# ⚛  L1 Principle — GW Approximation for Quasiparticle Energies

**ID:** `L1-319` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *Many-body perturbation theory*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** quasiparticle energies
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554080

---

## 🧠 1. Introduction

**GW Approximation for Quasiparticle Energies** is a **nonlinear inverse problem** whose unknown lives in **quasiparticle energies** space, within the **Many-body perturbation theory** sub-domain of **Quantum Mechanics**.

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

The forward operator applies, in order: E · G function operator; E · W screening operator; applies a smooth nonlinear function element-wise; O · quasiparticle operator.

Observations are corrupted by additive Gaussian noise. Moderately ill-posed; self-consistent GW (scGW) partially removes G_0W_0 starting-point dependence.

## ⚙ 2. Forward Model

Physical chain: **x** → E · G function → E · W screening → Pointwise nonlinearity → O · quasiparticle → **y** (detector).

```
y = `O.quasiparticle` f(·) `E.W_screening` `E.G_function` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.G_function` | E · g function operator |
| `E.W_screening` | E · w screening operator |
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.quasiparticle` | O · quasiparticle operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Quantum Mechanics |
| Sub domain | Many-body perturbation theory |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | quasiparticle_energies |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 10 |
| L dag | 4 |

## 📡 4. Measurement Model

Moderately ill-posed; self-consistent GW (scGW) partially removes G_0W_0 starting-point dependence.

| Metric | Value |
|---|---|
| Metric | band_gap_error_eV |
| Secondary | energy_dispersion_L2_eV |

## 📏 5. Operating Range (Ω)

**Center problem class:** `gw_approximation` · **Forward operator:** `GW_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N omega | — | 20 |
| Species | — | Si |
| N kpoints | — | 8 |
| Cutoff ev | — | 400 |
| N bands gw | — | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N omega | — | 10 – 200 |
| N kpoints | — | 1 – 1000 |
| Cutoff ev | — | 200 – 2000 |
| N bands gw | — | 20 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** gap error <= 0.07 eV

| Metric | Range |
|---|---|
| Band gap error ev | 0.02 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **band_gap_error_eV**, with κ = `2500` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x82cd9e6737865a0e949c907619b01c3673a8f485b825c68dad90e511e017f853`
- **Chain tx hash:** `0xdf4152101dc83ff1b363b6a2ef186a434778a31de966c6964b2e023ce395ae77`
- **Chain block:** `41554080`

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## File Mapping

This bundle consists of: `L1-319.md`, `L1-319.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-319.md` | Source of truth — edit this | Human or LLM |
| `L1-319.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 `L1-319`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._