# ⚛  L1 Principle — CMB Gravitational Lensing Reconstruction

**ID:** `L1-367` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Astrophysics — *CMB lensing*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D lensing convergence
> **📡 Carrier:** photon · **🌫 Noise:** thermal gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555176

---

## 🧠 1. Introduction

**CMB Gravitational Lensing Reconstruction** is a **nonlinear inverse problem** whose unknown lives in **2D lensing convergence** space, within the **CMB lensing** sub-domain of **Astrophysics**.

Measurements consist of photons collected by an optical detector via a **quadratic lensing estimator** sensing mechanism.

The forward operator applies, in order: F · fourier · harmonic operator; S · quadratic · estimator operator; K · filter operator.

Observations are corrupted by thermally-driven Gaussian fluctuations. Existence of the recovered 2D_lensing_convergence 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); foreground_bias 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 · harmonic → S · quadratic · estimator → **y** (detector).

```
y = `S.quadratic.estimator` `F.fourier.harmonic` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `F.fourier.harmonic` | F · fourier · harmonic operator |
| `S.quadratic.estimator` | S · quadratic · estimator operator |

**🛰 Estimator components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `K.filter` | K · filter operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Astrophysics |
| Sub domain | CMB lensing |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 2D_lensing_convergence |
| Noise model | thermal_gaussian |
| Integration axis | angular_multipole |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

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

| Metric | Value |
|---|---|
| Metric | lensing_power_spectrum_C_kk_chi2 |
| Secondary | reconstruction_SNR |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N pix side | — | 2048 |
| Noise uk arcmin | — | 5 |
| Beam fwhm arcmin | — | 1.5 |
| Survey area deg2 | — | 10000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N pix side | — | 512 – 8192 |
| Noise uk arcmin | — | 1.0 – 50.0 |
| Beam fwhm arcmin | — | 0.5 – 10.0 |
| Survey area deg2 | — | 1000 – 40000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.05 lensing_power_spectrum_C_kk_chi2

| Metric | Range |
|---|---|
| Lensing power spectrum c kk chi2 | 0.9 – 2.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **lensing_power_spectrum_C_kk_chi2**, with κ = `100000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x3deb3f685300556ed7f8cfc9ca10983f283197b819c691782086109f0fe9ea25`
- **Chain tx hash:** `0x759b9b0c9a8be2cf518b8a556f4d52b15ebd637238c7d290b7513bcc4b6743d8`
- **Chain block:** `41555176`

---

## File Mapping

This bundle consists of: `L1-367.md`, `L1-367.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-367.md` | Source of truth — edit this | Human or LLM |
| `L1-367.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 `L1-367`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._