L1 ⚛ L1-258 ⊙ Testnet

Helmholtz Equation — frequency-domain acoustic scattering / interior field

Acoustics · Frequency-domain acoustic inverse scattering

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⬇ L1-258.md

# ⚛ L1 Principle — Helmholtz Equation — frequency-domain acoustic scattering / interior field **ID:** `L1-258` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Acoustics — *Frequency-domain acoustic inverse scattering* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex pressure field 2D > **📡 Carrier:** acoustic · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554035 --- ## 🧠 1. Introduction **Helmholtz Equation — frequency-domain acoustic scattering / interior field** is a **linear inverse problem** whose unknown lives in **complex pressure field 2D** space, within the **Frequency-domain acoustic inverse scattering** sub-domain of **Acoustics**. Measurements consist of acoustic pressure waves recorded by transducers via a **pressure receivers** sensing mechanism. The forward operator applies, in order: L · source injection operator; L · helmholtz solve operator; S · record · receivers 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 · source injection → L · helmholtz solve → S · record · receivers → Frequency integration → **y** (detector). ``` y = ∫dω `S.record.receivers` `L.helmholtz_solve` `L.source_injection` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.source_injection` | L · source injection operator | | `L.helmholtz_solve` | L · helmholtz solve operator | | `S.record.receivers` | S · record · receivers operator | | `int.frequency` | Integration over a band of frequencies | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Acoustics | | Sub domain | Frequency-domain acoustic inverse scattering | | Carrier | acoustic | | Problem class | linear_inverse | | Solution space | complex_pressure_field_2D | | Noise model | gaussian | | Integration axis | frequency | | Difficulty delta | 5 | | L dag | 3.3 | ## 📡 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` · **Forward operator:** `helmholtz_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Snr db | dB | 30 | | K mean | — | 20 | | N freqs | — | 16 | | K error | — | 0 | | N receivers | — | 32 | | Geometry error | — | 0 | | Impedance bc error | — | 0 | | Receiver pos error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 64 – 1024 | | W | px | 64 – 1024 | | Snr db | dB | 0 – 40 | | K mean | — | 1 – 200 | | N freqs | — | 4 – 128 | | K error | — | 0.0 – 0.05 | | N receivers | — | 8 – 256 | | Geometry error | — | 0.0 – 0.05 | | Impedance bc error | — | 0.0 – 0.3 | | Receiver pos error | — | 0.0 – 0.02 | ## 🎯 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 = `1000` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xf0c96d119c0fb83452aab765fecad67b9eab4a4edd1df308232ed2a41a19bea7` - **Chain tx hash:** `0x494a4b6d46a007355770f52f90848ff6989453012689871ee1552dd9530b91e9` - **Chain block:** `41554035` --- ## File Mapping This bundle consists of: `L1-258.md`, `L1-258.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-258.md` | Source of truth — edit this | Human or LLM | | `L1-258.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 `L1-258`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-258.md).
Regenerate L1-258.json so every field matches the Markdown.
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L2-258-001 Helmholtz Equation — frequency-domain acoustic scattering / interior field Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-258",
  "chain_block": 41554035,
  "chain_hash": "0xf0c96d119c0fb83452aab765fecad67b9eab4a4edd1df308232ed2a41a19bea7",
  "chain_tx_hash": "0x494a4b6d46a007355770f52f90848ff6989453012689871ee1552dd9530b91e9",
  "domain": "Acoustics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 1000,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "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": 3.3,
    "carrier": "acoustic",
    "difficulty_delta": 5,
    "domain": "Acoustics",
    "integration_axis": "frequency",
    "noise_model": "gaussian",
    "primitives": [
      "L.source_injection",
      "L.helmholtz_solve",
      "S.record.receivers",
      "int.frequency"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "pressure_receivers",
    "solution_space": "complex_pressure_field_2D",
    "sub_domain": "Frequency-domain acoustic inverse scattering",
    "title": "Helmholtz Equation \u2014 frequency-domain acoustic scattering / interior field"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "helmholtz_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "N_receivers",
      "N_freqs",
      "k_mean",
      "SNR_dB",
      "k_error",
      "geometry_error",
      "impedance_BC_error",
      "receiver_pos_error",
      "k_error",
      "geometry_error",
      "impedance_BC_error",
      "receiver_pos_error"
    ],
    "allowed_problem_classes": [
      "helmholtz"
    ],
    "center_spec": {
      "epsilon_fn_center": "24.0",
      "forward_operator": "helmholtz_forward",
      "input_format": "measurement_only",
      "omega": {
        "H": 256,
        "N_freqs": 16,
        "N_receivers": 32,
        "SNR_dB": 30,
        "W": 256,
        "geometry_error": 0.0,
        "impedance_BC_error": 0.0,
        "k_error": 0.0,
        "k_mean": 20,
        "receiver_pos_error": 0.0
      },
      "problem_class": "helmholtz"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "H": [
        64,
        1024
      ],
      "N_freqs": [
        4,
        128
      ],
      "N_receivers": [
        8,
        256
      ],
      "SNR_dB": [
        0,
        40
      ],
      "W": [
        64,
        1024
      ],
      "geometry_error": [
        0.0,
        0.05
      ],
      "impedance_BC_error": [
        0.0,
        0.3
      ],
      "k_error": [
        0.0,
        0.05
      ],
      "k_mean": [
        1,
        200
      ],
      "receiver_pos_error": [
        0.0,
        0.02
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Frequency-domain acoustic inverse scattering",
  "title": "Helmholtz Equation \u2014 frequency-domain acoustic scattering / interior field"
}

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