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    "boundary": {
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  "internally_reviewed": false,
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      "sources",
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  "assumptions": [],
  "limitations": [
    "Ix and Iy are supplied about each part centroid",
    "Axes stay parallel to x and y",
    "Negative area is a hole",
    "Total area must not be zero",
    "Not product of inertia, principal axes, section modulus, or beam deflection"
  ],
  "boundary_conditions": [
    "no parts → MISSING_REQUIRED_INPUT",
    "a row that is not Ix,Iy,A,x,y → INVALID_INPUT",
    "total area zero → INVALID_INPUT"
  ],
  "sources": [
    {
      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Moments of inertia",
      "supports": "Parallel-axis theorem I = Ī + A d² for a composite area",
      "kind": "reference"
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  "fingerprint": "sha256:526a59ffdaec42ede7544b272049552d795b8ebd52e5838a27311023acdb08e4",
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      "kind": "golden",
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        "y_bar": 1,
        "x_bar": 0,
        "Ix": 4,
        "Iy": 0,
        "model": "area_moment"
      },
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      "actual": {
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        "area": 2,
        "x_bar": 0,
        "y_bar": 1,
        "Ix": 4,
        "Iy": 0,
        "kx": 1.4142135623730951,
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        "n": 2,
        "solver": "plane centroid + parallel axis",
        "convention": "Ix and Iy are centroidal and parallel to x,y. Ix = Σ(Ix + A (y−ȳ)²). Negative area is a hole."
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      "id": "golden-single",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "16,4,4,0,0"
      },
      "expect": {
        "area": 4,
        "Ix": 16,
        "Iy": 4,
        "kx": 2,
        "ky": 1
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      "actual": {
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        "x_bar": 0,
        "y_bar": 0,
        "Ix": 16,
        "Iy": 4,
        "kx": 2,
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        "n": 1,
        "solver": "plane centroid + parallel axis",
        "convention": "Ix and Iy are centroidal and parallel to x,y. Ix = Σ(Ix + A (y−ȳ)²). Negative area is a hole."
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      "tol": 1e-12
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      "ok": true,
      "kind": "boundary",
      "inputs": {
        "parts": "8,8,4,0,0; -2,-2,-1,0,0"
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        "area": 3,
        "Ix": 6,
        "Iy": 6,
        "x_bar": 0,
        "y_bar": 0
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      "error": "Enter at least one part as Ix,Iy,A,x,y."
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        "parts": "0,0,1,0,0; 0,0,-1,1,0"
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      "code": "INVALID_INPUT",
      "error": "Total area must not be zero."
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