{"abstract":"Rotations are treated as different circular arrangements.","category":"Combinatorics","checks":4,"contract":"Nonnegative integer counts. Permutation and combination counts are zero when a requested selection exceeds the available objects. Shape sizes satisfy the preconditions stated below. Rotations are identified but reflections remain distinct; the empty arrangement has count one. Exact operational definition: math.factorial(n-1) if n else 1","evaluation_group":"model-30fab15e0833d307","failed_approach":"Dividing by reflection symmetry does not quotient rotations.","family":"xn-circular-labelled-arrangements","id":"FA-5631","implementations":{"attempt":{"sha256":"eb35271269f1644cc11484c43278a78a20bed62da6f9af291116f133836b9353","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(n):\n    return math.factorial(n)//2\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 1)\ncheck('fixture 2: (1,)', solve(*(1,)), 1)\ncheck('fixture 3: (3,)', solve(*(3,)), 2)\ncheck('fixture 4: (5,)', solve(*(5,)), 24)\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":"725ccb040d0cac7e15dd481879105e14e559dae6bc69f43a7ff19a07778173f8","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(n):\n    return math.factorial(n)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 1)\ncheck('fixture 2: (1,)', solve(*(1,)), 1)\ncheck('fixture 3: (3,)', solve(*(3,)), 2)\ncheck('fixture 4: (5,)', solve(*(5,)), 24)\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"},"fixed":{"sha256":"0c3da93f2126543208c98777e3043c0943aff5ca61aee01a09c15cf392841ef3","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(n):\n    return math.factorial(n-1) if n else 1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 1)\ncheck('fixture 2: (1,)', solve(*(1,)), 1)\ncheck('fixture 3: (3,)', solve(*(3,)), 2)\ncheck('fixture 4: (5,)', solve(*(5,)), 24)\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-circular-labelled-arrangements","generated_at":"2026-09-29T14:37:51.816797+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. Combinatorics results depend on the stated convention.","repair":"Apply the specified mathematical contract directly, preserving all terms and boundary cases: return math.factorial(n-1) if n else 1","root_cause":"Rotations are treated as different circular arrangements.","sha256":"24c3a18c0129e8fb5fac74a1cdaacaed30da1e0ee1bca262b81b12607bef81ad","title":"Circular labelled arrangements · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":133.201,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":1,"passed":false},{"actual":0,"check":"fixture 2: (1,)","expected":1,"passed":false},{"actual":3,"check":"fixture 3: (3,)","expected":2,"passed":false},{"actual":60,"check":"fixture 4: (5,)","expected":24,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 2: (1,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 3: (3,)\", \"actual\": 3, \"expected\": 2, \"passed\": false}, {\"check\": \"fixture 4: (5,)\", \"actual\": 60, \"expected\": 24, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":53.246,"exit_code":1,"observations":[{"actual":1,"check":"fixture 1: (0,)","expected":1,"passed":true},{"actual":1,"check":"fixture 2: (1,)","expected":1,"passed":true},{"actual":6,"check":"fixture 3: (3,)","expected":2,"passed":false},{"actual":120,"check":"fixture 4: (5,)","expected":24,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 2: (1,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 3: (3,)\", \"actual\": 6, \"expected\": 2, \"passed\": false}, {\"check\": \"fixture 4: (5,)\", \"actual\": 120, \"expected\": 24, \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":41.942,"exit_code":0,"observations":[{"actual":1,"check":"fixture 1: (0,)","expected":1,"passed":true},{"actual":1,"check":"fixture 2: (1,)","expected":1,"passed":true},{"actual":2,"check":"fixture 3: (3,)","expected":2,"passed":true},{"actual":24,"check":"fixture 4: (5,)","expected":24,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 2: (1,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 3: (3,)\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"fixture 4: (5,)\", \"actual\": 24, \"expected\": 24, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}