# ⚛  L1 Principle — Structured Illumination Microscopy (SIM) — lateral 2x resolution

**ID:** `L1-005` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Super-resolution via spatial-frequency demodulation*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D intensity
> **📡 Carrier:** photon · **🌫 Noise:** poisson gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554153

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## 🧠 1. Introduction

**Structured Illumination Microscopy (SIM) — lateral 2x resolution** is a **linear inverse problem** whose unknown lives in **2D intensity** space, within the **Super-resolution via spatial-frequency demodulation** sub-domain of **Microscopy**.

Measurements consist of photons collected by an optical detector via a **structured illumination** sensing mechanism.

The forward operator applies, in order: L · modulation · grating 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 moderately conditioned (kappa_eff ~= 25); pattern_phase_error dominates the stability cliff; pattern_angle_error and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise 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 · modulation · grating → Airy PSF convolution → Temporal integration → **y** (detector).

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

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.modulation.grating` | L · modulation · grating 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 | Super-resolution via spatial-frequency demodulation |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 2D_intensity |
| Noise model | poisson_gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 2.8 |

## 📡 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 ~= 25); pattern_phase_error dominates the stability cliff; pattern_angle_error and the remaining mismatch parameters contribute higher-order bias terms. Poisson signal noise + gaussian read noise 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:** `sim_3phase_3angle` · **Forward operator:** `sim_forward_3phase_3angle`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Na | — | 1.4 |
| N angle | — | 3 |
| N phase | — | 3 |
| Pixel nm | nm | 32 |
| Peak photons | photons | 500 |
| Pattern angle error | — | 0 |
| Pattern phase error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 128 – 2048 |
| W | px | 128 – 2048 |
| Na | — | 1.0 – 1.49 |
| N angle | — | 3 – 5 |
| N phase | — | 3 – 5 |
| Pixel nm | nm | 20 – 100 |
| Peak photons | photons | 50 – 5000 |
| Pattern angle error | — | 0.0 – 0.05 |
| Pattern phase error | — | 0.0 – 0.15 |
| Pattern frequency drift | — | 0.0 – 0.02 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x9c04bb004ae98936d313b23e64ea0a714d44319c71f5f92685c46ab0a3cbd04a`
- **Chain tx hash:** `0x5f5c300ca855e9e2e7ab5252c8ec44edc8a8181d76c3f9286c90a296d53d14e1`
- **Chain block:** `41554153`

---

## File Mapping

This bundle consists of: `L1-005.md`, `L1-005.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-005.md` | Source of truth — edit this | Human or LLM |
| `L1-005.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 `L1-005`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._