# ⚛  L1 Principle — Adaptive Filtering (LMS / RLS time-varying system identification)

**ID:** `L1-395` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Signal Processing — *Online stochastic-gradient / recursive-least-squares*
> **🎯 Problem class:** online linear regression · **🧮 Solution space:** time varying filter
> **📡 Carrier:** electromagnetic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 2 · **⛓ Block:** 41555199

---

## 🧠 1. Introduction

**Adaptive Filtering (LMS / RLS time-varying system identification)** is a **online linear regression** whose unknown lives in **time varying filter** space, within the **Online stochastic-gradient / recursive-least-squares** sub-domain of **Signal Processing**.

Measurements consist of electromagnetic field measurements via a **reference probe online** sensing mechanism.

The forward operator applies, in order: L · project · reference operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. LMS converges in mean for 0 < mu < 2/lambda_max; RLS converges in one step for stationary deterministic input; tracking error scales with non-stationarity rate.

## ⚙ 2. Forward Model

Physical chain: **x** → L · project · reference → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.project.reference` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.project.reference` | L · project · reference operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Signal Processing |
| Sub domain | Online stochastic-gradient / recursive-least-squares |
| Carrier | electromagnetic |
| Problem class | online_linear_regression |
| Solution space | time_varying_filter |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 2 |
| L dag | 2.3 |

## 📡 4. Measurement Model

LMS converges in mean for 0 < mu < 2/lambda_max; RLS converges in one step for stationary deterministic input; tracking error scales with non-stationarity rate.

| Metric | Value |
|---|---|
| Metric | tracking_MSE_dB |
| Secondary | convergence_time_samples |

## 📏 5. Operating Range (Ω)

**Center problem class:** `lms_system_identification` · **Forward operator:** `fir_filter_online`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| L | — | 32 |
| Mu | — | 0.01 |
| Sigma v | — | 0.01 |
| Eigenspread | — | 10 |
| Non stationarity rate | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| L | — | 4 – 4096 |
| Mu | — | 1e-05 – 1.0 |
| Sigma v | — | 0.0001 – 1.0 |
| Eigenspread | — | 1 – 1000 |
| Step size miscal | — | 0.0 – 5.0 |
| Non stationarity rate | — | 0.0 – 0.1 |
| Impulsive noise fraction | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** -30 dB tracking MSE

| Metric | Range |
|---|---|
| Mse db | -60 – 10 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **tracking_MSE_dB**, with κ = `500` and δ = `2`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x8b6ed2bd7c400d6066681e6c1b8a9eb9205e749ba055bce04eed3eefe408bed0`
- **Chain tx hash:** `0x26bc3c4359680388e625126cecdd2373db19a7c5a70ce5fb781abf7f5d7917fb`
- **Chain block:** `41555199`

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## File Mapping

This bundle consists of: `L1-395.md`, `L1-395.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-395.md` | Source of truth — edit this | Human or LLM |
| `L1-395.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 `L1-395`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._