# ⚛  L1 Principle — LED and Laser Rate Equations

**ID:** `L1-494` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Semiconductor Physics — *Optoelectronic devices*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** N S time trajectories
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554142

---

## 🧠 1. Introduction

**LED and Laser Rate Equations** is a **nonlinear inverse problem** whose unknown lives in **N S time trajectories** space, within the **Optoelectronic devices** sub-domain of **Semiconductor Physics**.

Measurements consist of electrons collected by an electron detector via a **electroluminescence modulation** sensing mechanism.

The forward operator applies, in order: E · carrier rate operator; E · photon rate operator; D · time · implicit operator; O · LI modulation operator.

Observations are corrupted by additive Gaussian noise. Well-posed for typical semiconductor lasers; threshold bifurcation well-resolved; multi-mode regimes need spatial extension.

## ⚙ 2. Forward Model

Physical chain: **x** → E · carrier rate → E · photon rate → D · time · implicit → O · LI modulation → **y** (detector).

```
y = `O.LI_modulation` `D.time.implicit` `E.photon_rate` `E.carrier_rate` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.carrier_rate` | E · carrier rate operator |
| `E.photon_rate` | E · photon rate operator |
| `D.time.implicit` | D · time · implicit operator |
| `O.LI_modulation` | O · li modulation operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Semiconductor Physics |
| Sub domain | Optoelectronic devices |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | N_S_time_trajectories |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Well-posed for typical semiconductor lasers; threshold bifurcation well-resolved; multi-mode regimes need spatial extension.

| Metric | Value |
|---|---|
| Metric | LI_relative_error |
| Secondary | modulation_bandwidth_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `laser_rate_parameter_inversion` · **Forward operator:** `rate_eq_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 298 |
| I ma | — | 20 |
| V um3 | — | 100 |
| N tr cm3 | — | 1000000000000000000 |
| Lambda nm | nm | 850 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 200 – 400 |
| I ma | — | 0.1 – 1000 |
| V um3 | — | 1 – 10000 |
| N tr cm3 | — | 100000000000000000 – 10000000000000000000 |
| Lambda nm | nm | 400 – 1550 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** L-I relative error <= 0.03

| Metric | Range |
|---|---|
| Li relative error | 0.01 – 0.4 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **LI_relative_error**, with κ = `100` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x774268abb12879adc3f1f27401fb91cce67f6065f3dd6600fc0f7d9afc4adced`
- **Chain tx hash:** `0x97d42d45519fb670b38c993e1743df8e600dccb76be2725224c60a2b69206446`
- **Chain block:** `41554142`

---

## File Mapping

This bundle consists of: `L1-494.md`, `L1-494.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-494.md` | Source of truth — edit this | Human or LLM |
| `L1-494.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.
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> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-494`._
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