{"abstract":"The empty divisor loop falsely accepts zero and one.","category":"Number theory","checks":5,"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. n is any nonnegative integer; zero and one are not prime. Exact operational definition: n>=2 and all(n%d for d in range(2,math.isqrt(n)+1))","contract_signature":"n","evaluation_group":"model-d53d34599f2f1f87","failed_approach":"Excluding the lower boundary also excludes the first prime.","family":"xn-prime-predicate-small-boundary","id":"FA-5506","implementations":{"attempt":{"sha256":"f9b0c21f357f77b8eb4a455d38692088edfb668f1897025381dba43ebb8cecd1","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>2 and all(n%d for d 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: (0,)', solve(*(0,)), False)\ncheck('fixture 2: (1,)', solve(*(1,)), False)\ncheck('fixture 3: (2,)', solve(*(2,)), True)\ncheck('fixture 4: (9,)', solve(*(9,)), False)\ncheck('fixture 5: (13,)', solve(*(13,)), True)\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":"1212f55986e33654049b30dd5178405df0e8998769144e1f9bf3257f90e2186d","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%d for d in range(2,math.isqrt(max(0,n))+1))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), False)\ncheck('fixture 2: (1,)', solve(*(1,)), False)\ncheck('fixture 3: (2,)', solve(*(2,)), True)\ncheck('fixture 4: (9,)', solve(*(9,)), False)\ncheck('fixture 5: (13,)', solve(*(13,)), True)\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-prime-predicate-small-boundary","generated_at":"2026-09-29T14:37:50.209831+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 empty divisor loop falsely accepts zero and one.","sha256":"207cd62bd51433d59f239fb427ad0fe7168d158cf39aa647531528ce61020128","title":"Prime predicate small boundary · 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":46.444,"exit_code":1,"observations":[{"actual":false,"check":"fixture 1: (0,)","expected":false,"passed":true},{"actual":false,"check":"fixture 2: (1,)","expected":false,"passed":true},{"actual":false,"check":"fixture 3: (2,)","expected":true,"passed":false},{"actual":false,"check":"fixture 4: (9,)","expected":false,"passed":true},{"actual":true,"check":"fixture 5: (13,)","expected":true,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 2: (1,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 3: (2,)\", \"actual\": false, \"expected\": true, \"passed\": false}, {\"check\": \"fixture 4: (9,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 5: (13,)\", \"actual\": true, \"expected\": true, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":50.03,"exit_code":1,"observations":[{"actual":true,"check":"fixture 1: (0,)","expected":false,"passed":false},{"actual":true,"check":"fixture 2: (1,)","expected":false,"passed":false},{"actual":true,"check":"fixture 3: (2,)","expected":true,"passed":true},{"actual":false,"check":"fixture 4: (9,)","expected":false,"passed":true},{"actual":true,"check":"fixture 5: (13,)","expected":true,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 2: (1,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 3: (2,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 4: (9,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 5: (13,)\", \"actual\": true, \"expected\": true, \"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."}}