# ⚛  L1 Principle — Particle Filter (Sequential Monte Carlo)

**ID:** `L1-430` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Control Theory — *Particle filtering*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** particle distribution
> **📡 Carrier:** N/A · **🌫 Noise:** non gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555240

---

## 🧠 1. Introduction

**Particle Filter (Sequential Monte Carlo)** is a **nonlinear inverse problem** whose unknown lives in **particle distribution** space, within the **Particle filtering** sub-domain of **Control Theory**.

Measurements consist of N/A via a **sequential monte carlo estimation** sensing mechanism.

The forward operator applies, in order: S · pf · importance sampling operator; adds a prior term that biases the solution toward smoothness/sparsity; O · ess · effective sample size operator.

Observations are corrupted by non gaussian. Existence of the recovered particle_distribution 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 ~= 200); particle_impoverishment dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Non gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → S · pf · importance sampling → O · ess · effective sample size → **y** (detector).

```
y = `O.ess.effective_sample_size` `S.pf.importance_sampling` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.pf.importance_sampling` | S · pf · importance sampling operator |
| `O.ess.effective_sample_size` | O · ess · effective sample size operator |

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

| Primitive | What it does |
|---|---|
| `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Control Theory |
| Sub domain | Particle filtering |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | particle_distribution |
| Noise model | non_gaussian |
| Integration axis | discrete_time |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Existence of the recovered particle_distribution 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 ~= 200); particle_impoverishment dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Non gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | RMSE_state_particle_filter |
| Secondary | effective_sample_size_ESS |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N particles | — | 1000 |
| N state dim | — | 4 |
| N time steps | — | 100 |
| Nonlinearity degree | — | 1.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N particles | — | 100 – 100000 |
| N state dim | — | 1 – 20 |
| N time steps | — | 10 – 1000 |
| Nonlinearity degree | — | 0.5 – 5.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.15 RMSE_state_particle_filter

| Metric | Range |
|---|---|
| Rmse state particle filter | 0.02 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **RMSE_state_particle_filter**, with κ = `5000` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x337199767be602e792232154c65bbdb7fd49cfa09311d427d20322790539e8e5`
- **Chain tx hash:** `0x6093ffa085a046fed0a4b9b7d2b99ff23c5144cb3aaa2dd2c9fdfdaf670c39ff`
- **Chain block:** `41555240`

---

## File Mapping

This bundle consists of: `L1-430.md`, `L1-430.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-430.md` | Source of truth — edit this | Human or LLM |
| `L1-430.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 `L1-430`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._