# ⚛  L1 Principle — Lattice QCD Path Integral

**ID:** `L1-480` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Particle Physics — *Lattice field theory*
> **🎯 Problem class:** statistical inverse · **🧮 Solution space:** QCD observable spectrum
> **📡 Carrier:** particle · **🌫 Noise:** monte carlo statistical
> **⚖ Difficulty (δ):** 50 · **⛓ Block:** 41555301

---

## 🧠 1. Introduction

**Lattice QCD Path Integral** is a **statistical inverse problem** whose unknown lives in **QCD observable spectrum** space, within the **Lattice field theory** sub-domain of **Particle Physics**.

Measurements consist of particle interactions in a detector via a **lattice gauge mc sampling** sensing mechanism.

The forward operator applies, in order: S · hmc · hybrid monte carlo operator; gradient / divergence with respect to position; O · correlator · hadron spectrum operator.

Observations are corrupted by monte carlo statistical. Existence of the recovered QCD_observable_spectrum is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 1000000.0); discretization_error_O_a2 dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Monte carlo statistical sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → S · hmc · hybrid monte carlo → Spatial derivative → O · correlator · hadron spectrum → **y** (detector).

```
y = `O.correlator.hadron_spectrum` ∇ `S.hmc.hybrid_monte_carlo` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.hmc.hybrid_monte_carlo` | S · hmc · hybrid monte carlo operator |
| `D.space` | Gradient / divergence with respect to position |
| `O.correlator.hadron_spectrum` | O · correlator · hadron spectrum operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Particle Physics |
| Sub domain | Lattice field theory |
| Carrier | particle |
| Problem class | statistical_inverse |
| Solution space | QCD_observable_spectrum |
| Noise model | monte_carlo_statistical |
| Integration axis | spacetime_lattice |
| Difficulty delta | 50 |
| L dag | 6 |

## 📡 4. Measurement Model

Existence of the recovered QCD_observable_spectrum is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 1000000.0); discretization_error_O_a2 dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Monte carlo statistical sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | hadron_mass_RMSE_MeV |
| Secondary | continuum_extrapolation_chi2 |

## 📏 5. Operating Range (Ω)

**Center problem class:** `statistical_inverse` · **Forward operator:** `lattice_gauge_mc_sampling`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N configs | — | 500 |
| N l spatial | — | 32 |
| N t temporal | — | 64 |
| Beta coupling | — | 6.2 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N configs | — | 100 – 10000 |
| N l spatial | — | 8 – 64 |
| N t temporal | — | 16 – 128 |
| Beta coupling | — | 5.5 – 7.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 5 hadron_mass_RMSE_MeV

| Metric | Range |
|---|---|
| Hadron mass rmse mev | 0.5 – 50 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **hadron_mass_RMSE_MeV**, with κ = `10000000000.0` and δ = `50`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x328a83a9fb2a8e5db58e86bf5dce897a2fad750fb0eb332808a9313010e62593`
- **Chain tx hash:** `0xdc01e867600c259e0693960671052d94ea9616497a226a91ad446cb2b965ad65`
- **Chain block:** `41555301`

---

## File Mapping

This bundle consists of: `L1-480.md`, `L1-480.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-480.md` | Source of truth — edit this | Human or LLM |
| `L1-480.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 `L1-480`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._