{"abstract":"A nonsquare can still contain a squared prime factor.","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. Squarefree predicate. Exact operational definition: all(n%(p*p) for p in range(2,math.isqrt(n)+1))","evaluation_group":"model-90b3630c4dd73a74","failed_approach":"Rejecting every divisor confuses squarefree with prime.","family":"xn-squarefree-predicate","id":"FA-5561","implementations":{"attempt":{"sha256":"ed9e5b7372eb0f8cf5d627258e5f4514dfbd4858fef3962661096f32e3c468a6","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 all(n%p for p in range(2,math.isqrt(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,)), True)\ncheck('fixture 2: (12,)', solve(*(12,)), False)\ncheck('fixture 3: (6,)', solve(*(6,)), True)\ncheck('fixture 4: (25,)', solve(*(25,)), 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":"96cefa7f4162efd00a114848e193a19d5d75f322bff5e8f680daed61e698d3f4","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.isqrt(n)**2 != n\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (1,)', solve(*(1,)), True)\ncheck('fixture 2: (12,)', solve(*(12,)), False)\ncheck('fixture 3: (6,)', solve(*(6,)), True)\ncheck('fixture 4: (25,)', solve(*(25,)), 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":"7a32536ba54da8fcfa8aba3301a64522db5ff52937840c27a0af5936e3544333","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 all(n%(p*p) for p in range(2,math.isqrt(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,)), True)\ncheck('fixture 2: (12,)', solve(*(12,)), False)\ncheck('fixture 3: (6,)', solve(*(6,)), True)\ncheck('fixture 4: (25,)', solve(*(25,)), 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-squarefree-predicate","generated_at":"2026-09-29T14:37:50.942244+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 all(n%(p*p) for p in range(2,math.isqrt(n)+1))","root_cause":"A nonsquare can still contain a squared prime factor.","sha256":"340738b08ad08a004c9e632403460dacc00103c7f04e8607b26161f8823aa3ee","title":"Squarefree predicate · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":46.735,"exit_code":1,"observations":[{"actual":true,"check":"fixture 1: (1,)","expected":true,"passed":true},{"actual":false,"check":"fixture 2: (12,)","expected":false,"passed":true},{"actual":false,"check":"fixture 3: (6,)","expected":true,"passed":false},{"actual":false,"check":"fixture 4: (25,)","expected":false,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 2: (12,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 3: (6,)\", \"actual\": false, \"expected\": true, \"passed\": false}, {\"check\": \"fixture 4: (25,)\", \"actual\": false, \"expected\": false, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.703,"exit_code":1,"observations":[{"actual":false,"check":"fixture 1: (1,)","expected":true,"passed":false},{"actual":true,"check":"fixture 2: (12,)","expected":false,"passed":false},{"actual":true,"check":"fixture 3: (6,)","expected":true,"passed":true},{"actual":false,"check":"fixture 4: (25,)","expected":false,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": false, \"expected\": true, \"passed\": false}, {\"check\": \"fixture 2: (12,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 3: (6,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 4: (25,)\", \"actual\": false, \"expected\": false, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":41.763,"exit_code":0,"observations":[{"actual":true,"check":"fixture 1: (1,)","expected":true,"passed":true},{"actual":false,"check":"fixture 2: (12,)","expected":false,"passed":true},{"actual":true,"check":"fixture 3: (6,)","expected":true,"passed":true},{"actual":false,"check":"fixture 4: (25,)","expected":false,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 2: (12,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 3: (6,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 4: (25,)\", \"actual\": false, \"expected\": false, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}