# ⚛  L1 Principle — Bethe-Salpeter Equation for Excitons

**ID:** `L1-320` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *Optical excitations from GW+BSE*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** exciton spectrum
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554080

---

## 🧠 1. Introduction

**Bethe-Salpeter Equation for Excitons** is a **nonlinear inverse problem** whose unknown lives in **exciton spectrum** space, within the **Optical excitations from GW+BSE** sub-domain of **Quantum Mechanics**.

Measurements consist of photons collected by an optical detector via a **optical absorption** sensing mechanism.

The forward operator applies, in order: E · GW bands operator; E · kernel K eh operator; computes eigen-pairs of a linear operator; O · absorption operator.

Observations are corrupted by additive Gaussian noise. Tamm-Dancoff approximation often used; scaling as N^4 limits size.

## ⚙ 2. Forward Model

Physical chain: **x** → E · GW bands → E · kernel K eh → O · absorption → **y** (detector).

```
y = `O.absorption` `E.kernel_K_eh` `E.GW_bands` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.GW_bands` | E · gw bands operator |
| `E.kernel_K_eh` | E · kernel k eh operator |
| `O.absorption` | O · absorption 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 | Optical excitations from GW+BSE |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | exciton_spectrum |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 10 |
| L dag | 4.2 |

## 📡 4. Measurement Model

Tamm-Dancoff approximation often used; scaling as N^4 limits size.

| Metric | Value |
|---|---|
| Metric | exciton_binding_error_eV |
| Secondary | optical_spectrum_L2 |

## 📏 5. Operating Range (Ω)

**Center problem class:** `bethe_salpeter` · **Forward operator:** `BSE_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N kpoints | — | 64 |
| N cv pairs | — | 500 |
| Omega range ev | — | 0 – 10 |
| Dielectric screening | — | static |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N kpoints | — | 8 – 10000 |
| N cv pairs | — | 50 – 50000 |
| Dielectric screening | — | static, dynamic |

## 🎯 6. Tolerance (ε)

**Center tolerance:** binding error <= 0.05 eV

| Metric | Range |
|---|---|
| Exciton binding error ev | 0.01 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **exciton_binding_error_eV**, with κ = `3000` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x636afa279a36289b70b868a82409af8e16d1ae7c4f7cce219babbc46858b4b62`
- **Chain tx hash:** `0x8875687ca7bf4a1a6d61606f29dc32a63abb12b369d806032eaaadce12c1fc50`
- **Chain block:** `41554080`

---

## File Mapping

This bundle consists of: `L1-320.md`, `L1-320.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-320.md` | Source of truth — edit this | Human or LLM |
| `L1-320.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 `L1-320`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._