# ⚛  L1 Principle — Quantum Scattering Theory

**ID:** `L1-314` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *S-matrix and phase shifts*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** phase shifts scattering amplitude
> **📡 Carrier:** electron · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554079

---

## 🧠 1. Introduction

**Quantum Scattering Theory** is a **nonlinear inverse problem** whose unknown lives in **phase shifts scattering amplitude** space, within the **S-matrix and phase shifts** sub-domain of **Quantum Mechanics**.

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

The forward operator applies, in order: E · lippmann schwinger operator; computes eigen-pairs of a linear operator; O · cross section operator.

Observations are corrupted by Poisson counting noise. Well-posed forward; inverse scattering (reconstructing V from f(E,theta)) is Borg-Marchenko-style; unique with phase info.

## ⚙ 2. Forward Model

Physical chain: **x** → E · lippmann schwinger → O · cross section → **y** (detector).

```
y = `O.cross_section` `E.lippmann_schwinger` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.lippmann_schwinger` | E · lippmann schwinger operator |
| `O.cross_section` | O · cross section 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 | S-matrix and phase shifts |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | phase_shifts_scattering_amplitude |
| Noise model | poisson |
| Integration axis | angle_and_energy |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Well-posed forward; inverse scattering (reconstructing V from f(E,theta)) is Borg-Marchenko-style; unique with phase info.

| Metric | Value |
|---|---|
| Metric | sigma_total_relative_error |
| Secondary | phase_shift_error_deg |

## 📏 5. Operating Range (Ω)

**Center problem class:** `quantum_scattering` · **Forward operator:** `scattering_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| E ev | — | 10 |
| L max | — | 8 |
| N grid | — | 500 |
| V well ev | — | 5 |
| Target type | — | atomic |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| E ev | — | 0.001 – 1000000.0 |
| L max | — | 1 – 50 |
| N grid | — | 100 – 10000 |
| V well ev | — | 0.1 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** sigma error <= 0.03

| Metric | Range |
|---|---|
| Sigma total relative error | 0.005 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **sigma_total_relative_error**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x5eeb1cfc8baec60d6b737b7493896a6f892bbf0c686bce3979281e612c6a5823`
- **Chain tx hash:** `0x465854a0ec7d1dd8f0e40494d21f7caf07ba95d67b5ad6dd432bf33f4f0c6df4`
- **Chain block:** `41554079`

---

## File Mapping

This bundle consists of: `L1-314.md`, `L1-314.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-314.md` | Source of truth — edit this | Human or LLM |
| `L1-314.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 `L1-314`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._