{
  "capability_id": "math.sum",
  "tool_id": "add",
  "engine": "sum",
  "calculation_version": "1.0.2",
  "generated_at": "2026-09-16T22:03:43.781Z",
  "engine_build": "2026-09-16",
  "test_count": 31,
  "passed": 31,
  "failed": 0,
  "tests_summary": {
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      "total": 24,
      "passed": 24
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    "boundary": {
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      "passed": 7
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    "property": {
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      "passed": 0
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  "engine_tested": true,
  "expert_reviewed": false,
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  "source_checked": true,
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  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
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    "scope": [
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      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
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  "internal_review": {
    "status": "not_applicable",
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  "page": "https://www.calculatorx.com/calc/math/add",
  "api": "https://www.calculatorx.com/api/v1/calc/add",
  "schema_input": "https://www.calculatorx.com/schemas/math.sum.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.sum.output.json",
  "assumptions": [
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    "Units are not parsed. 5 kg + 3 kg is treated as 5 + 3; the sum keeps whatever unit the inputs shared.",
    "Thousands separators are not supported. A comma in 7,890 is a value separator (7 + 890), with a non-blocking warning.",
    "Long addition pads decimal places with zeros so columns align.",
    "Plain decimal tokens are summed in base-10 (exact). Any scientific-notation token switches the list to Neumaier compensated IEEE-754 summation."
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  "limitations": [
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    "Long mode requires non-negative decimals without scientific notation",
    "Too many values → TOO_MANY_VALUES"
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  "boundary_conditions": [
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    "Negative values are rejected in Long addition mode.",
    "Long addition rejects scientific notation."
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      "title": "OpenStax Prealgebra 2e — Add Whole Numbers",
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      "supports": "Place-value addition of a finite list of terms, including carrying.",
      "kind": "reference"
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    {
      "title": "OpenStax Prealgebra 2e — Add Integers",
      "section": "3.2 Signed addition",
      "url": "https://openstax.org/books/prealgebra-2e/pages/3-2-add-integers",
      "supports": "Signed addition of integers (and by extension rationals) as a finite sum.",
      "kind": "reference"
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    {
      "title": "CalculatorX mathematical conventions",
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      "kind": "definition"
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        "mode_label": "Sum",
        "explainer": "Count 2 · Mean 7.5",
        "inputs_summary": "10 + 5",
        "display": {
          "value": "15",
          "sum": "15",
          "mean": "7.5"
        },
        "work": null,
        "chart": [
          10,
          15
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "plus-signs",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "+3, +4"
      },
      "expect": 7,
      "expectWarning": null,
      "actual": {
        "mode": "sum",
        "algorithm": "exact-decimal",
        "count": 2,
        "values": [
          3,
          4
        ],
        "tokens": [
          "+3",
          "+4"
        ],
        "sum": 7,
        "value": 7,
        "sumStr": "7",
        "mean": 3.5,
        "mean_exact": true,
        "mean_rational_exact": true,
        "mean_numeric_approximate": false,
        "mean_num": "7",
        "mean_den": "2",
        "mean_display": "3.5",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 2 · Mean 3.5",
        "inputs_summary": "+3 + +4",
        "display": {
          "value": "7",
          "sum": "7",
          "mean": "3.5"
        },
        "work": null,
        "chart": [
          3,
          7
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "multiline",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "1\n2\n3"
      },
      "expect": 6,
      "expectWarning": null,
      "actual": {
        "mode": "sum",
        "algorithm": "exact-decimal",
        "count": 3,
        "values": [
          1,
          2,
          3
        ],
        "tokens": [
          "1",
          "2",
          "3"
        ],
        "sum": 6,
        "value": 6,
        "sumStr": "6",
        "mean": 2,
        "mean_exact": true,
        "mean_rational_exact": true,
        "mean_numeric_approximate": false,
        "mean_num": "6",
        "mean_den": "3",
        "mean_display": "2",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 3 · Mean 2",
        "inputs_summary": "1 + 2 + 3",
        "display": {
          "value": "6",
          "sum": "6",
          "mean": "2"
        },
        "work": null,
        "chart": [
          1,
          3,
          6
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "whitespace",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "  4   5  "
      },
      "expect": 9,
      "expectWarning": null,
      "actual": {
        "mode": "sum",
        "algorithm": "exact-decimal",
        "count": 2,
        "values": [
          4,
          5
        ],
        "tokens": [
          "4",
          "5"
        ],
        "sum": 9,
        "value": 9,
        "sumStr": "9",
        "mean": 4.5,
        "mean_exact": true,
        "mean_rational_exact": true,
        "mean_numeric_approximate": false,
        "mean_num": "9",
        "mean_den": "2",
        "mean_display": "4.5",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 2 · Mean 4.5",
        "inputs_summary": "4 + 5",
        "display": {
          "value": "9",
          "sum": "9",
          "mean": "4.5"
        },
        "work": null,
        "chart": [
          4,
          9
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "mixed-separators",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "1, 2; 3|4"
      },
      "expect": 10,
      "expectWarning": null,
      "actual": {
        "mode": "sum",
        "algorithm": "exact-decimal",
        "count": 4,
        "values": [
          1,
          2,
          3,
          4
        ],
        "tokens": [
          "1",
          "2",
          "3",
          "4"
        ],
        "sum": 10,
        "value": 10,
        "sumStr": "10",
        "mean": 2.5,
        "mean_exact": true,
        "mean_rational_exact": true,
        "mean_numeric_approximate": false,
        "mean_num": "10",
        "mean_den": "4",
        "mean_display": "2.5",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 4 · Mean 2.5",
        "inputs_summary": "1 + 2 + 3 + 4",
        "display": {
          "value": "10",
          "sum": "10",
          "mean": "2.5"
        },
        "work": null,
        "chart": [
          1,
          3,
          6,
          10
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "thousands-separator-warning",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "sumInput": "7,890"
      },
      "expect": {
        "value": 897,
        "sumStr": "897"
      },
      "expectWarning": "THOUSANDS_SEPARATOR_AMBIGUITY",
      "actual": {
        "mode": "sum",
        "algorithm": "exact-decimal",
        "count": 2,
        "values": [
          7,
          890
        ],
        "tokens": [
          "7",
          "890"
        ],
        "sum": 897,
        "value": 897,
        "sumStr": "897",
        "mean": 448.5,
        "mean_exact": true,
        "mean_rational_exact": true,
        "mean_numeric_approximate": false,
        "mean_num": "897",
        "mean_den": "2",
        "mean_display": "448.5",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 2 · Mean 448.5. “7,890” was interpreted as two values: 7 and 890. If you meant 7890 as one number, enter 7890.",
        "inputs_summary": "7 + 890",
        "display": {
          "value": "897",
          "sum": "897",
          "mean": "448.5"
        },
        "work": null,
        "chart": [
          7,
          897
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warning": "“7,890” was interpreted as two values: 7 and 890. If you meant 7890 as one number, enter 7890.",
        "warnings": [
          {
            "code": "THOUSANDS_SEPARATOR_AMBIGUITY",
            "message": "“7,890” was interpreted as two values: 7 and 890. If you meant 7890 as one number, enter 7890.",
            "example": "7,890",
            "compact": "7890",
            "parts": [
              "7",
              "890"
            ]
          }
        ]
      },
      "tol": 0
    },
    {
      "id": "leading-zeros",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "00.50, 1.50"
      },
      "expect": {
        "sumStr": "2"
      },
      "expectWarning": null,
      "actual": {
        "mode": "sum",
        "algorithm": "exact-decimal",
        "count": 2,
        "values": [
          0.5,
          1.5
        ],
        "tokens": [
          "00.50",
          "1.50"
        ],
        "sum": 2,
        "value": 2,
        "sumStr": "2",
        "mean": 1,
        "mean_exact": true,
        "mean_rational_exact": true,
        "mean_numeric_approximate": false,
        "mean_num": "200",
        "mean_den": "200",
        "mean_display": "1",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 2 · Mean 1",
        "inputs_summary": "00.50 + 1.50",
        "display": {
          "value": "2",
          "sum": "2",
          "mean": "1"
        },
        "work": null,
        "chart": [
          0.5,
          2
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "large-sci",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "1e8, 2e8"
      },
      "expect": 300000000,
      "expectWarning": null,
      "actual": {
        "mode": "sum",
        "algorithm": "neumaier",
        "count": 2,
        "values": [
          100000000,
          200000000
        ],
        "tokens": [
          "1e8",
          "2e8"
        ],
        "sum": 300000000,
        "value": 300000000,
        "sumStr": "300000000",
        "mean": 150000000,
        "mean_exact": false,
        "mean_rational_exact": false,
        "mean_numeric_approximate": true,
        "mean_num": null,
        "mean_den": null,
        "mean_display": "150000000",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 2 · Mean 150000000",
        "inputs_summary": "1e8 + 2e8",
        "display": {
          "value": "300000000",
          "sum": "300000000",
          "mean": "150000000"
        },
        "work": null,
        "chart": [
          100000000,
          300000000
        ],
        "chart_model": "binary64",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "long-list",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "values": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          9,
          10,
          11,
          12,
          13,
          14,
          15,
          16,
          17,
          18,
          19,
          20
        ]
      },
      "expect": 210,
      "expectWarning": null,
      "actual": {
        "mode": "sum",
        "algorithm": "exact-decimal",
        "count": 20,
        "values": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          9,
          10,
          11,
          12,
          13,
          14,
          15,
          16,
          17,
          18,
          19,
          20
        ],
        "tokens": [
          "1",
          "2",
          "3",
          "4",
          "5",
          "6",
          "7",
          "8",
          "9",
          "10",
          "11",
          "12",
          "13",
          "14",
          "15",
          "16",
          "17",
          "18",
          "19",
          "20"
        ],
        "sum": 210,
        "value": 210,
        "sumStr": "210",
        "mean": 10.5,
        "mean_exact": true,
        "mean_rational_exact": true,
        "mean_numeric_approximate": false,
        "mean_num": "210",
        "mean_den": "20",
        "mean_display": "10.5",
        "mean_approx": false,
        "formula": "Σ xᵢ",
        "heading": "Sum",
        "mode_label": "Sum",
        "explainer": "Count 20 · Mean 10.5",
        "inputs_summary": "1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20",
        "display": {
          "value": "210",
          "sum": "210",
          "mean": "10.5"
        },
        "work": null,
        "chart": [
          1,
          3,
          6,
          10,
          15,
          21,
          28,
          36,
          45,
          55,
          66,
          78,
          91,
          105,
          120,
          136,
          153,
          171,
          190,
          210
        ],
        "chart_model": "exact-decimal",
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "long-int",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "937, 129, 2248",
        "mode": "long"
      },
      "expect": {
        "value": 3314,
        "mode": "long"
      },
      "expectWarning": null,
      "actual": {
        "mode": "long",
        "algorithm": "long-addition",
        "count": 3,
        "values": [
          937,
          129,
          2248
        ],
        "tokens": [
          "937",
          "129",
          "2248"
        ],
        "sum": 3314,
        "value": 3314,
        "sumStr": "3314",
        "mean": 1104.6666666666667,
        "mean_exact": false,
        "mean_rational_exact": false,
        "mean_numeric_approximate": true,
        "formula": "column addition with carry",
        "heading": "Total",
        "mode_label": "Long addition",
        "explainer": "Added 3 numbers with long addition (place-value columns).",
        "inputs_summary": "937 + 129 + 2248",
        "display": {
          "value": "3314",
          "sum": "3314"
        },
        "work": "Long addition (align place values, add right → left):\n\n   937\n   129\n+ 2248\n  ----\n  3314\n\nColumn work:\n  thousands: 1 + 0 + 0 + 2 = 3 → write 3\n  hundreds: 1 + 9 + 1 + 2 = 13 → write 3, carry 1\n  tens: 2 + 3 + 2 + 4 = 11 → write 1, carry 1\n  ones: 7 + 9 + 8 = 24 → write 4, carry 2",
        "chart": null,
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "long-decimals",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "37.5, 129.44, 48.123",
        "mode": "long"
      },
      "expect": {
        "sumStr": "215.063",
        "mode": "long"
      },
      "expectWarning": null,
      "actual": {
        "mode": "long",
        "algorithm": "long-addition",
        "count": 3,
        "values": [
          37.5,
          129.44,
          48.123
        ],
        "tokens": [
          "37.5",
          "129.44",
          "48.123"
        ],
        "sum": 215.063,
        "value": 215.063,
        "sumStr": "215.063",
        "mean": 71.68766666666666,
        "mean_exact": false,
        "mean_rational_exact": false,
        "mean_numeric_approximate": true,
        "formula": "column addition with carry",
        "heading": "Total",
        "mode_label": "Long addition",
        "explainer": "Added 3 numbers with long addition (place-value columns).",
        "inputs_summary": "37.5 + 129.44 + 48.123",
        "display": {
          "value": "215.063",
          "sum": "215.063"
        },
        "work": "Long addition (align place values, add right → left):\n\n   37.500\n  129.440\n+  48.123\n  -------\n  215.063\n\nColumn work:\n  hundreds: 1 + 0 + 1 + 0 = 2 → write 2\n  tens: 2 + 3 + 2 + 4 = 11 → write 1, carry 1\n  ones: 1 + 7 + 9 + 8 = 25 → write 5, carry 2\n  10^−1: 5 + 4 + 1 = 10 → write 0, carry 1\n  10^−2: 0 + 4 + 2 = 6 → write 6\n  10^−3: 0 + 0 + 3 = 3 → write 3",
        "chart": null,
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "long-tenths",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "sumInput": "0.1, 0.2",
        "mode": "long"
      },
      "expect": {
        "sumStr": "0.3"
      },
      "expectWarning": null,
      "actual": {
        "mode": "long",
        "algorithm": "long-addition",
        "count": 2,
        "values": [
          0.1,
          0.2
        ],
        "tokens": [
          "0.1",
          "0.2"
        ],
        "sum": 0.3,
        "value": 0.3,
        "sumStr": "0.3",
        "mean": 0.15,
        "mean_exact": false,
        "mean_rational_exact": false,
        "mean_numeric_approximate": true,
        "formula": "column addition with carry",
        "heading": "Total",
        "mode_label": "Long addition",
        "explainer": "Added 2 numbers with long addition (place-value columns).",
        "inputs_summary": "0.1 + 0.2",
        "display": {
          "value": "0.3",
          "sum": "0.3"
        },
        "work": "Long addition (align place values, add right → left):\n\n  0.1\n+ 0.2\n  ---\n  0.3\n\nColumn work:\n  ones: 0 + 0 = 0 → write 0\n  10^−1: 1 + 2 = 3 → write 3",
        "chart": null,
        "calculation_version": "1.0.2",
        "warnings": []
      },
      "tol": 0
    },
    {
      "id": "empty",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "sumInput": ""
      },
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "Enter at least one number."
    },
    {
      "id": "invalid-token",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "sumInput": "1, two, 3"
      },
      "expectError": "INVALID_NUMBER",
      "code": "INVALID_NUMBER",
      "error": "Invalid number: “two”. Separate values with commas, spaces, or new lines. Do not use commas as thousands separators."
    },
    {
      "id": "long-negative",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "sumInput": "-1, 2",
        "mode": "long"
      },
      "expectError": "VALUE_MUST_BE_NON_NEGATIVE",
      "code": "VALUE_MUST_BE_NON_NEGATIVE",
      "error": "Long addition mode supports non-negative numbers only. Use Sum mode for negatives."
    },
    {
      "id": "long-scientific",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "sumInput": "1e3",
        "mode": "long"
      },
      "expectError": "UNSUPPORTED_FORMAT",
      "code": "UNSUPPORTED_FORMAT",
      "error": "Long addition does not support scientific notation. Expand the number first."
    },
    {
      "id": "invalid-mode",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "sumInput": "1",
        "mode": "nope"
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "Unknown sum mode: “nope”."
    },
    {
      "id": "boundary-sum-missing",
      "ok": true,
      "kind": "boundary",
      "inputs": {},
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "Enter at least one number."
    }
  ],
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 31,
      "total": 31,
      "golden": 24,
      "boundary": 7,
      "property": 0
    },
    "contract_checks": {
      "passed": 3,
      "total": 3
    }
  },
  "evidence_core_generated_at": "2026-09-16T06:41:00.884Z",
  "rendered_at": "2026-09-16T22:03:43.781Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.0.2",
    "artifacts": {
      "engine": "1.0.2",
      "page": "1.0.2",
      "capability": "1.0.2",
      "evidence": "1.0.2",
      "input_schema": "1.0.2",
      "output_schema": "1.0.2",
      "mcp_contract": "1.0.2"
    },
    "digests": {
      "input_schema_sha256": "bb22b7968bf29963c5e3912e6a6f9fa90f57f36d2d2830bfc206641805fc7fdb",
      "output_schema_sha256": "9ad9d0010c16804531f21dc94fea93194a027fe033a3dd265e149e96d4a0c37a",
      "evidence_sha256": "42b19eb5b57350e467933f57ae4fddd60b536ecb948f0f8f79719f1eb08de3cb",
      "evidence_core_sha256": "42b19eb5b57350e467933f57ae4fddd60b536ecb948f0f8f79719f1eb08de3cb",
      "capability_sha256": "10322f3848a4331a9305d9166e99552513def02c718798372da89caeae49796e",
      "engine_sha256": "c8d4ccdbfeaf61d7b024d4e1babb0c2ba3b324c1b37bd0cb456a526d8d2b1bd1",
      "mcp_contract_sha256": "863098b311a7b17dc4b229a6302b90b3545b175f716e5d84772c7be8e9716b86"
    },
    "contract_checks": {
      "total": 3,
      "passed": 3,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.0.2"
        },
        {
          "id": "schema-valid-input-executes",
          "status": "pass"
        },
        {
          "id": "engine-success-output-validates-schema",
          "status": "pass"
        }
      ]
    },
    "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json"
  }
}