L1 ⚛ L1-479 ⊙ Testnet

Landau-de Gennes Liquid Crystal Theory

Polymer Physics · Nematic liquid crystal field theory

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⬇ L1-479.md

# ⚛ 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` --- ## 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._

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Read the attached Markdown (L1-479.md).
Regenerate L1-479.json so every field matches the Markdown.
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📋 JSON metadata

{
  "artifact_id": "L1-479",
  "chain_block": 41555300,
  "chain_hash": "0x55cdab476edea2dfa4c63e65a14318c289c9be15748ca70ce061fab85f156c89",
  "chain_tx_hash": "0xbe9e8806a2b39e90fffd7215e64ec02379aa3911ad5927dc854a327a86cee934",
  "domain": "Polymer Physics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 900,
    "metric": "Q_tensor_RMSE",
    "type": "epsilon_fn"
  },
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "Q_tensor_RMSE",
    "regime": "Non-unique globally due to topological sector; locally unique in each sector. Defect cores are singular in one-constant approximation.",
    "secondary": "defect_position_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.8,
    "carrier": "photon",
    "difficulty_delta": 5,
    "domain": "Polymer Physics",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "E.free_energy_LdG",
      "O.regularize",
      "O.Q_tensor_field",
      "O.optical_image"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "polarization_microscopy",
    "solution_space": "Q_tensor_5component_field",
    "sub_domain": "Nematic liquid crystal field theory",
    "title": "Landau-de Gennes Liquid Crystal Theory"
  },
  "size_tiers": {
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      "ldg_forward"
    ],
    "allowed_omega_dimensions": [
      "T_over_Tc",
      "L1_pN",
      "anchoring_W_J_m2",
      "cell_gap_um",
      "mesh_N"
    ],
    "allowed_problem_classes": [
      "lc_texture_inversion"
    ],
    "center_spec": {
      "epsilon_fn_center": "Q RMSE \u003c= 0.015",
      "forward_operator": "ldg_forward",
      "input_format": "measurement_only",
      "omega": {
        "L1_pN": 5.0,
        "T_over_Tc": 0.98,
        "anchoring_W_J_m2": 0.0001,
        "cell_gap_um": 10,
        "mesh_N": 128
      },
      "problem_class": "lc_texture_inversion"
    },
    "epsilon_bounds": {
      "Q_tensor_RMSE": [
        0.005,
        0.1
      ]
    },
    "omega_bounds": {
      "L1_pN": [
        1.0,
        30.0
      ],
      "T_over_Tc": [
        0.85,
        1.05
      ],
      "anchoring_W_J_m2": [
        1e-06,
        0.001
      ],
      "cell_gap_um": [
        1,
        100
      ],
      "mesh_N": [
        32,
        512
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Nematic liquid crystal field theory",
  "title": "Landau-de Gennes Liquid Crystal Theory"
}

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