# ⚛  L1 Principle — Phonon Boltzmann Transport Equation

**ID:** `L1-497` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Semiconductor Physics — *Thermal phonon transport*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** phonon distribution n q s
> **📡 Carrier:** phonon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554142

---

## 🧠 1. Introduction

**Phonon Boltzmann Transport Equation** is a **linear inverse problem** whose unknown lives in **phonon distribution n q s** space, within the **Thermal phonon transport** sub-domain of **Semiconductor Physics**.

Measurements consist of lattice phonons probed indirectly via a **thermal conductivity measurement** sensing mechanism.

The forward operator applies, in order: E · phonon dispersion operator; E · scattering rates operator; O · composite method operator; O · kappa operator.

Observations are corrupted by additive Gaussian noise. Well-posed; inversion for branch-resolved tau is ill-conditioned (only integrated kappa measured).

## ⚙ 2. Forward Model

Physical chain: **x** → E · phonon dispersion → E · scattering rates → O · composite method → O · kappa → **y** (detector).

```
y = `O.kappa` `O.composite_method` `E.scattering_rates` `E.phonon_dispersion` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.phonon_dispersion` | E · phonon dispersion operator |
| `E.scattering_rates` | E · scattering rates operator |
| `O.composite_method` | O · composite method operator |
| `O.kappa` | O · kappa operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Semiconductor Physics |
| Sub domain | Thermal phonon transport |
| Carrier | phonon |
| Problem class | linear_inverse |
| Solution space | phonon_distribution_n_q_s |
| Noise model | gaussian |
| Integration axis | wavevector |
| Difficulty delta | 5 |
| L dag | 3.6 |

## 📡 4. Measurement Model

Well-posed; inversion for branch-resolved tau is ill-conditioned (only integrated kappa measured).

| Metric | Value |
|---|---|
| Metric | kappa_relative_error |
| Secondary | kappa_T_curve_RMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `phonon_bte_inversion` · **Forward operator:** `phonon_bte_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 300 |
| N q grid | — | 16 |
| Material | — | Si |
| Grain size um | µm | 10 |
| Isotope naturality | — | 1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 10 – 1500 |
| N q grid | — | 8 – 64 |
| Grain size um | µm | 0.001 – 1000 |
| Isotope naturality | — | 0.0 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** kappa rel error <= 0.03

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **kappa_relative_error**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xa90c41129839d479a6f9f714c56d115fb02474e988cafe143aff0d0f20a41892`
- **Chain tx hash:** `0xe42c883903794eac325d3f907d95f786b069bc9352d658bbd7df71d98e1ecd80`
- **Chain block:** `41554142`

---

## File Mapping

This bundle consists of: `L1-497.md`, `L1-497.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-497.md` | Source of truth — edit this | Human or LLM |
| `L1-497.json` | Structured metadata for the registry | LLM regenerates from the sections above |

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> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
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