# ⚛  L1 Principle — Linear Elasticity — Navier-Cauchy 3D solid mechanics

**ID:** `L1-207` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Infinitesimal-strain 3D linear elasticity*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D displacement field
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554009

---

## 🧠 1. Introduction

**Linear Elasticity — Navier-Cauchy 3D solid mechanics** is a **linear inverse problem** whose unknown lives in **3D displacement field** space, within the **Infinitesimal-strain 3D linear elasticity** sub-domain of **Structural Mechanics**.

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

The forward operator applies, in order: L · fem assemble operator; L · constitutive operator; L · solve 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 · fem assemble → L · constitutive → L · solve → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.solve` `L.constitutive` `L.fem_assemble` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.fem_assemble` | L · fem assemble operator |
| `L.constitutive` | L · constitutive operator |
| `L.solve` | L · solve operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Infinitesimal-strain 3D linear elasticity |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | 3D_displacement_field |
| Noise model | gaussian |
| Integration axis | spatial |
| 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:** `linear_elasticity` · **Forward operator:** `linear_elasticity_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 10000 |
| Snr db | dB | 30 |
| E error | — | 0 |
| Nu error | — | 0 |
| N sensors | — | 32 |
| Load error | — | 0 |
| Boundary error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 100 – 1000000 |
| Snr db | dB | 0 – 40 |
| E error | — | 0.0 – 0.1 |
| Nu error | — | 0.0 – 0.05 |
| N sensors | — | 4 – 4096 |
| Load error | — | 0.0 – 0.1 |
| Boundary error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 26.0

| 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:** `0x47b3232750d828d3cb44e609e076325904fc4d5284ffb63a27c5198bfbcf0baa`
- **Chain tx hash:** `0xe8ecc4d396d026d5f9a2bfdffcc59c6c07a39c2e2a2184aa71720b8c698e2656`
- **Chain block:** `41554009`

---

## File Mapping

This bundle consists of: `L1-207.md`, `L1-207.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-207.md` | Source of truth — edit this | Human or LLM |
| `L1-207.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 `L1-207`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._