# ⚛  L1 Principle — GNSS Positioning and Navigation

**ID:** `L1-466` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geodesy — *Satellite positioning*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D position velocity
> **📡 Carrier:** RF · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555298

---

## 🧠 1. Introduction

**GNSS Positioning and Navigation** is a **linear inverse problem** whose unknown lives in **3D position velocity** space, within the **Satellite positioning** sub-domain of **Geodesy**.

Measurements consist of RF coil signals (typical of MRI) via a **pseudorange doppler measurement** sensing mechanism.

The forward operator applies, in order: Pi · geometry operator; S · kalman · navigation operator; O · least squares · weighted operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered 3D_position_velocity is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 10); ionospheric_model_error_m dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Pi · geometry → O · least squares · weighted → **y** (detector).

```
y = `O.least_squares.weighted` `Pi.geometry` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `Pi.geometry` | Pi · geometry operator |
| `O.least_squares.weighted` | O · least squares · weighted operator |

**🛰 Estimator components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `S.kalman.navigation` | S · kalman · navigation operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geodesy |
| Sub domain | Satellite positioning |
| Carrier | RF |
| Problem class | linear_inverse |
| Solution space | 3D_position_velocity |
| Noise model | gaussian |
| Integration axis | temporal_navigation |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 4. Measurement Model

Existence of the recovered 3D_position_velocity is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 10); ionospheric_model_error_m dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | position_RMSE_m |
| Secondary | PDOP |

## 📏 5. Operating Range (Ω)

**Center problem class:** `linear_inverse` · **Forward operator:** `pseudorange_doppler_measurement`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N sat | — | 8 |
| Ionospheric tecu | — | 5 |
| Multipath sigma m | m | 0.5 |
| Elevation cutoff deg | deg | 15 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N sat | — | 4 – 30 |
| Ionospheric tecu | — | 0.5 – 50 |
| Multipath sigma m | m | 0.1 – 5.0 |
| Elevation cutoff deg | deg | 5 – 30 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.5 position_RMSE_m

| Metric | Range |
|---|---|
| Position rmse m | 0.01 – 10.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **position_RMSE_m**, with κ = `100` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x0a9120762927c26e70273d9422745920b65641572cdabb858e1e6c38fb79dce6`
- **Chain tx hash:** `0xa3fb986ef0e11cc0baca6524e8d2d21e9d13700aed0b33f989e747e94b1db584`
- **Chain block:** `41555298`

---

## File Mapping

This bundle consists of: `L1-466.md`, `L1-466.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-466.md` | Source of truth — edit this | Human or LLM |
| `L1-466.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 `L1-466`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._