# ⚛  L1 Principle — Method of Moments (MoM) — integral-equation EM

**ID:** `L1-235` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Surface/volume integral-equation MoM*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** surface currents and rcs
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553987

---

## 🧠 1. Introduction

**Method of Moments (MoM) — integral-equation EM** is a **linear inverse problem** whose unknown lives in **surface currents and rcs** space, within the **Surface/volume integral-equation MoM** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **far field antenna** sensing mechanism.

The forward operator applies, in order: L · efie mfie operator; L · rwg expand operator; L · linear solve operator; integration over a band of frequencies.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · efie mfie → L · rwg expand → L · linear solve → Frequency integration → **y** (detector).

```
y = ∫dω `L.linear_solve` `L.rwg_expand` `L.efie_mfie` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.efie_mfie` | L · efie mfie operator |
| `L.rwg_expand` | L · rwg expand operator |
| `L.linear_solve` | L · linear solve operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Surface/volume integral-equation MoM |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | surface_currents_and_rcs |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 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:** `mom` · **Forward operator:** `mom_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N mesh | — | 10000 |
| Snr db | dB | 30 |
| N freqs | — | 16 |
| Bc error | — | 0 |
| N angles | — | 256 |
| Mesh error | — | 0 |
| Material error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N mesh | — | 100 – 1000000 |
| Snr db | dB | 0 – 40 |
| N freqs | — | 1 – 256 |
| Bc error | — | 0.0 – 0.2 |
| N angles | — | 16 – 4096 |
| Mesh error | — | 0.0 – 0.1 |
| Material error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 24.0

| 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:** `0xac55197b2796857a918394a278f96b1393e62a9abf97b5ef91c933988f63eb89`
- **Chain tx hash:** `0x3e6b80bcaf774329b736d26a28e3bd016474af7295377d7a6fee61a0148aa730`
- **Chain block:** `41553987`

---

## File Mapping

This bundle consists of: `L1-235.md`, `L1-235.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-235.md` | Source of truth — edit this | Human or LLM |
| `L1-235.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 `L1-235`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._