# ⚛  L1 Principle — Kalman Filter State Estimation

**ID:** `L1-427` · **Status:** ⊙ Testnet (genesis catalog)

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

---

## 🧠 1. Introduction

**Kalman Filter State Estimation** is a **parameter-estimation problem** whose unknown lives in **state estimate covariance** space, within the **Optimal estimation** sub-domain of **Control Theory**.

Measurements consist of N/A via a **linear optimal state estimation** sensing mechanism.

The forward operator applies, in order: S · kalman · predict update operator; computes eigen-pairs of a linear operator; O · innov · sequence whiteness operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered state_estimate_covariance 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 ~= 20); Q_R_mismatch 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** → O · innov · sequence whiteness → **y** (detector).

```
y = `O.innov.sequence_whiteness` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `O.innov.sequence_whiteness` | O · innov · sequence whiteness operator |

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

| Primitive | What it does |
|---|---|
| `S.kalman.predict_update` | S · kalman · predict update operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `E.eigensolve` | Computes eigen-pairs of a linear operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Control Theory |
| Sub domain | Optimal estimation |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | state_estimate_covariance |
| Noise model | gaussian |
| Integration axis | discrete_time |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 4. Measurement Model

Existence of the recovered state_estimate_covariance 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 ~= 20); Q_R_mismatch dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | NEES_normalized_estimation_error_squared |
| Secondary | NIS_normalized_innovation_squared |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `linear_optimal_state_estimation`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Q r ratio | — | 1 |
| P obs dim | — | 2 |
| N state dim | — | 4 |
| N time steps | — | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Q r ratio | — | 0.01 – 100 |
| P obs dim | — | 1 – 10 |
| N state dim | — | 1 – 20 |
| N time steps | — | 10 – 10000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.05 NEES_normalized_estimation_error_squared

| Metric | Range |
|---|---|
| Nees normalized estimation error squared | 0.5 – 3.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **NEES_normalized_estimation_error_squared**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xf5cd46a31ed61e959fa019a451146c1450dd7a58403e6fee6bd587f387f79ab5`
- **Chain tx hash:** `0x11d69e8fba7258e0d52bb6102493e3eedc3ff24fac63196e993c1f646f3780a4`
- **Chain block:** `41555240`

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## File Mapping

This bundle consists of: `L1-427.md`, `L1-427.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-427.md` | Source of truth — edit this | Human or LLM |
| `L1-427.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 `L1-427`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._