L1 ⚛ L1-475 ⊙ Testnet

Self-Consistent Field Theory (SCFT)

Polymer Physics · Block copolymer microphase separation

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⬇ L1-475.md

# ⚛ L1 Principle — Self-Consistent Field Theory (SCFT) **ID:** `L1-475` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Polymer Physics — *Block copolymer microphase separation* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D composition field > **📡 Carrier:** none · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555300 --- ## 🧠 1. Introduction **Self-Consistent Field Theory (SCFT)** is a **nonlinear inverse problem** whose unknown lives in **3D composition field** space, within the **Block copolymer microphase separation** sub-domain of **Polymer Physics**. Measurements consist of none via a **scattering forward** sensing mechanism. The forward operator applies, in order: E · propagator diffusion operator; a fixed-point or gradient iteration on the unknown; O · density field operator; O · scattering operator. Observations are corrupted by additive Gaussian noise. Multiple local minima (lamellae, gyroid, hexagonal) → non-unique inversion; phase diagram navigation requires morphology seeding. ## ⚙ 2. Forward Model Physical chain: **x** → E · propagator diffusion → O · density field → O · scattering → **y** (detector). ``` y = `O.scattering` `O.density_field` `E.propagator_diffusion` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.propagator_diffusion` | E · propagator diffusion operator | | `O.density_field` | O · density field operator | | `O.scattering` | O · scattering operator | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `O.iter` | A fixed-point or gradient iteration on the unknown | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Polymer Physics | | Sub domain | Block copolymer microphase separation | | Carrier | none | | Problem class | nonlinear_inverse | | Solution space | 3D_composition_field | | Noise model | gaussian | | Integration axis | chain_contour | | Difficulty delta | 5 | | L dag | 3.8 | ## 📡 4. Measurement Model Multiple local minima (lamellae, gyroid, hexagonal) → non-unique inversion; phase diagram navigation requires morphology seeding. | Metric | Value | |---|---| | Metric | field_RMSE | | Secondary | S_q_peak_position_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `scft_morphology_inversion` · **Forward operator:** `scft_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | F | — | 0.5 | | N ds | — | 100 | | Chi n | N | 15 | | Grid n | N | 64 | | L box r g | g | 4 | | Mixing alpha | — | 0.05 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | F | — | 0.2 – 0.8 | | N ds | — | 50 – 400 | | Chi n | N | 10 – 40 | | Grid n | N | 32 – 128 | | L box r g | g | 2.0 – 10.0 | | Mixing alpha | — | 0.01 – 0.2 | ## 🎯 6. Tolerance (ε) **Center tolerance:** phi_A RMSE <= 0.025 | Metric | Range | |---|---| | Field rmse | 0.01 – 0.2 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **field_RMSE**, with κ = `1200` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xc56bbf2b12bad462c8eb1df69631cf5ddf19be662a2224471a85d2cf5c85a6cd` - **Chain tx hash:** `0x6cb7d134443dc1a082d20e89d16c4df0be62d806a2efcda2597e62905da09678` - **Chain block:** `41555300` --- ## File Mapping This bundle consists of: `L1-475.md`, `L1-475.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-475.md` | Source of truth — edit this | Human or LLM | | `L1-475.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 `L1-475`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-475.md).
Regenerate L1-475.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

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L2-475-001 Self-Consistent Field Theory (SCFT) Mismatch-Only Spec

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📋 JSON metadata

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  "artifact_id": "L1-475",
  "chain_block": 41555300,
  "chain_hash": "0xc56bbf2b12bad462c8eb1df69631cf5ddf19be662a2224471a85d2cf5c85a6cd",
  "chain_tx_hash": "0x6cb7d134443dc1a082d20e89d16c4df0be62d806a2efcda2597e62905da09678",
  "domain": "Polymer Physics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 1200,
    "metric": "field_RMSE",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "field_RMSE",
    "regime": "Multiple local minima (lamellae, gyroid, hexagonal) \u2192 non-unique inversion; phase diagram navigation requires morphology seeding.",
    "secondary": "S_q_peak_position_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.8,
    "carrier": "none",
    "difficulty_delta": 5,
    "domain": "Polymer Physics",
    "integration_axis": "chain_contour",
    "noise_model": "gaussian",
    "primitives": [
      "E.propagator_diffusion",
      "O.iter",
      "O.density_field",
      "O.scattering"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "scattering_forward",
    "solution_space": "3D_composition_field",
    "sub_domain": "Block copolymer microphase separation",
    "title": "Self-Consistent Field Theory (SCFT)"
  },
  "size_tiers": {
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      "scft_forward"
    ],
    "allowed_omega_dimensions": [
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      "f",
      "L_box_R_g",
      "grid_N",
      "N_ds",
      "mixing_alpha"
    ],
    "allowed_problem_classes": [
      "scft_morphology_inversion"
    ],
    "center_spec": {
      "epsilon_fn_center": "phi_A RMSE \u003c= 0.025",
      "forward_operator": "scft_forward",
      "input_format": "measurement_only",
      "omega": {
        "L_box_R_g": 4.0,
        "N_ds": 100,
        "chi_N": 15,
        "f": 0.5,
        "grid_N": 64,
        "mixing_alpha": 0.05
      },
      "problem_class": "scft_morphology_inversion"
    },
    "epsilon_bounds": {
      "field_RMSE": [
        0.01,
        0.2
      ]
    },
    "omega_bounds": {
      "L_box_R_g": [
        2.0,
        10.0
      ],
      "N_ds": [
        50,
        400
      ],
      "chi_N": [
        10,
        40
      ],
      "f": [
        0.2,
        0.8
      ],
      "grid_N": [
        32,
        128
      ],
      "mixing_alpha": [
        0.01,
        0.2
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Block copolymer microphase separation",
  "title": "Self-Consistent Field Theory (SCFT)"
}

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