# ⚛  L1 Principle — Linear State-Space System Analysis

**ID:** `L1-426` · **Status:** ⊙ Testnet (genesis catalog)

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

---

## 🧠 1. Introduction

**Linear State-Space System Analysis** is a **parameter-estimation problem** whose unknown lives in **state vector** space, within the **Linear systems** sub-domain of **Control Theory**.

Measurements consist of N/A via a **state space output measurement** sensing mechanism.

The forward operator applies, in order: computes eigen-pairs of a linear operator; S · observability · gramian operator; O · lyapunov · stability operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered state_vector 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 ~= 50); model_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 · observability · gramian → **y** (detector).

```
y = `S.observability.gramian` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.observability.gramian` | S · observability · gramian 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 |

**🛡 Analytical properties** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.lyapunov.stability` | O · lyapunov · stability operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Control Theory |
| Sub domain | Linear systems |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | state_vector |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered state_vector 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 ~= 50); model_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 | state_RMSE_normalized |
| Secondary | output_prediction_RMSE |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Noise snr db | dB | 30 |
| System order n | N | 5 |
| Sampling rate hz | Hz | 1000 |
| Condition number kappa | — | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Noise snr db | dB | 0 – 60 |
| System order n | N | 2 – 50 |
| Sampling rate hz | Hz | 10 – 100000.0 |
| Condition number kappa | — | 10 – 1000000.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.05 state_RMSE_normalized

| Metric | Range |
|---|---|
| State rmse normalized | 0.001 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **state_RMSE_normalized**, with κ = `1000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x0f32999d9f39215efc5d2aa3148591be18774780792deff2c9d41328f738731f`
- **Chain tx hash:** `0x4b41989773fb14f4ca09a6dfdcc5b2cbb43d2a59112bc58e84d1b3bf4441114e`
- **Chain block:** `41555240`

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## File Mapping

This bundle consists of: `L1-426.md`, `L1-426.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-426.md` | Source of truth — edit this | Human or LLM |
| `L1-426.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 `L1-426`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._