# ⚛  L1 Principle — Charged Particle Track Reconstruction

**ID:** `L1-484` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Particle Physics — *Tracking detectors*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** track parameter vector
> **📡 Carrier:** particle · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555317

---

## 🧠 1. Introduction

**Charged Particle Track Reconstruction** is a **nonlinear inverse problem** whose unknown lives in **track parameter vector** space, within the **Tracking detectors** sub-domain of **Particle Physics**.

Measurements consist of particle interactions in a detector via a **silicon tracker hit pattern** sensing mechanism.

The forward operator applies, in order: S · kalman · track filter operator; time evolution of the state; O · chi2 · hit residuals operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered track_parameter_vector 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 ~= 200); alignment_uncertainty_um dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Time derivative → O · chi2 · hit residuals → **y** (detector).

```
y = `O.chi2.hit_residuals` ∂_t x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `O.chi2.hit_residuals` | O · chi2 · hit residuals operator |

**🛰 Estimator components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `S.kalman.track_filter` | S · kalman · track filter operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Particle Physics |
| Sub domain | Tracking detectors |
| Carrier | particle |
| Problem class | nonlinear_inverse |
| Solution space | track_parameter_vector |
| Noise model | gaussian |
| Integration axis | track_helix |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered track_parameter_vector 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 ~= 200); alignment_uncertainty_um dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | track_parameter_resolution_sigma_pt_over_pt |
| Secondary | tracking_efficiency_percent |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N layers | — | 10 |
| B field t | — | 2 |
| Pileup mu | — | 50 |
| Material x0 percent | — | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N layers | — | 4 – 20 |
| B field t | — | 0.5 – 4.0 |
| Pileup mu | — | 0 – 200 |
| Material x0 percent | — | 0.1 – 3.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.001 track_parameter_resolution_sigma_pt_over_pt

| Metric | Range |
|---|---|
| Track parameter resolution sigma pt over pt | 0.0001 – 0.01 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **track_parameter_resolution_sigma_pt_over_pt**, with κ = `10000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x751db39c2e389f94308495e5a28df961aff9eed62bff526231f292a7ead5f943`
- **Chain tx hash:** `0x0a900f5a04283e365f732dcd071354524a5e14b22bd4fec638f3257acdf17f3b`
- **Chain block:** `41555317`

---

## File Mapping

This bundle consists of: `L1-484.md`, `L1-484.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-484.md` | Source of truth — edit this | Human or LLM |
| `L1-484.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 `L1-484`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._