# ⚛  L1 Principle — Waveguide Modes — dispersion and S-parameter analysis

**ID:** `L1-239` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Rectangular/coax/photonic waveguide eigenmodes*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** modal fields and S params
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553988

---

## 🧠 1. Introduction

**Waveguide Modes — dispersion and S-parameter analysis** is a **linear inverse problem** whose unknown lives in **modal fields and S params** space, within the **Rectangular/coax/photonic waveguide eigenmodes** sub-domain of **Electromagnetics**.

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

The forward operator applies, in order: L · eigenmode solve operator; L · propagate operator; L · S param extract 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 · eigenmode solve → L · propagate → L · S param extract → Frequency integration → **y** (detector).

```
y = ∫dω `L.S_param_extract` `L.propagate` `L.eigenmode_solve` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.eigenmode_solve` | L · eigenmode solve operator |
| `L.propagate` | L · propagate operator |
| `L.S_param_extract` | L · s param extract operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Rectangular/coax/photonic waveguide eigenmodes |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | modal_fields_and_S_params |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 3 |
| L dag | 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:** `waveguide` · **Forward operator:** `waveguide_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 40 |
| N freqs | — | 401 |
| N modes | — | 5 |
| Dim error | — | 0 |
| Freq ghz max | — | 40 |
| Permittivity error | — | 0 |
| Conductor loss error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 10 – 60 |
| N freqs | — | 11 – 10001 |
| N modes | — | 1 – 50 |
| Dim error | — | 0.0 – 0.02 |
| Freq ghz max | — | 0.1 – 500 |
| Permittivity error | — | 0.0 – 0.1 |
| Conductor loss error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 28.0

| 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:** `0x2541b4c36b163d361b5e9cb6075371dc9f8930b168893777df997918e12d003c`
- **Chain tx hash:** `0x16f6b3d47dd165786fe984a72ea9c4af02e4698f8e96e47df1e10e43fb8760b4`
- **Chain block:** `41553988`

---

## File Mapping

This bundle consists of: `L1-239.md`, `L1-239.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-239.md` | Source of truth — edit this | Human or LLM |
| `L1-239.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 `L1-239`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._