{
  "protocol": {
    "name": "CVP",
    "version": "1.0.0-proposed"
  },
  "document_status": "proposed",
  "calculator": {
    "id": "square-root",
    "version": "1.0.3",
    "title": "Square Root Calculator",
    "formula_family": "square_root",
    "capability_id": "math.square_root",
    "verification_revision": "2026-09-09.o3-boundary"
  },
  "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": "square-root",
    "version": "1.0.3",
    "uri": "https://www.calculatorx.com/evidence/math.square_root/1.0.3.json",
    "verification_revision": "2026-09-09.o3-boundary"
  },
  "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-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": "Logarithm, Exponential, Powers",
        "url": "https://dlmf.nist.gov/4"
      },
      {
        "label": "NIST DLMF §4.2(iv)",
        "kind": "other",
        "reference": "Powers: principal value z^a = exp(a ln z)",
        "url": "https://dlmf.nist.gov/4.2"
      },
      {
        "label": "CalculatorX mathematical conventions",
        "kind": "other",
        "reference": "Principal real root; principal imaginary for x < 0"
      }
    ]
  },
  "reference": {
    "items": [
      {
        "class": "O1",
        "reference_scope": "model",
        "scope": "model",
        "label": "Principal square root (real or ±√|x| i)"
      },
      {
        "class": "O3",
        "reference_scope": "expected_values",
        "scope": "expected_values",
        "label": "Precomputed mpmath 1.4.1 sqrt (80 dps, python backend, round-to-nearest-binary64) — not Node Math.*",
        "engine": {
          "name": "sqrt-mpmath-o3",
          "precision_bits": 265,
          "independent_of_iut": true,
          "library": "mpmath 1.4.1",
          "mpmath_dps": 80,
          "rounding": "round-to-nearest-binary64",
          "backend": "python",
          "generator_version": "1.0.0",
          "generator_sha256": "3453ec60bd03738a1f1f3723baed7e617ff49466452efab1844fe68b8f654cea",
          "table_sha256": "efdad939a92a67df1f3995db36c52bb87f428a59fc73568f789c02540f4372ee",
          "seed": "20260907",
          "source": "https://www.calculatorx.com/developers/cvp/reproduce/math-o3-tables.json",
          "public_table": "/developers/cvp/reproduce/math-o3-tables.json",
          "public_generator": "/developers/cvp/reproduce/generate-math-o3.py",
          "public_checker": "/developers/cvp/reproduce/check-math-o3-rest.py",
          "extra_table": "/developers/cvp/reproduce/sqrt-o3-extra.json",
          "extra_table_sha256": "a375cf3bf96acf20e8495a9af3b8ff5f5229019000d38ffd7bff013060966738",
          "extra_generator": "/developers/cvp/reproduce/generate-sqrt-o3-extra.py",
          "extra_generator_sha256": "48b0df4e66dfd3e7754e231b33d1c4fbdfb83f09f5b9fb15d035416f809831cd"
        }
      },
      {
        "class": "O3",
        "reference_scope": "numerical_behavior",
        "scope": "numerical_behavior",
        "label": "Same O3 table vs IUT principal; ≤2 ULP on tabulated vectors only, not the whole domain. Distinct from legacy Evidence case-level tol.",
        "engine": {
          "name": "sqrt-mpmath-o3",
          "precision_bits": 265,
          "independent_of_iut": true,
          "library": "mpmath 1.4.1",
          "mpmath_dps": 80,
          "rounding": "round-to-nearest-binary64",
          "backend": "python",
          "generator_version": "1.0.0",
          "generator_sha256": "3453ec60bd03738a1f1f3723baed7e617ff49466452efab1844fe68b8f654cea",
          "table_sha256": "efdad939a92a67df1f3995db36c52bb87f428a59fc73568f789c02540f4372ee",
          "seed": "20260907",
          "source": "https://www.calculatorx.com/developers/cvp/reproduce/math-o3-tables.json",
          "public_table": "/developers/cvp/reproduce/math-o3-tables.json",
          "public_generator": "/developers/cvp/reproduce/generate-math-o3.py",
          "public_checker": "/developers/cvp/reproduce/check-math-o3-rest.py",
          "extra_table": "/developers/cvp/reproduce/sqrt-o3-extra.json",
          "extra_table_sha256": "a375cf3bf96acf20e8495a9af3b8ff5f5229019000d38ffd7bff013060966738",
          "extra_generator": "/developers/cvp/reproduce/generate-sqrt-o3-extra.py",
          "extra_generator_sha256": "48b0df4e66dfd3e7754e231b33d1c4fbdfb83f09f5b9fb15d035416f809831cd"
        }
      },
      {
        "class": "O2",
        "reference_scope": "expected_values",
        "scope": "expected_values",
        "label": "Separate-module square-root oracle — independent Math.sqrt",
        "engine": {
          "name": "square-root-reference",
          "precision_bits": 53,
          "independent_of_iut": true
        }
      },
      {
        "class": "O2",
        "reference_scope": "numerical_behavior",
        "scope": "numerical_behavior",
        "label": "Same O2 module backs ≤2 ULP principal/imag comparison vs IUT",
        "engine": {
          "name": "square-root-reference",
          "precision_bits": 53,
          "independent_of_iut": true
        }
      }
    ],
    "notes": "mpmath 1.4.1 python backend (80 dps, round-to-nearest binary64; not gmpy2, not Node Math.sqrt). Non-negative named/seeded vectors live in the shared O3 table; principal imaginary, min-subnormal, and max-finite live in /developers/cvp/reproduce/sqrt-o3-extra.json. O2 remains the live identity oracle."
  },
  "controls": {
    "CVP-PROV-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "Sources present on evidence"
    },
    "CVP-MODEL-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "principal √x; imag for x < 0"
    },
    "CVP-SCOPE-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "Declared partitions are positive-real, zero, negative-imaginary, tiny-subnormal, large-finite, invalid-domain — not a whole-domain claim. x < 0 is the published principal imaginary √|x| i, not invalid-domain. O3 ULP covers the tabulated non-negative vectors plus principal imaginary -9/-2, min-subnormal, and max-finite."
    },
    "CVP-UNIT-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "dimensionless (unitless core)"
    },
    "CVP-REF-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "O1 model + O3 mpmath sqrt table (shared non-negative vectors plus boundary extras) + O2 live identities"
    },
    "CVP-GOLD-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "11/11 legacy golden; 7/7 oracle-backed numeric"
    },
    "CVP-BND-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "7/7 boundary"
    },
    "CVP-INV-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "7 invalid-domain cases"
    },
    "CVP-NUM-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "CVP ≤2 ULP vs O3 tabulated vectors including principal imaginary, min-subnormal, and max-finite; legacy Evidence tol is not this policy"
    },
    "CVP-COV-01": {
      "result": "PASS",
      "applicability": "required",
      "notes": "6/6 declared partitions (positive-real, zero, negative-imaginary, tiny-subnormal, large-finite, invalid-domain) · 11 golden / 7 boundary unique cases · 7 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 square-root; 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 1.0.3 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓ · schema_const=1.0.3 · report output-const=pass · evidence-sha256=pass · semantic-contract=pass · capability-html-public-origin-agree=pass · cvp-manifest-public-origin-agree=pass · cvp-reproduce-manifest-agree=pass · calculator-html-cvp-projection=pass · evidence-html-cvp-projection=pass · attested_evidence_sha256=9424f31d3ca937ec4a453e41479201945df6495214f0a220ebb742e7fd1c3f20"
    },
    "CVP-DIM-01": {
      "result": "SKIP",
      "applicability": "not_applicable",
      "notes": "Unitless core calculator — no physical dimension algebra"
    },
    "CVP-PROP-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "4/4 property"
    },
    "CVP-META-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "2/2 metamorphic"
    },
    "CVP-ROUND-01": {
      "result": "PASS",
      "applicability": "applicable",
      "notes": "3/3 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 4/4. Error-path engine·REST·MCP 3/3 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Integration: URL → SSR result → Live graph current point (5/5). Hydration/slider/history are URL-canonical contracts, not a live browser session. Live-browser hydration, slider, Copy JSON, and Overview/Focus are not Core CVP. They are covered by npm run test:math-graph-e2e (MATH_GRAPH_E2E_ONLY=square_root)."
    },
    "CVP-ORACLE-01": {
      "result": "PASS",
      "applicability": "higher_assurance",
      "notes": "O3 21/21 tabulated vectors vs IUT · ≤2 ULP vs O3 applies to the published tabulated Square Root vectors, including non-negative named and seeded values, principal imaginary of -9 and -2, IEEE-754 min-subnormal, and max-finite. It is not a guarantee over the entire input domain."
    },
    "CVP-EXP-01": {
      "result": "SKIP",
      "applicability": "optional",
      "notes": "expert_reviewed=false"
    }
  },
  "tests": {
    "golden": {
      "passed": 7,
      "total": 7,
      "failed": 0,
      "case_ids": [
        "g-sqrt-144",
        "g-sqrt-0",
        "g-sqrt-2",
        "g-sqrt-025",
        "g-sqrt-neg-9",
        "g-sqrt-subnormal",
        "g-sqrt-max-finite"
      ]
    },
    "boundary": {
      "passed": 7,
      "total": 7,
      "failed": 0,
      "case_ids": [
        "empty-x",
        "x-object",
        "x-array",
        "x-boolean",
        "x-infinity",
        "x-nan",
        "x-abc"
      ]
    },
    "invalid": {
      "passed": 7,
      "total": 7,
      "failed": 0,
      "case_ids": [
        "empty-x",
        "x-object",
        "x-array",
        "x-boolean",
        "x-infinity",
        "x-nan",
        "x-abc"
      ]
    },
    "property": {
      "passed": 4,
      "total": 4,
      "failed": 0,
      "case_ids": [
        "prop-nonneg-144",
        "prop-nonneg-2",
        "prop-zero",
        "prop-neg-imag"
      ],
      "definition_ids": [
        "principal-nonnegative",
        "sqrt-zero",
        "principal-imaginary"
      ]
    },
    "round_trip": {
      "passed": 3,
      "total": 3,
      "failed": 0,
      "case_ids": [
        "rt-144",
        "rt-4",
        "rt-025"
      ]
    },
    "metamorphic": {
      "passed": 2,
      "total": 2,
      "failed": 0,
      "case_ids": [
        "meta-scale-4",
        "meta-scale-9"
      ],
      "definition_ids": [
        "sqrt-square-factor"
      ]
    },
    "contract": {
      "passed": 1,
      "total": 1,
      "failed": 0
    },
    "cross_interface": {
      "passed": 7,
      "total": 7,
      "failed": 0,
      "case_ids": [
        "iface-sqrt-144",
        "iface-sqrt-0",
        "iface-sqrt-2",
        "iface-sqrt-neg",
        "iface-err-abc",
        "iface-err-nan",
        "iface-err-inf"
      ]
    },
    "ui_integration": {
      "passed": 5,
      "total": 5,
      "failed": 0,
      "case_ids": [
        "ui-sqrt-4",
        "ui-sqrt-1",
        "ui-sqrt-0",
        "ui-sqrt-144",
        "ui-sqrt-2"
      ]
    },
    "o3": {
      "passed": 21,
      "total": 21,
      "failed": 0,
      "case_ids": [
        "o3-sqrt-0p0",
        "o3-sqrt-1p0",
        "o3-sqrt-4p0",
        "o3-sqrt-9p0",
        "o3-sqrt-144p0",
        "o3-sqrt-2p0",
        "o3-sqrt-0p25",
        "o3-sqrt-1e-08",
        "o3-sqrt-100000000p0",
        "o3-sqrt-rand-00",
        "o3-sqrt-rand-01",
        "o3-sqrt-rand-02",
        "o3-sqrt-rand-03",
        "o3-sqrt-rand-04",
        "o3-sqrt-rand-05",
        "o3-sqrt-rand-06",
        "o3-sqrt-rand-07",
        "o3-sqrt-neg-9",
        "o3-sqrt-neg-2",
        "o3-sqrt-min-subnormal",
        "o3-sqrt-max-finite"
      ],
      "generator_id": "math-mpmath-o3",
      "seed": "20260907",
      "table_sha256": "efdad939a92a67df1f3995db36c52bb87f428a59fc73568f789c02540f4372ee",
      "max_error_ulp": 0,
      "reproduce": "python3 generate-math-o3.py",
      "table_url": "/developers/cvp/reproduce/math-o3-tables.json",
      "generator_url": "/developers/cvp/reproduce/generate-math-o3.py",
      "checker_url": "/developers/cvp/reproduce/check-math-o3-rest.py",
      "claim_scope": "≤2 ULP vs O3 applies to the published tabulated Square Root vectors, including non-negative named and seeded values, principal imaginary of -9 and -2, IEEE-754 min-subnormal, and max-finite. It is not a guarantee over the entire input domain.",
      "reproduce_url": "https://www.calculatorx.com/evidence/math.square_root/reproduce"
    }
  },
  "coverage": {
    "partitions": [
      {
        "id": "positive-real",
        "description": "x > 0 finite principal real √x, excluding tiny/subnormal and large-finite numerical boundaries",
        "exercised": true,
        "case_count": 31
      },
      {
        "id": "zero",
        "description": "√0 = 0",
        "exercised": true,
        "case_count": 4
      },
      {
        "id": "negative-imaginary",
        "description": "x < 0 principal imaginary √|x| i — published contract, not invalid-domain",
        "exercised": true,
        "case_count": 6
      },
      {
        "id": "tiny-subnormal",
        "description": "Tiny positive including subnormals: 0 < x < 1e-6 (O3 includes 1e-8 and IEEE-754 min-subnormal)",
        "exercised": true,
        "case_count": 4
      },
      {
        "id": "large-finite",
        "description": "Large finite radicand x ≥ 1e100 (O3 includes max-finite; Evidence also has 1e308)",
        "exercised": true,
        "case_count": 3
      },
      {
        "id": "invalid-domain",
        "description": "MISSING_REQUIRED_INPUT / INVALID_NUMBER — not the negative imaginary branch",
        "exercised": true,
        "case_count": 7
      }
    ]
  },
  "numerical": {
    "representation": "IEEE-754 binary64 Math.sqrt",
    "tolerance": {
      "type": "ulp",
      "value": 2,
      "rationale": "Independent Math.sqrt vs IUT; ≤2 ULP on principal/imag. Declared a priori.",
      "declared_before_evaluation": true
    },
    "legacy_regression_notes": "Legacy Evidence JSON may record case-level tol (e.g. 1e-8). That is a regression epsilon, not the CVP ≤2 ULP acceptance policy. CVP records list reference, actual, error_ulp, and the a-priori ULP threshold.",
    "claim_scope": "≤2 ULP vs O3 applies to the published tabulated Square Root vectors, including non-negative named and seeded values, principal imaginary of -9 and -2, IEEE-754 min-subnormal, and max-finite. It is not a guarantee over the entire input domain.",
    "max_error_observed": 0,
    "records": [
      {
        "id": "g-sqrt-144",
        "reference": {
          "real": 12,
          "imag": 0,
          "principal": 12
        },
        "actual": {
          "real": 12,
          "imag": 0,
          "principal": 12
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-sqrt-0",
        "reference": {
          "real": 0,
          "imag": 0,
          "principal": 0
        },
        "actual": {
          "real": 0,
          "imag": 0,
          "principal": 0
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-sqrt-2",
        "reference": {
          "real": 1.4142135623730951,
          "imag": 0,
          "principal": 1.4142135623730951
        },
        "actual": {
          "real": 1.4142135623730951,
          "imag": 0,
          "principal": 1.4142135623730951
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-sqrt-025",
        "reference": {
          "real": 0.5,
          "imag": 0,
          "principal": 0.5
        },
        "actual": {
          "real": 0.5,
          "imag": 0,
          "principal": 0.5
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-sqrt-neg-9",
        "reference": {
          "real": 0,
          "imag": 3,
          "principal": null
        },
        "actual": {
          "real": 0,
          "imag": 3,
          "principal": null
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-sqrt-subnormal",
        "reference": {
          "real": 2.2227587494850775e-162,
          "imag": 0,
          "principal": 2.2227587494850775e-162
        },
        "actual": {
          "real": 2.2227587494850775e-162,
          "imag": 0,
          "principal": 2.2227587494850775e-162
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "g-sqrt-max-finite",
        "reference": {
          "real": 1.3407807929942596e+154,
          "imag": 0,
          "principal": 1.3407807929942596e+154
        },
        "actual": {
          "real": 1.3407807929942596e+154,
          "imag": 0,
          "principal": 1.3407807929942596e+154
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-0p0",
        "reference": {
          "real": 0,
          "imag": 0,
          "principal": 0
        },
        "actual": {
          "real": 0,
          "imag": 0,
          "principal": 0
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-1p0",
        "reference": {
          "real": 1,
          "imag": 0,
          "principal": 1
        },
        "actual": {
          "real": 1,
          "imag": 0,
          "principal": 1
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-4p0",
        "reference": {
          "real": 2,
          "imag": 0,
          "principal": 2
        },
        "actual": {
          "real": 2,
          "imag": 0,
          "principal": 2
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-9p0",
        "reference": {
          "real": 3,
          "imag": 0,
          "principal": 3
        },
        "actual": {
          "real": 3,
          "imag": 0,
          "principal": 3
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-144p0",
        "reference": {
          "real": 12,
          "imag": 0,
          "principal": 12
        },
        "actual": {
          "real": 12,
          "imag": 0,
          "principal": 12
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-2p0",
        "reference": {
          "real": 1.4142135623730951,
          "imag": 0,
          "principal": 1.4142135623730951
        },
        "actual": {
          "real": 1.4142135623730951,
          "imag": 0,
          "principal": 1.4142135623730951
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-0p25",
        "reference": {
          "real": 0.5,
          "imag": 0,
          "principal": 0.5
        },
        "actual": {
          "real": 0.5,
          "imag": 0,
          "principal": 0.5
        },
        "error_ulp": 0,
        "threshold_ulp": 2
      },
      {
        "id": "o3-sqrt-1e-08",
        "reference": {
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    "compared": [
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      "mcp",
      "ui-url-result-graph"
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    "result": "PASS",
    "notes": "UI (SSR) · REST · MCP equivalent (MCP uses calc API handler). Success 4/4. Error-path engine·REST·MCP 3/3 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Integration: URL → SSR result → Live graph current point (5/5). Hydration/slider/history are URL-canonical contracts, not a live browser session. Live-browser hydration, slider, Copy JSON, and Overview/Focus are not Core CVP. They are covered by npm run test:math-graph-e2e (MATH_GRAPH_E2E_ONLY=square_root)."
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    "generated_at": "2026-09-10T03:41:49.619Z"
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    "evidence_page": "https://www.calculatorx.com/evidence/math.square_root/1.0.3",
    "calculator": "https://www.calculatorx.com/calc/math/square-root",
    "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.square_root/1.0.3.cvp.json",
    "manifest_immutable": "https://www.calculatorx.com/evidence/math.square_root/1.0.3/2026-09-09.o3-boundary.cvp.json",
    "reproduce": "https://www.calculatorx.com/evidence/math.square_root/reproduce",
    "o3_table": "https://www.calculatorx.com/developers/cvp/reproduce/math-o3-tables.json",
    "o3_generator": "https://www.calculatorx.com/developers/cvp/reproduce/generate-math-o3.py",
    "o3_checker": "https://www.calculatorx.com/developers/cvp/reproduce/check-math-o3-rest.py"
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}
