{
  "capability_id": "math.linear_algebra",
  "tool_id": "linear-system",
  "engine": "linear-algebra",
  "calculation_version": "1.0.0",
  "generated_at": "2026-09-16T22:38:08.876Z",
  "engine_build": "2026-09-16",
  "test_count": 10,
  "passed": 10,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 3,
      "passed": 3
    },
    "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/linear-system",
  "api": "https://www.calculatorx.com/api/v1/calc/linear-system",
  "schema_input": "https://www.calculatorx.com/schemas/math.linear_algebra.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.linear_algebra.output.json",
  "assumptions": [
    "Real matrices",
    "Order ≤ 4",
    "Not a CAS"
  ],
  "limitations": [
    "Order ≤ 4 (local small-matrix seed)",
    "Singular inverse → SINGULAR_MATRIX",
    "Not eigenvalue / SVD / large-matrix server compute"
  ],
  "boundary_conditions": [
    "singular inverse → SINGULAR_MATRIX",
    "unknown mode (including eigen) → INVALID_MODE"
  ],
  "sources": [
    "ISO 80000-2:2019 Quantities and units — Mathematics"
  ],
  "methods": [],
  "fingerprint": "sha256:33d693b4211f8c5cee8499739c2c5f65a9faf7db9639323db671f5823873efab",
  "tests": [
    {
      "id": "golden-2x2",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "solve",
        "order": 2,
        "a11": 2,
        "a12": 1,
        "a21": 1,
        "a22": 1,
        "b1": 5,
        "b2": 3
      },
      "expect": {
        "x": 2,
        "y": 1,
        "status": "unique"
      },
      "expectWarning": null,
      "actual": {
        "mode": "solve",
        "order": 2,
        "A": [
          [
            2,
            1
          ],
          [
            1,
            1
          ]
        ],
        "rhs": [
          5,
          3
        ],
        "x": 2,
        "y": 1,
        "vector": [
          2,
          1
        ],
        "status": "unique",
        "det": 1,
        "formula": "Ax = b"
      },
      "tol": null
    },
    {
      "id": "golden-det",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "det",
        "order": 2,
        "a11": 1,
        "a12": 2,
        "a21": 3,
        "a22": 4
      },
      "expect": {
        "det": -2
      },
      "expectWarning": null,
      "actual": {
        "mode": "det",
        "order": 2,
        "A": [
          [
            1,
            2
          ],
          [
            3,
            4
          ]
        ],
        "det": -2,
        "formula": "det(A)"
      },
      "tol": null
    },
    {
      "id": "golden-identity-3",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "solve",
        "order": 3,
        "a11": 1,
        "a12": 0,
        "a13": 0,
        "a21": 0,
        "a22": 1,
        "a23": 0,
        "a31": 0,
        "a32": 0,
        "a33": 1,
        "b1": 4,
        "b2": 5,
        "b3": 6
      },
      "expect": {
        "x": 4,
        "y": 5,
        "z": 6
      },
      "expectWarning": null,
      "actual": {
        "mode": "solve",
        "order": 3,
        "A": [
          [
            1,
            0,
            0
          ],
          [
            0,
            1,
            0
          ],
          [
            0,
            0,
            1
          ]
        ],
        "rhs": [
          4,
          5,
          6
        ],
        "x": 4,
        "y": 5,
        "z": 6,
        "vector": [
          4,
          5,
          6
        ],
        "status": "unique",
        "det": 1,
        "formula": "Ax = b"
      },
      "tol": null
    },
    {
      "id": "boundary-singular",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "inverse",
        "order": 2,
        "a11": 1,
        "a12": 2,
        "a21": 2,
        "a22": 4
      },
      "expectError": "SINGULAR_MATRIX",
      "code": "SINGULAR_MATRIX",
      "error": "Matrix is singular; inverse does not exist."
    },
    {
      "id": "boundary-eigen",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "eigen",
        "a11": 1,
        "a12": 0,
        "a21": 0,
        "a22": 1
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "mode must be solve, det, inverse, multiply, or transpose."
    },
    {
      "id": "property-la-solve",
      "ok": true,
      "kind": "property",
      "detail": "2x+y=5",
      "expect": 2,
      "actual": 2
    },
    {
      "id": "property-la-det",
      "ok": true,
      "kind": "property",
      "detail": "det",
      "expect": -2,
      "actual": -2
    },
    {
      "id": "property-la-inv",
      "ok": true,
      "kind": "property",
      "detail": "diag inverse",
      "expect": 0.5,
      "actual": 0.5
    },
    {
      "id": "property-la-t",
      "ok": true,
      "kind": "property",
      "detail": "transpose",
      "expect": 3,
      "actual": 3
    },
    {
      "id": "property-la-id3",
      "ok": true,
      "kind": "property",
      "detail": "I₃ solve",
      "expect": 9,
      "actual": 9
    }
  ],
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 10,
      "total": 10,
      "golden": 3,
      "boundary": 2,
      "property": 5
    },
    "contract_checks": {
      "passed": 3,
      "total": 3
    }
  },
  "evidence_core_generated_at": "2026-09-16T15:14:48.779Z",
  "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": "0e17c89cf4e428a60059791c8e69c2f8a75fb9fecef59d23328bf7e0640d4e58",
      "output_schema_sha256": "bf1a78398b0e21e520a3c3ae48d341cdac2d38a2cab6f653731789d9f7183082",
      "evidence_sha256": "96a9dec93994e3d6258ecd6d9063295496f110b557723882ec6ce96a32f7deea",
      "evidence_core_sha256": "96a9dec93994e3d6258ecd6d9063295496f110b557723882ec6ce96a32f7deea",
      "capability_sha256": "92dfd3070979b4769a6f0c0ed50ca91b4b3a83c11ed43c119c6f074e50480d20",
      "engine_sha256": "c8d4ccdbfeaf61d7b024d4e1babb0c2ba3b324c1b37bd0cb456a526d8d2b1bd1",
      "mcp_contract_sha256": "863098b311a7b17dc4b229a6302b90b3545b175f716e5d84772c7be8e9716b86"
    },
    "contract_checks": {
      "total": 3,
      "passed": 3,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.0.0"
        },
        {
          "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"
  }
}