# ⚛  L1 Principle — Radiative Transfer Equation — radiative heat exchange

**ID:** `L1-199` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Heat Transfer — *RTE inversion for surface emissivity / participating media*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D radiance field spectral
> **📡 Carrier:** thermal · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554034

---

## 🧠 1. Introduction

**Radiative Transfer Equation — radiative heat exchange** is a **linear inverse problem** whose unknown lives in **3D radiance field spectral** space, within the **RTE inversion for surface emissivity / participating media** sub-domain of **Heat Transfer**.

Measurements consist of thermal radiation / temperature measurements via a **ir spectroradiometer** sensing mechanism.

The forward operator applies, in order: isotropic point-source emission model; L · absorb scatter operator; L · surface reflect operator; detector sums all spectral bands.

Observations are corrupted by additive Gaussian noise. Volume RTE becomes ill-posed for optically thick media; well-posed for optically thin.

## ⚙ 2. Forward Model

Physical chain: **x** → Source emission → L · absorb scatter → L · surface reflect → Spectral integration → **y** (detector).

```
y = Σ_λ `L.surface_reflect` `L.absorb_scatter` δ_source x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.source_emit` | Isotropic point-source emission model |
| `L.absorb_scatter` | L · absorb scatter operator |
| `L.surface_reflect` | L · surface reflect operator |
| `int.spectral` | Detector sums all spectral bands |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Heat Transfer |
| Sub domain | RTE inversion for surface emissivity / participating media |
| Carrier | thermal |
| Problem class | linear_inverse |
| Solution space | 3D_radiance_field_spectral |
| Noise model | gaussian |
| Integration axis | spectral |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Volume RTE becomes ill-posed for optically thick media; well-posed for optically thin.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `rte` · **Forward operator:** `rte_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 32 |
| N dirs | — | 16 |
| Snr db | dB | 30 |
| N freqs | — | 16 |
| Kappa error | — | 0 |
| Emissivity error | — | 0 |
| Phase function error | — | 0 |
| Scatter albedo error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Z | — | 8 – 256 |
| N dirs | — | 1 – 256 |
| Snr db | dB | 0 – 40 |
| N freqs | — | 1 – 256 |
| Kappa error | — | 0.0 – 0.1 |
| Emissivity error | — | 0.0 – 0.3 |
| Phase function error | — | 0.0 – 0.3 |
| Scatter albedo error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `600` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x697e60188454474c9e570631c22778318398976c10ae7b6215f774f82aff924b`
- **Chain tx hash:** `0x37198c0141c7f2396c52d44dd987daee9f5874d68798ade54dadc3828a2683bf`
- **Chain block:** `41554034`

---

## File Mapping

This bundle consists of: `L1-199.md`, `L1-199.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-199.md` | Source of truth — edit this | Human or LLM |
| `L1-199.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 `L1-199`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._