{
  "capability_id": "engineering.tolerance.worst_case",
  "tool_id": "worst-case-analysis",
  "engine": "worst-case-analysis",
  "calculation_version": "1.3.0",
  "generated_at": "2026-09-16T22:03:43.730Z",
  "engine_build": "2026-09-12",
  "test_count": 27,
  "passed": 27,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 7,
      "passed": 7
    },
    "boundary": {
      "total": 7,
      "passed": 7
    },
    "property": {
      "total": 13,
      "passed": 13
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": true,
  "review_kind": "internal",
  "source_checked": true,
  "reviewed_by": "CalculatorX tolerance / mechanical engineering review",
  "reviewer_role": "tolerance-engineer",
  "last_reviewed": "2026-09-12",
  "expert_review": {
    "status": "not_applicable",
    "reviewer": null,
    "note": "Internal domain review completed; not a named external expert review.",
    "kind": "internal"
  },
  "internal_review": {
    "status": "approved",
    "reviewer": {
      "name": "CalculatorX tolerance / mechanical engineering review",
      "role": "tolerance-engineer"
    },
    "reviewed_at": "2026-09-12",
    "scope": [
      "formula",
      "assumptions",
      "limitations",
      "sources",
      "input semantics",
      "error codes",
      "REST compatibility",
      "unit policy",
      "tests"
    ],
    "layers": [
      {
        "id": "model",
        "label": "Model / formula",
        "reviewed_at": "2026-08-12",
        "status": "current",
        "note": "Formula and engineering assumptions unchanged in v1.3.0.",
        "scope": [
          "formula",
          "assumptions",
          "limitations",
          "sources"
        ]
      },
      {
        "id": "interface",
        "label": "Interface / input semantics",
        "reviewed_at": "2026-09-12",
        "status": "current",
        "note": "Canonical non-negative magnitudes, REST signed compatibility, overflow, and unit label policy.",
        "scope": [
          "input semantics",
          "error codes",
          "REST compatibility",
          "unit policy",
          "tests"
        ]
      }
    ],
    "kind": "internal"
  },
  "page": "https://www.calculatorx.com/calc/engineering/worst-case-analysis",
  "api": "https://www.calculatorx.com/api/v1/calc/worst-case-analysis",
  "schema_input": "https://www.calculatorx.com/schemas/engineering.tolerance.worst_case.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/engineering.tolerance.worst_case.output.json",
  "assumptions": [
    "Linear 1-D magnitude stack only",
    "Unit sensitivity |cᵢ|=1",
    "Worst-case assumes every included contributor can reach its declared limit",
    "RSS comparison assumes consistent statistical basis and independence",
    "Canonical/UI magnitudes are non-negative; REST signed values ignore sign"
  ],
  "limitations": [
    "1-D linear magnitude stack only — not a full nominal min/max dimensional chain",
    "RSS comparison assumes consistent statistical basis and independence",
    "MCP/agent input schema is canonical (tolerances[] ≥ 0); REST still accepts tol[] / tol1…tol8",
    "Canonical/UI magnitudes must be ≥ 0 (NEGATIVE_TOLERANCE_MAGNITUDE). REST signed values ignore sign (SIGN_IGNORED_FOR_TOLERANCE_MAGNITUDE)",
    "Unit is a shared label; all contributors must already use the same unit — no conversion"
  ],
  "boundary_conditions": [
    "No tol → MISSING_REQUIRED_INPUT",
    "Canonical negative → NEGATIVE_TOLERANCE_MAGNITUDE",
    "binary64 overflow → RESULT_OVERFLOW"
  ],
  "sources": [
    {
      "title": "NISTIR 6223 — Design for Tolerance of Electromechanical Assemblies",
      "section": "Worst-case tolerance analysis",
      "url": "https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir6223.pdf",
      "supports": "Deterministic analysis using component max/min limits for assembly requirements",
      "kind": "reference"
    },
    {
      "title": "ASME Y14.5 — Dimensioning and Tolerancing",
      "section": "Dimensional tolerancing / GD&T semantics",
      "url": "https://www.asme.org/codes-standards/find-codes-standards/y14-5-dimensioning-tolerancing",
      "supports": "Background language for dimensions and tolerances (not the Σ|t| algorithm itself)",
      "kind": "standard"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:385eede741ce878fa115231e8c7a6758469899f23e10067431a6df315b7c33ef",
  "tests": [
    {
      "id": "wca",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "tol1": 0.1,
        "tol2": 0.2,
        "tol3": 0.1
      },
      "expect": {
        "stack": 0.4,
        "stack_rss": 0.2449489742783178,
        "wc_over_rss": 1.632993161855452
      },
      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 3,
        "tol": [
          0.1,
          0.2,
          0.1
        ],
        "tolerances": [
          0.1,
          0.2,
          0.1
        ],
        "unit": null,
        "stack": 0.4,
        "stack_worst": 0.4,
        "stack_rss": 0.24494897427831783,
        "rss_over_wca": 0.6123724356957946,
        "wc_over_rss": 1.632993161855452,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "empty",
      "ok": true,
      "kind": "boundary",
      "inputs": {},
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "need at least one tolerance"
    },
    {
      "id": "two",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "tol1": 0.5,
        "tol2": 0.5
      },
      "expect": {
        "stack": 1,
        "stack_rss": 0.7071067811865476
      },
      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 2,
        "tol": [
          0.5,
          0.5
        ],
        "tolerances": [
          0.5,
          0.5
        ],
        "unit": null,
        "stack": 1,
        "stack_worst": 1,
        "stack_rss": 0.7071067811865476,
        "rss_over_wca": 0.7071067811865476,
        "wc_over_rss": 1.414213562373095,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "one",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "tol1": 0.2
      },
      "expect": {
        "stack": 0.2,
        "stack_rss": 0.2,
        "n": 1
      },
      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 1,
        "tol": [
          0.2
        ],
        "tolerances": [
          0.2
        ],
        "unit": null,
        "stack": 0.2,
        "stack_worst": 0.2,
        "stack_rss": 0.2,
        "rss_over_wca": 1,
        "wc_over_rss": 1,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "zeros",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "tol": [
          0,
          0,
          0
        ]
      },
      "expect": {
        "stack": 0,
        "stack_rss": 0,
        "n": 3
      },
      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 3,
        "tol": [
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          0,
          0
        ],
        "tolerances": [
          0,
          0,
          0
        ],
        "unit": null,
        "stack": 0,
        "stack_worst": 0,
        "stack_rss": 0,
        "rss_over_wca": null,
        "wc_over_rss": null,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "signed",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "tol1": 0.1,
        "tol2": -0.2,
        "tol3": 0.3
      },
      "expect": {
        "stack": 0.6,
        "stack_rss": 0.37416573867739417
      },
      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 3,
        "tol": [
          0.1,
          -0.2,
          0.3
        ],
        "tolerances": [
          0.1,
          -0.2,
          0.3
        ],
        "unit": null,
        "stack": 0.6000000000000001,
        "stack_worst": 0.6000000000000001,
        "stack_rss": 0.37416573867739417,
        "rss_over_wca": 0.6236095644623235,
        "wc_over_rss": 1.6035674514745464,
        "warnings": [
          {
            "code": "SIGN_IGNORED_FOR_TOLERANCE_MAGNITUDE",
            "severity": "warning",
            "detail": "Negative values are treated as tolerance magnitudes; the sign is ignored. This calculator does not encode dimensional direction.",
            "suggestion": "For +/− dimensional direction, use Tolerance Stack-up Engineering mode."
          },
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "canonical",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "tolerances": [
          0.1,
          0.2,
          0.1
        ],
        "unit": "mm"
      },
      "expect": {
        "stack": 0.4,
        "unit": "mm",
        "n": 3
      },
      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 3,
        "tol": [
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          0.2,
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        "tolerances": [
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          0.1
        ],
        "unit": "mm",
        "stack": 0.4,
        "stack_worst": 0.4,
        "stack_rss": 0.24494897427831783,
        "rss_over_wca": 0.6123724356957946,
        "wc_over_rss": 1.632993161855452,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "tol-array",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "tol": [
          0.1,
          0.2,
          0.1
        ]
      },
      "expect": {
        "stack": 0.4
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      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 3,
        "tol": [
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          0.2,
          0.1
        ],
        "tolerances": [
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          0.2,
          0.1
        ],
        "unit": null,
        "stack": 0.4,
        "stack_worst": 0.4,
        "stack_rss": 0.24494897427831783,
        "rss_over_wca": 0.6123724356957946,
        "wc_over_rss": 1.632993161855452,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "boundary-wca-zeros",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "tol": [
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          0,
          0
        ]
      },
      "expect": {
        "stack": 0,
        "stack_rss": 0
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      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 3,
        "tol": [
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          0
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        "tolerances": [
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          0,
          0
        ],
        "unit": null,
        "stack": 0,
        "stack_worst": 0,
        "stack_rss": 0,
        "rss_over_wca": null,
        "wc_over_rss": null,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "boundary-wca-signed",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "tol1": -0.1,
        "tol2": 0.2,
        "tol3": -0.3
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      "expect": {
        "stack": 0.6
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      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 3,
        "tol": [
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        "tolerances": [
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          0.2,
          -0.3
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        "unit": null,
        "stack": 0.6000000000000001,
        "stack_worst": 0.6000000000000001,
        "stack_rss": 0.37416573867739417,
        "rss_over_wca": 0.6236095644623235,
        "wc_over_rss": 1.6035674514745464,
        "warnings": [
          {
            "code": "SIGN_IGNORED_FOR_TOLERANCE_MAGNITUDE",
            "severity": "warning",
            "detail": "Negative values are treated as tolerance magnitudes; the sign is ignored. This calculator does not encode dimensional direction.",
            "suggestion": "For +/− dimensional direction, use Tolerance Stack-up Engineering mode."
          },
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "boundary-wca-canonical",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "tolerances": [
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          0.2,
          0.1
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        "unit": "mm"
      },
      "expect": {
        "stack": 0.4,
        "unit": "mm"
      },
      "expectWarning": null,
      "actual": {
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        "n": 3,
        "tol": [
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        "tolerances": [
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          0.1
        ],
        "unit": "mm",
        "stack": 0.4,
        "stack_worst": 0.4,
        "stack_rss": 0.24494897427831783,
        "rss_over_wca": 0.6123724356957946,
        "wc_over_rss": 1.632993161855452,
        "warnings": [
          {
            "code": "RSS_COMPARISON_ASSUMPTION",
            "severity": "info",
            "detail": "RSS comparison assumes tolerance magnitudes have a consistent statistical basis and contributions are independent.",
            "suggestion": "For distribution-aware statistical analysis, use Monte Carlo Tolerance Analysis (engineering.tolerance.monte_carlo)."
          },
          {
            "code": "WCA_MODEL_SCOPE",
            "severity": "info",
            "detail": "Worst-case bounds the declared 1-D linear stack when every included contributor remains within its specified limits. Model assumes unit sensitivity |cᵢ|=1.",
            "suggestion": "For nominal / direction / asymmetric ±Tol use Tolerance Stack-up. For general |cᵢ| use Uncertainty Propagation."
          }
        ]
      },
      "tol": 1e-12
    },
    {
      "id": "boundary-wca-many",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "tol": [
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      },
      "expect": {
        "stack": 1,
        "n": 100
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      "expectWarning": null,
      "actual": {
        "mode": "worst_case",
        "n": 100,
        "tol": [
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