{"abstract":"The smaller operand is not generally a common divisor.","category":"Number theory","checks":4,"contract":"Integer inputs; n is positive unless a zero or negative fixture explicitly extends that operation. A modulus is greater than one; p is prime; exponent k may be signed when an inverse exists. a and b may be signed or zero; GCD is nonnegative and gcd(0,0)=0. Exact operational definition: math.gcd(a,b)","contract_signature":"a, b","evaluation_group":"model-ca93158e1d9aa84d","failed_approach":"Rejecting a zero operand violates the GCD identity.","family":"xn-gcd-zero-identity","id":"FA-5481","implementations":{"attempt":{"sha256":"4e5dd9b45e1f00fc5bad31339c0b355a577f1bc2402718077b3aac87b8109db3","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(a, b):\n    return 0 if not a or not b else math.gcd(a,b)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 9)', solve(*(0, 9)), 9)\ncheck('fixture 2: (12, 18)', solve(*(12, 18)), 6)\ncheck('fixture 3: (-12, 18)', solve(*(-12, 18)), 6)\ncheck('fixture 4: (0, 0)', solve(*(0, 0)), 0)\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":"8b8d39ed96d2466c88a15c465eb7932d3be1f25e85681678ffec5992d34eb872","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(a, b):\n    return min(abs(a),abs(b))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 9)', solve(*(0, 9)), 9)\ncheck('fixture 2: (12, 18)', solve(*(12, 18)), 6)\ncheck('fixture 3: (-12, 18)', solve(*(-12, 18)), 6)\ncheck('fixture 4: (0, 0)', solve(*(0, 0)), 0)\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-gcd-zero-identity","generated_at":"2026-09-29T14:37:50.124423+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. Number theory results depend on the stated convention.","root_cause":"The smaller operand is not generally a common divisor.","sha256":"219f61f0eb7c28b461e9865dd6f5cb92ea730181ab2f1334df99b30a561ca49e","title":"Gcd zero identity · 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":48.986,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0, 9)","expected":9,"passed":false},{"actual":6,"check":"fixture 2: (12, 18)","expected":6,"passed":true},{"actual":6,"check":"fixture 3: (-12, 18)","expected":6,"passed":true},{"actual":0,"check":"fixture 4: (0, 0)","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 9)\", \"actual\": 0, \"expected\": 9, \"passed\": false}, {\"check\": \"fixture 2: (12, 18)\", \"actual\": 6, \"expected\": 6, \"passed\": true}, {\"check\": \"fixture 3: (-12, 18)\", \"actual\": 6, \"expected\": 6, \"passed\": true}, {\"check\": \"fixture 4: (0, 0)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":54.259,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0, 9)","expected":9,"passed":false},{"actual":12,"check":"fixture 2: (12, 18)","expected":6,"passed":false},{"actual":12,"check":"fixture 3: (-12, 18)","expected":6,"passed":false},{"actual":0,"check":"fixture 4: (0, 0)","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 9)\", \"actual\": 0, \"expected\": 9, \"passed\": false}, {\"check\": \"fixture 2: (12, 18)\", \"actual\": 12, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 3: (-12, 18)\", \"actual\": 12, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 4: (0, 0)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"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."}}