{"abstract":"Repeated prime powers are retained in the radical.","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. Radical distinct prime product. Exact operational definition: math.prod(p for p in range(2,n+1) if n%p==0 and all(p%d for d in range(2,math.isqrt(p)+1)))","contract_signature":"n","evaluation_group":"model-4f11352ce25be282","failed_approach":"Multiplying all divisors counts composite factors and repeated prime contributions.","family":"xn-radical-distinct-prime-product","id":"FA-5526","implementations":{"attempt":{"sha256":"b03713ce475d3d4d90eb004ff8799e9051025155f0e0018e9297ab99bfd445a6","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 math.prod(p for p in range(2,n+1) if n%p==0)\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: (12,)', solve(*(12,)), 6)\ncheck('fixture 3: (8,)', solve(*(8,)), 2)\ncheck('fixture 4: (15,)', solve(*(15,)), 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":"72e0f2113902e4c8e57cad448d5e0575e230a4bfccb23f64a9c4c30bb3624454","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\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: (12,)', solve(*(12,)), 6)\ncheck('fixture 3: (8,)', solve(*(8,)), 2)\ncheck('fixture 4: (15,)', solve(*(15,)), 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-radical-distinct-prime-product","generated_at":"2026-09-29T14:37:50.479709+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":"Repeated prime powers are retained in the radical.","sha256":"a2a5f92cd28f606f46c4719e7bd9cdf52bf643a37b750a827de6c0496b37a612","title":"Radical distinct prime product · 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":54.072,"exit_code":1,"observations":[{"actual":1,"check":"fixture 1: (1,)","expected":1,"passed":true},{"actual":1728,"check":"fixture 2: (12,)","expected":6,"passed":false},{"actual":64,"check":"fixture 3: (8,)","expected":2,"passed":false},{"actual":225,"check":"fixture 4: (15,)","expected":15,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 2: (12,)\", \"actual\": 1728, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 3: (8,)\", \"actual\": 64, \"expected\": 2, \"passed\": false}, {\"check\": \"fixture 4: (15,)\", \"actual\": 225, \"expected\": 15, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":105.215,"exit_code":1,"observations":[{"actual":1,"check":"fixture 1: (1,)","expected":1,"passed":true},{"actual":12,"check":"fixture 2: (12,)","expected":6,"passed":false},{"actual":8,"check":"fixture 3: (8,)","expected":2,"passed":false},{"actual":15,"check":"fixture 4: (15,)","expected":15,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 2: (12,)\", \"actual\": 12, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 3: (8,)\", \"actual\": 8, \"expected\": 2, \"passed\": false}, {\"check\": \"fixture 4: (15,)\", \"actual\": 15, \"expected\": 15, \"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."}}