# ⚛  L1 Principle — Darcy Flow — low-Re porous-medium flow

**ID:** `L1-188` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Linear Darcy's law in porous media*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** pressure velocity in porous
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553977

---

## 🧠 1. Introduction

**Darcy Flow — low-Re porous-medium flow** is a **linear inverse problem** whose unknown lives in **pressure velocity in porous** space, within the **Linear Darcy's law in porous media** sub-domain of **Fluid Dynamics**.

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

The forward operator applies, in order: L · darcy linear operator; L · mass conservation operator; L · bc operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · darcy linear → L · mass conservation → L · bc → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.bc` `L.mass_conservation` `L.darcy_linear` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.darcy_linear` | L · darcy linear operator |
| `L.mass_conservation` | L · mass conservation operator |
| `L.bc` | L · bc operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Linear Darcy's law in porous media |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | pressure_velocity_in_porous |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 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:** `darcy` · **Forward operator:** `darcy_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 32 |
| Snr db | dB | 30 |
| Bc error | — | 0 |
| Viscosity error | — | 0 |
| Permeability error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 16 – 1024 |
| W | px | 16 – 1024 |
| Z | — | 8 – 512 |
| Snr db | dB | 0 – 40 |
| Bc error | — | 0.0 – 0.2 |
| Viscosity error | — | 0.0 – 0.1 |
| Permeability error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 26.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x32fed19be36cc2c5ae115920fc3d4ccd17d55f6e3a9c5135006586191ce5b8aa`
- **Chain tx hash:** `0x8306ffe18a481beb6d6bd02c9afb6a7e714305e098805c3ab04b95fc403e3053`
- **Chain block:** `41553977`

---

## File Mapping

This bundle consists of: `L1-188.md`, `L1-188.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-188.md` | Source of truth — edit this | Human or LLM |
| `L1-188.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 `L1-188`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._