# ⚛  L1 Principle — Parton Distribution Functions (DGLAP)

**ID:** `L1-481` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Particle Physics — *QCD phenomenology*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** PDF function space
> **📡 Carrier:** particle · **🌫 Noise:** experimental gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41555301

---

## 🧠 1. Introduction

**Parton Distribution Functions (DGLAP)** is a **nonlinear inverse problem** whose unknown lives in **PDF function space** space, within the **QCD phenomenology** sub-domain of **Particle Physics**.

Measurements consist of particle interactions in a detector via a **deep inelastic scattering** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; F · mellin · transform operator; O · chi2 · cross section data operator.

Observations are corrupted by experimental gaussian. Existence of the recovered PDF_function_space 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 ~= 10000.0); nuclear_correction_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Experimental gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Pointwise nonlinearity → F · mellin · transform → O · chi2 · cross section data → **y** (detector).

```
y = `O.chi2.cross_section_data` `F.mellin.transform` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `F.mellin.transform` | F · mellin · transform operator |
| `O.chi2.cross_section_data` | O · chi2 · cross section data operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Particle Physics |
| Sub domain | QCD phenomenology |
| Carrier | particle |
| Problem class | nonlinear_inverse |
| Solution space | PDF_function_space |
| Noise model | experimental_gaussian |
| Integration axis | bjorken_x |
| Difficulty delta | 10 |
| L dag | 5 |

## 📡 4. Measurement Model

Existence of the recovered PDF_function_space 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 ~= 10000.0); nuclear_correction_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Experimental gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | chi2_per_dof_global_fit |
| Secondary | PDF_uncertainty_band_percent |

## 📏 5. Operating Range (Ω)

**Center problem class:** `nonlinear_inverse` · **Forward operator:** `deep_inelastic_scattering`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Q range gev | — | 1000 |
| N data points | — | 3000 |
| Alpha s value | — | 0.118 |
| X min bjorken | — | 1e-05 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Q range gev | — | 10 – 10000 |
| N data points | — | 100 – 10000 |
| Alpha s value | — | 0.11 – 0.125 |
| X min bjorken | — | 1e-06 – 0.01 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.05 chi2_per_dof_global_fit

| Metric | Range |
|---|---|
| Chi2 per dof global fit | 0.8 – 2.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **chi2_per_dof_global_fit**, with κ = `1000000.0` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xaeffe6d3b9db61dc28582c8613e73539a794a3004fe5d1df599e576f6b4bf17a`
- **Chain tx hash:** `0x41a04ee30d49fe320e0465af284462b14e987ba2a60d2009e6e0f883f8e766af`
- **Chain block:** `41555301`

---

## File Mapping

This bundle consists of: `L1-481.md`, `L1-481.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-481.md` | Source of truth — edit this | Human or LLM |
| `L1-481.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 `L1-481`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._