# ⚛  L1 Principle — Electron-Phonon Coupling and Eliashberg Theory

**ID:** `L1-325` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *Phonon-mediated superconductivity*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** eliashberg spectral function
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554081

---

## 🧠 1. Introduction

**Electron-Phonon Coupling and Eliashberg Theory** is a **nonlinear inverse problem** whose unknown lives in **eliashberg spectral function** space, within the **Phonon-mediated superconductivity** sub-domain of **Quantum Mechanics**.

Measurements consist of electrons collected by an electron detector via a **Tc measurement IXS** sensing mechanism.

The forward operator applies, in order: E · phonon dispersion operator; E · electron phonon operator; O · composite method operator; O · Tc operator.

Observations are corrupted by additive Gaussian noise. Non-linear; T_c sensitive to spectral function shape; inversion ill-conditioned.

## ⚙ 2. Forward Model

Physical chain: **x** → E · phonon dispersion → E · electron phonon → O · composite method → O · Tc → **y** (detector).

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

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.phonon_dispersion` | E · phonon dispersion operator |
| `E.electron_phonon` | E · electron phonon operator |
| `O.composite_method` | O · composite method operator |
| `O.Tc` | O · tc operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Quantum Mechanics |
| Sub domain | Phonon-mediated superconductivity |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | eliashberg_spectral_function |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 10 |
| L dag | 3.9 |

## 📡 4. Measurement Model

Non-linear; T_c sensitive to spectral function shape; inversion ill-conditioned.

| Metric | Value |
|---|---|
| Metric | Tc_relative_error |
| Secondary | alpha2F_L2_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `electron_phonon_coupling` · **Forward operator:** `eliashberg_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N bands | bands | 20 |
| Mu star | — | 0.1 |
| Species | — | Pb |
| N kpoints | — | 32 |
| N qpoints | — | 16 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N bands | bands | 5 – 200 |
| Mu star | — | 0.05 – 0.3 |
| N kpoints | — | 8 – 1000 |
| N qpoints | — | 4 – 256 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** T_c rel error <= 0.05

| Metric | Range |
|---|---|
| Tc relative error | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **Tc_relative_error**, with κ = `1500` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xa3bfae629bea0395757305e590a06d442b84dec6abc40dcea12dc9c9cea588f0`
- **Chain tx hash:** `0xbc5c6d23ecdd75360cee6049d543746a9d626ff56173749251831649a7996d5e`
- **Chain block:** `41554081`

---

## File Mapping

This bundle consists of: `L1-325.md`, `L1-325.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-325.md` | Source of truth — edit this | Human or LLM |
| `L1-325.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 `L1-325`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._