# ⚛  L1 Principle — Fluorescence Lifetime Imaging Microscopy (FLIM)

**ID:** `L1-007` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Time-correlated single photon counting*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D lifetime
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554154

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## 🧠 1. Introduction

**Fluorescence Lifetime Imaging Microscopy (FLIM)** is a **nonlinear inverse problem** whose unknown lives in **2D lifetime** space, within the **Time-correlated single photon counting** sub-domain of **Microscopy**.

Measurements consist of photons collected by an optical detector via a **tcspc lifetime** sensing mechanism.

The forward operator applies, in order: L · excitation · pulsed operator; D · tcspc histogram operator; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D lifetime 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 ~= 12); IRF_shape_drift dominates the stability cliff; afterpulsing_rate and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) 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 · excitation · pulsed → D · tcspc histogram → Temporal integration → **y** (detector).

```
y = ∫_t dt `D.tcspc_histogram` `L.excitation.pulsed` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.excitation.pulsed` | L · excitation · pulsed operator |
| `D.tcspc_histogram` | D · tcspc histogram operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | Time-correlated single photon counting |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 2D_lifetime |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered 2D lifetime 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 ~= 12); IRF_shape_drift dominates the stability cliff; afterpulsing_rate and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) 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:** `flim_tcspc` · **Forward operator:** `flim_tcspc_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| N t | — | 256 |
| N tau | — | 1 |
| Photon count | — | 500 |
| Tau range ns | ns | 0.5 – 8.0 |
| Irf shape drift | — | 0 |
| Ambient fluorescence | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 – 1024 |
| W | px | 64 – 1024 |
| N t | — | 64 – 1024 |
| N tau | — | 1 – 3 |
| Photon count | — | 20 – 10000 |
| Irf shape drift | — | 0.0 – 0.2 |
| Afterpulsing rate | — | 0.0 – 0.05 |
| Ambient fluorescence | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.1

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `240` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x946340703e52739d464cc09d82470a4295ddff89f84d1fc4ed17d58b600a6cba`
- **Chain tx hash:** `0x6de3035aa50fca6421f6d8bd3eb46a1e6a4abfc80bb1b54be566893ed7faaa3c`
- **Chain block:** `41554154`

---

## File Mapping

This bundle consists of: `L1-007.md`, `L1-007.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-007.md` | Source of truth — edit this | Human or LLM |
| `L1-007.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 `L1-007`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._