{"abstract":"Multiplication retains shared factors twice.","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; LCM is nonnegative and is zero if either input is zero. Exact operational definition: math.lcm(a,b)","evaluation_group":"model-42b1e2daf8a8621a","failed_approach":"Removing shared factors without magnitude normalization yields negative LCMs.","family":"xn-least-common-multiple-sign","id":"FA-5486","implementations":{"attempt":{"sha256":"29684dacd137abff659ff4b68f5de58875fe4988d6117800d54d9af13c3e379e","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 a*b//math.gcd(a,b) if a and b else 0\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 6)', solve(*(4, 6)), 12)\ncheck('fixture 2: (-4, 6)', solve(*(-4, 6)), 12)\ncheck('fixture 3: (0, 7)', solve(*(0, 7)), 0)\ncheck('fixture 4: (-3, -5)', solve(*(-3, -5)), 15)\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":"c67897d0b1a3c437b90c1b97a03e06d490d3f1c9c966f925d8110c629f9e031e","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 a*b\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 6)', solve(*(4, 6)), 12)\ncheck('fixture 2: (-4, 6)', solve(*(-4, 6)), 12)\ncheck('fixture 3: (0, 7)', solve(*(0, 7)), 0)\ncheck('fixture 4: (-3, -5)', solve(*(-3, -5)), 15)\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":"e344478650f0f044783152927463fe99b532518706b6b1a90b739d57a9ae0ce1","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 math.lcm(a,b)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 6)', solve(*(4, 6)), 12)\ncheck('fixture 2: (-4, 6)', solve(*(-4, 6)), 12)\ncheck('fixture 3: (0, 7)', solve(*(0, 7)), 0)\ncheck('fixture 4: (-3, -5)', solve(*(-3, -5)), 15)\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-least-common-multiple-sign","generated_at":"2026-09-29T14:37:50.126851+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.","repair":"Apply the specified mathematical contract directly, preserving all terms and boundary cases: return math.lcm(a,b)","root_cause":"Multiplication retains shared factors twice.","sha256":"adfec7f601d910927649a6ba1c96ae19fda36fcc73b3e2c994e2a44768497636","title":"Least common multiple sign · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":47.328,"exit_code":1,"observations":[{"actual":12,"check":"fixture 1: (4, 6)","expected":12,"passed":true},{"actual":-12,"check":"fixture 2: (-4, 6)","expected":12,"passed":false},{"actual":0,"check":"fixture 3: (0, 7)","expected":0,"passed":true},{"actual":15,"check":"fixture 4: (-3, -5)","expected":15,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 6)\", \"actual\": 12, \"expected\": 12, \"passed\": true}, {\"check\": \"fixture 2: (-4, 6)\", \"actual\": -12, \"expected\": 12, \"passed\": false}, {\"check\": \"fixture 3: (0, 7)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 4: (-3, -5)\", \"actual\": 15, \"expected\": 15, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":56.153,"exit_code":1,"observations":[{"actual":24,"check":"fixture 1: (4, 6)","expected":12,"passed":false},{"actual":-24,"check":"fixture 2: (-4, 6)","expected":12,"passed":false},{"actual":0,"check":"fixture 3: (0, 7)","expected":0,"passed":true},{"actual":15,"check":"fixture 4: (-3, -5)","expected":15,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 6)\", \"actual\": 24, \"expected\": 12, \"passed\": false}, {\"check\": \"fixture 2: (-4, 6)\", \"actual\": -24, \"expected\": 12, \"passed\": false}, {\"check\": \"fixture 3: (0, 7)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 4: (-3, -5)\", \"actual\": 15, \"expected\": 15, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.613,"exit_code":0,"observations":[{"actual":12,"check":"fixture 1: (4, 6)","expected":12,"passed":true},{"actual":12,"check":"fixture 2: (-4, 6)","expected":12,"passed":true},{"actual":0,"check":"fixture 3: (0, 7)","expected":0,"passed":true},{"actual":15,"check":"fixture 4: (-3, -5)","expected":15,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 6)\", \"actual\": 12, \"expected\": 12, \"passed\": true}, {\"check\": \"fixture 2: (-4, 6)\", \"actual\": 12, \"expected\": 12, \"passed\": true}, {\"check\": \"fixture 3: (0, 7)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 4: (-3, -5)\", \"actual\": 15, \"expected\": 15, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}