# ⚛  L1 Principle — Landau-de Gennes Liquid Crystal Theory

**ID:** `L1-479` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Polymer Physics — *Nematic liquid crystal field theory*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** Q tensor 5component field
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555300

---

## 🧠 1. Introduction

**Landau-de Gennes Liquid Crystal Theory** is a **nonlinear inverse problem** whose unknown lives in **Q tensor 5component field** space, within the **Nematic liquid crystal field theory** sub-domain of **Polymer Physics**.

Measurements consist of photons collected by an optical detector via a **polarization microscopy** sensing mechanism.

The forward operator applies, in order: E · free energy LdG operator; adds a prior term that biases the solution toward smoothness/sparsity; O · Q tensor field operator; O · optical image operator.

Observations are corrupted by additive Gaussian noise. Non-unique globally due to topological sector; locally unique in each sector. Defect cores are singular in one-constant approximation.

## ⚙ 2. Forward Model

Physical chain: **x** → E · free energy LdG → O · Q tensor field → O · optical image → **y** (detector).

```
y = `O.optical_image` `O.Q_tensor_field` `E.free_energy_LdG` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.free_energy_LdG` | E · free energy ldg operator |
| `O.Q_tensor_field` | O · q tensor field operator |
| `O.optical_image` | O · optical image operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Polymer Physics |
| Sub domain | Nematic liquid crystal field theory |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | Q_tensor_5component_field |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Non-unique globally due to topological sector; locally unique in each sector. Defect cores are singular in one-constant approximation.

| Metric | Value |
|---|---|
| Metric | Q_tensor_RMSE |
| Secondary | defect_position_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `lc_texture_inversion` · **Forward operator:** `ldg_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| L1 pn | — | 5 |
| Mesh n | N | 128 |
| T over tc | — | 0.98 |
| Cell gap um | µm | 10 |
| Anchoring w j m2 | — | 0.0001 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| L1 pn | — | 1.0 – 30.0 |
| Mesh n | N | 32 – 512 |
| T over tc | — | 0.85 – 1.05 |
| Cell gap um | µm | 1 – 100 |
| Anchoring w j m2 | — | 1e-06 – 0.001 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** Q RMSE <= 0.015

| Metric | Range |
|---|---|
| Q tensor rmse | 0.005 – 0.1 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **Q_tensor_RMSE**, with κ = `900` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x55cdab476edea2dfa4c63e65a14318c289c9be15748ca70ce061fab85f156c89`
- **Chain tx hash:** `0xbe9e8806a2b39e90fffd7215e64ec02379aa3911ad5927dc854a327a86cee934`
- **Chain block:** `41555300`

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## File Mapping

This bundle consists of: `L1-479.md`, `L1-479.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-479.md` | Source of truth — edit this | Human or LLM |
| `L1-479.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 `L1-479`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._