# ⚛  L1 Principle — Eulerian-Lagrangian — dispersed-phase particle tracking

**ID:** `L1-185` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Euler-Lagrange DPM / two-way coupling*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** particle trajectories in flow
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553967

---

## 🧠 1. Introduction

**Eulerian-Lagrangian — dispersed-phase particle tracking** is a **linear inverse problem** whose unknown lives in **particle trajectories in flow** space, within the **Euler-Lagrange DPM / two-way coupling** sub-domain of **Fluid Dynamics**.

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

The forward operator applies, in order: L · fluid eulerian operator; L · particle drag operator; L · two way coupling 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 · fluid eulerian → L · particle drag → L · two way coupling → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.two_way_coupling` `L.particle_drag` `L.fluid_eulerian` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.fluid_eulerian` | L · fluid eulerian operator |
| `L.particle_drag` | L · particle drag operator |
| `L.two_way_coupling` | L · two way coupling operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Euler-Lagrange DPM / two-way coupling |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | particle_trajectories_in_flow |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 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:** `dpm` · **Forward operator:** `dpm_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Snr db | dB | 30 |
| N times | — | 500 |
| N particles | — | 100000 |
| Collision error | — | 0 |
| Drag model error | — | 0 |
| Turbulent dispersion error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 2048 |
| W | px | 32 – 2048 |
| Snr db | dB | 0 – 40 |
| N times | — | 50 – 100000 |
| N particles | — | 100.0 – 1000000000.0 |
| Collision error | — | 0.0 – 0.2 |
| Drag model error | — | 0.0 – 0.3 |
| Turbulent dispersion error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xe6c2a26b531749858168d5f474fa96ae174190495f203807403902091def0129`
- **Chain tx hash:** `0x946909d4015b665976193a48b7519586e540f29a19824649be451a889025bb23`
- **Chain block:** `41553967`

---

## File Mapping

This bundle consists of: `L1-185.md`, `L1-185.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-185.md` | Source of truth — edit this | Human or LLM |
| `L1-185.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 `L1-185`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._