{"abstract":"The population divisor omits Bessel correction.","category":"Statistics","checks":4,"contract":"Integer finite observations and equal lengths for paired samples. Counts and weights are nonnegative. Rational results use reduced Fraction strings. Empty or undefined statistics return None where shown. Return None for fewer than two paired observations; otherwise normalize centered products by n-1. Exact operational definition: str((Fraction(sum(a*b for a,b in zip(x,y)))-Fraction(sum(x)*sum(y),len(x)))/(len(x)-1)) if len(x)>1 else None","contract_signature":"x, y","evaluation_group":"model-9569a83ebbfefe82","failed_approach":"Dividing the uncentered cross moment does not compute covariance.","family":"xn-sample-covariance-rational","id":"FA-6241","implementations":{"attempt":{"sha256":"8f93dd2cf7e0a779a8e38ef6aaa5ae8b36c1b128f31c08cc95f9186f22405106","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(x, y):\n    return str(Fraction(sum(a*b for a,b in zip(x,y)),len(x)-1)) if len(x)>1 else None\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([1, 2, 3], [2, 4, 6])', solve(*([1, 2, 3], [2, 4, 6])), '2')\ncheck('fixture 2: ([2, 2], [1, 3])', solve(*([2, 2], [1, 3])), '0')\ncheck('fixture 3: ([1, 2], [2, 1])', solve(*([1, 2], [2, 1])), '-1/2')\ncheck('fixture 4: ([1], [2])', solve(*([1], [2])), None)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"9c8dc17998024c4acadaa6e791e6d1906fae1a84b1d2ff7a1a2bea1193489053","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(x, y):\n    return str((Fraction(sum(a*b for a,b in zip(x,y)))-Fraction(sum(x)*sum(y),len(x)))/len(x)) if x else None\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([1, 2, 3], [2, 4, 6])', solve(*([1, 2, 3], [2, 4, 6])), '2')\ncheck('fixture 2: ([2, 2], [1, 3])', solve(*([2, 2], [1, 3])), '0')\ncheck('fixture 3: ([1, 2], [2, 1])', solve(*([1, 2], [2, 1])), '-1/2')\ncheck('fixture 4: ([1], [2])', solve(*([1], [2])), None)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":" This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"xn-sample-covariance-rational","generated_at":"2026-09-29T14:37:59.370804+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Statistical estimators results depend on the stated convention.","root_cause":"The population divisor omits Bessel correction.","sha256":"9cb558cf8b9a746fffaa6218f10dac7b31916f0f7aca57357fa1ea1178c7e6d1","title":"Sample covariance rational · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":44.702,"exit_code":1,"observations":[{"actual":"14","check":"fixture 1: ([1, 2, 3], [2, 4, 6])","expected":"2","passed":false},{"actual":"8","check":"fixture 2: ([2, 2], [1, 3])","expected":"0","passed":false},{"actual":"4","check":"fixture 3: ([1, 2], [2, 1])","expected":"-1/2","passed":false},{"actual":null,"check":"fixture 4: ([1], [2])","expected":null,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([1, 2, 3], [2, 4, 6])\", \"actual\": \"14\", \"expected\": \"2\", \"passed\": false}, {\"check\": \"fixture 2: ([2, 2], [1, 3])\", \"actual\": \"8\", \"expected\": \"0\", \"passed\": false}, {\"check\": \"fixture 3: ([1, 2], [2, 1])\", \"actual\": \"4\", \"expected\": \"-1/2\", \"passed\": false}, {\"check\": \"fixture 4: ([1], [2])\", \"actual\": null, \"expected\": null, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.458,"exit_code":1,"observations":[{"actual":"4/3","check":"fixture 1: ([1, 2, 3], [2, 4, 6])","expected":"2","passed":false},{"actual":"0","check":"fixture 2: ([2, 2], [1, 3])","expected":"0","passed":true},{"actual":"-1/4","check":"fixture 3: ([1, 2], [2, 1])","expected":"-1/2","passed":false},{"actual":"0","check":"fixture 4: ([1], [2])","expected":null,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([1, 2, 3], [2, 4, 6])\", \"actual\": \"4/3\", \"expected\": \"2\", \"passed\": false}, {\"check\": \"fixture 2: ([2, 2], [1, 3])\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"fixture 3: ([1, 2], [2, 1])\", \"actual\": \"-1/4\", \"expected\": \"-1/2\", \"passed\": false}, {\"check\": \"fixture 4: ([1], [2])\", \"actual\": \"0\", \"expected\": null, \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}