# ⚛  L1 Principle — GARCH Volatility Estimation

**ID:** `L1-445` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Finance — *Time series volatility*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** conditional vol series
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian student t
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555260

---

## 🧠 1. Introduction

**GARCH Volatility Estimation** is a **parameter-estimation problem** whose unknown lives in **conditional vol series** space, within the **Time series volatility** sub-domain of **Computational Finance**.

Measurements consist of N/A via a **mle garch estimation** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; O · mle · log likelihood operator; S · recursion · garch update operator.

Observations are corrupted by gaussian student t. Existence of the recovered conditional_vol_series 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); structural_break_in_vol_regime dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian student t sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Pointwise nonlinearity → O · mle · log likelihood → S · recursion · garch update → **y** (detector).

```
y = `S.recursion.garch_update` `O.mle.log_likelihood` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.mle.log_likelihood` | O · mle · log likelihood operator |
| `S.recursion.garch_update` | S · recursion · garch update operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Finance |
| Sub domain | Time series volatility |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | conditional_vol_series |
| Noise model | gaussian_student_t |
| Integration axis | temporal_financial |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

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

| Metric | Value |
|---|---|
| Metric | volatility_forecast_QLIKE |
| Secondary | log_likelihood_per_obs |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Beta true | — | 0.85 |
| Alpha true | — | 0.1 |
| T observations | — | 2000 |
| Structural break prob | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Beta true | — | 0.5 – 0.99 |
| Alpha true | — | 0.01 – 0.3 |
| T observations | — | 100 – 10000 |
| Structural break prob | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.08 volatility_forecast_QLIKE

| Metric | Range |
|---|---|
| Volatility forecast qlike | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x63e4a670d4c960d88f4f1b72e68b06c602a82438ce4019dd857bae8c6e2a3066`
- **Chain tx hash:** `0xe8d9657c4300310590ddb58876216d6cc0e310ba01a7f34ecd24ebb96ac10a74`
- **Chain block:** `41555260`

---

## File Mapping

This bundle consists of: `L1-445.md`, `L1-445.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-445.md` | Source of truth — edit this | Human or LLM |
| `L1-445.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 `L1-445`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._