# ⚛  L1 Principle — Three-Photon Microscopy (chi-5 deep-tissue imaging)

**ID:** `L1-023` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Higher-order NLO microscopy for thick scattering tissue*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D intensity
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554169

---

## 🧠 1. Introduction

**Three-Photon Microscopy (chi-5 deep-tissue imaging)** is a **nonlinear inverse problem** whose unknown lives in **3D intensity** space, within the **Higher-order NLO microscopy for thick scattering tissue** sub-domain of **Microscopy**.

Measurements consist of photons collected by an optical detector via a **three photon nlo** sensing mechanism.

The forward operator applies, in order: L · excitation · nonlinear chi5 operator; ordered pixel-by-pixel sampling; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 3D intensity 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 ~= 22); pulse_compression_drift dominates the stability cliff; sample_scattering 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 · excitation · nonlinear chi5 → Raster scan → Temporal integration → **y** (detector).

```
y = ∫_t dt S_raster `L.excitation.nonlinear_chi5` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.excitation.nonlinear_chi5` | L · excitation · nonlinear chi5 operator |
| `S.scan.raster` | Ordered pixel-by-pixel sampling |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | Higher-order NLO microscopy for thick scattering tissue |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 3D_intensity |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 10 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Existence of the recovered 3D intensity 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 ~= 22); pulse_compression_drift dominates the stability cliff; sample_scattering 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:** `three_photon` · **Forward operator:** `three_photon_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Z | — | 100 |
| Na | — | 1 |
| Pixel nm | nm | 400 |
| Lambda ex nm | nm | 1300 |
| Peak photons | photons | 30 |
| Pulse width fs | — | 60 |
| Sample scattering | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 128 – 2048 |
| W | px | 128 – 2048 |
| Z | — | 16 – 512 |
| Na | — | 0.5 – 1.1 |
| Pixel nm | nm | 200 – 1000 |
| Lambda ex nm | nm | 1200 – 1700 |
| Peak photons | photons | 5 – 500 |
| Pulse width fs | — | 30 – 300 |
| Sample scattering | — | 0.0 – 0.8 |
| Long wavelength absorption | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `440` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x6006f879b51c8780e420c3954e99bc17c13956a58980560d0d0fe2508eaa644a`
- **Chain tx hash:** `0xe772075299607a0c14e359f93aad3b5b4757c04e22c383ee8a3054094792b0e2`
- **Chain block:** `41554169`

---

## File Mapping

This bundle consists of: `L1-023.md`, `L1-023.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-023.md` | Source of truth — edit this | Human or LLM |
| `L1-023.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 `L1-023`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._