# ⚛  L1 Principle — Extended Kalman Filter (EKF)

**ID:** `L1-428` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Control Theory — *Nonlinear estimation*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** nonlinear state estimate
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555240

---

## 🧠 1. Introduction

**Extended Kalman Filter (EKF)** is a **nonlinear inverse problem** whose unknown lives in **nonlinear state estimate** space, within the **Nonlinear estimation** sub-domain of **Control Theory**.

Measurements consist of N/A via a **linearized nonlinear estimation** sensing mechanism.

The forward operator applies, in order: S · ekf · jacobian linearization operator; D · time · explicit operator; O · innov · consistency check operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered nonlinear_state_estimate 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 ~= 100); strong_nonlinearity 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** → S · ekf · jacobian linearization → D · time · explicit → O · innov · consistency check → **y** (detector).

```
y = `O.innov.consistency_check` `D.time.explicit` `S.ekf.jacobian_linearization` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.ekf.jacobian_linearization` | S · ekf · jacobian linearization operator |
| `D.time.explicit` | D · time · explicit operator |
| `O.innov.consistency_check` | O · innov · consistency check operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Control Theory |
| Sub domain | Nonlinear estimation |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | nonlinear_state_estimate |
| Noise model | gaussian |
| Integration axis | discrete_time |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered nonlinear_state_estimate 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 ~= 100); strong_nonlinearity dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | RMSE_state_estimation |
| Secondary | filter_consistency_NEES |

## 📏 5. Operating Range (Ω)

**Center problem class:** `nonlinear_inverse` · **Forward operator:** `linearized_nonlinear_estimation`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N steps | — | 200 |
| Q r ratio | — | 1 |
| N state dim | — | 6 |
| Nonlinearity degree | — | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N steps | — | 20 – 5000 |
| Q r ratio | — | 0.01 – 100 |
| N state dim | — | 2 – 20 |
| Nonlinearity degree | — | 0.1 – 3.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.08 RMSE_state_estimation

| Metric | Range |
|---|---|
| Rmse state estimation | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **RMSE_state_estimation**, with κ = `2000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xc51b690c1f98e27dfc3162de4531fb67d8322b788a352e92951a47d10a1b8c75`
- **Chain tx hash:** `0xb17130492a353b9b7fcd21bedeaee0aea2e1ee1db791159919a62f10fbb64d8b`
- **Chain block:** `41555240`

---

## File Mapping

This bundle consists of: `L1-428.md`, `L1-428.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-428.md` | Source of truth — edit this | Human or LLM |
| `L1-428.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 `L1-428`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._