{"abstract":"The prime-only totient shortcut is applied to composite numbers.","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. Euler totient. Exact operational definition: sum(math.gcd(k,n)==1 for k in range(1,n+1))","contract_signature":"n","evaluation_group":"model-ac2e230b773998bc","failed_approach":"Numbers that do not divide n can still share a factor with n.","family":"xn-euler-totient","id":"FA-5516","implementations":{"attempt":{"sha256":"23d5761cb4730ea7d2b4d17e4f6175e28a17d1c4e9d807694c87e90826e86104","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(n%k!=0 for k in range(1,n))+1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (1,)', solve(*(1,)), 1)\ncheck('fixture 2: (9,)', solve(*(9,)), 6)\ncheck('fixture 3: (12,)', solve(*(12,)), 4)\ncheck('fixture 4: (7,)', solve(*(7,)), 6)\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":"fb7d2b3685240c21bc6b22c59b29355b748e8c4a1f2356979041751acf850fd6","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-1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (1,)', solve(*(1,)), 1)\ncheck('fixture 2: (9,)', solve(*(9,)), 6)\ncheck('fixture 3: (12,)', solve(*(12,)), 4)\ncheck('fixture 4: (7,)', solve(*(7,)), 6)\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-euler-totient","generated_at":"2026-09-29T14:37:50.219495+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.","root_cause":"The prime-only totient shortcut is applied to composite numbers.","sha256":"ab95fb16bb55e50361031c076786b5ea05ef6fa1814217921aab08c26719eb3e","title":"Euler totient · 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":43.12,"exit_code":1,"observations":[{"actual":1,"check":"fixture 1: (1,)","expected":1,"passed":true},{"actual":7,"check":"fixture 2: (9,)","expected":6,"passed":false},{"actual":7,"check":"fixture 3: (12,)","expected":4,"passed":false},{"actual":6,"check":"fixture 4: (7,)","expected":6,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 2: (9,)\", \"actual\": 7, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 3: (12,)\", \"actual\": 7, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 4: (7,)\", \"actual\": 6, \"expected\": 6, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":47.252,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (1,)","expected":1,"passed":false},{"actual":8,"check":"fixture 2: (9,)","expected":6,"passed":false},{"actual":11,"check":"fixture 3: (12,)","expected":4,"passed":false},{"actual":6,"check":"fixture 4: (7,)","expected":6,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 2: (9,)\", \"actual\": 8, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 3: (12,)\", \"actual\": 11, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 4: (7,)\", \"actual\": 6, \"expected\": 6, \"passed\": true}], \"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."}}