L1 ⚛ L1-333 ⊙ Testnet

Solid-State Diffusion (Fick + Onsager)

Materials Science · Multi-component diffusion

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⬇ L1-333.md

# ⚛ L1 Principle — Solid-State Diffusion (Fick + Onsager) **ID:** `L1-333` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Materials Science — *Multi-component diffusion* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** concentration field c i > **📡 Carrier:** none · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554128 --- ## 🧠 1. Introduction **Solid-State Diffusion (Fick + Onsager)** is a **linear inverse problem** whose unknown lives in **concentration field c i** space, within the **Multi-component diffusion** sub-domain of **Materials Science**. Measurements consist of none via a **concentration profile measurement** sensing mechanism. The forward operator applies, in order: E · onsager flux operator; time evolution of the state; O · concentration profile operator. Observations are corrupted by additive Gaussian noise. Parabolic; well-posed; inversion of off-diagonal D_ij ill-conditioned for small gradients. ## ⚙ 2. Forward Model Physical chain: **x** → E · onsager flux → Time derivative → O · concentration profile → **y** (detector). ``` y = `O.concentration_profile` ∂_t `E.onsager_flux` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.onsager_flux` | E · onsager flux operator | | `D.time` | Time evolution of the state | | `O.concentration_profile` | O · concentration profile operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Materials Science | | Sub domain | Multi-component diffusion | | Carrier | none | | Problem class | linear_inverse | | Solution space | concentration_field_c_i | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 3 | ## 📡 4. Measurement Model Parabolic; well-posed; inversion of off-diagonal D_ij ill-conditioned for small gradients. | Metric | Value | |---|---| | Metric | concentration_profile_L2_error | | Secondary | D_ij_relative_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `solid_state_diffusion` · **Forward operator:** `diffusion_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | T k | — | 1000 | | T s | s | 3600 | | Dx um | µm | 1 | | Grid n | N | 256 | | N species | — | 3 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | T k | — | 300 – 2500 | | T s | s | 1 – 10000000.0 | | Dx um | µm | 0.01 – 100 | | Grid n | N | 64 – 2048 | | N species | — | 2 – 10 | ## 🎯 6. Tolerance (ε) **Center tolerance:** profile L2 <= 0.02 | Metric | Range | |---|---| | Concentration profile l2 error | 0.005 – 0.2 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **concentration_profile_L2_error**, with κ = `100` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x85a9cab7dc4e5b17b598593ad14c0755bf9c524f0dcf5145ff8eb2779fef0198` - **Chain tx hash:** `0x816a06882aa267d452e0e2c8a58ee216411dcb58e24fbd453b10d956484ee16e` - **Chain block:** `41554128` --- ## File Mapping This bundle consists of: `L1-333.md`, `L1-333.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-333.md` | Source of truth — edit this | Human or LLM | | `L1-333.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 `L1-333`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-333.md).
Regenerate L1-333.json so every field matches the Markdown.
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L2-333-001 Solid-State Diffusion (Fick + Onsager) Mismatch-Only Spec

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📋 JSON metadata

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  "artifact_id": "L1-333",
  "chain_block": 41554128,
  "chain_hash": "0x85a9cab7dc4e5b17b598593ad14c0755bf9c524f0dcf5145ff8eb2779fef0198",
  "chain_tx_hash": "0x816a06882aa267d452e0e2c8a58ee216411dcb58e24fbd453b10d956484ee16e",
  "domain": "Materials Science",
  "hardness_fn": {
    "delta": 3,
    "kappa": 100,
    "metric": "concentration_profile_L2_error",
    "type": "epsilon_fn"
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      "license_hash": null,
      "name": "primary",
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  "layer": "L1",
  "observable_profile": {
    "metric": "concentration_profile_L2_error",
    "regime": "Parabolic; well-posed; inversion of off-diagonal D_ij ill-conditioned for small gradients.",
    "secondary": "D_ij_relative_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.0,
    "carrier": "none",
    "difficulty_delta": 3,
    "domain": "Materials Science",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "E.onsager_flux",
      "D.time",
      "O.concentration_profile"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "concentration_profile_measurement",
    "solution_space": "concentration_field_c_i",
    "sub_domain": "Multi-component diffusion",
    "title": "Solid-State Diffusion (Fick + Onsager)"
  },
  "size_tiers": {
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      "diffusion_forward"
    ],
    "allowed_omega_dimensions": [
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      "dx_um",
      "t_s",
      "T_K",
      "N_species"
    ],
    "allowed_problem_classes": [
      "solid_state_diffusion"
    ],
    "center_spec": {
      "epsilon_fn_center": "profile L2 \u003c= 0.02",
      "forward_operator": "diffusion_forward",
      "input_format": "measurement_only",
      "omega": {
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        "T_K": 1000,
        "dx_um": 1,
        "grid_N": 256,
        "t_s": 3600
      },
      "problem_class": "solid_state_diffusion"
    },
    "epsilon_bounds": {
      "concentration_profile_L2_error": [
        0.005,
        0.2
      ]
    },
    "omega_bounds": {
      "N_species": [
        2,
        10
      ],
      "T_K": [
        300,
        2500
      ],
      "dx_um": [
        0.01,
        100
      ],
      "grid_N": [
        64,
        2048
      ],
      "t_s": [
        1,
        10000000.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Multi-component diffusion",
  "title": "Solid-State Diffusion (Fick + Onsager)"
}

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