L1 ⚛ L1-327 ⊙ Testnet

Phase-Field Allen-Cahn Dynamics

Materials Science · Non-conserved order parameter dynamics

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⬇ principle.md ⬇ principle.json

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⬇ L1-327.md ⬇ L1-327.json

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You are helping me submit a MODIFICATION of L1-327 (Phase-Field Allen-Cahn Dynamics) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-327.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-327.json so EVERY field matches the Markdown.
3. Keep the schema in the "File Mapping" footer at the bottom of the MD.
4. Keep the parent reference unchanged unless I ask otherwise.
Rules: the Markdown is the source of truth; use SI units; do NOT invent benchmark scores.
Output each file in its own fenced code block tagged with the filename.

Here is my template:
[PASTE THE .md HERE]

⚙ Forward Model

y = `O.domain_map` ∂_t `E.gradient_flow` `E.free_energy` x + n,    n ~ 𝒩(0, σ²)

world state x (order parameter field phi)

→

E · free energy `E.free_energy`

→

E · gradient flow `E.gradient_flow`

→

Time derivative ∂_t

→

O · domain map `O.domain_map`

→

observation y (none detector)

Noise: additive Gaussian noise

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-327.md

# ⚛ L1 Principle — Phase-Field Allen-Cahn Dynamics **ID:** `L1-327` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Materials Science — *Non-conserved order parameter dynamics* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** order parameter field phi > **📡 Carrier:** none · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554127 --- ## 🧠 1. Introduction **Phase-Field Allen-Cahn Dynamics** is a **nonlinear inverse problem** whose unknown lives in **order parameter field phi** space, within the **Non-conserved order parameter dynamics** sub-domain of **Materials Science**. Measurements consist of none via a **electron or optical micrograph** sensing mechanism. The forward operator applies, in order: E · free energy operator; E · gradient flow operator; time evolution of the state; O · domain map operator. Observations are corrupted by additive Gaussian noise. Well-posed parabolic PDE; inversion for (kappa, M, W) is ill-conditioned due to kinematic degeneracy between mobility and barrier. ## ⚙ 2. Forward Model Physical chain: **x** → E · free energy → E · gradient flow → Time derivative → O · domain map → **y** (detector). ``` y = `O.domain_map` ∂_t `E.gradient_flow` `E.free_energy` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.free_energy` | E · free energy operator | | `E.gradient_flow` | E · gradient flow operator | | `D.time` | Time evolution of the state | | `O.domain_map` | O · domain map operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Materials Science | | Sub domain | Non-conserved order parameter dynamics | | Carrier | none | | Problem class | nonlinear_inverse | | Solution space | order_parameter_field_phi | | Noise model | gaussian | | Integration axis | time | | Difficulty delta | 5 | | L dag | 3.5 | ## 📡 4. Measurement Model Well-posed parabolic PDE; inversion for (kappa, M, W) is ill-conditioned due to kinematic degeneracy between mobility and barrier. | Metric | Value | |---|---| | Metric | domain_velocity_relative_error | | Secondary | interface_width_error_nm | ## 📏 5. Operating Range (Ω) **Center problem class:** `phase_field_allen_cahn` · **Forward operator:** `allen_cahn_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | T k | — | 300 | | Dt s | s | 0.001 | | M phi | — | 1 | | Dx nm | nm | 10 | | Kappa | — | 1 | | Grid n | N | 256 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | T k | — | 77 – 1500 | | M phi | — | 0.1 – 100 | | Dx nm | nm | 0.5 – 100 | | Kappa | — | 0.1 – 10 | | Grid n | N | 64 – 1024 | ## 🎯 6. Tolerance (ε) **Center tolerance:** velocity rel error <= 0.04 | Metric | Range | |---|---| | Domain velocity relative error | 0.01 – 0.4 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **domain_velocity_relative_error**, with κ = `300` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xa7ed16ca675d449b1b4d7c13c443255793c5e53a53f79b8b7b6ad88350d23e13` - **Chain tx hash:** `0x43384ce95987b9b98eb749d5a726d02a6509a29ec829feb110525c3298ccb60b` - **Chain block:** `41554127` --- ## File Mapping This bundle consists of: `L1-327.md`, `L1-327.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-327.md` | Source of truth — edit this | Human or LLM | | `L1-327.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 `L1-327`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-327-001 Phase-Field Allen-Cahn Dynamics Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-327",
  "chain_block": 41554127,
  "chain_hash": "0xa7ed16ca675d449b1b4d7c13c443255793c5e53a53f79b8b7b6ad88350d23e13",
  "chain_tx_hash": "0x43384ce95987b9b98eb749d5a726d02a6509a29ec829feb110525c3298ccb60b",
  "domain": "Materials Science",
  "hardness_fn": {
    "delta": 5,
    "kappa": 300,
    "metric": "domain_velocity_relative_error",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "domain_velocity_relative_error",
    "regime": "Well-posed parabolic PDE; inversion for (kappa, M, W) is ill-conditioned due to kinematic degeneracy between mobility and barrier.",
    "secondary": "interface_width_error_nm"
  },
  "physics_fingerprint": {
    "L_DAG": 3.5,
    "carrier": "none",
    "difficulty_delta": 5,
    "domain": "Materials Science",
    "integration_axis": "time",
    "noise_model": "gaussian",
    "primitives": [
      "E.free_energy",
      "E.gradient_flow",
      "D.time",
      "O.domain_map"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "electron_or_optical_micrograph",
    "solution_space": "order_parameter_field_phi",
    "sub_domain": "Non-conserved order parameter dynamics",
    "title": "Phase-Field Allen-Cahn Dynamics"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "allen_cahn_forward"
    ],
    "allowed_omega_dimensions": [
      "grid_N",
      "dx_nm",
      "dt_s",
      "M_phi",
      "kappa",
      "T_K"
    ],
    "allowed_problem_classes": [
      "phase_field_allen_cahn"
    ],
    "center_spec": {
      "epsilon_fn_center": "velocity rel error \u003c= 0.04",
      "forward_operator": "allen_cahn_forward",
      "input_format": "measurement_only",
      "omega": {
        "M_phi": 1.0,
        "T_K": 300,
        "dt_s": 0.001,
        "dx_nm": 10,
        "grid_N": 256,
        "kappa": 1.0
      },
      "problem_class": "phase_field_allen_cahn"
    },
    "epsilon_bounds": {
      "domain_velocity_relative_error": [
        0.01,
        0.4
      ]
    },
    "omega_bounds": {
      "M_phi": [
        0.1,
        100
      ],
      "T_K": [
        77,
        1500
      ],
      "dx_nm": [
        0.5,
        100
      ],
      "grid_N": [
        64,
        1024
      ],
      "kappa": [
        0.1,
        10
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Non-conserved order parameter dynamics",
  "title": "Phase-Field Allen-Cahn Dynamics"
}

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