L1 ⚛ L1-251 ⊙ Testnet

Magnetostatics — coil design and field inversion

Electromagnetics · Biot-Savart / vector potential magnetostatics

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⬇ L1-251.md

# ⚛ L1 Principle — Magnetostatics — coil design and field inversion **ID:** `L1-251` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Electromagnetics — *Biot-Savart / vector potential magnetostatics* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D magnetic field > **📡 Carrier:** em · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554000 --- ## 🧠 1. Introduction **Magnetostatics — coil design and field inversion** is a **linear inverse problem** whose unknown lives in **3D magnetic field** space, within the **Biot-Savart / vector potential magnetostatics** sub-domain of **Electromagnetics**. Measurements consist of electromagnetic field measurements via a **hall probes** sensing mechanism. The forward operator applies, in order: L · current source operator; L · vector potential operator; L · curl A operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds. ## ⚙ 2. Forward Model Physical chain: **x** → L · current source → L · vector potential → L · curl A → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.curl_A` `L.vector_potential` `L.current_source` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.current_source` | L · current source operator | | `L.vector_potential` | L · vector potential operator | | `L.curl_A` | L · curl a operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Electromagnetics | | Sub domain | Biot-Savart / vector potential magnetostatics | | Carrier | em | | Problem class | linear_inverse | | Solution space | 3D_magnetic_field | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 2.8 | ## 📡 4. Measurement Model Conditional stability; mismatch parameters dominate at Omega bounds. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `magnetostatic` · **Forward operator:** `magnetostatic_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 32 | | Snr db | dB | 30 | | N probes | — | 64 | | Mu r error | — | 0 | | Current error | — | 0 | | Coil pos error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 32 – 1024 | | W | px | 32 – 1024 | | Z | — | 8 – 512 | | Snr db | dB | 0 – 40 | | N probes | — | 4 – 4096 | | Mu r error | — | 0.0 – 0.1 | | Current error | — | 0.0 – 0.02 | | Coil pos error | — | 0.0 – 0.05 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 26.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x2eec4b847f183fd74ec25d2a21f30ce4fea175ff7f57921e90516b2c80fd18d3` - **Chain tx hash:** `0x46dba085c0497efc3eb6e67f6c98bdc9e6b581d521d4b49cd7bd52c94599ae00` - **Chain block:** `41554000` --- ## File Mapping This bundle consists of: `L1-251.md`, `L1-251.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-251.md` | Source of truth — edit this | Human or LLM | | `L1-251.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 `L1-251`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-251.md).
Regenerate L1-251.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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📋 JSON metadata

{
  "artifact_id": "L1-251",
  "chain_block": 41554000,
  "chain_hash": "0x2eec4b847f183fd74ec25d2a21f30ce4fea175ff7f57921e90516b2c80fd18d3",
  "chain_tx_hash": "0x46dba085c0497efc3eb6e67f6c98bdc9e6b581d521d4b49cd7bd52c94599ae00",
  "domain": "Electromagnetics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 200,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Conditional stability; mismatch parameters dominate at Omega bounds.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 2.8,
    "carrier": "em",
    "difficulty_delta": 3,
    "domain": "Electromagnetics",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.current_source",
      "L.vector_potential",
      "L.curl_A",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "hall_probes",
    "solution_space": "3D_magnetic_field",
    "sub_domain": "Biot-Savart / vector potential magnetostatics",
    "title": "Magnetostatics \u2014 coil design and field inversion"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "magnetostatic_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "Z",
      "N_probes",
      "SNR_dB",
      "coil_pos_error",
      "mu_r_error",
      "current_error",
      "coil_pos_error",
      "mu_r_error",
      "current_error"
    ],
    "allowed_problem_classes": [
      "magnetostatic"
    ],
    "center_spec": {
      "epsilon_fn_center": "26.0",
      "forward_operator": "magnetostatic_forward",
      "input_format": "measurement_only",
      "omega": {
        "H": 128,
        "N_probes": 64,
        "SNR_dB": 30,
        "W": 128,
        "Z": 32,
        "coil_pos_error": 0.0,
        "current_error": 0.0,
        "mu_r_error": 0.0
      },
      "problem_class": "magnetostatic"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "H": [
        32,
        1024
      ],
      "N_probes": [
        4,
        4096
      ],
      "SNR_dB": [
        0,
        40
      ],
      "W": [
        32,
        1024
      ],
      "Z": [
        8,
        512
      ],
      "coil_pos_error": [
        0.0,
        0.05
      ],
      "current_error": [
        0.0,
        0.02
      ],
      "mu_r_error": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Biot-Savart / vector potential magnetostatics",
  "title": "Magnetostatics \u2014 coil design and field inversion"
}

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