# ⚛  L1 Principle — Quantum Tunneling Through Barriers

**ID:** `L1-315` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *WKB and transfer matrix*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** V x barrier profile
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554079

---

## 🧠 1. Introduction

**Quantum Tunneling Through Barriers** is a **nonlinear inverse problem** whose unknown lives in **V x barrier profile** space, within the **WKB and transfer matrix** sub-domain of **Quantum Mechanics**.

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

The forward operator applies, in order: E · barrier profile operator; computes eigen-pairs of a linear operator; O · T E operator.

Observations are corrupted by additive Gaussian noise. Well-posed; inversion of V(x) from T(E) is classic inverse problem (unique for real potentials with smoothness).

## ⚙ 2. Forward Model

Physical chain: **x** → E · barrier profile → O · T E → **y** (detector).

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

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.barrier_profile` | E · barrier profile operator |
| `O.T_E` | O · t e 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 | WKB and transfer matrix |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | V_x_barrier_profile |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Well-posed; inversion of V(x) from T(E) is classic inverse problem (unique for real potentials with smoothness).

| Metric | Value |
|---|---|
| Metric | log_T_error |
| Secondary | barrier_height_error_meV |

## 📏 5. Operating Range (Ω)

**Center problem class:** `quantum_tunneling` · **Forward operator:** `tunneling_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| V0 ev | — | 1 |
| N layers | — | 3 |
| Width nm | nm | 5 |
| E range ev | — | 0.01 – 2 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| V0 ev | — | 0.01 – 10 |
| N layers | — | 1 – 100 |
| Width nm | nm | 0.5 – 100 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** log10(T) error <= 0.05

| Metric | Range |
|---|---|
| Log t error | 0.02 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **log_T_error**, with κ = `100` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x218160b93759f9d9236e1f64ad51175e850cb346a79e6adc232292aa38b36630`
- **Chain tx hash:** `0x1d0735dad7ebc2b6e6113f1305fa5e075a52ed220508268daa63349b11d28adb`
- **Chain block:** `41554079`

---

## File Mapping

This bundle consists of: `L1-315.md`, `L1-315.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-315.md` | Source of truth — edit this | Human or LLM |
| `L1-315.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 `L1-315`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._