L1 ⚛ L1-233 ⊙ Testnet

Helmholtz Scattering — frequency-domain EM scattering

Electromagnetics · Frequency-domain vector Helmholtz scattering

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⬇ L1-233.md

# ⚛ L1 Principle — Helmholtz Scattering — frequency-domain EM scattering **ID:** `L1-233` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Electromagnetics — *Frequency-domain vector Helmholtz scattering* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex E field freq > **📡 Carrier:** em · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553987 --- ## 🧠 1. Introduction **Helmholtz Scattering — frequency-domain EM scattering** is a **linear inverse problem** whose unknown lives in **complex E field freq** space, within the **Frequency-domain vector Helmholtz scattering** sub-domain of **Electromagnetics**. Measurements consist of electromagnetic field measurements via a **far field antenna** sensing mechanism. The forward operator applies, in order: L · incident wave operator; L · scatter boundary operator; L · green function operator; integration over a band of frequencies. Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds. ## ⚙ 2. Forward Model Physical chain: **x** → L · incident wave → L · scatter boundary → L · green function → Frequency integration → **y** (detector). ``` y = ∫dω `L.green_function` `L.scatter_boundary` `L.incident_wave` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.incident_wave` | L · incident wave operator | | `L.scatter_boundary` | L · scatter boundary operator | | `L.green_function` | L · green function operator | | `int.frequency` | Integration over a band of frequencies | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Electromagnetics | | Sub domain | Frequency-domain vector Helmholtz scattering | | Carrier | em | | Problem class | linear_inverse | | Solution space | complex_E_field_freq | | Noise model | gaussian | | Integration axis | frequency | | Difficulty delta | 5 | | L dag | 3.2 | ## 📡 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:** `helmholtz_em` · **Forward operator:** `helmholtz_em_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 64 | | Snr db | dB | 30 | | N freqs | — | 16 | | K error | — | 0 | | Bc error | — | 0 | | N angles | — | 256 | | Surface error | — | 0 | | Material error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 32 – 2048 | | W | px | 32 – 2048 | | Z | — | 32 – 1024 | | Snr db | dB | 0 – 40 | | N freqs | — | 1 – 256 | | K error | — | 0.0 – 0.05 | | Bc error | — | 0.0 – 0.2 | | N angles | — | 16 – 4096 | | Surface error | — | 0.0 – 0.1 | | Material error | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 24.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `800` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xf0218231898026cd11536b7b79fe90409ca7abbef8bd2f81d3acad7b479e021d` - **Chain tx hash:** `0x3348d75b5c4133103dea3a01afb716e100de7e200388754b90860b9614fb20e7` - **Chain block:** `41553987` --- ## File Mapping This bundle consists of: `L1-233.md`, `L1-233.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-233.md` | Source of truth — edit this | Human or LLM | | `L1-233.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 `L1-233`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-233.md).
Regenerate L1-233.json so every field matches the Markdown.
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L2-233-001 Helmholtz Scattering — frequency-domain EM scattering Mismatch-Only Spec

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📋 JSON metadata

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  "artifact_id": "L1-233",
  "chain_block": 41553987,
  "chain_hash": "0xf0218231898026cd11536b7b79fe90409ca7abbef8bd2f81d3acad7b479e021d",
  "chain_tx_hash": "0x3348d75b5c4133103dea3a01afb716e100de7e200388754b90860b9614fb20e7",
  "domain": "Electromagnetics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 800,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
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  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Conditional stability; mismatch parameters dominate at Omega bounds.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.2,
    "carrier": "em",
    "difficulty_delta": 5,
    "domain": "Electromagnetics",
    "integration_axis": "frequency",
    "noise_model": "gaussian",
    "primitives": [
      "L.incident_wave",
      "L.scatter_boundary",
      "L.green_function",
      "int.frequency"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "far_field_antenna",
    "solution_space": "complex_E_field_freq",
    "sub_domain": "Frequency-domain vector Helmholtz scattering",
    "title": "Helmholtz Scattering \u2014 frequency-domain EM scattering"
  },
  "size_tiers": {
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      "helmholtz_em_forward"
    ],
    "allowed_omega_dimensions": [
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      "W",
      "Z",
      "N_freqs",
      "N_angles",
      "SNR_dB",
      "k_error",
      "surface_error",
      "material_error",
      "BC_error",
      "k_error",
      "surface_error",
      "material_error",
      "BC_error"
    ],
    "allowed_problem_classes": [
      "helmholtz_em"
    ],
    "center_spec": {
      "epsilon_fn_center": "24.0",
      "forward_operator": "helmholtz_em_forward",
      "input_format": "measurement_only",
      "omega": {
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        "H": 256,
        "N_angles": 256,
        "N_freqs": 16,
        "SNR_dB": 30,
        "W": 256,
        "Z": 64,
        "k_error": 0.0,
        "material_error": 0.0,
        "surface_error": 0.0
      },
      "problem_class": "helmholtz_em"
    },
    "epsilon_bounds": {
      "psnr_db": [
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        45.0
      ]
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    "omega_bounds": {
      "BC_error": [
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        0.2
      ],
      "H": [
        32,
        2048
      ],
      "N_angles": [
        16,
        4096
      ],
      "N_freqs": [
        1,
        256
      ],
      "SNR_dB": [
        0,
        40
      ],
      "W": [
        32,
        2048
      ],
      "Z": [
        32,
        1024
      ],
      "k_error": [
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      ],
      "material_error": [
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        0.1
      ],
      "surface_error": [
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        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Frequency-domain vector Helmholtz scattering",
  "title": "Helmholtz Scattering \u2014 frequency-domain EM scattering"
}

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