# ⚛  L1 Principle — Neutron Diffusion — multi-group diffusion approximation

**ID:** `L1-355` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Nuclear Engineering — *Multi-group diffusion (PWR / BWR core physics)*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** core flux map
> **📡 Carrier:** neutron · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554097

---

## 🧠 1. Introduction

**Neutron Diffusion — multi-group diffusion approximation** is a **linear inverse problem** whose unknown lives in **core flux map** space, within the **Multi-group diffusion (PWR / BWR core physics)** sub-domain of **Nuclear Engineering**.

Measurements consist of neutrons transmitted through the sample via a **flux map incore** sensing mechanism.

The forward operator applies, in order: L · diffusion operator; L · group coupling operator; L · fission source operator; int · energy operator.

Observations are corrupted by Poisson counting noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · diffusion → L · group coupling → L · fission source → int · energy → **y** (detector).

```
y = `int.energy` `L.fission_source` `L.group_coupling` `L.diffusion` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.diffusion` | L · diffusion operator |
| `L.group_coupling` | L · group coupling operator |
| `L.fission_source` | L · fission source operator |
| `int.energy` | Int · energy operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Nuclear Engineering |
| Sub domain | Multi-group diffusion (PWR / BWR core physics) |
| Carrier | neutron |
| Problem class | linear_inverse |
| Solution space | core_flux_map |
| Noise model | poisson |
| Integration axis | energy |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | flux_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `diffusion` · **Forward operator:** `diffusion_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 64 |
| Snr db | dB | 30 |
| N groups | — | 2 |
| N detectors | — | 64 |
| D coeff error | — | 0 |
| Sigma a error | — | 0 |
| Nu sigma f error | — | 0 |
| Extrapol dist error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 512 |
| W | px | 32 – 512 |
| Z | — | 32 – 512 |
| Snr db | dB | 0 – 40 |
| N groups | — | 1 – 10 |
| N detectors | — | 8 – 512 |
| D coeff error | — | 0.0 – 0.05 |
| Sigma a error | — | 0.0 – 0.05 |
| Nu sigma f error | — | 0.0 – 0.05 |
| Extrapol dist error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 3.0

| Metric | Range |
|---|---|
| Flux rel error | 0.5 – 30.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **flux_rel_error_percent**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x587c24a0168e9b20d41842ecc720aa0f295f759892b564244887143fac66b69e`
- **Chain tx hash:** `0x790956123eb436b8e42585bd77ffc50d1b82a741448e167ac12626322c1ef2c4`
- **Chain block:** `41554097`

---

## File Mapping

This bundle consists of: `L1-355.md`, `L1-355.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-355.md` | Source of truth — edit this | Human or LLM |
| `L1-355.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 `L1-355`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._