L1 ⚛ L1-250 ⊙ Testnet

Electrostatics — Poisson equation inversion

Electromagnetics · Poisson equation for electrostatic potential

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⬇ L1-250.md

# ⚛ L1 Principle — Electrostatics — Poisson equation inversion **ID:** `L1-250` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Electromagnetics — *Poisson equation for electrostatic potential* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D potential field > **📡 Carrier:** em · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554000 --- ## 🧠 1. Introduction **Electrostatics — Poisson equation inversion** is a **linear inverse problem** whose unknown lives in **3D potential field** space, within the **Poisson equation for electrostatic potential** sub-domain of **Electromagnetics**. Measurements consist of electromagnetic field measurements via a **e field probes** sensing mechanism. The forward operator applies, in order: L · charge source operator; L · poisson operator; L · boundary match 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 · charge source → L · poisson → L · boundary match → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.boundary_match` `L.poisson` `L.charge_source` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.charge_source` | L · charge source operator | | `L.poisson` | L · poisson operator | | `L.boundary_match` | L · boundary match operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Electromagnetics | | Sub domain | Poisson equation for electrostatic potential | | Carrier | em | | Problem class | linear_inverse | | Solution space | 3D_potential_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:** `electrostatic` · **Forward operator:** `electrostatic_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 32 | | Snr db | dB | 30 | | N probes | — | 64 | | Eps r error | — | 0 | | Boundary error | — | 0 | | Sensor 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 | | Eps r error | — | 0.0 – 0.1 | | Boundary error | — | 0.0 – 0.2 | | Sensor pos error | — | 0.0 – 0.02 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 25.0 | 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:** `0x75b95302856b368ca1e67824c6a406fbe89ecbea8909af56b297c44c374168d3` - **Chain tx hash:** `0x50c0de0159c2e99287fef4f63edc1a88888475a337b851bb91b1d8206f088a69` - **Chain block:** `41554000` --- ## File Mapping This bundle consists of: `L1-250.md`, `L1-250.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-250.md` | Source of truth — edit this | Human or LLM | | `L1-250.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 `L1-250`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-250.md).
Regenerate L1-250.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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L2-250-001 Electrostatics — Poisson equation inversion Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41554000,
  "chain_hash": "0x75b95302856b368ca1e67824c6a406fbe89ecbea8909af56b297c44c374168d3",
  "chain_tx_hash": "0x50c0de0159c2e99287fef4f63edc1a88888475a337b851bb91b1d8206f088a69",
  "domain": "Electromagnetics",
  "hardness_fn": {
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    "kappa": 200,
    "metric": "PSNR_dB",
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  "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.charge_source",
      "L.poisson",
      "L.boundary_match",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "e_field_probes",
    "solution_space": "3D_potential_field",
    "sub_domain": "Poisson equation for electrostatic potential",
    "title": "Electrostatics \u2014 Poisson equation inversion"
  },
  "size_tiers": {
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    "allowed_omega_dimensions": [
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      "W",
      "Z",
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      "boundary_error",
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    ],
    "allowed_problem_classes": [
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    "center_spec": {
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      "forward_operator": "electrostatic_forward",
      "input_format": "measurement_only",
      "omega": {
        "H": 128,
        "N_probes": 64,
        "SNR_dB": 30,
        "W": 128,
        "Z": 32,
        "boundary_error": 0.0,
        "eps_r_error": 0.0,
        "sensor_pos_error": 0.0
      },
      "problem_class": "electrostatic"
    },
    "epsilon_bounds": {
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    },
    "omega_bounds": {
      "H": [
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        1024
      ],
      "N_probes": [
        4,
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      ],
      "SNR_dB": [
        0,
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      ],
      "W": [
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      ],
      "Z": [
        8,
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      ],
      "boundary_error": [
        0.0,
        0.2
      ],
      "eps_r_error": [
        0.0,
        0.1
      ],
      "sensor_pos_error": [
        0.0,
        0.02
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Poisson equation for electrostatic potential",
  "title": "Electrostatics \u2014 Poisson equation inversion"
}

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