# ⚛  L1 Principle — THz Time-Domain Spectroscopy — material parameter extraction

**ID:** `L1-256` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *THz-TDS complex refractive-index extraction*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex n vs freq
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554001

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## 🧠 1. Introduction

**THz Time-Domain Spectroscopy — material parameter extraction** is a **linear inverse problem** whose unknown lives in **complex n vs freq** space, within the **THz-TDS complex refractive-index extraction** sub-domain of **Electromagnetics**.

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

The forward operator applies, in order: L · thz pulse operator; L · transmit reflect operator; L · fft extract operator; detector accumulates flux over the exposure window.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · thz pulse → L · transmit reflect → L · fft extract → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.fft_extract` `L.transmit_reflect` `L.thz_pulse` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.thz_pulse` | L · thz pulse operator |
| `L.transmit_reflect` | L · transmit reflect operator |
| `L.fft_extract` | L · fft extract operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | THz-TDS complex refractive-index extraction |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | complex_n_vs_freq |
| Noise model | gaussian |
| Integration axis | temporal |
| 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:** `thz_tds` · **Forward operator:** `thz_tds_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| Fs thz | — | 0.1 |
| N samples | — | 4096 |
| Etaloning error | — | 0 |
| Beam align error | — | 0 |
| Sample thickness error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| Fs thz | — | 0.001 – 100 |
| N samples | — | 256 – 1048576 |
| Etaloning error | — | 0.0 – 0.3 |
| Beam align error | — | 0.0 – 0.05 |
| Sample thickness error | — | 0.0 – 0.05 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `300` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xe26bc7ee9dea43480d380ec0fe92eb7549dd3ff923e34c3be6fa759ae3c09144`
- **Chain tx hash:** `0x2c90128db37dbd158dd58c24a5eaf6cb59019f600318cae13b1bd6a1dabdb171`
- **Chain block:** `41554001`

---

## File Mapping

This bundle consists of: `L1-256.md`, `L1-256.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-256.md` | Source of truth — edit this | Human or LLM |
| `L1-256.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 `L1-256`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._