# ⚛  L1 Principle — Incompressible Navier-Stokes — low-Mach viscous flow

**ID:** `L1-170` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Incompressible NS at moderate Re*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D velocity pressure fields
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553965

---

## 🧠 1. Introduction

**Incompressible Navier-Stokes — low-Mach viscous flow** is a **nonlinear inverse problem** whose unknown lives in **3D velocity pressure fields** space, within the **Incompressible NS at moderate Re** sub-domain of **Fluid Dynamics**.

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

The forward operator applies, in order: momentum balance (Navier-Stokes / Euler); mass-balance: ∂_t ρ + ∇·(ρv) = 0; L · pressure projection 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** → Momentum equation → Continuity equation → L · pressure projection → Temporal integration → **y** (detector).

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

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.momentum` | Momentum balance (navier-stokes / euler) |
| `L.continuity` | Mass-balance: ∂_t ρ + ∇·(ρv) = 0 |
| `L.pressure_projection` | L · pressure projection operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Incompressible NS at moderate Re |
| Carrier | mechanical |
| Problem class | nonlinear_inverse |
| Solution space | 3D_velocity_pressure_fields |
| 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:** `incomp_ns` · **Forward operator:** `incomp_ns_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 128 |
| Re | — | 1000 |
| Snr db | dB | 30 |
| N times | — | 500 |
| Bc error | — | 0 |
| Nu error | — | 0 |
| Initial condition error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 4096 |
| W | px | 32 – 4096 |
| Z | — | 16 – 1024 |
| Re | — | 1 – 100000000.0 |
| Snr db | dB | 0 – 40 |
| N times | — | 50 – 100000 |
| Bc error | — | 0.0 – 0.2 |
| Nu error | — | 0.0 – 0.1 |
| Initial condition 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 = `600` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xeac424843de2c14dd8b64269029a8866cfe3d7db8510c686570778618df557d0`
- **Chain tx hash:** `0xf77b9f0ff6ab90d53e466e80e4466a541d856deab3c2cd4982eb8ae594f1444c`
- **Chain block:** `41553965`

---

## File Mapping

This bundle consists of: `L1-170.md`, `L1-170.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-170.md` | Source of truth — edit this | Human or LLM |
| `L1-170.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 `L1-170`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._