{"abstract":"A base-two Fermat congruence admits primes as Carmichael numbers.","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 a nonnegative integer; return true precisely for composite n satisfying a**(n-1)=1 modulo n for every a coprime to n. Exact operational definition: n>2 and any(n%d==0 for d in range(2,math.isqrt(n)+1)) and all(pow(a,n-1,n)==1 for a in range(2,n) if math.gcd(a,n)==1)","contract_signature":"n","evaluation_group":"model-4d0987d2afea7eaf","failed_approach":"Requiring compositeness still accepts pseudoprimes that fail other coprime bases.","family":"xn-carmichael-number-predicate","id":"FA-5511","implementations":{"attempt":{"sha256":"a4ead187758294440d6813ee03dee681c850be6c1925632a8deadcf00115172e","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 any(n%d==0 for d in range(2,math.isqrt(n)+1)) and pow(2,n-1,n)==1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (561,)', solve(*(561,)), True)\ncheck('fixture 2: (1105,)', solve(*(1105,)), True)\ncheck('fixture 3: (341,)', solve(*(341,)), False)\ncheck('fixture 4: (13,)', solve(*(13,)), False)\ncheck('fixture 5: (1,)', solve(*(1,)), 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":"07a3892c720bbe09b1a0b0ac763b5b9eb237878b8843e54515a88b62ae62e33b","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 and pow(2,n-1,n)==1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (561,)', solve(*(561,)), True)\ncheck('fixture 2: (1105,)', solve(*(1105,)), True)\ncheck('fixture 3: (341,)', solve(*(341,)), False)\ncheck('fixture 4: (13,)', solve(*(13,)), False)\ncheck('fixture 5: (1,)', solve(*(1,)), 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-carmichael-number-predicate","generated_at":"2026-09-29T14:37:50.209965+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":"A base-two Fermat congruence admits primes as Carmichael numbers.","sha256":"6f2cb2bcf8e3745ec95368d75e6c0616e2cc2534d96ef18006592d8e94bafa53","title":"Carmichael number predicate · 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":43.105,"exit_code":1,"observations":[{"actual":true,"check":"fixture 1: (561,)","expected":true,"passed":true},{"actual":true,"check":"fixture 2: (1105,)","expected":true,"passed":true},{"actual":true,"check":"fixture 3: (341,)","expected":false,"passed":false},{"actual":false,"check":"fixture 4: (13,)","expected":false,"passed":true},{"actual":false,"check":"fixture 5: (1,)","expected":false,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (561,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 2: (1105,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 3: (341,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 4: (13,)\", \"actual\": false, \"expected\": false, \"passed\": true}, {\"check\": \"fixture 5: (1,)\", \"actual\": false, \"expected\": false, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.401,"exit_code":1,"observations":[{"actual":true,"check":"fixture 1: (561,)","expected":true,"passed":true},{"actual":true,"check":"fixture 2: (1105,)","expected":true,"passed":true},{"actual":true,"check":"fixture 3: (341,)","expected":false,"passed":false},{"actual":true,"check":"fixture 4: (13,)","expected":false,"passed":false},{"actual":false,"check":"fixture 5: (1,)","expected":false,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (561,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 2: (1105,)\", \"actual\": true, \"expected\": true, \"passed\": true}, {\"check\": \"fixture 3: (341,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 4: (13,)\", \"actual\": true, \"expected\": false, \"passed\": false}, {\"check\": \"fixture 5: (1,)\", \"actual\": false, \"expected\": false, \"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."}}