# ⚛  L1 Principle — Neutron Transport — linear Boltzmann equation

**ID:** `L1-354` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Nuclear Engineering — *Discrete-ordinates / SN neutron transport*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** angular flux phase space
> **📡 Carrier:** neutron · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554096

---

## 🧠 1. Introduction

**Neutron Transport — linear Boltzmann equation** is a **linear inverse problem** whose unknown lives in **angular flux phase space** space, within the **Discrete-ordinates / SN neutron transport** sub-domain of **Nuclear Engineering**.

Measurements consist of neutrons transmitted through the sample via a **flux foils detectors** sensing mechanism.

The forward operator applies, in order: L · streaming operator; L · collision scatter operator; L · source fission operator; int · angular energy operator.

Observations are corrupted by Poisson counting noise. 7D phase space (x, y, z, E, mu, phi, t); curse of dimensionality; ill-posed for sparse detector data.

## ⚙ 2. Forward Model

Physical chain: **x** → L · streaming → L · collision scatter → L · source fission → int · angular energy → **y** (detector).

```
y = `int.angular_energy` `L.source_fission` `L.collision_scatter` `L.streaming` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.streaming` | L · streaming operator |
| `L.collision_scatter` | L · collision scatter operator |
| `L.source_fission` | L · source fission operator |
| `int.angular_energy` | Int · angular energy operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Nuclear Engineering |
| Sub domain | Discrete-ordinates / SN neutron transport |
| Carrier | neutron |
| Problem class | linear_inverse |
| Solution space | angular_flux_phase_space |
| Noise model | poisson |
| Integration axis | angular_energy |
| Difficulty delta | 10 |
| L dag | 3.8 |

## 📡 4. Measurement Model

7D phase space (x, y, z, E, mu, phi, t); curse of dimensionality; ill-posed for sparse detector data.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `sn_transport` · **Forward operator:** `sn_transport_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| Z | — | 64 |
| Snr db | dB | 25 |
| N detectors | — | 16 |
| N angular dir | — | 24 |
| Xs data error | — | 0 |
| N energy groups | — | 47 |
| Material homo error | — | 0 |
| Boundary albedo error | — | 0 |
| Source spectrum error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 16 – 512 |
| W | px | 16 – 512 |
| Z | — | 16 – 512 |
| Snr db | dB | 0 – 40 |
| N detectors | — | 4 – 256 |
| N angular dir | — | 4 – 288 |
| Xs data error | — | 0.0 – 0.05 |
| N energy groups | — | 1 – 500 |
| Material homo error | — | 0.0 – 0.1 |
| Boundary albedo error | — | 0.0 – 0.2 |
| Source spectrum error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 5.0

| Metric | Range |
|---|---|
| Flux rel error | 1.0 – 50.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **flux_rel_error_percent**, with κ = `1200` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x06dd2b6aaaea158e4fd9975ba33aca7df674c73ad57dea914026c7adb6dbf8f6`
- **Chain tx hash:** `0x9621272528a0152743f7a09dcb16fe3e5da95987b53097e333dcea79764f1867`
- **Chain block:** `41554096`

---

## File Mapping

This bundle consists of: `L1-354.md`, `L1-354.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-354.md` | Source of truth — edit this | Human or LLM |
| `L1-354.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 `L1-354`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._