{"abstract":"The exact balanced odd radix encoding result violates the stated contract at balanced carry threshold.","category":"Numerics","checks":8,"contract":"Input [integer, odd radix>=3]; return least-significant-first balanced digits between -(base//2) and base//2, zero=[0]. Bounds: |integer|<=100 and radix<=9.","evaluation_group":"s3-numerics-balanced-odd-radix-encoding","failed_approach":"The partial repair r>half+1 still violates the balanced carry threshold invariant.","family":"s3-numerics-balanced-odd-radix-encoding-balanced-carry-threshold","id":"FA-15026","implementations":{"attempt":{"sha256":"9dc8a0e88818a05d44536e47427a740ce12ab4d6ac7e0146655438918f357e31","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,b=x\n    if n==0:return [0]\n    out=[];half=b//2\n    for _ in range(60):\n     if n==0:break\n     r=n%b\n     if r>half+1:r-=b\n     out.append(r)\n     n=(n-r)//b\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-100, 5], [0, 0, 1, -1]), ([-100, 7], [-2, 0, -2]), ([-100, 9], [-1, -2, -1]), ([-99, 3], [0, 0, 1, -1, -1]), ([-99, 5], [1, 0, 1, -1]), ([-99, 7], [-1, 0, -2])], [([-99, 3], [0, 0, 1, -1, -1]), ([-97, 3], [-1, 1, 1, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-96, 5], [-1, 1, 1, -1]), ([-96, 7], [2, 0, -2]), ([-96, 9], [3, -2, -1]), ([-95, 3], [1, 1, 1, -1, -1])], [([-98, 3], [1, 0, 1, -1, -1]), ([-95, 3], [1, 1, 1, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-92, 7], [-1, 1, -2]), ([-92, 9], [-2, -1, -1]), ([-91, 3], [-1, 0, -1, 0, -1]), ([-91, 5], [-1, 2, 1, -1])], [([-98, 5], [2, 0, 1, -1]), ([-94, 9], [-4, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-88, 9], [2, -1, -1]), ([-87, 3], [0, 1, -1, 0, -1]), ([-87, 5], [-2, -2, 2, -1]), ([-87, 7], [-3, 2, -2])], [([-97, 3], [-1, 1, 1, -1, -1]), ([-92, 5], [-2, 2, 1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-83, 3], [1, -1, 0, 0, -1]), ([-83, 5], [2, -2, 2, -1]), ([-83, 7], [1, 2, -2]), ([-83, 9], [-2, 0, -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":"bd7bee1d7abe358625efa6e0b48bfaa40b2060420afc3ffc64ed4dbd59e4d2a1","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,b=x\n    if n==0:return [0]\n    out=[];half=b//2\n    for _ in range(60):\n     if n==0:break\n     r=n%b\n     if r>=half:r-=b\n     out.append(r)\n     n=(n-r)//b\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-100, 5], [0, 0, 1, -1]), ([-100, 7], [-2, 0, -2]), ([-100, 9], [-1, -2, -1]), ([-99, 3], [0, 0, 1, -1, -1]), ([-99, 5], [1, 0, 1, -1]), ([-99, 7], [-1, 0, -2])], [([-99, 3], [0, 0, 1, -1, -1]), ([-97, 3], [-1, 1, 1, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-96, 5], [-1, 1, 1, -1]), ([-96, 7], [2, 0, -2]), ([-96, 9], [3, -2, -1]), ([-95, 3], [1, 1, 1, -1, -1])], [([-98, 3], [1, 0, 1, -1, -1]), ([-95, 3], [1, 1, 1, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-92, 7], [-1, 1, -2]), ([-92, 9], [-2, -1, -1]), ([-91, 3], [-1, 0, -1, 0, -1]), ([-91, 5], [-1, 2, 1, -1])], [([-98, 5], [2, 0, 1, -1]), ([-94, 9], [-4, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-88, 9], [2, -1, -1]), ([-87, 3], [0, 1, -1, 0, -1]), ([-87, 5], [-2, -2, 2, -1]), ([-87, 7], [-3, 2, -2])], [([-97, 3], [-1, 1, 1, -1, -1]), ([-92, 5], [-2, 2, 1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-83, 3], [1, -1, 0, 0, -1]), ([-83, 5], [2, -2, 2, -1]), ([-83, 7], [1, 2, -2]), ([-83, 9], [-2, 0, -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"},"fixed":{"sha256":"687d3c028ddfa89edad80c6677f5853d12c7b440d5ae52d15eadffc211c04ca4","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,b=x\n    if n==0:return [0]\n    out=[];half=b//2\n    for _ in range(60):\n     if n==0:break\n     r=n%b\n     if r>half:r-=b\n     out.append(r)\n     n=(n-r)//b\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-100, 5], [0, 0, 1, -1]), ([-100, 7], [-2, 0, -2]), ([-100, 9], [-1, -2, -1]), ([-99, 3], [0, 0, 1, -1, -1]), ([-99, 5], [1, 0, 1, -1]), ([-99, 7], [-1, 0, -2])], [([-99, 3], [0, 0, 1, -1, -1]), ([-97, 3], [-1, 1, 1, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-96, 5], [-1, 1, 1, -1]), ([-96, 7], [2, 0, -2]), ([-96, 9], [3, -2, -1]), ([-95, 3], [1, 1, 1, -1, -1])], [([-98, 3], [1, 0, 1, -1, -1]), ([-95, 3], [1, 1, 1, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-92, 7], [-1, 1, -2]), ([-92, 9], [-2, -1, -1]), ([-91, 3], [-1, 0, -1, 0, -1]), ([-91, 5], [-1, 2, 1, -1])], [([-98, 5], [2, 0, 1, -1]), ([-94, 9], [-4, -1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-88, 9], [2, -1, -1]), ([-87, 3], [0, 1, -1, 0, -1]), ([-87, 5], [-2, -2, 2, -1]), ([-87, 7], [-3, 2, -2])], [([-97, 3], [-1, 1, 1, -1, -1]), ([-92, 5], [-2, 2, 1, -1]), ([-100, 3], [-1, 0, 1, -1, -1]), ([100, 9], [1, 2, 1]), ([-83, 3], [1, -1, 0, 0, -1]), ([-83, 5], [2, -2, 2, -1]), ([-83, 7], [1, 2, -2]), ([-83, 9], [-2, 0, -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-balanced-odd-radix-encoding-balanced-carry-threshold","generated_at":"2026-09-29T14:39:22.616383+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.","repair":"Use r>half at the balanced carry threshold step.","root_cause":"The balanced carry threshold step uses r>=half instead of r>half.","sha256":"7eaa16e13a700f381e30a3417274780519d9d6292c2bfee7ea43d4943744e934","title":"Balanced odd radix encoding: balanced carry threshold · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":42.701,"exit_code":1,"observations":[{"actual":[2,2,0,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2],"check":"explicit oracle 0","expected":[-1,0,1,-1,-1],"passed":false},{"actual":[1,2,1],"check":"explicit oracle 1","expected":[1,2,1],"passed":true},{"actual":[0,0,1,-1],"check":"explicit oracle 2","expected":[0,0,1,-1],"passed":true},{"actual":[-2,0,-2],"check":"explicit oracle 3","expected":[-2,0,-2],"passed":true},{"actual":[-1,-2,-1],"check":"explicit oracle 4","expected":[-1,-2,-1],"passed":true},{"actual":[0,0,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2],"check":"explicit oracle 5","expected":[0,0,1,-1,-1],"passed":false},{"actual":[1,0,1,-1],"check":"explicit oracle 6","expected":[1,0,1,-1],"passed":true},{"actual":[-1,0,-2],"check":"explicit oracle 7","expected":[-1,0,-2],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [2, 2, 0, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], \"expected\": [-1, 0, 1, -1, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 2, 1], \"expected\": [1, 2, 1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 0, 1, -1], \"expected\": [0, 0, 1, -1], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [-2, 0, -2], \"expected\": [-2, 0, -2], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -2, -1], \"expected\": [-1, -2, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], \"expected\": [0, 0, 1, -1, -1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 0, 1, -1], \"expected\": [1, 0, 1, -1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, 0, -2], \"expected\": [-1, 0, -2], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":41.246,"exit_code":1,"observations":[{"actual":[-1,0,-2,0,-1],"check":"explicit oracle 0","expected":[-1,0,1,-1,-1],"passed":false},{"actual":[1,2,1],"check":"explicit oracle 1","expected":[1,2,1],"passed":true},{"actual":[0,0,1,-1],"check":"explicit oracle 2","expected":[0,0,1,-1],"passed":true},{"actual":[-2,0,-2],"check":"explicit oracle 3","expected":[-2,0,-2],"passed":true},{"actual":[-1,-2,-1],"check":"explicit oracle 4","expected":[-1,-2,-1],"passed":true},{"actual":[0,0,-2,0,-1],"check":"explicit oracle 5","expected":[0,0,1,-1,-1],"passed":false},{"actual":[1,0,1,-1],"check":"explicit oracle 6","expected":[1,0,1,-1],"passed":true},{"actual":[-1,0,-2],"check":"explicit oracle 7","expected":[-1,0,-2],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [-1, 0, -2, 0, -1], \"expected\": [-1, 0, 1, -1, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 2, 1], \"expected\": [1, 2, 1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 0, 1, -1], \"expected\": [0, 0, 1, -1], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [-2, 0, -2], \"expected\": [-2, 0, -2], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -2, -1], \"expected\": [-1, -2, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0, -2, 0, -1], \"expected\": [0, 0, 1, -1, -1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 0, 1, -1], \"expected\": [1, 0, 1, -1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, 0, -2], \"expected\": [-1, 0, -2], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":41.2,"exit_code":0,"observations":[{"actual":[-1,0,1,-1,-1],"check":"explicit oracle 0","expected":[-1,0,1,-1,-1],"passed":true},{"actual":[1,2,1],"check":"explicit oracle 1","expected":[1,2,1],"passed":true},{"actual":[0,0,1,-1],"check":"explicit oracle 2","expected":[0,0,1,-1],"passed":true},{"actual":[-2,0,-2],"check":"explicit oracle 3","expected":[-2,0,-2],"passed":true},{"actual":[-1,-2,-1],"check":"explicit oracle 4","expected":[-1,-2,-1],"passed":true},{"actual":[0,0,1,-1,-1],"check":"explicit oracle 5","expected":[0,0,1,-1,-1],"passed":true},{"actual":[1,0,1,-1],"check":"explicit oracle 6","expected":[1,0,1,-1],"passed":true},{"actual":[-1,0,-2],"check":"explicit oracle 7","expected":[-1,0,-2],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [-1, 0, 1, -1, -1], \"expected\": [-1, 0, 1, -1, -1], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 2, 1], \"expected\": [1, 2, 1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 0, 1, -1], \"expected\": [0, 0, 1, -1], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [-2, 0, -2], \"expected\": [-2, 0, -2], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -2, -1], \"expected\": [-1, -2, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0, 1, -1, -1], \"expected\": [0, 0, 1, -1, -1], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 0, 1, -1], \"expected\": [1, 0, 1, -1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, 0, -2], \"expected\": [-1, 0, -2], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}