{"abstract":"Balanced totals are accepted without the prefix-balance constraint.","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. Catalan balanced parentheses. Exact operational definition: math.comb(2*n,n)//(n+1)","evaluation_group":"model-ced87f33690b74ee","failed_approach":"A constant symmetry divisor does not remove invalid prefixes.","family":"xn-catalan-balanced-parentheses","id":"FA-5656","implementations":{"attempt":{"sha256":"96b63cb8512a3953e088f57471adf14e2807397fc02345bef5f1730937990154","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.comb(2*n,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,)), 5)\ncheck('fixture 4: (4,)', solve(*(4,)), 14)\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":"6045d258deeeec196cd34c9c31440138634b225b00b0babc74b11ff493e6aa26","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.comb(2*n,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,)), 5)\ncheck('fixture 4: (4,)', solve(*(4,)), 14)\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":"5442c046ae64713d84fbd04ad963e31f8973035ae8c446e75b6077b9891a459b","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.comb(2*n,n)//(n+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,)), 5)\ncheck('fixture 4: (4,)', solve(*(4,)), 14)\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-catalan-balanced-parentheses","generated_at":"2026-09-29T14:37:51.912161+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.comb(2*n,n)//(n+1)","root_cause":"Balanced totals are accepted without the prefix-balance constraint.","sha256":"784dbaaf391b510fafa11c97a8004fc46ba09dd966150f53a28623836016118c","title":"Catalan balanced parentheses · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":45.635,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":1,"passed":false},{"actual":1,"check":"fixture 2: (1,)","expected":1,"passed":true},{"actual":10,"check":"fixture 3: (3,)","expected":5,"passed":false},{"actual":35,"check":"fixture 4: (4,)","expected":14,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 2: (1,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 3: (3,)\", \"actual\": 10, \"expected\": 5, \"passed\": false}, {\"check\": \"fixture 4: (4,)\", \"actual\": 35, \"expected\": 14, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.199,"exit_code":1,"observations":[{"actual":1,"check":"fixture 1: (0,)","expected":1,"passed":true},{"actual":2,"check":"fixture 2: (1,)","expected":1,"passed":false},{"actual":20,"check":"fixture 3: (3,)","expected":5,"passed":false},{"actual":70,"check":"fixture 4: (4,)","expected":14,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 2: (1,)\", \"actual\": 2, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 3: (3,)\", \"actual\": 20, \"expected\": 5, \"passed\": false}, {\"check\": \"fixture 4: (4,)\", \"actual\": 70, \"expected\": 14, \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.604,"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":5,"check":"fixture 3: (3,)","expected":5,"passed":true},{"actual":14,"check":"fixture 4: (4,)","expected":14,"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\": 5, \"expected\": 5, \"passed\": true}, {\"check\": \"fixture 4: (4,)\", \"actual\": 14, \"expected\": 14, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}