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      "sources",
      "tests"
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    "Moments are supplied about each part centroid",
    "θ is the I1 axis, CCW from +x, in (−90°, 90°]",
    "Equal principal moments report θ = 0",
    "Negative area is a hole",
    "Total area must not be zero",
    "Not a Mohr circle drawing, section modulus, or beam deflection"
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  "boundary_conditions": [
    "no parts → MISSING_REQUIRED_INPUT",
    "a row that is not Ix,Iy,Ixy,A,x,y → INVALID_INPUT",
    "total area zero → INVALID_INPUT"
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      "section": "Product of inertia and principal moments",
      "supports": "Ixy = Īxy + A x' y' and the principal values of the plane inertia tensor",
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      "kind": "golden",
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        "Iy": 1,
        "Ixy": 0,
        "I1": 4,
        "I2": 1,
        "theta_deg": 0,
        "model": "principal_moment"
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        "y_bar": 0,
        "Ix": 4,
        "Iy": 1,
        "Ixy": 0,
        "I1": 4,
        "I2": 1,
        "theta_deg": 0,
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      "tol": 1e-12
    },
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      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "0,0,0,1,0,0; 0,0,0,1,2,2"
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        "x_bar": 1,
        "y_bar": 1,
        "Ix": 2,
        "Iy": 2,
        "Ixy": 2,
        "I1": 4,
        "I2": 0,
        "theta_deg": -45
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        "x_bar": 1,
        "y_bar": 1,
        "Ix": 2,
        "Iy": 2,
        "Ixy": 2,
        "I1": 4,
        "I2": 0,
        "theta_deg": -45,
        "n": 2,
        "solver": "area moment + math.linear_algebra eigen",
        "convention": "Ixy = Σ(Īxy + A (x−x̄)(y−ȳ)). θ is the I1 axis, CCW from +x, in (−90°, 90°]."
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      "tol": 1e-9
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      "id": "boundary-swap",
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        "parts": "1,4,0,2,0,0"
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        "I2": 1,
        "theta_deg": 90
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        "convention": "Ixy = Σ(Īxy + A (x−x̄)(y−ȳ)). θ is the I1 axis, CCW from +x, in (−90°, 90°]."
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      "id": "boundary-cancel",
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      "error": "Total area must not be zero."
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