{
  "capability_id": "math.numerical.root",
  "tool_id": "numerical-root",
  "engine": "numerical-root",
  "calculation_version": "1.0.0",
  "generated_at": "2026-09-16T22:38:08.876Z",
  "engine_build": "2026-09-16",
  "test_count": 9,
  "passed": 9,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 2,
      "passed": 2
    },
    "boundary": {
      "total": 2,
      "passed": 2
    },
    "property": {
      "total": 5,
      "passed": 5
    }
  },
  "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/numerical-root",
  "api": "https://www.calculatorx.com/api/v1/calc/numerical-root",
  "schema_input": "https://www.calculatorx.com/schemas/math.numerical.root.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.numerical.root.output.json",
  "assumptions": [
    "Real polynomials",
    "Not CAS / not complex roots listed"
  ],
  "limitations": [
    "Not ⁿ√a — that remains /calc/math/root",
    "Not three independent Newton/bisection/secant product pages",
    "Degree > 8 → INVALID_INPUT",
    "No convergence → NO_CONVERGENCE"
  ],
  "boundary_conditions": [
    "no sign change → INVALID_INPUT",
    "unknown method → INVALID_MODE",
    "no convergence → NO_CONVERGENCE"
  ],
  "sources": [
    "ISO 80000-2:2019 Quantities and units — Mathematics"
  ],
  "methods": [],
  "fingerprint": "sha256:fa2831c926669b8a418d15ca943fe489d99082f20cabac6f1a00aa56a751eddd",
  "tests": [
    {
      "id": "golden-newton-sqrt2",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "method": "newton",
        "coeffs": [
          -2,
          0,
          1
        ],
        "x0": 1.5
      },
      "expect": {
        "root": 1.4142135623730951
      },
      "expectWarning": null,
      "actual": {
        "method": "newton",
        "coeffs": [
          -2,
          0,
          1
        ],
        "degree": 2,
        "root": 1.41421356237,
        "residual": 4.51061410445e-12,
        "iterations": 4,
        "tol": 1e-10,
        "formula": "x ← x − f(x)/f′(x)"
      },
      "tol": 1e-8
    },
    {
      "id": "golden-bisection-sqrt2",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "method": "bisection",
        "coeffs": [
          -2,
          0,
          1
        ],
        "a": 0,
        "b": 2
      },
      "expect": {
        "root": 1.4142135623730951
      },
      "expectWarning": null,
      "actual": {
        "method": "bisection",
        "coeffs": [
          -2,
          0,
          1
        ],
        "degree": 2,
        "root": 1.41421356238,
        "residual": 3.15445447541e-11,
        "iterations": 30,
        "tol": 1e-10,
        "formula": "midpoint until sign change bracket shrinks"
      },
      "tol": 1e-8
    },
    {
      "id": "boundary-no-sign-change",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "method": "bisection",
        "coeffs": [
          -2,
          0,
          1
        ],
        "a": 2,
        "b": 3
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "Bisection needs f(a) and f(b) of opposite sign."
    },
    {
      "id": "boundary-halley",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "method": "halley",
        "coeffs": [
          -2,
          0,
          1
        ],
        "x0": 1
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "method must be newton, bisection, or secant."
    },
    {
      "id": "property-root-newton",
      "ok": true,
      "kind": "property",
      "detail": "√2 newton",
      "expect": 1.4142135623730951,
      "actual": 1.41421356237
    },
    {
      "id": "property-root-bisect",
      "ok": true,
      "kind": "property",
      "detail": "√2 bisection",
      "expect": 1.4142135623730951,
      "actual": 1.41421356238
    },
    {
      "id": "property-root-secant",
      "ok": true,
      "kind": "property",
      "detail": "√2 secant",
      "expect": 1.4142135623730951,
      "actual": 1.41421356237
    },
    {
      "id": "property-root-lin",
      "ok": true,
      "kind": "property",
      "detail": "x-4=0",
      "expect": 4,
      "actual": 4
    },
    {
      "id": "property-root-zero",
      "ok": true,
      "kind": "property",
      "detail": "x=0",
      "expect": 0,
      "actual": 0
    }
  ],
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 9,
      "total": 9,
      "golden": 2,
      "boundary": 2,
      "property": 5
    },
    "contract_checks": {
      "passed": 1,
      "total": 1
    }
  },
  "evidence_core_generated_at": "2026-09-16T15:17:27.422Z",
  "rendered_at": "2026-09-16T22:38:08.876Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.0.0",
    "artifacts": {
      "engine": "1.0.0",
      "page": "1.0.0",
      "capability": "1.0.0",
      "evidence": "1.0.0",
      "input_schema": "1.0.0",
      "output_schema": "1.0.0",
      "mcp_contract": "1.0.0"
    },
    "digests": {
      "input_schema_sha256": "b18eb4a15dcc4a82cb1b73be6c893a0ff5312c1d88b5c03367d8ad7d03302cb6",
      "output_schema_sha256": "9196bde7e9884fab5a62cd21ec2d6b2424ca620749bb995fc5035c3b93eb22d6",
      "evidence_sha256": "6a1646c2cb91903f8442475730ca0e62f4b44b74bba274dbf403f029cf4761ce",
      "evidence_core_sha256": "6a1646c2cb91903f8442475730ca0e62f4b44b74bba274dbf403f029cf4761ce",
      "capability_sha256": "42bd79658e2916d70c50fa01ae3eaf1bf327fb54024b7fb2ea6355b985614374",
      "engine_sha256": "c8d4ccdbfeaf61d7b024d4e1babb0c2ba3b324c1b37bd0cb456a526d8d2b1bd1",
      "mcp_contract_sha256": "863098b311a7b17dc4b229a6302b90b3545b175f716e5d84772c7be8e9716b86"
    },
    "contract_checks": {
      "total": 1,
      "passed": 1,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.0.0"
        }
      ]
    },
    "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json"
  }
}
