# ⚛  L1 Principle — Kinetic Monte Carlo (KMC)

**ID:** `L1-309` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Chemistry — *Rare-event stochastic dynamics*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** kmc trajectory ensemble
> **📡 Carrier:** none · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554115

---

## 🧠 1. Introduction

**Kinetic Monte Carlo (KMC)** is a **nonlinear inverse problem** whose unknown lives in **kmc trajectory ensemble** space, within the **Rare-event stochastic dynamics** sub-domain of **Computational Chemistry**.

Measurements consist of none via a **event sequence observable** sensing mechanism.

The forward operator applies, in order: E · rate catalog operator; int · stochastic operator; O · trajectory distribution operator.

Observations are corrupted by Poisson counting noise. Well-posed; Markov chain analysis gives eigenvalues of rate matrix.

## ⚙ 2. Forward Model

Physical chain: **x** → E · rate catalog → int · stochastic → O · trajectory distribution → **y** (detector).

```
y = `O.trajectory_distribution` `int.stochastic` `E.rate_catalog` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.rate_catalog` | E · rate catalog operator |
| `int.stochastic` | Int · stochastic operator |
| `O.trajectory_distribution` | O · trajectory distribution operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Chemistry |
| Sub domain | Rare-event stochastic dynamics |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | kmc_trajectory_ensemble |
| Noise model | poisson |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Well-posed; Markov chain analysis gives eigenvalues of rate matrix.

| Metric | Value |
|---|---|
| Metric | rate_distribution_KL |
| Secondary | first_passage_time_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `kinetic_monte_carlo` · **Forward operator:** `kmc_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 600 |
| N events | — | 1e+06 |
| N states | — | 1000 |
| N rates per state | — | 10 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 100 – 3000 |
| N events | — | 1000 – 1000000000000.0 |
| N states | — | 10 – 1000000000.0 |
| N rates per state | — | 2 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** KL <= 0.03

| Metric | Range |
|---|---|
| Rate distribution kl | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **rate_distribution_KL**, with κ = `100` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xaf6a09ab0a5c2131e4cf52d2256a0f21f3ab0da3861795c2743a92ca4506ffad`
- **Chain tx hash:** `0xe61c632d17c8e852f738552e7867bf6b46a58dc3c10a8d2f5fcf74ace6496826`
- **Chain block:** `41554115`

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## File Mapping

This bundle consists of: `L1-309.md`, `L1-309.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-309.md` | Source of truth — edit this | Human or LLM |
| `L1-309.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 `L1-309`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._