# ⚛  L1 Principle — Linear Buckling — eigenvalue stability analysis

**ID:** `L1-214` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Linear buckling load factor*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** buckling load factor mode
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554011

---

## 🧠 1. Introduction

**Linear Buckling — eigenvalue stability analysis** is a **linear inverse problem** whose unknown lives in **buckling load factor mode** space, within the **Linear buckling load factor** sub-domain of **Structural Mechanics**.

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

The forward operator applies, in order: L · geom stiffness operator; L · eigenvalue operator; L · critical mode operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Sensitive to imperfections; Koiter analysis required for post-buckling.

## ⚙ 2. Forward Model

Physical chain: **x** → L · geom stiffness → L · eigenvalue → L · critical mode → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.critical_mode` `L.eigenvalue` `L.geom_stiffness` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.geom_stiffness` | L · geom stiffness operator |
| `L.eigenvalue` | L · eigenvalue operator |
| `L.critical_mode` | L · critical mode operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Linear buckling load factor |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | buckling_load_factor_mode |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Sensitive to imperfections; Koiter analysis required for post-buckling.

| Metric | Value |
|---|---|
| Metric | lambda_cr_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `buckling` · **Forward operator:** `buckling_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 5000 |
| Snr db | dB | 30 |
| N modes | — | 10 |
| Bc error | — | 0 |
| Material error | — | 0 |
| Load path error | — | 0 |
| Geom imperfect error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 100 – 500000 |
| Snr db | dB | 0 – 40 |
| N modes | — | 1 – 50 |
| Bc error | — | 0.0 – 0.2 |
| Material error | — | 0.0 – 0.1 |
| Load path error | — | 0.0 – 0.1 |
| Geom imperfect error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 5.0

| Metric | Range |
|---|---|
| Lambda cr rel | 0.5 – 50.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **lambda_cr_rel_error_percent**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x145c91aff13842f69d45746260520c520ff8826493b9f28d286fd7a66d6fd85e`
- **Chain tx hash:** `0xf7887ba572be443409eaabf70023d6a9cff4db227c80393881b806d0adc6e833`
- **Chain block:** `41554011`

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## File Mapping

This bundle consists of: `L1-214.md`, `L1-214.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-214.md` | Source of truth — edit this | Human or LLM |
| `L1-214.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 `L1-214`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._