# ⚛  L1 Principle — Electronic Band Structure

**ID:** `L1-316` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *Crystal momentum eigenproblem*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** band structure E n k
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554079

---

## 🧠 1. Introduction

**Electronic Band Structure** is a **nonlinear inverse problem** whose unknown lives in **band structure E n k** space, within the **Crystal momentum eigenproblem** sub-domain of **Quantum Mechanics**.

Measurements consist of electrons collected by an electron detector via a **arpes optical absorption** sensing mechanism.

The forward operator applies, in order: E · crystal potential operator; computes eigen-pairs of a linear operator; O · band structure operator.

Observations are corrupted by additive Gaussian noise. Well-posed; DFT gap underestimate (Kohn-Sham gap problem); inverse structure→band mapping requires phase info.

## ⚙ 2. Forward Model

Physical chain: **x** → E · crystal potential → O · band structure → **y** (detector).

```
y = `O.band_structure` `E.crystal_potential` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.crystal_potential` | E · crystal potential operator |
| `O.band_structure` | O · band structure 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 | Crystal momentum eigenproblem |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | band_structure_E_n_k |
| Noise model | gaussian |
| Integration axis | k_space |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Well-posed; DFT gap underestimate (Kohn-Sham gap problem); inverse structure→band mapping requires phase info.

| Metric | Value |
|---|---|
| Metric | band_gap_error_eV |
| Secondary | effective_mass_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `band_structure` · **Forward operator:** `dft_band_structure_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Xc | — | PBE |
| Ecut ev | — | 400 |
| N atoms | — | 2 |
| Species | — | Si |
| N kpoints | — | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Xc | — | LDA, PBE, HSE06 |
| Ecut ev | — | 100 – 2000 |
| N atoms | — | 1 – 100 |
| N kpoints | — | 10 – 10000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** band gap error <= 0.1 eV

| Metric | Range |
|---|---|
| Band gap error ev | 0.02 – 2.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **band_gap_error_eV**, with κ = `600` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x2e034436660d1d9528fcf4496770ad9a044d166f70d9fa0099c908850700c36c`
- **Chain tx hash:** `0xa8a4570fb71fc030771d010711633f48b6ae995429fa695257eb502fd1bb2135`
- **Chain block:** `41554079`

---

## File Mapping

This bundle consists of: `L1-316.md`, `L1-316.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-316.md` | Source of truth — edit this | Human or LLM |
| `L1-316.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 `L1-316`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._