{"abstract":"Starting at exponent zero returns the trivial identity.","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. Multiplicative order. Exact operational definition: next((k for k in range(1,m+1) if pow(a,k,m)==1),None)","evaluation_group":"model-7099d5adc5956464","failed_approach":"The group size is an upper bound rather than every element order.","family":"xn-multiplicative-order","id":"FA-5566","implementations":{"attempt":{"sha256":"46e39b607887318b8cb2158e2e1b5cbc6a189a995b16ea25b97c5d369d8f1ac2","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, m):\n    return m-1 if math.gcd(a,m)==1 else None\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (2, 7)', solve(*(2, 7)), 3)\ncheck('fixture 2: (3, 7)', solve(*(3, 7)), 6)\ncheck('fixture 3: (2, 4)', solve(*(2, 4)), None)\ncheck('fixture 4: (1, 9)', solve(*(1, 9)), 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":"5a7abdc0d1ba383e776d1628b455461341c3b3526e96f4dfa56b14ae6272893d","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, m):\n    return next((k for k in range(m+1) if pow(a,k,m)==1),None)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (2, 7)', solve(*(2, 7)), 3)\ncheck('fixture 2: (3, 7)', solve(*(3, 7)), 6)\ncheck('fixture 3: (2, 4)', solve(*(2, 4)), None)\ncheck('fixture 4: (1, 9)', solve(*(1, 9)), 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":"3eadcafc315806222fbe09b222cbf0d6ace61547b015ec9d474941fce0c5d5ca","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, m):\n    return next((k for k in range(1,m+1) if pow(a,k,m)==1),None)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (2, 7)', solve(*(2, 7)), 3)\ncheck('fixture 2: (3, 7)', solve(*(3, 7)), 6)\ncheck('fixture 3: (2, 4)', solve(*(2, 4)), None)\ncheck('fixture 4: (1, 9)', solve(*(1, 9)), 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-multiplicative-order","generated_at":"2026-09-29T14:37:50.979653+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 next((k for k in range(1,m+1) if pow(a,k,m)==1),None)","root_cause":"Starting at exponent zero returns the trivial identity.","sha256":"b8645307ee84c6680591df9c6e49df85a21c9753928d1580e32961ec53d1c70b","title":"Multiplicative order · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":45.0,"exit_code":1,"observations":[{"actual":6,"check":"fixture 1: (2, 7)","expected":3,"passed":false},{"actual":6,"check":"fixture 2: (3, 7)","expected":6,"passed":true},{"actual":null,"check":"fixture 3: (2, 4)","expected":null,"passed":true},{"actual":8,"check":"fixture 4: (1, 9)","expected":1,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (2, 7)\", \"actual\": 6, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 2: (3, 7)\", \"actual\": 6, \"expected\": 6, \"passed\": true}, {\"check\": \"fixture 3: (2, 4)\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"fixture 4: (1, 9)\", \"actual\": 8, \"expected\": 1, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":46.617,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (2, 7)","expected":3,"passed":false},{"actual":0,"check":"fixture 2: (3, 7)","expected":6,"passed":false},{"actual":0,"check":"fixture 3: (2, 4)","expected":null,"passed":false},{"actual":0,"check":"fixture 4: (1, 9)","expected":1,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (2, 7)\", \"actual\": 0, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 2: (3, 7)\", \"actual\": 0, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 3: (2, 4)\", \"actual\": 0, \"expected\": null, \"passed\": false}, {\"check\": \"fixture 4: (1, 9)\", \"actual\": 0, \"expected\": 1, \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.504,"exit_code":0,"observations":[{"actual":3,"check":"fixture 1: (2, 7)","expected":3,"passed":true},{"actual":6,"check":"fixture 2: (3, 7)","expected":6,"passed":true},{"actual":null,"check":"fixture 3: (2, 4)","expected":null,"passed":true},{"actual":1,"check":"fixture 4: (1, 9)","expected":1,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (2, 7)\", \"actual\": 3, \"expected\": 3, \"passed\": true}, {\"check\": \"fixture 2: (3, 7)\", \"actual\": 6, \"expected\": 6, \"passed\": true}, {\"check\": \"fixture 3: (2, 4)\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"fixture 4: (1, 9)\", \"actual\": 1, \"expected\": 1, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}