{"abstract":"An exact-cardinality count replaces a cumulative count.","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. Cumulative binomial count. Exact operational definition: sum(math.comb(n,i) for i in range(min(n,k)+1))","evaluation_group":"model-046fd81e01069031","failed_approach":"The allowed maximum cardinality is excluded.","family":"xn-cumulative-binomial-count","id":"FA-5741","implementations":{"attempt":{"sha256":"1553e3b72de69fb02d9b866b1e079f83677aa0ad2e1a298c54624cf2e19c1022","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, k):\n    return sum(math.comb(n,i) for i in range(min(n,k)))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 2)', solve(*(4, 2)), 11)\ncheck('fixture 2: (3, 0)', solve(*(3, 0)), 1)\ncheck('fixture 3: (2, 5)', solve(*(2, 5)), 4)\ncheck('fixture 4: (0, 0)', solve(*(0, 0)), 1)\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":"49f4655c63edaa5864f7dd82270bf8675136e1293f17d18f4e4892008b87d3fd","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, k):\n    return math.comb(n,k)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 2)', solve(*(4, 2)), 11)\ncheck('fixture 2: (3, 0)', solve(*(3, 0)), 1)\ncheck('fixture 3: (2, 5)', solve(*(2, 5)), 4)\ncheck('fixture 4: (0, 0)', solve(*(0, 0)), 1)\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":"e6cd62a55b8b0f37d498466248af9202362f91cd6b286150ba69144b4866ad92","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, k):\n    return sum(math.comb(n,i) for i in range(min(n,k)+1))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 2)', solve(*(4, 2)), 11)\ncheck('fixture 2: (3, 0)', solve(*(3, 0)), 1)\ncheck('fixture 3: (2, 5)', solve(*(2, 5)), 4)\ncheck('fixture 4: (0, 0)', solve(*(0, 0)), 1)\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-cumulative-binomial-count","generated_at":"2026-09-29T14:37:53.093521+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 sum(math.comb(n,i) for i in range(min(n,k)+1))","root_cause":"An exact-cardinality count replaces a cumulative count.","sha256":"14c3a984673986d17b2f8c743473d3e54ba36ce8d574795adbde3961c874da97","title":"Cumulative binomial count · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":49.354,"exit_code":1,"observations":[{"actual":5,"check":"fixture 1: (4, 2)","expected":11,"passed":false},{"actual":0,"check":"fixture 2: (3, 0)","expected":1,"passed":false},{"actual":3,"check":"fixture 3: (2, 5)","expected":4,"passed":false},{"actual":0,"check":"fixture 4: (0, 0)","expected":1,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 2)\", \"actual\": 5, \"expected\": 11, \"passed\": false}, {\"check\": \"fixture 2: (3, 0)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 3: (2, 5)\", \"actual\": 3, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 4: (0, 0)\", \"actual\": 0, \"expected\": 1, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":49.109,"exit_code":1,"observations":[{"actual":6,"check":"fixture 1: (4, 2)","expected":11,"passed":false},{"actual":1,"check":"fixture 2: (3, 0)","expected":1,"passed":true},{"actual":0,"check":"fixture 3: (2, 5)","expected":4,"passed":false},{"actual":1,"check":"fixture 4: (0, 0)","expected":1,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 2)\", \"actual\": 6, \"expected\": 11, \"passed\": false}, {\"check\": \"fixture 2: (3, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 3: (2, 5)\", \"actual\": 0, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 4: (0, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":50.018,"exit_code":0,"observations":[{"actual":11,"check":"fixture 1: (4, 2)","expected":11,"passed":true},{"actual":1,"check":"fixture 2: (3, 0)","expected":1,"passed":true},{"actual":4,"check":"fixture 3: (2, 5)","expected":4,"passed":true},{"actual":1,"check":"fixture 4: (0, 0)","expected":1,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 2)\", \"actual\": 11, \"expected\": 11, \"passed\": true}, {\"check\": \"fixture 2: (3, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 3: (2, 5)\", \"actual\": 4, \"expected\": 4, \"passed\": true}, {\"check\": \"fixture 4: (0, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}