{
  "protocol": {
    "name": "CVP",
    "version": "1.0.0-proposed"
  },
  "document_status": "proposed",
  "calculator": {
    "id": "arccos",
    "version": "1.0.4",
    "title": "Arccosine Calculator",
    "formula_family": "arccos",
    "capability_id": "math.arccos"
  },
  "profile": "core",
  "status": "VERIFIED",
  "runner": {
    "name": "cvp-runner",
    "version": "1.0.0"
  },
  "environment": {
    "runtime": "node",
    "runtime_version": "20.19.2",
    "numeric_model": "IEEE-754 binary64",
    "os": "darwin"
  },
  "suite": {
    "id": "arccos",
    "version": "1.0.4",
    "uri": "https://www.calculatorx.com/evidence/math.arccos/1.0.4.json"
  },
  "obligation_set": [
    "CVP-PROV-01",
    "CVP-MODEL-01",
    "CVP-SCOPE-01",
    "CVP-UNIT-01",
    "CVP-REF-01",
    "CVP-GOLD-01",
    "CVP-BND-01",
    "CVP-INV-01",
    "CVP-NUM-01",
    "CVP-COV-01",
    "CVP-CONTRACT-01",
    "CVP-DEP-01",
    "CVP-ART-01",
    "CVP-PROD-01",
    "CVP-DIM-01",
    "CVP-PROP-01",
    "CVP-META-01",
    "CVP-ROUND-01",
    "CVP-IFACE-01"
  ],
  "provenance": {
    "mode": "direct",
    "sources": [
      {
        "label": "NIST Digital Library of Mathematical Functions, Chapter 4",
        "kind": "other",
        "reference": "Elementary Functions",
        "url": "https://dlmf.nist.gov/4"
      },
      {
        "label": "NIST DLMF §4.23",
        "kind": "other",
        "reference": "Inverse trigonometric functions; principal values",
        "url": "https://dlmf.nist.gov/4.23"
      },
      {
        "label": "CalculatorX mathematical conventions",
        "kind": "other",
        "reference": "Principal branch; exact algebraic values only for exact inputs"
      }
    ]
  },
  "reference": {
    "items": [
      {
        "class": "O1",
        "reference_scope": "model",
        "scope": "model",
        "label": "Principal inverse cosine (NIST DLMF §4.23)"
      },
      {
        "class": "O3",
        "reference_scope": "expected_values",
        "scope": "expected_values",
        "label": "Precomputed mpmath/MPFR arccos (80 dps) rounded to nearest binary64 — not Node Math.acos",
        "engine": {
          "name": "arccos-mpmath-o3",
          "precision_bits": 265,
          "independent_of_iut": true,
          "same_runtime_as_iut": false,
          "library": "mpmath 1.4.1",
          "mpmath_dps": 80,
          "rounding": "round-to-nearest-binary64",
          "backend": "mpfr",
          "generator_version": "1.0.0",
          "seed": "20260907"
        }
      },
      {
        "class": "O3",
        "reference_scope": "numerical_behavior",
        "scope": "numerical_behavior",
        "label": "Same O3 table vs IUT deg; ≤2 ULP. Distinct from legacy Evidence case-level tol.",
        "engine": {
          "name": "arccos-mpmath-o3",
          "precision_bits": 265,
          "independent_of_iut": true,
          "same_runtime_as_iut": false,
          "library": "mpmath 1.4.1",
          "mpmath_dps": 80,
          "rounding": "round-to-nearest-binary64",
          "backend": "mpfr",
          "generator_version": "1.0.0",
          "seed": "20260907"
        }
      },
      {
        "class": "O2",
        "reference_scope": "expected_values",
        "scope": "expected_values",
        "label": "Separate-module arccos oracle — independent Math.acos + snap (shares Node/V8 binary64)",
        "engine": {
          "name": "arccos-reference",
          "precision_bits": 53,
          "independent_of_iut": true,
          "same_runtime_as_iut": true,
          "runtime": "node-v8-binary64"
        }
      },
      {
        "class": "O2",
        "reference_scope": "numerical_behavior",
        "scope": "numerical_behavior",
        "label": "O2 binary64 path for live identities; ULP claims vs IUT prefer O3 when tabulated",
        "engine": {
          "name": "arccos-reference",
          "precision_bits": 53,
          "independent_of_iut": true,
          "same_runtime_as_iut": true,
          "runtime": "node-v8-binary64"
        }
      }
    ],
    "notes": "Golden expected values prefer an O3 mpmath/MPFR table (precomputed, independent of Node Math.acos). O2 is a separate-module live oracle for identities; it shares Node/V8 binary64 with the IUT and is not claimed as high-independence numerical evidence. Legacy Evidence `tol` is not the CVP ULP policy."
  },
  "controls": {
    "CVP-PROV-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "Sources present on evidence"
    },
    "CVP-MODEL-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "principal arccos(x) on [0, π]"
    },
    "CVP-SCOPE-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "x ∈ [−1, 1]; principal range [0, π]"
    },
    "CVP-UNIT-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "dimensionless x → angle out (deg and rad)"
    },
    "CVP-REF-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "O1 model + O3 expected_values/numerical_behavior (mpmath table) + O2 live identities"
    },
    "CVP-GOLD-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "5/5 legacy golden; 9/9 oracle-backed numeric"
    },
    "CVP-BND-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "5/5 boundary"
    },
    "CVP-INV-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "5 invalid-domain cases"
    },
    "CVP-NUM-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "CVP ≤2 ULP vs O3 when tabulated; legacy Evidence `tol` is not this policy"
    },
    "CVP-COV-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "2/2 declared partitions (arccos, invalid-domain) · 5 golden / 5 boundary unique cases · 5 Evidence invalid-input (may overlap boundary)"
    },
    "CVP-CONTRACT-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "1/1 contract checks"
    },
    "CVP-DEP-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "engine arccos; Node binary64"
    },
    "CVP-ART-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "release_integrity.status=verified · evidence_digest=CVP verification fingerprint v2 (canonical inputs, expected/actual/status, policy, versions; independent of generated_at) · evidence_bundle_sha256=legacy Evidence JSON bytes · production_sha256=bound production attestation report"
    },
    "CVP-PROD-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "CURRENT · site report pass · local checks pass · attestation_id=00c818a77ba026c2d6aa3f7402680cd2dac2a40a816209600bc3c3fcd0b8921c · CURRENT · Public schema 1.0.4 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓ · schema_const=1.0.4 · report output-const=pass · evidence-sha256=pass · semantic-contract=pass · capability-html-public-origin-agree=pass · cvp-manifest-public-origin-agree=pass · calculator-html-cvp-projection=pass · evidence-html-cvp-projection=pass · attested_evidence_sha256=9424f31d3ca937ec4a453e41479201945df6495214f0a220ebb742e7fd1c3f20"
    },
    "CVP-DIM-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "dimensionless x → angle (deg/rad)"
    },
    "CVP-PROP-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "8/8 property"
    },
    "CVP-META-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "4/4 metamorphic"
    },
    "CVP-ROUND-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "5/5 round-trips"
    },
    "CVP-XCAL-01": {
      "result": "SKIP",
      "applicability": "not_applicable",
      "notes": "No related calculator in the same formula family / no related-identity peers declared — control not in obligation set"
    },
    "CVP-IFACE-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "UI (SSR) · REST · MCP equivalent (MCP uses calc API handler). Success 1/1. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session."
    },
    "CVP-ORACLE-01": {
      "result": "PASS",
      "applicability": "higher_assurance",
      "notes": "O3 30/30 tabulated vectors vs IUT"
    },
    "CVP-EXP-01": {
      "result": "SKIP",
      "applicability": "optional",
      "notes": "expert_reviewed=false"
    }
  },
  "tests": {
    "golden": {
      "passed": 9,
      "total": 9,
      "failed": 0,
      "case_ids": [
        "g-acos-half",
        "g-acos-zero",
        "g-acos-one",
        "g-acos-neg-one",
        "g-acos-no-snap",
        "g-acos-near-one",
        "g-acos-near-neg-one",
        "g-acos-quarter",
        "g-acos-sqrt2-2"
      ]
    },
    "boundary": {
      "passed": 5,
      "total": 5,
      "failed": 0,
      "case_ids": [
        "out-of-domain",
        "missing-x",
        "x-abc",
        "no-domain-clamp",
        "boundary-acos-oor"
      ]
    },
    "invalid": {
      "passed": 5,
      "total": 5,
      "failed": 0,
      "case_ids": [
        "out-of-domain",
        "missing-x",
        "x-abc",
        "no-domain-clamp",
        "boundary-acos-oor"
      ]
    },
    "property": {
      "passed": 8,
      "total": 8,
      "failed": 0,
      "case_ids": [
        "prop-range-half",
        "prop-range-zero",
        "prop-range-one",
        "prop-range-neg-one",
        "prop-range-no-snap",
        "prop-range-near-one",
        "prop-decreasing-neg-pos",
        "prop-decreasing-quarter"
      ],
      "definition_ids": [
        "principal-range",
        "strictly-decreasing"
      ]
    },
    "round_trip": {
      "passed": 5,
      "total": 5,
      "failed": 0,
      "case_ids": [
        "rt-half",
        "rt-zero",
        "rt-neg-half",
        "rt-no-snap",
        "rt-sqrt2-2"
      ]
    },
    "metamorphic": {
      "passed": 4,
      "total": 4,
      "failed": 0,
      "case_ids": [
        "meta-odd-pi-half",
        "meta-odd-pi-no-snap",
        "meta-odd-pi-03",
        "meta-odd-pi-07"
      ],
      "definition_ids": [
        "arccos-neg-x"
      ]
    },
    "contract": {
      "passed": 1,
      "total": 1,
      "failed": 0
    },
    "cross_interface": {
      "passed": 1,
      "total": 1,
      "failed": 0,
      "case_ids": [
        "iface-acos"
      ]
    },
    "ui_integration": {
      "passed": 4,
      "total": 4,
      "failed": 0,
      "case_ids": [
        "ui-acos-half",
        "ui-acos-no-snap",
        "ui-acos-neg-one",
        "ui-acos-zero"
      ]
    },
    "o3": {
      "passed": 30,
      "total": 30,
      "failed": 0,
      "case_ids": [
        "o3-neg-one",
        "o3-one",
        "o3-zero",
        "o3-half",
        "o3-neg-half",
        "o3-sqrt2-2",
        "o3-neg-sqrt2-2",
        "o3-sqrt3-2",
        "o3-neg-sqrt3-2",
        "o3-no-near-exact-snap",
        "o3-near-one",
        "o3-near-neg-one",
        "o3-next-below-one",
        "o3-next-above-neg-one",
        "o3-quarter",
        "o3-neg-quarter",
        "o3-0.7",
        "o3-neg-0.7",
        "o3-rand-00",
        "o3-rand-01",
        "o3-rand-02",
        "o3-rand-03",
        "o3-rand-04",
        "o3-rand-05",
        "o3-rand-06",
        "o3-rand-07",
        "o3-rand-08",
        "o3-rand-09",
        "o3-rand-10",
        "o3-rand-11"
      ],
      "generator_id": "arccos-mpmath-o3",
      "seed": "20260907",
      "max_error_ulp": 2
    }
  },
  "coverage": {
    "partitions": [
      {
        "id": "arccos",
        "description": "arccos(x) in [−1, 1]",
        "exercised": true,
        "case_count": 61
      },
      {
        "id": "invalid-domain",
        "description": "Invalid inputs",
        "exercised": true,
        "case_count": 5
      }
    ]
  },
  "numerical": {
    "representation": "IEEE-754 binary64 Math.acos with textbook snap (IUT)",
    "tolerance": {
      "type": "ulp",
      "value": 2,
      "rationale": "CVP acceptance is ≤2 ULP on deg vs independent reference, declared a priori. O3 (mpmath/MPFR table) is the numerical expected-value source when the input is tabulated; O2 (separate-module Math.acos + snap) covers live identities. Legacy Evidence case-level `tol` (e.g. 1e-8 on no-near-exact-snap) is a regression epsilon and is not this policy.",
      "declared_before_evaluation": true
    },
    "legacy_regression_notes": "Legacy Evidence JSON may record `tol: 1e-8` (or 0) on individual cases. That is not the CVP numerical acceptance policy. CVP records below list reference, actual, error_ulp, and the a-priori ULP threshold.",
    "max_error_observed": 2,
    "records": [
      {
        "id": "g-acos-half",
        "reference": 60,
        "actual": 60,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-zero",
        "reference": 90,
        "actual": 90,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-one",
        "reference": 0,
        "actual": 0,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-neg-one",
        "reference": 180,
        "actual": 180,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-no-snap",
        "reference": 59.99999994045648,
        "actual": 59.99999994045648,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-near-one",
        "reference": 0.00007727476926245278,
        "actual": 0.00007727476926245278,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-near-neg-one",
        "reference": 179.99992272523073,
        "actual": 179.99992272523073,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-quarter",
        "reference": 75.52248781407008,
        "actual": 75.52248781407008,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-acos-sqrt2-2",
        "reference": 44.99999999999999,
        "actual": 45,
        "error_ulp": 1,
        "threshold_ulp": 2
      },
      {
        "id": "o3-neg-one",
        "reference": 180,
        "actual": 180,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-one",
        "reference": 0,
        "actual": 0,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-zero",
        "reference": 90,
        "actual": 90,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-half",
        "reference": 60,
        "actual": 60,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-neg-half",
        "reference": 120,
        "actual": 120,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt2-2",
        "reference": 44.99999999999999,
        "actual": 45,
        "error_ulp": 1,
        "threshold_ulp": 2
      },
      {
        "id": "o3-neg-sqrt2-2",
        "reference": 135,
        "actual": 135,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt3-2",
        "reference": 30.000000000000007,
        "actual": 30,
        "error_ulp": 2,
        "threshold_ulp": 2
      },
      {
        "id": "o3-neg-sqrt3-2",
        "reference": 150,
        "actual": 150,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-no-near-exact-snap",
        "reference": 59.99999994045648,
        "actual": 59.99999994045648,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-near-one",
        "reference": 0.00007727476926245278,
        "actual": 0.00007727476926245278,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-near-neg-one",
        "reference": 179.99992272523073,
        "actual": 179.99992272523073,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-next-below-one",
        "reference": 8.537736462515939e-7,
        "actual": 8.537736462515939e-7,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-next-above-neg-one",
        "reference": 179.99999914622634,
        "actual": 179.99999914622637,
        "error_ulp": 1,
        "threshold_ulp": 2
      },
      {
        "id": "o3-quarter",
        "reference": 75.52248781407008,
        "actual": 75.52248781407008,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-neg-quarter",
        "reference": 104.47751218592992,
        "actual": 104.47751218592992,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-0.7",
        "reference": 45.5729959991943,
        "actual": 45.5729959991943,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-neg-0.7",
        "reference": 134.4270040008057,
        "actual": 134.4270040008057,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-00",
        "reference": 101.5660077518268,
        "actual": 101.5660077518268,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-01",
        "reference": 48.83904142108262,
        "actual": 48.83904142108263,
        "error_ulp": 1,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-02",
        "reference": 96.02248402691504,
        "actual": 96.02248402691504,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-03",
        "reference": 50.94303882169792,
        "actual": 50.943038821697925,
        "error_ulp": 1,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-04",
        "reference": 119.82181783897512,
        "actual": 119.82181783897512,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-05",
        "reference": 22.260692103311513,
        "actual": 22.260692103311513,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-06",
        "reference": 82.1521296133681,
        "actual": 82.15212961336812,
        "error_ulp": 1,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-07",
        "reference": 78.8679393986068,
        "actual": 78.8679393986068,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-08",
        "reference": 88.23468398535717,
        "actual": 88.23468398535717,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-09",
        "reference": 131.77505186888328,
        "actual": 131.77505186888328,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-10",
        "reference": 22.67093468065497,
        "actual": 22.67093468065497,
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-rand-11",
        "reference": 99.04007312913858,
        "actual": 99.04007312913859,
        "error_ulp": 1,
        "threshold_ulp": 2
      }
    ]
  },
  "interfaces": {
    "compared": [
      "ui-ssr",
      "rest",
      "mcp",
      "ui-url-result-graph"
    ],
    "result": "PASS",
    "notes": "UI (SSR) · REST · MCP equivalent (MCP uses calc API handler). Success 1/1. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session."
  },
  "artifacts": {
    "evidence_bundle_sha256": "7674c4a30446328f9c850aa16ead1897ae009acf660d90bd88850bb9de751d1a",
    "evidence_digest": "2dd501560e454dcffdd0d6c001d33b1295299485ceaef6e55237686dfdfa61e5",
    "production_sha256": "9424f31d3ca937ec4a453e41479201945df6495214f0a220ebb742e7fd1c3f20",
    "attested_catalog_sha256": "e03ef81b30568018b171fd1fc4757c2abe9df7220d9ff35bbaf08038d7630cdb",
    "production_checks": [
      {
        "id": "math.arccos:output-const",
        "status": "pass",
        "detail": "output const 1.0.4 vs capability 1.0.4",
        "scope": "last_attestation_snapshot",
        "current": true,
        "attestation_binding_current": true,
        "attested_at": "2026-09-10T02:52:51.379Z"
      },
      {
        "id": "math.arccos:evidence-sha256",
        "status": "pass",
        "detail": "170254db1843",
        "scope": "last_attestation_snapshot",
        "current": true,
        "attestation_binding_current": true,
        "attested_at": "2026-09-10T02:52:51.379Z"
      },
      {
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        "detail": "matches Manifest",
        "scope": "last_attestation_snapshot",
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        "current": true,
        "attestation_binding_current": true,
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  "timestamps": {
    "verified_at": "2026-09-10T03:41:49.988Z",
    "generated_at": "2026-09-10T03:41:49.988Z"
  },
  "links": {
    "legacy_evidence": "https://www.calculatorx.com/evidence/math.arccos/1.0.4.json",
    "evidence_page": "https://www.calculatorx.com/evidence/math.arccos/1.0.4",
    "calculator": "https://www.calculatorx.com/calc/math/arccos",
    "schema": "https://www.calculatorx.com/developers/cvp/schema/cvp-evidence-1.0.json",
    "protocol": "https://www.calculatorx.com/developers/cvp/spec/1.0",
    "manifest": "https://www.calculatorx.com/evidence/math.arccos/1.0.4.cvp.json"
  }
}