# ⚛  L1 Principle — Piezoelectricity — coupled mechanical-electrical

**ID:** `L1-229` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Linear piezoelectric direct/inverse effect*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** coupled u phi electric fields
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554023

---

## 🧠 1. Introduction

**Piezoelectricity — coupled mechanical-electrical** is a **linear inverse problem** whose unknown lives in **coupled u phi electric fields** space, within the **Linear piezoelectric direct/inverse effect** sub-domain of **Structural Mechanics**.

Measurements consist of mechanical vibrations or strain signals via a **strain electrode** sensing mechanism.

The forward operator applies, in order: L · piezo stress operator; L · electric displacement operator; L · coupled solve operator; integration over a band of frequencies.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · piezo stress → L · electric displacement → L · coupled solve → Frequency integration → **y** (detector).

```
y = ∫dω `L.coupled_solve` `L.electric_displacement` `L.piezo_stress` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.piezo_stress` | L · piezo stress operator |
| `L.electric_displacement` | L · electric displacement operator |
| `L.coupled_solve` | L · coupled solve operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Linear piezoelectric direct/inverse effect |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | coupled_u_phi_electric_fields |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `piezoelectric` · **Forward operator:** `piezoelectric_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 10000 |
| Snr db | dB | 30 |
| E error | — | 0 |
| N freqs | — | 16 |
| Eps r error | — | 0 |
| D coeff error | — | 0 |
| Electrode error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 100 – 1000000 |
| Snr db | dB | 0 – 40 |
| E error | — | 0.0 – 0.1 |
| N freqs | — | 1 – 256 |
| Eps r error | — | 0.0 – 0.1 |
| D coeff error | — | 0.0 – 0.1 |
| Electrode error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 25.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `240` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xceca7f19c0877499be484bb1c1da317dd0fa216120ab88a72b728c4451de68c4`
- **Chain tx hash:** `0xe1f4d2d30a2adfc142eb103437a5d729c4a0a5689806c89adcdfc02e3e61a42e`
- **Chain block:** `41554023`

---

## File Mapping

This bundle consists of: `L1-229.md`, `L1-229.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-229.md` | Source of truth — edit this | Human or LLM |
| `L1-229.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 `L1-229`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._