# ⚛  L1 Principle — Metropolis Monte Carlo

**ID:** `L1-310` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Chemistry — *Equilibrium configurational sampling*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** Boltzmann ensemble
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554115

---

## 🧠 1. Introduction

**Metropolis Monte Carlo** is a **linear inverse problem** whose unknown lives in **Boltzmann ensemble** space, within the **Equilibrium configurational sampling** sub-domain of **Computational Chemistry**.

Measurements consist of none via a **ensemble observable** sensing mechanism.

The forward operator applies, in order: E · potential U operator; int · stochastic operator; O · ensemble average operator.

Observations are corrupted by additive Gaussian noise. Well-posed; ergodicity + detailed balance ensure convergence.

## ⚙ 2. Forward Model

Physical chain: **x** → E · potential U → int · stochastic → O · ensemble average → **y** (detector).

```
y = `O.ensemble_average` `int.stochastic` `E.potential_U` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.potential_U` | E · potential u operator |
| `int.stochastic` | Int · stochastic operator |
| `O.ensemble_average` | O · ensemble average operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Chemistry |
| Sub domain | Equilibrium configurational sampling |
| Carrier | none |
| Problem class | linear_inverse |
| Solution space | Boltzmann_ensemble |
| Noise model | gaussian |
| Integration axis | mc_steps |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Well-posed; ergodicity + detailed balance ensure convergence.

| Metric | Value |
|---|---|
| Metric | observable_relative_error |
| Secondary | autocorrelation_time |

## 📏 5. Operating Range (Ω)

**Center problem class:** `monte_carlo_metropolis` · **Forward operator:** `metropolis_mc_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 300 |
| N sweeps | — | 1e+06 |
| Ensemble | — | NVT |
| N particles | — | 1000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 1 – 10000 |
| N sweeps | — | 10000 – 10000000000.0 |
| Ensemble | — | NVT, NPT, GCMC, Gibbs |
| N particles | — | 10 – 10000000.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** observable rel error <= 0.02

| Metric | Range |
|---|---|
| Observable relative error | 0.005 – 0.3 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x98eab4dc52f93d074d6742458d1cb041fe58fed198ce87d0cdd6431f0b113b88`
- **Chain tx hash:** `0x7af301dfb644e5a15bc354a486da48fa9a2c41047d5413a7648d3813653e8b98`
- **Chain block:** `41554115`

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## File Mapping

This bundle consists of: `L1-310.md`, `L1-310.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-310.md` | Source of truth — edit this | Human or LLM |
| `L1-310.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 `L1-310`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._