{
  "capability_id": "math.numerical.root",
  "tool_id": "numerical-root",
  "engine": "numerical-root",
  "calculation_version": "1.1.2",
  "generated_at": "2026-10-05T23:27:20.241Z",
  "engine_build": "2026-10-05",
  "test_count": 16,
  "passed": 16,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "passed": 4,
      "total": 4
    },
    "boundary": {
      "passed": 5,
      "total": 5
    },
    "property": {
      "passed": 7,
      "total": 7
    }
  },
  "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": [
    "The Result Card includes an engine-linked figure of f around the published root (marker is pack.root on y=0; Newton/secant seeds and the bisection bracket are read from inputs, not REST fields). Same fnValue as REST/MCP. Iterate history is not a REST field.",
    "Not CAS / not complex roots listed",
    "Not a system of equations",
    "Not all-roots of a polynomial",
    "Not ⁿ√a",
    "Polynomial degree ≤ 8",
    "Polynomial convergence uses |f(x)| divided by Σ|cᵢ||x|ⁱ; sin uses |sin x|. A small residual alone does not prove a nearby root for every polynomial."
  ],
  "limitations": [
    "Not ⁿ√a — that remains /calc/math/root",
    "Not three independent Newton/bisection/secant product pages",
    "Not CAS / not a nonlinear system",
    "Degree > 8 → INVALID_INPUT",
    "No convergence → NO_CONVERGENCE"
  ],
  "boundary_conditions": [
    "no sign change → INVALID_INPUT",
    "Bisection requires opposite endpoint signs even for tiny finite function values → INVALID_INPUT",
    "unknown method (including cas, halley) → INVALID_MODE",
    "no convergence → NO_CONVERGENCE",
    "eˣ=0 has no finite real root → NO_CONVERGENCE"
  ],
  "sources": [
    {
      "title": "NIST Digital Library of Mathematical Functions §3.8 — Nonlinear Equations",
      "url": "https://dlmf.nist.gov/3.8",
      "supports": "Newton's iteration, bisection with opposite-sign endpoints, and the secant method for approximate zeros",
      "kind": "reference",
      "reference_scope": "model",
      "oracle_class": "O1"
    },
    {
      "title": "MIT OpenCourseWare — Numerical Fluid Mechanics, Lecture 3",
      "url": "https://ocw.mit.edu/courses/2-29-numerical-fluid-mechanics-spring-2015/pages/lecture-notes-and-references/",
      "supports": "Lecture 3 covers bisection, Newton-Raphson, and secant methods for roots of nonlinear equations",
      "kind": "reference",
      "reference_scope": "model",
      "oracle_class": "O1"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:87cedd3fe3b2f56febcbce359b779e1301bd45313a83854e725e2a1f354536f8",
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 16,
      "total": 16,
      "golden": 4,
      "boundary": 5,
      "property": 7
    },
    "contract_checks": {
      "passed": 5,
      "total": 5
    }
  },
  "evidence_core_generated_at": "2026-10-05T23:25:56.117Z",
  "rendered_at": "2026-10-05T23:27:20.241Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.1.2",
    "artifacts": {
      "engine": "1.1.2",
      "page": "1.1.2",
      "capability": "1.1.2",
      "evidence": "1.1.2",
      "input_schema": "1.1.2",
      "output_schema": "1.1.2",
      "mcp_contract": "1.1.2"
    },
    "digests": {
      "input_schema_sha256": "75ad0ce3abdaff3d6e801f35205e5ecf56dc902a9102c30479b1352d793b5729",
      "output_schema_sha256": "607741277b1f93a72e494439e0e1fc6af2ca649ffffcffcd9f0205b8a24983aa",
      "evidence_sha256": "305f4f29e0bb43c477dc1e35533d6515f47aa718075a74735312078d8e0f0cfd",
      "evidence_core_sha256": "305f4f29e0bb43c477dc1e35533d6515f47aa718075a74735312078d8e0f0cfd",
      "capability_sha256": "71ed7ea7cace9bc2917f4c09f9ce47c321466a6934137752d38aa275cd50cf9d",
      "engine_sha256": "de164a39feabc37997e09354949becc3182b03a314edbb82d01fddc018716fc0",
      "mcp_contract_sha256": "f69c14c7d2344aad1cdc4aaa154251f93b423cccde393f92084f619a0de88701"
    },
    "contract_checks": {
      "total": 5,
      "passed": 5,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.1.2"
        },
        {
          "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"
  }
}
