# ⚛  L1 Principle — Cryo-Electron Tomography (cryoET) — in-situ 3D biology at nm scale

**ID:** `L1-092` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electron Microscopy — *In-situ structural biology via cryo tilt-series*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D density
> **📡 Carrier:** electron · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554185

---

## 🧠 1. Introduction

**Cryo-Electron Tomography (cryoET) — in-situ 3D biology at nm scale** is a **linear inverse problem** whose unknown lives in **3D density** space, within the **In-situ structural biology via cryo tilt-series** sub-domain of **Electron Microscopy**.

Measurements consist of electrons collected by an electron detector via a **cryo electron transmission** sensing mechanism.

The forward operator applies, in order: L · transmit sample operator; rotates source / detector to acquire different projections; spreads measurements back along source rays (adjoint operator); L · subtomogram average operator; integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 3D density 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 ~= 35); tilt_angle_error dominates the stability cliff; CTF_per_tilt 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 · transmit sample → Angular scan → L · subtomogram average → Angular integration → **y** (detector).

```
y = ∫dΩ `L.subtomogram_average` R(θ) `L.transmit_sample` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.transmit_sample` | L · transmit sample operator |
| `S.scan.angular` | Rotates source / detector to acquire different projections |
| `L.subtomogram_average` | L · subtomogram average operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `L.backproject` | Spreads measurements back along source rays (adjoint operator) |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electron Microscopy |
| Sub domain | In-situ structural biology via cryo tilt-series |
| Carrier | electron |
| Problem class | linear_inverse |
| Solution space | 3D_density |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 10 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered 3D density 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 ~= 35); tilt_angle_error dominates the stability cliff; CTF_per_tilt 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:** `cryoet_tilt_series` · **Forward operator:** `cryoet_tilt_series_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 4096 |
| W | px | 4096 |
| Z | — | 256 |
| Kv | — | 300 |
| N tilts | — | 61 |
| Beam damage | — | 0.2 |
| Total dose e | — | 60 |
| Missing wedge | — | 0.33 |
| Tilt range deg | deg | 60 |
| Tilt angle error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 1024 – 8192 |
| W | px | 1024 – 8192 |
| Z | — | 64 – 1024 |
| Kv | — | 300 – 300 |
| N tilts | — | 20 – 200 |
| Beam damage | — | 0.0 – 0.8 |
| Total dose e | — | 20 – 200 |
| Missing wedge | — | 0.1 – 0.5 |
| Dose weighting | — | 0.0 – 1.0 |
| Tilt range deg | deg | 40 – 80 |
| Tilt angle error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `700` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x0dbf3b1a3249338f26ada5f995e0eff81af86c94902cebc8b873a412a75666d1`
- **Chain tx hash:** `0x2b24a247c36da2878087519cc99e669d3cb0cc60ad41af578be47e17fbc2f306`
- **Chain block:** `41554185`

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## File Mapping

This bundle consists of: `L1-092.md`, `L1-092.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-092.md` | Source of truth — edit this | Human or LLM |
| `L1-092.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.
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_This Markdown was auto-synthesized from the catalog row for `L1-092`._
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