{"abstract":"The exact josephus elimination order result violates the stated contract at elimination cardinality.","category":"Numerics","checks":8,"contract":"Input [n,k,start], n>0 k>0 0<=start<n; return removal order counting current position as one.","contract_signature":"x","evaluation_group":"s3-numerics-josephus-elimination-order","failed_approach":"The partial repair range(1,n) still violates the elimination cardinality invariant.","family":"s3-numerics-josephus-elimination-order-elimination-cardinality","id":"FA-15301","implementations":{"attempt":{"sha256":"557f8237758abf7e381ac15452d41876a4dfde1cc34b72bfd9b4dc8c6d8d8731","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    n,k,start=x\n    ring=list(range(n));index=start;out=[]\n    for _ in range(1,n):\n     if not ring:break\n     index=(index+k-1)%len(ring)\n     out.append(ring.pop(index))\n     index=index\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([1, 2, 0], [0]), ([1, 3, 0], [0]), ([1, 4, 0], [0]), ([1, 5, 0], [0]), ([1, 6, 0], [0]), ([1, 7, 0], [0])], [([1, 2, 0], [0]), ([1, 4, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([2, 5, 1], [1, 0]), ([2, 6, 0], [1, 0]), ([2, 6, 1], [0, 1]), ([2, 7, 0], [0, 1])], [([1, 3, 0], [0]), ([1, 7, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([3, 4, 1], [1, 0, 2]), ([3, 4, 2], [2, 1, 0]), ([3, 5, 0], [1, 2, 0]), ([3, 5, 1], [2, 0, 1])], [([1, 4, 0], [0]), ([2, 1, 1], [1, 0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 1, 3], [3, 0, 1, 2]), ([4, 2, 0], [1, 3, 2, 0]), ([4, 2, 1], [2, 0, 3, 1]), ([4, 2, 2], [3, 1, 0, 2])], [([1, 5, 0], [0]), ([2, 3, 0], [0, 1]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 6, 0], [1, 0, 3, 2]), ([4, 6, 1], [2, 1, 0, 3]), ([4, 6, 2], [3, 2, 1, 0]), ([4, 6, 3], [0, 3, 2, 1])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %d\" % i, solve(args), expected)\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":"39b0a7ee8d3c9d1c0ba32e9c2e5e154ec1b2ff0bd4a83412be2767eeabb678c8","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    n,k,start=x\n    ring=list(range(n));index=start;out=[]\n    for _ in range(n-1):\n     if not ring:break\n     index=(index+k-1)%len(ring)\n     out.append(ring.pop(index))\n     index=index\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([1, 2, 0], [0]), ([1, 3, 0], [0]), ([1, 4, 0], [0]), ([1, 5, 0], [0]), ([1, 6, 0], [0]), ([1, 7, 0], [0])], [([1, 2, 0], [0]), ([1, 4, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([2, 5, 1], [1, 0]), ([2, 6, 0], [1, 0]), ([2, 6, 1], [0, 1]), ([2, 7, 0], [0, 1])], [([1, 3, 0], [0]), ([1, 7, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([3, 4, 1], [1, 0, 2]), ([3, 4, 2], [2, 1, 0]), ([3, 5, 0], [1, 2, 0]), ([3, 5, 1], [2, 0, 1])], [([1, 4, 0], [0]), ([2, 1, 1], [1, 0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 1, 3], [3, 0, 1, 2]), ([4, 2, 0], [1, 3, 2, 0]), ([4, 2, 1], [2, 0, 3, 1]), ([4, 2, 2], [3, 1, 0, 2])], [([1, 5, 0], [0]), ([2, 3, 0], [0, 1]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 6, 0], [1, 0, 3, 2]), ([4, 6, 1], [2, 1, 0, 3]), ([4, 6, 2], [3, 2, 1, 0]), ([4, 6, 3], [0, 3, 2, 1])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %d\" % i, solve(args), expected)\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":"A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. 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":"s3-numerics-josephus-elimination-order-elimination-cardinality","generated_at":"2026-09-29T14:39:25.331340+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.","root_cause":"The elimination cardinality step uses range(n-1) instead of range(n).","sha256":"e80da65f1bfdd4433d0a89b8003c4fb22344436de435060377e9066722907230","title":"Josephus elimination order: elimination cardinality · 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":43.162,"exit_code":1,"observations":[{"actual":[],"check":"explicit oracle 0","expected":[0],"passed":false},{"actual":[6,3,1,0,2,5,9,8,4,10],"check":"explicit oracle 1","expected":[6,3,1,0,2,5,9,8,4,10,7],"passed":false},{"actual":[],"check":"explicit oracle 2","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 3","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 4","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 5","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 6","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 7","expected":[0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [6, 3, 1, 0, 2, 5, 9, 8, 4, 10], \"expected\": [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [], \"expected\": [0], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.59,"exit_code":1,"observations":[{"actual":[],"check":"explicit oracle 0","expected":[0],"passed":false},{"actual":[6,3,1,0,2,5,9,8,4,10],"check":"explicit oracle 1","expected":[6,3,1,0,2,5,9,8,4,10,7],"passed":false},{"actual":[],"check":"explicit oracle 2","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 3","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 4","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 5","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 6","expected":[0],"passed":false},{"actual":[],"check":"explicit oracle 7","expected":[0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [6, 3, 1, 0, 2, 5, 9, 8, 4, 10], \"expected\": [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [], \"expected\": [0], \"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."}}