{
  "capability_id": "math.complex",
  "tool_id": "complex",
  "engine": "complex",
  "calculation_version": "1.3.0",
  "generated_at": "2026-10-29T17:04:55.749Z",
  "engine_build": "2026-10-29",
  "test_count": 18,
  "passed": 18,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "passed": 6,
      "total": 6
    },
    "boundary": {
      "passed": 4,
      "total": 4
    },
    "property": {
      "passed": 8,
      "total": 8
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "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."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
  },
  "page": "https://www.calculatorx.com/calc/math/complex-number",
  "api": "https://www.calculatorx.com/api/v1/calc/complex",
  "schema_input": "https://www.calculatorx.com/schemas/math.complex.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.complex.output.json",
  "assumptions": [
    "atan2 argument in degrees",
    "Not a CAS"
  ],
  "limitations": [
    "Finite decimal inputs; matching aliases only. Missing rectangular components default0; explicit empty inputs are errors.",
    "Power integer |n|≤1000; roots integer2≤n≤64; zero has one distinct root;0^0 undefined.",
    "Argument in degrees (−180,180]; negative real axis assigned+180 including signed zero; origin0 by convention (undefined mathematically).",
    "Nonzero input/output components and modulus must be normal finite binary64; unrepresentable scaling/intermediates return NUMERIC_RANGE.",
    "Vector error budget2e-11 times modulus, argument2e-9 degrees away from ill-conditioned origin/cut; cancellation can lose relative component accuracy.",
    "No symbolic CAS or complex noninteger powers."
  ],
  "boundary_conditions": [
    "divide by 0+0i → DIVISION_BY_ZERO",
    "unknown mode (including simplify) → INVALID_MODE"
  ],
  "sources": [
    {
      "title": "ISO 80000-2:2019 Quantities and units — Mathematics",
      "url": "https://www.iso.org/standard/64973.html",
      "supports": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.",
      "kind": "standard",
      "publisher": "International Organization for Standardization",
      "reference_scope": "all"
    },
    {
      "title": "CRC Standard Mathematical Tables — De Moivre and Euler form",
      "supports": "z^n = r^n (cos nθ + i sin nθ); z = r e^{iθ}; nth roots equally spaced on a circle",
      "kind": "reference",
      "reference_scope": "model",
      "oracle_class": "O1"
    },
    {
      "title": "NIST DLMF — 4.2 Definitions: Logarithm and Exponential",
      "url": "https://dlmf.nist.gov/4.2",
      "supports": "Complex exponential and logarithm definitions, including the exponential’s trigonometric representation and branch conventions. Mathematical reference only; not validation or endorsement of this implementation.",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
    },
    {
      "title": "Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare)",
      "section": "§9.4 Complex Numbers",
      "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch09.pdf",
      "supports": "Complex arithmetic, conjugates, modulus and argument, polar/exponential multiplication and division, de Moivre’s formula and complex roots. Noninteger powers require a specified branch; those branch and numerical policies remain CalculatorX’s own contract. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.",
      "kind": "reference",
      "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:9ed446af773095d59575406506a3f486648164ef855ef2494ac522ad231d2d3f",
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 18,
      "total": 18,
      "golden": 6,
      "boundary": 4,
      "property": 8
    },
    "contract_checks": {
      "passed": 5,
      "total": 5
    }
  },
  "evidence_core_generated_at": "2026-10-29T08:55:15.563Z",
  "rendered_at": "2026-10-29T17:04:55.749Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.3.0",
    "artifacts": {
      "engine": "1.3.0",
      "page": "1.3.0",
      "capability": "1.3.0",
      "evidence": "1.3.0",
      "input_schema": "1.3.0",
      "output_schema": "1.3.0",
      "mcp_contract": "1.3.0"
    },
    "digests": {
      "input_schema_sha256": "72afe573472e73e6f011c632f9ba19c5edd4f712c2e983310dc52e3dd218b996",
      "output_schema_sha256": "a0b90070ae3b09aca4baf9b9c4e2a2f01794d0c9b72b8118e386cd4a8a99eb82",
      "evidence_sha256": "bac843081a1f0529afcb820871c4629d8556f76f8e6edd10bc263c936c610005",
      "evidence_core_sha256": "bac843081a1f0529afcb820871c4629d8556f76f8e6edd10bc263c936c610005",
      "capability_sha256": "8d2e0d64f1403e606dd95ebc6b4d29d34246cf0053c0920fa4aa26fa5c2413b7",
      "engine_sha256": "c6ebd8436f08b413e1ac6317b2d48550f2e6639a90dff254244e3aa562158b8d",
      "mcp_contract_sha256": "1cafec71298a0deb89bd361494db4322a690d14b61e801243407475112bdacbd"
    },
    "contract_checks": {
      "total": 5,
      "passed": 5,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.3.0"
        },
        {
          "id": "schema-output-required-declared",
          "status": "pass"
        },
        {
          "id": "schema-valid-input-executes",
          "status": "pass"
        },
        {
          "id": "engine-success-output-validates-schema",
          "status": "pass"
        },
        {
          "id": "rest-success-payload-validates-schema",
          "status": "pass"
        }
      ]
    },
    "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json"
  },
  "tests": [],
  "disclosure": {
    "policy": "calculatorx-public-evidence-v1",
    "full_validation": "private",
    "representation": "summary"
  }
}