# ⚛  L1 Principle — Elastodynamics — dynamic wave propagation in solids

**ID:** `L1-230` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Elastic wave equation (P/S waves)*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D displacement spacetime
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554023

---

## 🧠 1. Introduction

**Elastodynamics — dynamic wave propagation in solids** is a **linear inverse problem** whose unknown lives in **3D displacement spacetime** space, within the **Elastic wave equation (P/S waves)** sub-domain of **Structural Mechanics**.

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

The forward operator applies, in order: L · initial condition operator; L · hyperbolic wave operator; L · record · sensors operator; detector accumulates flux over the exposure window.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · initial condition → L · hyperbolic wave → L · record · sensors → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.record.sensors` `L.hyperbolic_wave` `L.initial_condition` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.initial_condition` | L · initial condition operator |
| `L.hyperbolic_wave` | L · hyperbolic wave operator |
| `L.record.sensors` | L · record · sensors operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Elastic wave equation (P/S waves) |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | 3D_displacement_spacetime |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 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:** `elastodynamics` · **Forward operator:** `elastodynamics_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 50000 |
| Snr db | dB | 30 |
| E error | — | 0 |
| N times | — | 500 |
| Bc error | — | 0 |
| Nu error | — | 0 |
| N sensors | — | 32 |
| Rho error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 1000 – 10000000 |
| Snr db | dB | 0 – 40 |
| E error | — | 0.0 – 0.1 |
| N times | — | 100 – 100000 |
| Bc error | — | 0.0 – 0.2 |
| Nu error | — | 0.0 – 0.05 |
| N sensors | — | 4 – 4096 |
| Rho error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 23.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x3973d0cb866f99bf32f6fe74a6c965bbe9067cc293197d8e14fe2d3f8c0545e9`
- **Chain tx hash:** `0xde8a7b13c70d6ad0d9dbbb0cd68393e95b5ff6254f8d8ec855c6172952421a6f`
- **Chain block:** `41554023`

---

## File Mapping

This bundle consists of: `L1-230.md`, `L1-230.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-230.md` | Source of truth — edit this | Human or LLM |
| `L1-230.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 `L1-230`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._