{"abstract":"The final term is omitted from the arithmetic-series identity.","category":"Integer arithmetic","checks":4,"contract":"Integer arguments; denominators, steps, moduli, and bit widths are positive. Unsigned word inputs fit the specified width except where masking is explicitly the operation. Sequence sizes are nonnegative. Return sum(1..n), with n nonnegative. Exact operational definition: n*(n+1)//2","contract_signature":"n","evaluation_group":"model-adbb0758544d4b80","failed_approach":"Counting unordered pairs computes the preceding triangular number.","family":"xn-triangular-number","id":"FA-5446","implementations":{"attempt":{"sha256":"577acf079a82f3a3bbc7f3dbd5695508f3a9d8cdd727ebc250c69f1df2b29ec3","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 n*(n-1)//2\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 0)\ncheck('fixture 2: (1,)', solve(*(1,)), 1)\ncheck('fixture 3: (4,)', solve(*(4,)), 10)\ncheck('fixture 4: (7,)', solve(*(7,)), 28)\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":"2359763df00354364fabb43b2826ef672c349acce2bfec45b7a0cf7de4abe600","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 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,)), 0)\ncheck('fixture 2: (1,)', solve(*(1,)), 1)\ncheck('fixture 3: (4,)', solve(*(4,)), 10)\ncheck('fixture 4: (7,)', solve(*(7,)), 28)\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-triangular-number","generated_at":"2026-09-29T14:37:49.502254+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. Integer mathematics results depend on the stated convention.","root_cause":"The final term is omitted from the arithmetic-series identity.","sha256":"fdabfb85d97b95d1a798ad7934fc8071484288bbb1fb0788536c0950a6f281e5","title":"Triangular number · 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":43.715,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":0,"passed":true},{"actual":0,"check":"fixture 2: (1,)","expected":1,"passed":false},{"actual":6,"check":"fixture 3: (4,)","expected":10,"passed":false},{"actual":21,"check":"fixture 4: (7,)","expected":28,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 2: (1,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 3: (4,)\", \"actual\": 6, \"expected\": 10, \"passed\": false}, {\"check\": \"fixture 4: (7,)\", \"actual\": 21, \"expected\": 28, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.159,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":0,"passed":true},{"actual":0,"check":"fixture 2: (1,)","expected":1,"passed":false},{"actual":8,"check":"fixture 3: (4,)","expected":10,"passed":false},{"actual":24,"check":"fixture 4: (7,)","expected":28,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 2: (1,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 3: (4,)\", \"actual\": 8, \"expected\": 10, \"passed\": false}, {\"check\": \"fixture 4: (7,)\", \"actual\": 24, \"expected\": 28, \"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."}}