# ⚛  L1 Principle — System Identification (N4SID/ARX)

**ID:** `L1-431` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Control Theory — *System identification*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** system matrices ABCD
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555240

---

## 🧠 1. Introduction

**System Identification (N4SID/ARX)** is a **parameter-estimation problem** whose unknown lives in **system matrices ABCD** space, within the **System identification** sub-domain of **Control Theory**.

Measurements consist of N/A via a **input output system identification** sensing mechanism.

The forward operator applies, in order: S · n4sid · subspace id operator; applies a smooth nonlinear function element-wise; O · chi2 · one step ahead operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered system_matrices_ABCD 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); undermodeling_order_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** → S · n4sid · subspace id → Pointwise nonlinearity → O · chi2 · one step ahead → **y** (detector).

```
y = `O.chi2.one_step_ahead` f(·) `S.n4sid.subspace_id` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.n4sid.subspace_id` | S · n4sid · subspace id operator |
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.chi2.one_step_ahead` | O · chi2 · one step ahead operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Control Theory |
| Sub domain | System identification |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | system_matrices_ABCD |
| Noise model | gaussian |
| Integration axis | time_series |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered system_matrices_ABCD 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); undermodeling_order_mismatch dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | model_fit_percent |
| Secondary | VRMS_validation |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Input snr db | dB | 30 |
| N data points | — | 1000 |
| Model order n | N | 4 |
| Bandwidth fraction | — | 0.3 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Input snr db | dB | 10 – 60 |
| N data points | — | 100 – 100000 |
| Model order n | N | 1 – 20 |
| Bandwidth fraction | — | 0.1 – 0.9 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 85 model_fit_percent

| Metric | Range |
|---|---|
| Model fit percent | 50 – 99 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xb4f52cafc9f4e894c5b66fa7ceff50fa60401595be22250bc135c329cfc60b6a`
- **Chain tx hash:** `0x49e90e1df5a572b0082bb7997fd503f8ae9f9b718508029fbad92aa6c1e38f61`
- **Chain block:** `41555240`

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## File Mapping

This bundle consists of: `L1-431.md`, `L1-431.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-431.md` | Source of truth — edit this | Human or LLM |
| `L1-431.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 `L1-431`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._