# ⚛  L1 Principle — Accretion Disk Structure Inversion

**ID:** `L1-375` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Astrophysics — *Accretion physics*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 1D disk emissivity profile
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555178

---

## 🧠 1. Introduction

**Accretion Disk Structure Inversion** is a **nonlinear inverse problem** whose unknown lives in **1D disk emissivity profile** space, within the **Accretion physics** sub-domain of **Astrophysics**.

Measurements consist of photons collected by an optical detector via a **x ray spectral timing** sensing mechanism.

The forward operator applies, in order: S · disk · relxill reflection operator; F · fourier · power spectrum operator; O · chi2 · x ray spectrum operator.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 1D_disk_emissivity_profile 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 ~= 500); pile_up_contamination dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Shot poisson sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → S · disk · relxill reflection → F · fourier · power spectrum → O · chi2 · x ray spectrum → **y** (detector).

```
y = `O.chi2.x_ray_spectrum` `F.fourier.power_spectrum` `S.disk.relxill_reflection` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.disk.relxill_reflection` | S · disk · relxill reflection operator |
| `F.fourier.power_spectrum` | F · fourier · power spectrum operator |
| `O.chi2.x_ray_spectrum` | O · chi2 · x ray spectrum operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Astrophysics |
| Sub domain | Accretion physics |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 1D_disk_emissivity_profile |
| Noise model | shot_poisson |
| Integration axis | radial_disk |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered 1D_disk_emissivity_profile 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 ~= 500); pile_up_contamination dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Shot poisson sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | spin_constraint_sigma |
| Secondary | spectral_chi2_reduced |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr per bin | — | 20 |
| Spin a star | — | 0.7 |
| N counts total | — | 1e+06 |
| Inclination deg | deg | 40 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr per bin | — | 5 – 100 |
| Spin a star | — | 0.0 – 0.998 |
| N counts total | — | 10000.0 – 100000000.0 |
| Inclination deg | deg | 5 – 80 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.1 spectral_chi2_reduced

| Metric | Range |
|---|---|
| Spectral chi2 reduced | 0.8 – 3.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **spin_constraint_sigma**, with κ = `10000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xda64759d65ad666578c30849ca0830255f49522683040122a3dd95cdff3ba7cd`
- **Chain tx hash:** `0xf0cbfede7c123078bbf410e26c6cbbfa5aa4997af7e4567074cd08595556984c`
- **Chain block:** `41555178`

---

## File Mapping

This bundle consists of: `L1-375.md`, `L1-375.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-375.md` | Source of truth — edit this | Human or LLM |
| `L1-375.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 `L1-375`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._