# 📐  L2 Spec — Waveguide Modes — dispersion and S-parameter analysis Mismatch-Only Spec

**ID:** `L2-239-001` · **Status:** ⊙ Testnet (genesis catalog)
**Parents:** ⬆ Principle [`L1-239`](/artifact/l1/L1-239)

> **🎯 Tolerance (ε):** expression
> **⚖ Difficulty score:** 0.48 · **Delta tier:** 5
> **⛓ On-chain block:** 41553991

---

## 1. Domain Ω

**Type:** `range`

| 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 |

## 2. Tolerance (ε)

**Type:** `fn`

```
20.0 + 0.3*SNR_dB - 50.0*dim_error - 30.0*permittivity_error - 5.0*conductor_loss_error
```

## 3. Difficulty

| Property | Value |
|---|---|
| Difficulty score | 0.48 |
| Delta tier | 5 |

## 4. Fingerprint

`8f2eb7773dd6c41af609a54d60bb7400887b2e3e1409ece38c05e85417d1f7b0`

## 5. On-chain Registration

- **Chain hash:** `0xd34a25d7b33957b06c0c67dc4ef732117c8a3106d3902cd94a5cd1227c86c7ae`
- **Chain tx hash:** `0xdd536b3225808d7285c59e8e5e8e04c0f13539401a8e5a1826ffcfb9a12cc3a2`
- **Chain block:** `41553991`

---

## File Mapping

This bundle consists of: `L2-239-001.md`, `L2-239-001.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L2-239-001.md` | Source of truth — edit this | Human or LLM |
| `L2-239-001.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 `L2-239-001`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._