# ⚛  L1 Principle — Non-Equilibrium Green's Function (NEGF) Quantum Transport

**ID:** `L1-491` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Semiconductor Physics — *Nanoscale quantum transport*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** green functions complex matrix
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554141

---

## 🧠 1. Introduction

**Non-Equilibrium Green's Function (NEGF) Quantum Transport** is a **nonlinear inverse problem** whose unknown lives in **green functions complex matrix** space, within the **Nanoscale quantum transport** sub-domain of **Semiconductor Physics**.

Measurements consist of electrons collected by an electron detector via a **quantum tunneling spectroscopy** sensing mechanism.

The forward operator applies, in order: E · hamiltonian build operator; E · self energy operator; computes eigen-pairs of a linear operator; O · current LDOS operator.

Observations are corrupted by additive Gaussian noise. Dyson equation is numerically well-posed in principle but conditioning deteriorates with device size; self-consistent Born (SCBA) loops add non-linear coupling.

## ⚙ 2. Forward Model

Physical chain: **x** → E · hamiltonian build → E · self energy → O · current LDOS → **y** (detector).

```
y = `O.current_LDOS` `E.self_energy` `E.hamiltonian_build` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.hamiltonian_build` | E · hamiltonian build operator |
| `E.self_energy` | E · self energy operator |
| `O.current_LDOS` | O · current ldos 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 | Semiconductor Physics |
| Sub domain | Nanoscale quantum transport |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | green_functions_complex_matrix |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 10 |
| L dag | 4 |

## 📡 4. Measurement Model

Dyson equation is numerically well-posed in principle but conditioning deteriorates with device size; self-consistent Born (SCBA) loops add non-linear coupling.

| Metric | Value |
|---|---|
| Metric | IV_log_relative_error |
| Secondary | LDOS_KL_divergence |

## 📏 5. Operating Range (Ω)

**Center problem class:** `negf_inversion` · **Forward operator:** `negf_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 300 |
| V bias | — | 0.5 |
| N sites | — | 200 |
| N energy | — | 500 |
| Scba tol | — | 0.0001 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 4 – 400 |
| V bias | — | 0.001 – 2.0 |
| N sites | — | 20 – 5000 |
| N energy | — | 50 – 5000 |
| Scba tol | — | 1e-06 – 0.01 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** log10(I) error <= 0.04

| Metric | Range |
|---|---|
| Iv log relative error | 0.005 – 0.4 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **IV_log_relative_error**, with κ = `5000` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x4783c3f1fe60427384e4c98c762ee9d7203c7a31e74ba3c8d963fad1c911fead`
- **Chain tx hash:** `0xa956b1c693a701b2798fc9a1f72b05c007c92a9ad1437127cec6e3f35778567a`
- **Chain block:** `41554141`

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## File Mapping

This bundle consists of: `L1-491.md`, `L1-491.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-491.md` | Source of truth — edit this | Human or LLM |
| `L1-491.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 `L1-491`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._