{"abstract":"A square-root divisor is counted twice.","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. Divisor count square root pair. Exact operational definition: sum(n%d==0 for d in range(1,n+1))","contract_signature":"n","evaluation_group":"model-e702436250845283","failed_approach":"Excluding the root loses its one contribution and misses nonsquare boundary factors.","family":"xn-divisor-count-square-root-pair","id":"FA-5501","implementations":{"attempt":{"sha256":"7414f71cbbf31846c3b643468a8bb3135efb83854262098f21c32f6c180a33d5","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 2*sum(n%d==0 for d in range(1,math.isqrt(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: (9,)', solve(*(9,)), 3)\ncheck('fixture 3: (12,)', solve(*(12,)), 6)\ncheck('fixture 4: (25,)', solve(*(25,)), 3)\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":"d114d44315675f8cb7db06eddadd55042f3cc183f67bf1c99bc2949f60af3618","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 2*sum(n%d==0 for d in range(1,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,)), 1)\ncheck('fixture 2: (9,)', solve(*(9,)), 3)\ncheck('fixture 3: (12,)', solve(*(12,)), 6)\ncheck('fixture 4: (25,)', solve(*(25,)), 3)\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-divisor-count-square-root-pair","generated_at":"2026-09-29T14:37:50.176990+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 square-root divisor is counted twice.","sha256":"4b13d84454015d1c66394f1fad67affad1035dffa3557544deefa2511414f70d","title":"Divisor count square root pair · 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":44.904,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (1,)","expected":1,"passed":false},{"actual":2,"check":"fixture 2: (9,)","expected":3,"passed":false},{"actual":4,"check":"fixture 3: (12,)","expected":6,"passed":false},{"actual":2,"check":"fixture 4: (25,)","expected":3,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 2: (9,)\", \"actual\": 2, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 3: (12,)\", \"actual\": 4, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 4: (25,)\", \"actual\": 2, \"expected\": 3, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":48.596,"exit_code":1,"observations":[{"actual":2,"check":"fixture 1: (1,)","expected":1,"passed":false},{"actual":4,"check":"fixture 2: (9,)","expected":3,"passed":false},{"actual":6,"check":"fixture 3: (12,)","expected":6,"passed":true},{"actual":4,"check":"fixture 4: (25,)","expected":3,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (1,)\", \"actual\": 2, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 2: (9,)\", \"actual\": 4, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 3: (12,)\", \"actual\": 6, \"expected\": 6, \"passed\": true}, {\"check\": \"fixture 4: (25,)\", \"actual\": 4, \"expected\": 3, \"passed\": false}], \"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."}}