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      "sources",
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  "limitations": [
    "S1 = I1/c1 and S2 = I2/c2 along the principal axes",
    "c1 and c2 are distances from the composite centroid to the extreme fiber",
    "A blank distance omits that modulus",
    "θ is the I1 axis, CCW from +x, in (−90°, 90°]",
    "Not Sx or Sy, bending stress, plastic modulus, or beam deflection"
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  "boundary_conditions": [
    "no parts → MISSING_REQUIRED_INPUT",
    "neither c1 nor c2 → MISSING_REQUIRED_INPUT",
    "c1 or c2 ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "I1 < 0 when c1 is set, or I2 < 0 when c2 is set → INVALID_INPUT"
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  "sources": [
    {
      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Moments of inertia",
      "supports": "Elastic section modulus about a principal axis is I/c, with c measured from the centroid along that axis",
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      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "4,1,0,2,0,0",
        "c1": 2,
        "c2": 1
      },
      "expect": {
        "I1": 4,
        "I2": 1,
        "S1": 2,
        "S2": 1,
        "theta_deg": 0,
        "model": "principal_section_modulus"
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      "actual": {
        "model": "principal_section_modulus",
        "area": 2,
        "x_bar": 0,
        "y_bar": 0,
        "I1": 4,
        "I2": 1,
        "theta_deg": 0,
        "S1": 2,
        "S2": 1,
        "c1": 2,
        "c2": 1,
        "n": 1,
        "solver": "principal moment",
        "convention": "S1 = I1/c1 and S2 = I2/c2. c1 and c2 are distances from the centroid to the extreme fiber measured along the principal axes. θ is the I1 axis."
      },
      "tol": 1e-12
    },
    {
      "id": "golden-s1",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "0,0,0,1,0,0; 0,0,0,1,2,2",
        "c1": 1
      },
      "expect": {
        "I1": 4,
        "S1": 4,
        "S2": null,
        "theta_deg": -45
      },
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      "actual": {
        "model": "principal_section_modulus",
        "area": 2,
        "x_bar": 1,
        "y_bar": 1,
        "I1": 4,
        "I2": 0,
        "theta_deg": -45,
        "S1": 4,
        "S2": null,
        "c1": 1,
        "c2": null,
        "n": 2,
        "solver": "principal moment",
        "convention": "S1 = I1/c1 and S2 = I2/c2. c1 and c2 are distances from the centroid to the extreme fiber measured along the principal axes. θ is the I1 axis."
      },
      "tol": 1e-9
    },
    {
      "id": "boundary-zero-i2",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "parts": "0,0,0,1,0,0; 0,0,0,1,2,2",
        "c1": 1,
        "c2": 1
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      "expect": {
        "S2": 0,
        "I2": 0
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        "area": 2,
        "x_bar": 1,
        "y_bar": 1,
        "I1": 4,
        "I2": 0,
        "theta_deg": -45,
        "S1": 4,
        "S2": 0,
        "c1": 1,
        "c2": 1,
        "n": 2,
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        "convention": "S1 = I1/c1 and S2 = I2/c2. c1 and c2 are distances from the centroid to the extreme fiber measured along the principal axes. θ is the I1 axis."
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      "id": "boundary-empty",
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      "error": "Enter c1, c2, or both."
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      "id": "boundary-fiber",
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      "inputs": {
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        "c1": 0
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      "kind": "boundary",
      "inputs": {
        "parts": "-4,-1,0,2,0,0",
        "c1": 1
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "I1 must be non-negative to form S1."
    }
  ],
  "tier": "engine_tested",
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