# ⚛  L1 Principle — Lattice Boltzmann Method (LBM) — mesoscale fluid

**ID:** `L1-187` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Lattice-Boltzmann BGK for incompressible flow*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** distribution function lattice
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553977

---

## 🧠 1. Introduction

**Lattice Boltzmann Method (LBM) — mesoscale fluid** is a **linear inverse problem** whose unknown lives in **distribution function lattice** space, within the **Lattice-Boltzmann BGK for incompressible flow** sub-domain of **Fluid Dynamics**.

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

The forward operator applies, in order: L · stream operator; L · collide bgk operator; L · moment extract 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 · stream → L · collide bgk → L · moment extract → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.moment_extract` `L.collide_bgk` `L.stream` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.stream` | L · stream operator |
| `L.collide_bgk` | L · collide bgk operator |
| `L.moment_extract` | L · moment extract operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Lattice-Boltzmann BGK for incompressible flow |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | distribution_function_lattice |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 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:** `lbm` · **Forward operator:** `lbm_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 64 |
| Snr db | dB | 30 |
| N times | — | 5000 |
| Grid error | — | 0 |
| Compressibility error | — | 0 |
| Relaxation time error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 4096 |
| W | px | 32 – 4096 |
| Z | — | 8 – 1024 |
| Snr db | dB | 0 – 40 |
| N times | — | 100 – 1000000 |
| Grid error | — | 0.0 – 0.3 |
| Compressibility error | — | 0.0 – 0.2 |
| Relaxation time error | — | 0.0 – 0.2 |

## 🎯 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 = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xe163b1f5951dceee763a1ab6959e1d1d974f6acf2884de4c30b7ee47c8d4dbb7`
- **Chain tx hash:** `0x8019ccb6911183a9c551934c3be33ef2a93ebb84fcf539c11f801710eefa79ef`
- **Chain block:** `41553977`

---

## File Mapping

This bundle consists of: `L1-187.md`, `L1-187.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-187.md` | Source of truth — edit this | Human or LLM |
| `L1-187.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 `L1-187`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._