# ⚛  L1 Principle — Event Horizon Telescope (VLBI) Imaging

**ID:** `L1-168` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Astronomy — *Radio interferometry*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D brightness temperature
> **📡 Carrier:** radio_wave · **🌫 Noise:** thermal gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554244

---

## 🧠 1. Introduction

**Event Horizon Telescope (VLBI) Imaging** is a **nonlinear inverse problem** whose unknown lives in **2D brightness temperature** space, within the **Radio interferometry** sub-domain of **Astronomy**.

Measurements consist of radio-frequency electromagnetic waves via a **vlbi sparse aperture** sensing mechanism.

The forward operator applies, in order: F · fourier · nufft operator; S · closure · phase operator; adds a prior term that biases the solution toward smoothness/sparsity; O · chi2 · visibility operator.

Observations are corrupted by thermally-driven Gaussian fluctuations. Existence of the recovered 2D_brightness_temperature 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); station_gain_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Thermal gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → F · fourier · nufft → S · closure · phase → O · chi2 · visibility → **y** (detector).

```
y = `O.chi2.visibility` `S.closure.phase` `F.fourier.nufft` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `F.fourier.nufft` | F · fourier · nufft operator |
| `S.closure.phase` | S · closure · phase operator |
| `O.chi2.visibility` | O · chi2 · visibility 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 | Astronomy |
| Sub domain | Radio interferometry |
| Carrier | radio_wave |
| Problem class | nonlinear_inverse |
| Solution space | 2D_brightness_temperature |
| Noise model | thermal_gaussian |
| Integration axis | uv_plane |
| Difficulty delta | 5 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered 2D_brightness_temperature 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); station_gain_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Thermal gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | chi2_reduced |
| Secondary | image_fidelity_score |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| N baselines | — | 15 |
| Uv fraction | — | 0.001 |
| Noise sigma jy | — | 0.1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 512 |
| W | px | 32 – 512 |
| N baselines | — | 6 – 50 |
| Uv fraction | — | 0.0001 – 0.01 |
| Noise sigma jy | — | 0.01 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.2 chi2_reduced

| Metric | Range |
|---|---|
| Chi2 reduced | 0.8 – 3.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **chi2_reduced**, with κ = `1000000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xc117659285882e17ebe99e1e597e0872c212d4fd5da6b7b9bde35261efd32c2b`
- **Chain tx hash:** `0x6ce6ec74df271331b2949e1d1555e365520aa5620795915d1b2370904d5daa35`
- **Chain block:** `41554244`

---

## File Mapping

This bundle consists of: `L1-168.md`, `L1-168.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-168.md` | Source of truth — edit this | Human or LLM |
| `L1-168.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 `L1-168`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._