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          "precision": 9
        },
        {
          "label": "Distance CA",
          "value": 1.7346010553881064,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Angle A",
          "value": 1.5707963267948966,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Angle B",
          "value": 0.3297825101862239,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Angle C",
          "value": 0.3297825101862239,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Area method",
          "value": "Area derived from angle defect"
        },
        {
          "label": "Disk / Lorentz distance residual",
          "value": 4.440892098500626e-16,
          "precision": 4
        },
        {
          "label": "Radial distance B",
          "value": 1.7346010553881064,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Radial distance C",
          "value": 1.7346010553881064,
          "unit": "model units",
          "precision": 9
        }
      ],
      "diagnostics": {
        "method": "Disk: 2 asinh(chord / geometric mean of boundary factors); hyperboloid: Lorentz displacement norm; Area derived from angle defect. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.",
        "tolerance": 3.8339392950726147e-10,
        "residual": 4.440892098500626e-16,
        "warnings": []
      },
      "sourceIds": [
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      "jsonPath": "/studies/geometry/run-20260917003910210-62944-007.json",
      "csvPath": "/studies/geometry/run-20260917003910210-62944-007.csv",
      "csvRows": 333,
      "plotSampleCounts": [
        {
          "title": "Radial screen and metric distance",
          "series": "Hyperbolic distance",
          "samples": 95
        },
        {
          "title": "Radial screen and metric distance",
          "series": "Euclidean screen distance",
          "samples": 95
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      ],
      "scalarRange": null
    },
    {
      "runId": "run-20260917003910210-62944-008",
      "label": "Disk radial triangle · r=0.9",
      "parameters": {
        "model": "hyperbolic",
        "radius": 1,
        "points": [
          [
            0,
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          ],
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          [
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        ]
      },
      "createdAt": "2026-09-17T00:39:14.735Z",
      "elapsedMs": 2.68579199999931,
      "modelVersion": "geometry-1.0.0",
      "numericalMethod": "Disk: 2 asinh(chord / geometric mean of boundary factors); hyperboloid: Lorentz displacement norm; Area derived from angle defect. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.",
      "measurements": [
        {
          "label": "Angle sum",
          "value": 1.779974995758138,
          "unit": "rad",
          "precision": 10
        },
        {
          "label": "Triangle area",
          "value": 1.3616176578316552,
          "unit": "model units²",
          "precision": 10,
          "description": "Area derived from angle defect"
        },
        {
          "label": "Curvature K",
          "value": -1,
          "unit": "model units⁻²",
          "precision": 8
        },
        {
          "label": "Distance AB",
          "value": 2.944438979166441,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Distance BC",
          "value": 5.201232927686145,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Distance CA",
          "value": 2.944438979166441,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Angle A",
          "value": 1.5707963267948966,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Angle B",
          "value": 0.1045893344816207,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Angle C",
          "value": 0.1045893344816207,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Area method",
          "value": "Area derived from angle defect"
        },
        {
          "label": "Disk / Lorentz distance residual",
          "value": 4.440892098500626e-16,
          "precision": 4
        },
        {
          "label": "Radial distance B",
          "value": 2.944438979166441,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Radial distance C",
          "value": 2.944438979166441,
          "unit": "model units",
          "precision": 9
        }
      ],
      "diagnostics": {
        "method": "Disk: 2 asinh(chord / geometric mean of boundary factors); hyperboloid: Lorentz displacement norm; Area derived from angle defect. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.",
        "tolerance": 6.201232927686146e-10,
        "residual": 4.440892098500626e-16,
        "warnings": []
      },
      "sourceIds": [
        "M1",
        "M4",
        "M9",
        "M10",
        "M11"
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      "jsonPath": "/studies/geometry/run-20260917003910210-62944-008.json",
      "csvPath": "/studies/geometry/run-20260917003910210-62944-008.csv",
      "csvRows": 333,
      "plotSampleCounts": [
        {
          "title": "Radial screen and metric distance",
          "series": "Hyperbolic distance",
          "samples": 95
        },
        {
          "title": "Radial screen and metric distance",
          "series": "Euclidean screen distance",
          "samples": 95
        }
      ],
      "scalarRange": null
    },
    {
      "runId": "run-20260917003910210-62944-009",
      "label": "Disk radial triangle · r=0.94",
      "parameters": {
        "model": "hyperbolic",
        "radius": 1,
        "points": [
          [
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          ],
          [
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          ],
          [
            0,
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          ]
        ]
      },
      "createdAt": "2026-09-17T00:39:14.744Z",
      "elapsedMs": 2.604750000000422,
      "modelVersion": "geometry-1.0.0",
      "numericalMethod": "Disk: 2 asinh(chord / geometric mean of boundary factors); hyperboloid: Lorentz displacement norm; Area derived from angle defect. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.",
      "measurements": [
        {
          "label": "Angle sum",
          "value": 1.694232479441662,
          "unit": "rad",
          "precision": 10
        },
        {
          "label": "Triangle area",
          "value": 1.447360174148131,
          "unit": "model units²",
          "precision": 10,
          "description": "Area derived from angle defect"
        },
        {
          "label": "Curvature K",
          "value": -1,
          "unit": "model units⁻²",
          "precision": 8
        },
        {
          "label": "Distance AB",
          "value": 3.4760986898352724,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Distance BC",
          "value": 6.26095870084751,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Distance CA",
          "value": 3.4760986898352724,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Angle A",
          "value": 1.5707963267948966,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Angle B",
          "value": 0.061718076323382824,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Angle C",
          "value": 0.061718076323382824,
          "unit": "rad",
          "precision": 9
        },
        {
          "label": "Area method",
          "value": "Area derived from angle defect"
        },
        {
          "label": "Disk / Lorentz distance residual",
          "value": 4.440892098500626e-16,
          "precision": 4
        },
        {
          "label": "Radial distance B",
          "value": 3.4760986898352724,
          "unit": "model units",
          "precision": 9
        },
        {
          "label": "Radial distance C",
          "value": 3.4760986898352724,
          "unit": "model units",
          "precision": 9
        }
      ],
      "diagnostics": {
        "method": "Disk: 2 asinh(chord / geometric mean of boundary factors); hyperboloid: Lorentz displacement norm; Area derived from angle defect. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.",
        "tolerance": 7.260958700847511e-10,
        "residual": 4.440892098500626e-16,
        "warnings": [
          "A vertex is near the interaction cutoff. Disk screen lengths strongly understate metric lengths there."
        ]
      },
      "sourceIds": [
        "M1",
        "M4",
        "M9",
        "M10",
        "M11"
      ],
      "jsonPath": "/studies/geometry/run-20260917003910210-62944-009.json",
      "csvPath": "/studies/geometry/run-20260917003910210-62944-009.csv",
      "csvRows": 333,
      "plotSampleCounts": [
        {
          "title": "Radial screen and metric distance",
          "series": "Hyperbolic distance",
          "samples": 95
        },
        {
          "title": "Radial screen and metric distance",
          "series": "Euclidean screen distance",
          "samples": 95
        }
      ],
      "scalarRange": null
    }
  ],
  "results": [
    "This saved batch contains 9 completed calculations. The table and downloads below refer to those actual runs.",
    "The largest recorded absolute diagnostic residual was 4.44089e-16. 9 of 9 residual-bearing cases were within their own reported numerical tolerance.",
    "Across 9 recorded readings, Angle sum ranged from 1.694232 to 4.712389.",
    "Across 9 recorded readings, Triangle area ranged from 0.01999933 to 6.283185.",
    "Across 9 recorded readings, Distance AB ranged from 0.2006707 to 3.476099.",
    "Across 9 recorded readings, Distance BC ranged from 0.2847369 to 6.260959.",
    "Spherical radius 1: angle sum 4.712389 radians; area 1.570796 model units²; independent area residual 0. The area uses the vector expression, while the comparison uses the tangent angles.",
    "Spherical radius 2: angle sum 4.712389 radians; area 6.283185 model units²; independent area residual 0. The area uses the vector expression, while the comparison uses the tangent angles.",
    "Disk radius r = 0.1: radial metric distance 0.2006707; triangle area 0.01999933; disk/Lorentz distance residual 5.55112e-17. Area here is derived from angle defect.",
    "Disk radius r = 0.3: radial metric distance 0.6190392; triangle area 0.1795163; disk/Lorentz distance residual 0. Area here is derived from angle defect.",
    "Disk radius r = 0.5: radial metric distance 1.098612; triangle area 0.4899573; disk/Lorentz distance residual 2.22045e-16. Area here is derived from angle defect.",
    "Disk radius r = 0.7: radial metric distance 1.734601; triangle area 0.9112313; disk/Lorentz distance residual 4.44089e-16. Area here is derived from angle defect.",
    "Disk radius r = 0.9: radial metric distance 2.944439; triangle area 1.361618; disk/Lorentz distance residual 4.44089e-16. Area here is derived from angle defect.",
    "Disk radius r = 0.94: radial metric distance 3.476099; triangle area 1.44736; disk/Lorentz distance residual 4.44089e-16. Area here is derived from angle defect."
  ],
  "interpretation": [
    "The sphere cases keep the same three unit directions as radius changes. The disk cases keep the form (0, r, ir) while changing r. Compare values within those controlled families; the three model defaults do not represent an isometric triangle shared by all geometries.",
    "An area calculated directly from angle defect cannot independently validate the defect formula. The spherical cross-check uses a separate calculation; the disk area remains labeled as derived.",
    "The radial disk cases compare multiple coordinate representations of the same model, not measurements of physical space."
  ],
  "summaryCsvPath": "/studies/geometry/summary-20260917003910210-62944.csv"
}