# ⚛  L1 Principle — 4D-STEM — pixelated-detector scanning diffraction imaging

**ID:** `L1-088` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electron Microscopy — *Momentum-resolved electron ptychography*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** complex 2D
> **📡 Carrier:** electron · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554184

---

## 🧠 1. Introduction

**4D-STEM — pixelated-detector scanning diffraction imaging** is a **nonlinear inverse problem** whose unknown lives in **complex 2D** space, within the **Momentum-resolved electron ptychography** sub-domain of **Electron Microscopy**.

Measurements consist of electrons collected by an electron detector via a **4d stem diffraction** sensing mechanism.

The forward operator applies, in order: L · illumination · convergent probe operator; ordered pixel-by-pixel sampling; L · diffraction operator; phase is lost; only intensity is measured; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered complex 2D is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 40); probe_aberration dominates the stability cliff; scan_distortion and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

## ⚙ 2. Forward Model

Physical chain: **x** → L · illumination · convergent probe → Raster scan → L · diffraction → Magnitude-squared → Spatial integration → **y** (detector).

```
y = ∫_A dA |·|² `L.diffraction` S_raster `L.illumination.convergent_probe` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.illumination.convergent_probe` | L · illumination · convergent probe operator |
| `S.scan.raster` | Ordered pixel-by-pixel sampling |
| `L.diffraction` | L · diffraction operator |
| `L.modulus_squared` | Phase is lost; only intensity is measured |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electron Microscopy |
| Sub domain | Momentum-resolved electron ptychography |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | complex_2D |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 10 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered complex 2D is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 40); probe_aberration dominates the stability cliff; scan_distortion and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `stem_ptychography` · **Forward operator:** `stem_ptychography_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Kv | — | 200 |
| N q | — | 128 |
| Pixel a | — | 0.3 |
| Dose e per a2 | — | 10000 |
| Scan distortion | — | 0 |
| Probe aberration | — | 0 |
| Partial coherence | — | 1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Kv | — | 60 – 300 |
| N q | — | 32 – 1024 |
| Pixel a | — | 0.1 – 2.0 |
| Dose e per a2 | — | 100 – 100000 |
| Scan distortion | — | 0.0 – 0.05 |
| Probe aberration | — | 0.0 – 0.3 |
| Partial coherence | — | 0.6 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 24.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `800` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x7cd537f25ca9dedbb219a2a693607d677a0daf6fe75a8b693c8b436e3fa150cf`
- **Chain tx hash:** `0x0a49debd1fc8f2593e9823c630231c10af5a8eca5c4863f0cfd0dacee167530b`
- **Chain block:** `41554184`

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## File Mapping

This bundle consists of: `L1-088.md`, `L1-088.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-088.md` | Source of truth — edit this | Human or LLM |
| `L1-088.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 `L1-088`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._