L1 ⚛ L1-196 ⊙ Testnet

Steady Heat Conduction — conductivity imaging

Heat Transfer · Elliptic PDE conductivity inversion

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Markdown — human-readable source of truth

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⬇ L1-196.md

# ⚛ L1 Principle — Steady Heat Conduction — conductivity imaging **ID:** `L1-196` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Heat Transfer — *Elliptic PDE conductivity inversion* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D conductivity field > **📡 Carrier:** thermal · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554033 --- ## 🧠 1. Introduction **Steady Heat Conduction — conductivity imaging** is a **nonlinear inverse problem** whose unknown lives in **3D conductivity field** space, within the **Elliptic PDE conductivity inversion** sub-domain of **Heat Transfer**. Measurements consist of thermal radiation / temperature measurements via a **surface temp** sensing mechanism. The forward operator applies, in order: L · apply flux operator; L · elliptic solve operator; S · record · surface operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Calderon inverse conductivity problem — well-posed with full boundary data, ill-posed with sparse measurements. ## ⚙ 2. Forward Model Physical chain: **x** → L · apply flux → L · elliptic solve → S · record · surface → Spatial integration → **y** (detector). ``` y = ∫_A dA `S.record.surface` `L.elliptic_solve` `L.apply_flux` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.apply_flux` | L · apply flux operator | | `L.elliptic_solve` | L · elliptic solve operator | | `S.record.surface` | S · record · surface operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Heat Transfer | | Sub domain | Elliptic PDE conductivity inversion | | Carrier | thermal | | Problem class | nonlinear_inverse | | Solution space | 3D_conductivity_field | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 5 | | L dag | 3.2 | ## 📡 4. Measurement Model Calderon inverse conductivity problem — well-posed with full boundary data, ill-posed with sparse measurements. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `steady_heat` · **Forward operator:** `steady_heat_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 32 | | Snr db | dB | 30 | | N sensors | — | 128 | | Bc flux error | — | 0 | | Geometry 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 sensors | — | 16 – 4096 | | Bc flux error | — | 0.0 – 0.1 | | Geometry error | — | 0.0 – 0.05 | | Sensor 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 = `700` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x4ce9c5fb8aaee7799c39a7ec1b5ab147a8da7c21238a8a60b09dd47a843e73c1` - **Chain tx hash:** `0xec9304466a81833816079cb287d699d0b42955c372daede0f38fd527f4177d40` - **Chain block:** `41554033` --- ## File Mapping This bundle consists of: `L1-196.md`, `L1-196.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-196.md` | Source of truth — edit this | Human or LLM | | `L1-196.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 `L1-196`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-196.md).
Regenerate L1-196.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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L2-196-001 Steady Heat Conduction — conductivity imaging Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-196",
  "chain_block": 41554033,
  "chain_hash": "0x4ce9c5fb8aaee7799c39a7ec1b5ab147a8da7c21238a8a60b09dd47a843e73c1",
  "chain_tx_hash": "0xec9304466a81833816079cb287d699d0b42955c372daede0f38fd527f4177d40",
  "domain": "Heat Transfer",
  "hardness_fn": {
    "delta": 5,
    "kappa": 700,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
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  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Calderon inverse conductivity problem \u2014 well-posed with full boundary data, ill-posed with sparse measurements.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.2,
    "carrier": "thermal",
    "difficulty_delta": 5,
    "domain": "Heat Transfer",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.apply_flux",
      "L.elliptic_solve",
      "S.record.surface",
      "int.spatial"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "surface_temp",
    "solution_space": "3D_conductivity_field",
    "sub_domain": "Elliptic PDE conductivity inversion",
    "title": "Steady Heat Conduction \u2014 conductivity imaging"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "steady_heat_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "Z",
      "N_sensors",
      "SNR_dB",
      "bc_flux_error",
      "geometry_error",
      "sensor_pos_error",
      "bc_flux_error",
      "geometry_error",
      "sensor_pos_error"
    ],
    "allowed_problem_classes": [
      "steady_heat"
    ],
    "center_spec": {
      "epsilon_fn_center": "24.0",
      "forward_operator": "steady_heat_forward",
      "input_format": "measurement_only",
      "omega": {
        "H": 128,
        "N_sensors": 128,
        "SNR_dB": 30,
        "W": 128,
        "Z": 32,
        "bc_flux_error": 0.0,
        "geometry_error": 0.0,
        "sensor_pos_error": 0.0
      },
      "problem_class": "steady_heat"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "H": [
        32,
        1024
      ],
      "N_sensors": [
        16,
        4096
      ],
      "SNR_dB": [
        0,
        40
      ],
      "W": [
        32,
        1024
      ],
      "Z": [
        8,
        512
      ],
      "bc_flux_error": [
        0.0,
        0.1
      ],
      "geometry_error": [
        0.0,
        0.05
      ],
      "sensor_pos_error": [
        0.0,
        0.02
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Elliptic PDE conductivity inversion",
  "title": "Steady Heat Conduction \u2014 conductivity imaging"
}

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