# ⚛  L1 Principle — Expansion Microscopy (ExM) — physical hydrogel expansion

**ID:** `L1-014` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Sample-side super-resolution via isotropic swelling*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D intensity
> **📡 Carrier:** photon · **🌫 Noise:** poisson gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554155

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## 🧠 1. Introduction

**Expansion Microscopy (ExM) — physical hydrogel expansion** is a **linear inverse problem** whose unknown lives in **2D intensity** space, within the **Sample-side super-resolution via isotropic swelling** sub-domain of **Microscopy**.

Measurements consist of photons collected by an optical detector via a **fluorescence widefield** sensing mechanism.

The forward operator applies, in order: L · sample expansion operator; convolution with the Airy disk of a circular aperture; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise plus Gaussian read-out noise. 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 well-conditioned (kappa_eff ~= 9); expansion_anisotropy dominates the stability cliff; expansion_factor_drift and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

## ⚙ 2. Forward Model

Physical chain: **x** → L · sample expansion → Airy PSF convolution → Temporal integration → **y** (detector).

```
y = ∫_t dt K_Airy * `L.sample_expansion` x + n,    Poisson + 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.sample_expansion` | L · sample expansion operator |
| `K.psf.airy` | Convolution with the airy disk of a circular aperture |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | Sample-side super-resolution via isotropic swelling |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 2D_intensity |
| Noise model | poisson_gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 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 well-conditioned (kappa_eff ~= 9); expansion_anisotropy dominates the stability cliff; expansion_factor_drift and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `exm` · **Forward operator:** `exm_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1024 |
| W | px | 1024 |
| Na | — | 1.4 |
| F expand | — | 4 |
| Pixel nm | nm | 65 |
| Peak photons | photons | 500 |
| Expansion anisotropy | — | 0.01 |
| Expansion factor drift | — | 0.01 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 4096 |
| W | px | 256 – 4096 |
| Na | — | 1.0 – 1.49 |
| F expand | — | 2.0 – 20.0 |
| Pixel nm | nm | 30 – 200 |
| Peak photons | photons | 50 – 5000 |
| Labeling loss | — | 0.0 – 0.3 |
| Expansion anisotropy | — | 0.0 – 0.1 |
| Expansion factor drift | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 29.0

| Metric | Range |
|---|---|
| Psnr db | 12.0 – 42.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `180` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x58199cb8a72dcc774e6eb7374523c69ff971193ff50bef3f32a33ecc70dd08ec`
- **Chain tx hash:** `0xb92916a6b1edab33ba8493d43604696f92b49a5ce5efd0b938503ff0532b707d`
- **Chain block:** `41554155`

---

## File Mapping

This bundle consists of: `L1-014.md`, `L1-014.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-014.md` | Source of truth — edit this | Human or LLM |
| `L1-014.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 `L1-014`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._