# ⚛  L1 Principle — Pulsar Timing Array GW Background

**ID:** `L1-373` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Astrophysics — *Gravitational wave astronomy*
> **🎯 Problem class:** statistical inverse · **🧮 Solution space:** GW background spectrum
> **📡 Carrier:** radio_wave · **🌫 Noise:** timing noise red white
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555177

---

## 🧠 1. Introduction

**Pulsar Timing Array GW Background** is a **statistical inverse problem** whose unknown lives in **GW background spectrum** space, within the **Gravitational wave astronomy** sub-domain of **Astrophysics**.

Measurements consist of radio-frequency electromagnetic waves via a **pulsar timing residuals** sensing mechanism.

The forward operator applies, in order: S · pulsar · timing residuals operator; F · fourier · spectral density operator; O · entropy · bayesian operator.

Observations are corrupted by timing noise red white. Existence of the recovered GW_background_spectrum 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 ~= 1000); solar_system_ephemeris_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Timing noise red white sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → S · pulsar · timing residuals → F · fourier · spectral density → O · entropy · bayesian → **y** (detector).

```
y = `O.entropy.bayesian` `F.fourier.spectral_density` `S.pulsar.timing_residuals` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.pulsar.timing_residuals` | S · pulsar · timing residuals operator |
| `F.fourier.spectral_density` | F · fourier · spectral density operator |
| `O.entropy.bayesian` | O · entropy · bayesian operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Astrophysics |
| Sub domain | Gravitational wave astronomy |
| Carrier | radio_wave |
| Problem class | statistical_inverse |
| Solution space | GW_background_spectrum |
| Noise model | timing_noise_red_white |
| Integration axis | temporal_timing |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered GW_background_spectrum 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 ~= 1000); solar_system_ephemeris_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Timing noise red white sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | GWB_Bayes_factor_log10 |
| Secondary | A_15_constraint_sigma |

## 📏 5. Operating Range (Ω)

**Center problem class:** `statistical_inverse` · **Forward operator:** `pulsar_timing_residuals`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N psr | — | 25 |
| T obs yr | — | 15 |
| Cadence yr | — | 0.04 |
| Noise rms ns | ns | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N psr | — | 10 – 100 |
| T obs yr | — | 5 – 25 |
| Cadence yr | — | 0.01 – 0.1 |
| Noise rms ns | ns | 10 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 3.0 GWB_Bayes_factor_log10

| Metric | Range |
|---|---|
| Gwb bayes factor log10 | 0.5 – 6.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **GWB_Bayes_factor_log10**, with κ = `100000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xe0534c020796d29359f451a6360471ab64d3af89284fc8c7b563e6774a3b3f05`
- **Chain tx hash:** `0x9df456984c82fa2d5356189cade6f2d81c182a6e4844c914e3ae93f23ecad579`
- **Chain block:** `41555177`

---

## File Mapping

This bundle consists of: `L1-373.md`, `L1-373.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-373.md` | Source of truth — edit this | Human or LLM |
| `L1-373.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 `L1-373`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._