFL-stable explicit time stepping; ill-posed inverse for full 3D eps(x) reconstruction.

## ⚙ 2. Forward Model

Physical chain: **x** → L · source injection → L · curl curl → L · constitutive → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.constitutive` `L.curl_curl` `L.source_injection` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.source_injection` | L · source injection operator |
| `L.curl_curl` | L · curl curl operator |
| `L.constitutive` | L · constitutive operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | 3D vector Maxwell time-domain simulation |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | 3D_E_H_field_spacetime |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 10 |
| L dag | 3.8 |

## 📡 4. Measurement Model

CFL-stable explicit time stepping; ill-posed inverse for full 3D eps(x) reconstruction.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `maxwell_tm` · **Forward operator:** `maxwell_tm_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 128 |
| Snr db | dB | 30 |
| N probes | — | 16 |
| Mu error | — | 0 |
| N samples | — | 512 |
| Eps error | — | 0 |
| Sigma error | — | 0 |
| Boundary pml error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Z | — | 32 – 1024 |
| Snr db | dB | 0 – 40 |
| N probes | — | 4 – 1024 |
| Mu error | — | 0.0 – 0.05 |
| N samples | — | 64 – 10000 |
| Eps error | — | 0.0 – 0.1 |
| Sigma error | — | 0.0 – 0.1 |
| Boundary pml 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 = `1600` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x177babd070bb2e7be404fa90e2b0d96b63b48882e2d15ade5f0cc4a6167b8549`
- **Chain tx hash:** `0x57a5672d690bb3e38a9ddc1a2f7d34b146ca09665ce82986a7a2123cd863b89c`
- **Chain block:** `41553987`

---

## File Mapping

This bundle consists of: `L1-232.md`, `L1-232.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-232.md` | Source of truth — edit this | Human or LLM |
| `L1-232.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 `L1-232`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._