{"abstract":"A zero remainder represents nine for nonzero multiples.","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. n is any signed integer; repeatedly sum magnitude digits until one remains, preserving zero. Exact operational definition: 0 if n==0 else 1+(abs(n)-1)%9","evaluation_group":"model-5c940ab900fa0ea5","failed_approach":"One digit sum may still have more than one digit.","family":"xn-digital-root-base-ten","id":"FA-5571","implementations":{"attempt":{"sha256":"1e39318deacafb94afe25c41384d3a74c72cff042bc16ef731a1d26a49d82e03","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 sum(map(int,str(abs(n))))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 0)\ncheck('fixture 2: (18,)', solve(*(18,)), 9)\ncheck('fixture 3: (999,)', solve(*(999,)), 9)\ncheck('fixture 4: (-29,)', solve(*(-29,)), 2)\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":"81d8e791e81eb009777620bf7fff558f7bb5556669e1634686f6a5022d4ebb2d","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 abs(n)%9\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 0)\ncheck('fixture 2: (18,)', solve(*(18,)), 9)\ncheck('fixture 3: (999,)', solve(*(999,)), 9)\ncheck('fixture 4: (-29,)', solve(*(-29,)), 2)\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":"afd09070f0fe0463808c10c1e33280ccfe399235d8dd54dd84bf7027da7fe234","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 0 if n==0 else 1+(abs(n)-1)%9\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 0)\ncheck('fixture 2: (18,)', solve(*(18,)), 9)\ncheck('fixture 3: (999,)', solve(*(999,)), 9)\ncheck('fixture 4: (-29,)', solve(*(-29,)), 2)\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-digital-root-base-ten","generated_at":"2026-09-29T14:37:50.989451+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 0 if n==0 else 1+(abs(n)-1)%9","root_cause":"A zero remainder represents nine for nonzero multiples.","sha256":"1ccdcd847946d23a7a339151cfcd6f12a109a158e47dc209b7a9990def62b0d1","title":"Digital root base ten · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":43.138,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":0,"passed":true},{"actual":9,"check":"fixture 2: (18,)","expected":9,"passed":true},{"actual":27,"check":"fixture 3: (999,)","expected":9,"passed":false},{"actual":11,"check":"fixture 4: (-29,)","expected":2,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 2: (18,)\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"fixture 3: (999,)\", \"actual\": 27, \"expected\": 9, \"passed\": false}, {\"check\": \"fixture 4: (-29,)\", \"actual\": 11, \"expected\": 2, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":48.469,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":0,"passed":true},{"actual":0,"check":"fixture 2: (18,)","expected":9,"passed":false},{"actual":0,"check":"fixture 3: (999,)","expected":9,"passed":false},{"actual":2,"check":"fixture 4: (-29,)","expected":2,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 2: (18,)\", \"actual\": 0, \"expected\": 9, \"passed\": false}, {\"check\": \"fixture 3: (999,)\", \"actual\": 0, \"expected\": 9, \"passed\": false}, {\"check\": \"fixture 4: (-29,)\", \"actual\": 2, \"expected\": 2, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.038,"exit_code":0,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":0,"passed":true},{"actual":9,"check":"fixture 2: (18,)","expected":9,"passed":true},{"actual":9,"check":"fixture 3: (999,)","expected":9,"passed":true},{"actual":2,"check":"fixture 4: (-29,)","expected":2,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"fixture 2: (18,)\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"fixture 3: (999,)\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"fixture 4: (-29,)\", \"actual\": 2, \"expected\": 2, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}