# ⚛  L1 Principle — SPH — smoothed particle hydrodynamics

**ID:** `L1-186` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Lagrangian meshfree SPH*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** particle field meshfree
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553967

---

## 🧠 1. Introduction

**SPH — smoothed particle hydrodynamics** is a **nonlinear inverse problem** whose unknown lives in **particle field meshfree** space, within the **Lagrangian meshfree SPH** sub-domain of **Fluid Dynamics**.

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

The forward operator applies, in order: L · kernel approx operator; momentum balance (Navier-Stokes / Euler); L · density 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 · kernel approx → Momentum equation → L · density → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.density` ∂_t(ρv) + ∇·(ρv⊗v) `L.kernel_approx` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.kernel_approx` | L · kernel approx operator |
| `L.momentum` | Momentum balance (navier-stokes / euler) |
| `L.density` | L · density operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Lagrangian meshfree SPH |
| Carrier | mechanical |
| Problem class | nonlinear_inverse |
| Solution space | particle_field_meshfree |
| 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:** `sph` · **Forward operator:** `sph_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| N times | — | 2000 |
| N particles | — | 1e+06 |
| Boundary error | — | 0 |
| Pressure instability | — | 0 |
| Kernel bandwidth error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| N times | — | 100 – 1000000 |
| N particles | — | 1000.0 – 10000000000.0 |
| Boundary error | — | 0.0 – 0.3 |
| Pressure instability | — | 0.0 – 0.3 |
| Kernel bandwidth error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 20.0

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

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xde64bb22e0d6ef12d032f705fcffe61267d305e8d2ac054a851a5abe012e90cc`
- **Chain tx hash:** `0x6e3ec68a0dbeeec427d8fe4a9dd6f7203f050f360b9c10e1225839e7c9657caf`
- **Chain block:** `41553967`

---

## File Mapping

This bundle consists of: `L1-186.md`, `L1-186.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-186.md` | Source of truth — edit this | Human or LLM |
| `L1-186.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 `L1-186`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._