# ⚛  L1 Principle — Criticality Eigenvalue — k-effective calculation

**ID:** `L1-356` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Nuclear Engineering — *k-eff generalized eigenvalue*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** k eff scalar and fundamental mode
> **📡 Carrier:** neutron · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554097

---

## 🧠 1. Introduction

**Criticality Eigenvalue — k-effective calculation** is a **linear inverse problem** whose unknown lives in **k eff scalar and fundamental mode** space, within the **k-eff generalized eigenvalue** sub-domain of **Nuclear Engineering**.

Measurements consist of neutrons transmitted through the sample via a **reactivity meter** sensing mechanism.

The forward operator applies, in order: L · diffusion or transport operator; L · power iteration operator; L · fission source scale operator; int · eigenvalue operator.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · diffusion or transport → L · power iteration → L · fission source scale → int · eigenvalue → **y** (detector).

```
y = `int.eigenvalue` `L.fission_source_scale` `L.power_iteration` `L.diffusion_or_transport` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.diffusion_or_transport` | L · diffusion or transport operator |
| `L.power_iteration` | L · power iteration operator |
| `L.fission_source_scale` | L · fission source scale operator |
| `int.eigenvalue` | Int · eigenvalue operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Nuclear Engineering |
| Sub domain | k-eff generalized eigenvalue |
| Carrier | neutron |
| Problem class | linear_inverse |
| Solution space | k_eff_scalar_and_fundamental_mode |
| Noise model | poisson |
| Integration axis | eigenvalue |
| Difficulty delta | 5 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | k_eff_absolute_error |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `keff` · **Forward operator:** `keff_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 64 |
| Snr db | dB | 30 |
| N groups | — | 8 |
| Xs error | — | 0 |
| Burnup error | — | 0 |
| Reflector error | — | 0 |
| Power shape 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 – 47 |
| Xs error | — | 0.0 – 0.02 |
| Burnup error | — | 0.0 – 0.2 |
| Reflector error | — | 0.0 – 0.05 |
| Power shape error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.002

| Metric | Range |
|---|---|
| K eff delta | 0.0001 – 0.05 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **k_eff_absolute_error**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xa1e3a839b544bafd3a5f87d1fe9db23840096cf737ab7035c272670182d75064`
- **Chain tx hash:** `0x04d97db02a9cf14c0fb262121fa0cfb014ab1ab4d0b0dc50e11d92a47823877d`
- **Chain block:** `41554097`

---

## File Mapping

This bundle consists of: `L1-356.md`, `L1-356.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-356.md` | Source of truth — edit this | Human or LLM |
| `L1-356.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 `L1-356`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._