# ⚛  L1 Principle — Phonon Dispersion Relations

**ID:** `L1-317` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *Lattice dynamics*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** phonon dispersion omega q
> **📡 Carrier:** phonon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554080

---

## 🧠 1. Introduction

**Phonon Dispersion Relations** is a **linear inverse problem** whose unknown lives in **phonon dispersion omega q** space, within the **Lattice dynamics** sub-domain of **Quantum Mechanics**.

Measurements consist of lattice phonons probed indirectly via a **inelastic neutron scattering** sensing mechanism.

The forward operator applies, in order: E · force constants operator; E · dynamical matrix operator; computes eigen-pairs of a linear operator; O · phonon dispersion operator.

Observations are corrupted by additive Gaussian noise. Well-posed linear eigenproblem; inversion from INS is well-posed under symmetry constraints.

## ⚙ 2. Forward Model

Physical chain: **x** → E · force constants → E · dynamical matrix → O · phonon dispersion → **y** (detector).

```
y = `O.phonon_dispersion` `E.dynamical_matrix` `E.force_constants` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.force_constants` | E · force constants operator |
| `E.dynamical_matrix` | E · dynamical matrix operator |
| `O.phonon_dispersion` | O · phonon dispersion operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `E.eigensolve` | Computes eigen-pairs of a linear operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Quantum Mechanics |
| Sub domain | Lattice dynamics |
| Carrier | phonon |
| Problem class | linear_inverse |
| Solution space | phonon_dispersion_omega_q |
| Noise model | gaussian |
| Integration axis | q_space |
| Difficulty delta | 3 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Well-posed linear eigenproblem; inversion from INS is well-posed under symmetry constraints.

| Metric | Value |
|---|---|
| Metric | dispersion_L2_error_meV |
| Secondary | gamma_mode_frequency_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `phonon_dispersion` · **Forward operator:** `phonon_dispersion_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Q grid | — | 32 |
| N atoms | — | 2 |
| Species | — | Si |
| N neighbor shells | — | 5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Q grid | — | 8 – 256 |
| N atoms | — | 1 – 100 |
| N neighbor shells | — | 1 – 20 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** dispersion L2 <= 1.5 meV

| Metric | Range |
|---|---|
| Dispersion l2 error mev | 0.5 – 30 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **dispersion_L2_error_meV**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x77f8ec38900b0f0f1f1049c21151b57f20bddd224bce3abc62d20f52d69fd060`
- **Chain tx hash:** `0xbc873e74541376d59ca479a241a42be9ff541a017f79511bbb3f9f6016e4795a`
- **Chain block:** `41554080`

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## File Mapping

This bundle consists of: `L1-317.md`, `L1-317.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-317.md` | Source of truth — edit this | Human or LLM |
| `L1-317.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 `L1-317`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._