# ⚛  L1 Principle — Viscoelasticity — linear hereditary models

**ID:** `L1-218` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Maxwell / Kelvin-Voigt / prony-series viscoelasticity*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** prony series params
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554011

---

## 🧠 1. Introduction

**Viscoelasticity — linear hereditary models** is a **linear inverse problem** whose unknown lives in **prony series params** space, within the **Maxwell / Kelvin-Voigt / prony-series viscoelasticity** sub-domain of **Structural Mechanics**.

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

The forward operator applies, in order: L · hereditary integral operator; L · prony series operator; L · wlf shift 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 · hereditary integral → L · prony series → L · wlf shift → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.wlf_shift` `L.prony_series` `L.hereditary_integral` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.hereditary_integral` | L · hereditary integral operator |
| `L.prony_series` | L · prony series operator |
| `L.wlf_shift` | L · wlf shift operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Maxwell / Kelvin-Voigt / prony-series viscoelasticity |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | prony_series_params |
| Noise model | gaussian |
| Integration axis | temporal |
| 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:** `viscoelastic` · **Forward operator:** `viscoelastic_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| N freqs | — | 30 |
| N modes | — | 8 |
| G prony error | — | 0 |
| Tau prony error | — | 0 |
| Wlf c1 c2 error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| N freqs | — | 5 – 500 |
| N modes | — | 1 – 50 |
| G prony error | — | 0.0 – 0.1 |
| Tau prony error | — | 0.0 – 0.2 |
| Wlf c1 c2 error | — | 0.0 – 0.3 |

## 🎯 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:** `0x71385e60aeadb1de95bb23a4d88ed729702c56e46487d1ccfc1bbf14ecb612d8`
- **Chain tx hash:** `0x99d05cc644012e622809188444a8d11189ac16058712734b010b850c933ef32e`
- **Chain block:** `41554011`

---

## File Mapping

This bundle consists of: `L1-218.md`, `L1-218.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-218.md` | Source of truth — edit this | Human or LLM |
| `L1-218.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 `L1-218`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._