{
  "capability_id": "math.square_root",
  "tool_id": "square-root",
  "engine": "square-root",
  "calculation_version": "1.0.4",
  "generated_at": "2026-10-28T03:01:57.873Z",
  "engine_build": "2026-10-26",
  "test_count": 18,
  "passed": 18,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "passed": 11,
      "total": 11
    },
    "boundary": {
      "passed": 7,
      "total": 7
    },
    "property": {
      "passed": 0,
      "total": 0
    }
  },
  "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/square-root",
  "api": "https://www.calculatorx.com/api/v1/calc/square-root",
  "schema_input": "https://www.calculatorx.com/schemas/math.square_root.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.square_root.output.json",
  "assumptions": [
    "Principal branch: nonnegative real root when x ≥ 0.",
    "Unlike Log / Ln, this engine does not reject x < 0. It returns the principal imaginary √|x| i, not a real error.",
    "The other real root for x > 0 is −√x; the other imaginary root for x < 0 is −√|x| i. The published value is the principal one."
  ],
  "limitations": [
    "Principal branch on finite real radicands: x≥0 gives real=principal=√x,imag=0; x<0 gives real=0,imag=√|x|,principal=null. Both real/imag/principal are always present.",
    "Numbers or strict decimal/scientific strings of at most512 characters; x and xInput must agree. Unknown keys, nonfinite values, null, booleans, hex, commas and nonzero-to-zero underflow reject.",
    "Binary64 input rounding and Math.sqrt output rounding are distinct; at most2 ULP on declared independent Decimal2000 reference vectors, not an exhaustive or exact-decimal guarantee. Relative condition number1/2 away from zero; derivative unbounded at zero.",
    "12 significant digit display. Perfect-square claims require safe-integer radicands and roots. Scaled Babylonian working is an illustration, not the engine algorithm.",
    "Live graph shows the real branch in a bounded window; negative radicands have no real point. Typing computes locally; shared URLs and REST/MCP send inputs and may enter logs."
  ],
  "boundary_conditions": [
    "Missing x returns MISSING_REQUIRED_INPUT.",
    "Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.",
    "√0 = 0.",
    "x < 0 → imag = √(−x), principal = null (not NaN).",
    "Strict decimal/scientific strings are limited to 512 characters; hexadecimal, commas, booleans, null, arrays and objects reject. Unknown fields reject UNKNOWN_INPUT; x/xInput aliases must agree.",
    "Nonzero decimal text or raw JSON numbers that round to zero reject NUMERIC_UNDERFLOW. Overflowing inputs reject INVALID_NUMBER. Every finite normalized radicand has a finite representable square-root magnitude."
  ],
  "sources": [
    {
      "title": "NIST Digital Library of Mathematical Functions, Chapter 4",
      "section": "Logarithm, Exponential, Powers",
      "url": "https://dlmf.nist.gov/4",
      "supports": "Square root as the principal value of x^(1/2)",
      "kind": "reference"
    },
    {
      "title": "NIST DLMF §4.2(iv)",
      "section": "Powers: principal value z^a = exp(a ln z)",
      "url": "https://dlmf.nist.gov/4.2",
      "supports": "Principal square root is the principal branch of z^(1/2); nonnegative for real x > 0. The branch cut is the negative real axis; this engine reports the principal imaginary on that cut as √|x| i.",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
    },
    {
      "title": "CalculatorX mathematical conventions",
      "section": "Principal real root; principal imaginary for x < 0",
      "supports": "Matches the on-page contract. Cube and nth roots are the Root calculator, not this engine.",
      "kind": "definition"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:c5f09468ffa8ad03d95114a4a2621da17c7336834988fca78c1bfad2f9f3b341",
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 18,
      "total": 18,
      "golden": 11,
      "boundary": 7,
      "property": 0
    },
    "contract_checks": {
      "passed": 6,
      "total": 6
    }
  },
  "evidence_core_generated_at": "2026-10-26T05:34:16.140Z",
  "rendered_at": "2026-10-28T03:01:57.873Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.0.4",
    "artifacts": {
      "engine": "1.0.4",
      "page": "1.0.4",
      "capability": "1.0.4",
      "evidence": "1.0.4",
      "input_schema": "1.0.4",
      "output_schema": "1.0.4",
      "mcp_contract": "1.0.4"
    },
    "digests": {
      "input_schema_sha256": "69b9b78cd467274524000030ed067a5c87391b85c6af4b7d168f00914a774f79",
      "output_schema_sha256": "dfb5553c678fd2ff8916c23d13c5350a0fde5ceadb85438461be5492923fc37f",
      "evidence_sha256": "c2068787b531fa02649eb14ed6e9c22f5555014b7516f19a66143cc8a403e5f3",
      "evidence_core_sha256": "c2068787b531fa02649eb14ed6e9c22f5555014b7516f19a66143cc8a403e5f3",
      "capability_sha256": "ff8d7705eb5fcc1a11a8edd68fd42f620ff66df5dccd50cdf93df25a9fe82f3c",
      "engine_sha256": "c6ebd8436f08b413e1ac6317b2d48550f2e6639a90dff254244e3aa562158b8d",
      "mcp_contract_sha256": "5898d9426fe62990b5c5eed0b7ac18076acb83b29b083b5b4dd52f28efe79ae5"
    },
    "contract_checks": {
      "total": 6,
      "passed": 6,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.0.4"
        },
        {
          "id": "schema-output-required-declared",
          "status": "pass"
        },
        {
          "id": "schema-valid-input-executes",
          "status": "pass"
        },
        {
          "id": "engine-success-output-validates-schema",
          "status": "pass"
        },
        {
          "id": "schema-rejects-incomplete-input",
          "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"
  }
}