# ⚛  L1 Principle — Scanning TEM (STEM) — HAADF / BF imaging with convergent beam

**ID:** `L1-087` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electron Microscopy — *Atom-resolution Z-contrast imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D intensity
> **📡 Carrier:** electron · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554184

---

## 🧠 1. Introduction

**Scanning TEM (STEM) — HAADF / BF imaging with convergent beam** is a **linear inverse problem** whose unknown lives in **2D intensity** space, within the **Atom-resolution Z-contrast imaging** sub-domain of **Electron Microscopy**.

Measurements consist of electrons collected by an electron detector via a **convergent electron probe** sensing mechanism.

The forward operator applies, in order: L · illumination · convergent probe operator; ordered pixel-by-pixel sampling; D · annular detector operator; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D intensity is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 18); 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 → D · annular detector → Temporal integration → **y** (detector).

```
y = ∫_t dt `D.annular_detector` 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 |
| `D.annular_detector` | D · annular detector operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electron Microscopy |
| Sub domain | Atom-resolution Z-contrast imaging |
| Carrier | electron |
| Problem class | linear_inverse |
| Solution space | 2D_intensity |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Existence of the recovered 2D intensity is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 18); 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_haadf` · **Forward operator:** `stem_haadf_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 2048 |
| W | px | 2048 |
| Kv | — | 200 |
| Pixel a | — | 0.5 |
| Beam damage | — | 0 |
| Peak electrons | — | 200 |
| Scan distortion | — | 0 |
| Probe aberration | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 512 – 8192 |
| W | px | 512 – 8192 |
| Kv | — | 60 – 300 |
| Pixel a | — | 0.1 – 5.0 |
| Beam damage | — | 0.0 – 0.5 |
| Peak electrons | — | 5 – 5000 |
| Scan distortion | — | 0.0 – 0.05 |
| Probe aberration | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 28.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x3de9b42148752d3c7d632cad5be3635229b880185b8c96446c341163a0c20627`
- **Chain tx hash:** `0xad87a307345cbf69d564079e654446cd8b23316152b5bf0e7c765061f715d570`
- **Chain block:** `41554184`

---

## File Mapping

This bundle consists of: `L1-087.md`, `L1-087.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-087.md` | Source of truth — edit this | Human or LLM |
| `L1-087.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 `L1-087`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._