# ⚛  L1 Principle — Spalart-Allmaras — one-eq turbulence model

**ID:** `L1-178` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Single-eq SA for external aerodynamics*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** nu tilde field
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553966

---

## 🧠 1. Introduction

**Spalart-Allmaras — one-eq turbulence model** is a **nonlinear inverse problem** whose unknown lives in **nu tilde field** space, within the **Single-eq SA for external aerodynamics** sub-domain of **Fluid Dynamics**.

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

The forward operator applies, in order: L · mean flow ns operator; L · nu tilde transport operator; L · damping fn 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 · mean flow ns → L · nu tilde transport → L · damping fn → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.damping_fn` `L.nu_tilde_transport` `L.mean_flow_ns` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.mean_flow_ns` | L · mean flow ns operator |
| `L.nu_tilde_transport` | L · nu tilde transport operator |
| `L.damping_fn` | L · damping fn operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Single-eq SA for external aerodynamics |
| Carrier | mechanical |
| Problem class | nonlinear_inverse |
| Solution space | nu_tilde_field |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| 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:** `sa` · **Forward operator:** `sa_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 128 |
| Snr db | dB | 30 |
| C b error | — | 0 |
| Wall error | — | 0 |
| Damping error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 2048 |
| W | px | 32 – 2048 |
| Z | — | 16 – 1024 |
| Snr db | dB | 0 – 40 |
| C b error | — | 0.0 – 0.3 |
| Wall error | — | 0.0 – 0.3 |
| Damping 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 = `400` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x38e139678c0c3b8c84aec6ad5292680de1ca10fc74369f607305c46693179335`
- **Chain tx hash:** `0xbdc793eb065e21591197093d606635e4a4f6e99f1db3262174e2b0e1c9104ef4`
- **Chain block:** `41553966`

---

## File Mapping

This bundle consists of: `L1-178.md`, `L1-178.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-178.md` | Source of truth — edit this | Human or LLM |
| `L1-178.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 `L1-178`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._