# ⚛  L1 Principle — DNA-PAINT (transient DNA-hybridization localization microscopy)

**ID:** `L1-024` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Programmable-blinking single-molecule super-resolution*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** point list
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554169

---

## 🧠 1. Introduction

**DNA-PAINT (transient DNA-hybridization localization microscopy)** is a **nonlinear inverse problem** whose unknown lives in **point list** space, within the **Programmable-blinking single-molecule super-resolution** sub-domain of **Microscopy**.

Measurements consist of photons collected by an optical detector via a **single molecule localization** sensing mechanism.

The forward operator applies, in order: L · dna hybridization blinking operator; convolution with the Airy disk of a circular aperture; L · gaussian fit operator; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered point list 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 ~= 28); docking_strand_concentration dominates the stability cliff; drift_nm_per_frame 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 · dna hybridization blinking → Airy PSF convolution → L · gaussian fit → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.gaussian_fit` K_Airy * `L.dna_hybridization_blinking` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.dna_hybridization_blinking` | L · dna hybridization blinking operator |
| `K.psf.airy` | Convolution with the airy disk of a circular aperture |
| `L.gaussian_fit` | L · gaussian fit operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | Programmable-blinking single-molecule super-resolution |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | point_list |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 10 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered point list 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 ~= 28); docking_strand_concentration dominates the stability cliff; drift_nm_per_frame 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 | localization_precision_nm |
| Secondary | detection_recall |

## 📏 5. Operating Range (Ω)

**Center problem class:** `dnapaint` · **Forward operator:** `dnapaint_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Na | — | 1.49 |
| T frames | — | 30000 |
| Pixel nm | nm | 160 |
| Drift nm per frame | — | 0 |
| Photons per binding | — | 5000 |
| Non specific binding | — | 0.02 |
| Strand concentration nm | nm | 5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 – 2048 |
| W | px | 64 – 2048 |
| Na | — | 1.3 – 1.49 |
| T frames | — | 5000 – 200000 |
| Pixel nm | nm | 80 – 300 |
| Drift nm per frame | — | 0.0 – 3.0 |
| Photons per binding | — | 500 – 50000 |
| Non specific binding | — | 0.0 – 0.2 |
| Strand concentration nm | nm | 0.5 – 50 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 8.0

| Metric | Range |
|---|---|
| Localization nm | 0.5 – 50.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **localization_precision_nm**, with κ = `560` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x4edb8ecb50a6b1edb2f3b4f786698438c74036348e3bd3f3737100858e1b4fa6`
- **Chain tx hash:** `0xc1190d7b1d1a36d54875331e7c64136ada552c562d976632d12d21f58d60c84e`
- **Chain block:** `41554169`

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## File Mapping

This bundle consists of: `L1-024.md`, `L1-024.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-024.md` | Source of truth — edit this | Human or LLM |
| `L1-024.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 `L1-024`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._