{"abstract":"Perfect numbers are included at the abundance boundary.","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. Abundant number predicate. Exact operational definition: sum(d for d in range(1,n) if n%d==0)>n","evaluation_group":"model-6d3576abeb150835","failed_approach":"Including n makes deficient composites look abundant.","family":"xn-abundant-number-predicate","id":"FA-5556","implementations":{"attempt":{"sha256":"ef5e57f064d96db409d509784b00eb22a9a451b2ee1a8d268b760f2338691808","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 sum(d for d in range(1,n+1) if n%d==0)>n\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (12,)', solve(*(12,)), True)\ncheck('fixture 2: (6,)', solve(*(6,)), False)\ncheck('fixture 3: (1,)', solve(*(1,)), False)\ncheck('fixture 4: (8,)', solve(*(8,)), False)\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":"90834a1ec2fafa51d76a7491ea218f84d935fcf37968768ffe0e47d81b1dc8fd","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 sum(d for d in range(1,n) if n%d==0)>=n\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (12,)', solve(*(12,)), True)\ncheck('fixture 2: (6,)', solve(*(6,)), False)\ncheck('fixture 3: (1,)', solve(*(1,)), False)\ncheck('fixture 4: (8,)', solve(*(8,)), False)\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":"1f4a723d83145969b4867294fdf69cebaabee5e7f50a47057805390fd84ec4d8","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 sum(d for d in range(1,n) if n%d==0)>n\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (12,)', solve(*(12,)), True)\ncheck('fixture 2: (6,)', solve(*(6,)), False)\ncheck('fixture 3: (1,)', solve(*(1,)), False)\ncheck('fixture 4: (8,)', solve(*(8,)), False)\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-abundant-number-predicate","generated_at":"2026-09-29T14:37:50.909436+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 sum(d for d in range(1,n) if n%d==0)>n","root_cause":"Perfect numbers are included at the abundance boundary.","sha256":"05ef0dee0b396dbf946dfb9e0a669cc1bc3503c3d7d179a2413f6af97e622192","title":"Abundant number predicate · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":48.053,"exit_code":1,"observations":[{"actual":true,"check":"fixture 1: (12,)","expected":true,"passed":true},{"actual":true,"check":"fixture 2: (6,)","expected":false,"passed":false},{"actual":false,"check":"fixture 3: (1,)","expected":false,"passed":true},{"actual":true,"check":"fixture 4: (8,)","expected":false,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (12,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 2: (6,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 3: (1,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 4: (8,)\", \"actual\": true, \"expected\": false, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":47.832,"exit_code":1,"observations":[{"actual":true,"check":"fixture 1: (12,)","expected":true,"passed":true},{"actual":true,"check":"fixture 2: (6,)","expected":false,"passed":false},{"actual":false,"check":"fixture 3: (1,)","expected":false,"passed":true},{"actual":false,"check":"fixture 4: (8,)","expected":false,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (12,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 2: (6,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 3: (1,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 4: (8,)\", \"actual\": false, \"expected\": false, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.492,"exit_code":0,"observations":[{"actual":true,"check":"fixture 1: (12,)","expected":true,"passed":true},{"actual":false,"check":"fixture 2: (6,)","expected":false,"passed":true},{"actual":false,"check":"fixture 3: (1,)","expected":false,"passed":true},{"actual":false,"check":"fixture 4: (8,)","expected":false,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (12,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 2: (6,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 3: (1,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 4: (8,)\", \"actual\": false, \"expected\": false, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}