# ⚛  L1 Principle — Hertzian Contact — elastic sphere/cylinder contact

**ID:** `L1-225` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Classical Hertz elastic contact*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** contact pressure area
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554022

---

## 🧠 1. Introduction

**Hertzian Contact — elastic sphere/cylinder contact** is a **linear inverse problem** whose unknown lives in **contact pressure area** space, within the **Classical Hertz elastic contact** sub-domain of **Structural Mechanics**.

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

The forward operator applies, in order: L · hertz theory operator; L · contact pressure operator; L · elastic deformation operator; pixel-level spatial averaging on the detector.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · hertz theory → L · contact pressure → L · elastic deformation → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.elastic_deformation` `L.contact_pressure` `L.hertz_theory` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.hertz_theory` | L · hertz theory operator |
| `L.contact_pressure` | L · contact pressure operator |
| `L.elastic_deformation` | L · elastic deformation operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Classical Hertz elastic contact |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | contact_pressure_area |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 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:** `hertz_contact` · **Forward operator:** `hertz_contact_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| F n | N | 100 |
| R m | m | 0.01 |
| Snr db | dB | 30 |
| E error | — | 0 |
| Radius error | — | 0 |
| Friction error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| F n | N | 0.1 – 10000000.0 |
| R m | m | 1e-05 – 10 |
| Snr db | dB | 0 – 40 |
| E error | — | 0.0 – 0.1 |
| Radius error | — | 0.0 – 0.05 |
| Friction error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xaed109ce04833bd3110a8d40475aec5b928dc1483badc7f8c4a9a3fab57490e6`
- **Chain tx hash:** `0xc1bcd2acd904e7843b6cae1d49a2c792ff0664cdd2f6f72fa99b69d1b4e8965f`
- **Chain block:** `41554022`

---

## File Mapping

This bundle consists of: `L1-225.md`, `L1-225.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-225.md` | Source of truth — edit this | Human or LLM |
| `L1-225.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 `L1-225`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._