# ⚛ L1 Principle — k.p Perturbation Theory
**ID:** `L1-326` · **Status:** ⊙ Testnet (genesis catalog)
> **🌐 Domain:** Quantum Mechanics — *Near-gap band structure expansion*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** effective mass tensor
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554081
---
## 🧠 1. Introduction
**k.p Perturbation Theory** is a **linear inverse problem** whose unknown lives in **effective mass tensor** space, within the **Near-gap band structure expansion** sub-domain of **Quantum Mechanics**.
Measurements consist of electrons collected by an electron detector via a **cyclotron resonance** sensing mechanism.
The forward operator applies, in order: E · momentum matrix operator; E · perturbative expansion operator; O · effective mass operator.
Observations are corrupted by additive Gaussian noise. Well-posed near k_0; fails far from band edge.
## ⚙ 2. Forward Model
Physical chain: **x** → E · momentum matrix → E · perturbative expansion → O · effective mass → **y** (detector).
```
y = `O.effective_mass` `E.perturbative_expansion` `E.momentum_matrix` x + n, n ~ 𝒩(0, σ²)
```
**Measurement DAG:**
| Primitive | What it does |
|---|---|
| `E.momentum_matrix` | E · momentum matrix operator |
| `E.perturbative_expansion` | E · perturbative expansion operator |
| `O.effective_mass` | O · effective mass operator |
## 🔬 3. Physics Fingerprint
| Property | Value |
|---|---|
| Domain | Quantum Mechanics |
| Sub domain | Near-gap band structure expansion |
| Carrier | electron |
| Problem class | linear_inverse |
| Solution space | effective_mass_tensor |
| Noise model | gaussian |
| Integration axis | k_space |
| Difficulty delta | 3 |
| L dag | 3.1 |
## 📡 4. Measurement Model
Well-posed near k_0; fails far from band edge.
| Metric | Value |
|---|---|
| Metric | effective_mass_relative_error |
| Secondary | g_factor_error |
## 📏 5. Operating Range (Ω)
**Center problem class:** `kp_perturbation` · **Forward operator:** `kp_forward`
**Center point:**
| Parameter | Unit | Value |
|---|---|---|
| Species | — | GaAs |
| N bands kp | — | 8 |
| Include soc | — | True |
| K range a inv | — | 0.1 |
**Allowed bounds:**
| Parameter | Unit | Range |
|---|---|---|
| N bands kp | — | 4 – 30 |
| Include soc | — | True – False |
| K range a inv | — | 0.01 – 1.0 |
## 🎯 6. Tolerance (ε)
**Center tolerance:** m* rel error <= 0.03
| Metric | Range |
|---|---|
| Effective mass relative error | 0.01 – 0.3 |
## ⚖ 7. Hardness Function
Hardness scales as **`epsilon_fn`** on **effective_mass_relative_error**, with κ = `200` and δ = `3`.
## 💾 8. Reference Dataset
- **primary** · weight 1.0 · IPFS _(not pinned yet)_
## 9. On-chain Registration
- **Chain hash:** `0xcc48c3e2dc2376bee128bd0198bf23cfe1b34b8d39a5f43977df072efb1f7298`
- **Chain tx hash:** `0x940afd71a48c82fcf0988da2986bc7b5e2a2f5dbd57dcc7ca0912bfb4b32f6ca`
- **Chain block:** `41554081`
---
## File Mapping
This bundle consists of: `L1-326.md`, `L1-326.json`.
| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-326.md` | Source of truth — edit this | Human or LLM |
| `L1-326.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 `L1-326`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._