L1 ⚛ L1-328 ⊙ Testnet

Cahn-Hilliard Phase Separation

Materials Science · Conserved order parameter dynamics

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⬇ L1-328.md

# ⚛ L1 Principle — Cahn-Hilliard Phase Separation **ID:** `L1-328` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Materials Science — *Conserved order parameter dynamics* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** composition field phi > **📡 Carrier:** none · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554127 --- ## 🧠 1. Introduction **Cahn-Hilliard Phase Separation** is a **nonlinear inverse problem** whose unknown lives in **composition field phi** space, within the **Conserved order parameter dynamics** sub-domain of **Materials Science**. Measurements consist of none via a **scattering morphology** sensing mechanism. The forward operator applies, in order: E · chemical potential operator; E · conservative flux operator; time evolution of the state; O · morphology operator. Observations are corrupted by additive Gaussian noise. Fourth-order PDE; well-posed; inversion ill-conditioned (M-chi degeneracy). ## ⚙ 2. Forward Model Physical chain: **x** → E · chemical potential → E · conservative flux → Time derivative → O · morphology → **y** (detector). ``` y = `O.morphology` ∂_t `E.conservative_flux` `E.chemical_potential` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.chemical_potential` | E · chemical potential operator | | `E.conservative_flux` | E · conservative flux operator | | `D.time` | Time evolution of the state | | `O.morphology` | O · morphology operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Materials Science | | Sub domain | Conserved order parameter dynamics | | Carrier | none | | Problem class | nonlinear_inverse | | Solution space | composition_field_phi | | Noise model | gaussian | | Integration axis | time | | Difficulty delta | 5 | | L dag | 3.5 | ## 📡 4. Measurement Model Fourth-order PDE; well-posed; inversion ill-conditioned (M-chi degeneracy). | Metric | Value | |---|---| | Metric | S_q_time_L2_error | | Secondary | domain_size_error_nm | ## 📏 5. Operating Range (Ω) **Center problem class:** `phase_field_cahn_hilliard` · **Forward operator:** `cahn_hilliard_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | M | — | 1 | | T k | — | 300 | | Dx nm | nm | 5 | | Phi 0 | — | 0.5 | | Chi ab | — | 3 | | Grid n | N | 256 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | M | — | 0.01 – 100 | | T k | — | 77 – 1500 | | Dx nm | nm | 0.5 – 100 | | Phi 0 | — | 0.1 – 0.9 | | Chi ab | — | 1.5 – 20 | | Grid n | N | 64 – 2048 | ## 🎯 6. Tolerance (ε) **Center tolerance:** S(q,t) L2 <= 0.04 | Metric | Range | |---|---| | S q time l2 error | 0.01 – 0.5 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **S_q_time_L2_error**, with κ = `350` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x85f7bf568c22adc5dff5da9d39e3048af5b622de9fdb4ff57917723c89b5f063` - **Chain tx hash:** `0x5d78a235c117f3465ea34a83f617e04f3c129c0647b062ab0a9d9885bbb8dde6` - **Chain block:** `41554127` --- ## File Mapping This bundle consists of: `L1-328.md`, `L1-328.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-328.md` | Source of truth — edit this | Human or LLM | | `L1-328.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 `L1-328`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-328.md).
Regenerate L1-328.json so every field matches the Markdown.
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📋 JSON metadata

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  "artifact_id": "L1-328",
  "chain_block": 41554127,
  "chain_hash": "0x85f7bf568c22adc5dff5da9d39e3048af5b622de9fdb4ff57917723c89b5f063",
  "chain_tx_hash": "0x5d78a235c117f3465ea34a83f617e04f3c129c0647b062ab0a9d9885bbb8dde6",
  "domain": "Materials Science",
  "hardness_fn": {
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    "kappa": 350,
    "metric": "S_q_time_L2_error",
    "type": "epsilon_fn"
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "S_q_time_L2_error",
    "regime": "Fourth-order PDE; well-posed; inversion ill-conditioned (M-chi degeneracy).",
    "secondary": "domain_size_error_nm"
  },
  "physics_fingerprint": {
    "L_DAG": 3.5,
    "carrier": "none",
    "difficulty_delta": 5,
    "domain": "Materials Science",
    "integration_axis": "time",
    "noise_model": "gaussian",
    "primitives": [
      "E.chemical_potential",
      "E.conservative_flux",
      "D.time",
      "O.morphology"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "scattering_morphology",
    "solution_space": "composition_field_phi",
    "sub_domain": "Conserved order parameter dynamics",
    "title": "Cahn-Hilliard Phase Separation"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "cahn_hilliard_forward"
    ],
    "allowed_omega_dimensions": [
      "grid_N",
      "dx_nm",
      "chi_AB",
      "phi_0",
      "M",
      "T_K"
    ],
    "allowed_problem_classes": [
      "phase_field_cahn_hilliard"
    ],
    "center_spec": {
      "epsilon_fn_center": "S(q,t) L2 \u003c= 0.04",
      "forward_operator": "cahn_hilliard_forward",
      "input_format": "measurement_only",
      "omega": {
        "M": 1.0,
        "T_K": 300,
        "chi_AB": 3.0,
        "dx_nm": 5,
        "grid_N": 256,
        "phi_0": 0.5
      },
      "problem_class": "phase_field_cahn_hilliard"
    },
    "epsilon_bounds": {
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        0.01,
        0.5
      ]
    },
    "omega_bounds": {
      "M": [
        0.01,
        100
      ],
      "T_K": [
        77,
        1500
      ],
      "chi_AB": [
        1.5,
        20
      ],
      "dx_nm": [
        0.5,
        100
      ],
      "grid_N": [
        64,
        2048
      ],
      "phi_0": [
        0.1,
        0.9
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Conserved order parameter dynamics",
  "title": "Cahn-Hilliard Phase Separation"
}

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