{"abstract":"The exact negative radix encoding result violates the stated contract at negative base carry.","category":"Numerics","checks":8,"contract":"Input [integer, magnitude base>=2]; return most-significant-first digits 0..base-1 representing integer in radix -base. Bounds: |integer|<=100 and base<=8.","contract_signature":"x","evaluation_group":"s3-numerics-negative-radix-encoding","failed_approach":"The partial repair n//(-b) still violates the negative base carry invariant.","family":"s3-numerics-negative-radix-encoding-negative-base-carry","id":"FA-15051","implementations":{"attempt":{"sha256":"48e212294f24a1b3ff518d4f2f8ca0ab37055622b38985e4e68ee612f1638858","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=[]\n    for _ in range(60):\n     if n==0:break\n     r=n%b\n     out.append(r)\n     n=n//(-b)\n    return out[::-1] if bool(out) else []\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-100, 3], [1, 2, 1, 1, 1, 2]), ([-100, 4], [2, 2, 1, 0]), ([-100, 5], [1, 1, 0, 0]), ([-100, 6], [1, 4, 5, 2]), ([-100, 7], [1, 5, 1, 5]), ([-100, 8], [1, 7, 5, 4])], [([-100, 3], [1, 2, 1, 1, 1, 2]), ([-100, 5], [1, 1, 0, 0]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-98, 5], [1, 1, 0, 2]), ([-98, 6], [1, 4, 5, 4]), ([-98, 7], [1, 5, 0, 0]), ([-98, 8], [1, 7, 5, 6])], [([-100, 4], [2, 2, 1, 0]), ([-100, 8], [1, 7, 5, 4]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-96, 8], [1, 7, 4, 0]), ([-95, 2], [1, 1, 1, 0, 0, 0, 0, 1]), ([-95, 3], [1, 2, 1, 2, 2, 1]), ([-95, 4], [2, 2, 0, 1])], [([-100, 5], [1, 1, 0, 0]), ([-99, 4], [2, 2, 1, 1]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-93, 4], [2, 2, 0, 3]), ([-93, 5], [1, 2, 4, 2]), ([-93, 6], [1, 4, 4, 3]), ([-93, 7], [1, 5, 0, 5])], [([-100, 6], [1, 4, 5, 2]), ([-99, 7], [1, 5, 1, 6]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-91, 7], [1, 6, 6, 0]), ([-91, 8], [1, 7, 4, 5]), ([-90, 2], [1, 1, 1, 1, 1, 0, 1, 0]), ([-90, 3], [1, 2, 1, 2, 0, 0])]]\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":"9e43fd40dc8acd084220a202c8fe938d7fde29c5d6c64b9ef1d5fe60762c668e","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=[]\n    for _ in range(60):\n     if n==0:break\n     r=n%b\n     out.append(r)\n     n=(n-r)//b\n    return out[::-1] if bool(out) else []\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-100, 3], [1, 2, 1, 1, 1, 2]), ([-100, 4], [2, 2, 1, 0]), ([-100, 5], [1, 1, 0, 0]), ([-100, 6], [1, 4, 5, 2]), ([-100, 7], [1, 5, 1, 5]), ([-100, 8], [1, 7, 5, 4])], [([-100, 3], [1, 2, 1, 1, 1, 2]), ([-100, 5], [1, 1, 0, 0]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-98, 5], [1, 1, 0, 2]), ([-98, 6], [1, 4, 5, 4]), ([-98, 7], [1, 5, 0, 0]), ([-98, 8], [1, 7, 5, 6])], [([-100, 4], [2, 2, 1, 0]), ([-100, 8], [1, 7, 5, 4]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-96, 8], [1, 7, 4, 0]), ([-95, 2], [1, 1, 1, 0, 0, 0, 0, 1]), ([-95, 3], [1, 2, 1, 2, 2, 1]), ([-95, 4], [2, 2, 0, 1])], [([-100, 5], [1, 1, 0, 0]), ([-99, 4], [2, 2, 1, 1]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-93, 4], [2, 2, 0, 3]), ([-93, 5], [1, 2, 4, 2]), ([-93, 6], [1, 4, 4, 3]), ([-93, 7], [1, 5, 0, 5])], [([-100, 6], [1, 4, 5, 2]), ([-99, 7], [1, 5, 1, 6]), ([-100, 2], [1, 1, 1, 0, 1, 1, 0, 0]), ([100, 8], [2, 4, 4]), ([-91, 7], [1, 6, 6, 0]), ([-91, 8], [1, 7, 4, 5]), ([-90, 2], [1, 1, 1, 1, 1, 0, 1, 0]), ([-90, 3], [1, 2, 1, 2, 0, 0])]]\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-negative-radix-encoding-negative-base-carry","generated_at":"2026-09-29T14:39:22.965239+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 negative base carry step uses (n-r)//b instead of (n-r)//(-b).","sha256":"87215864897fdc3577a319238de11ab183c840a91095df0c8854e15816461eae","title":"Negative radix encoding: negative base carry · 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":48.686,"exit_code":1,"observations":[{"actual":[1,1,0,1,0,0,1,0,0],"check":"explicit oracle 0","expected":[1,1,1,0,1,1,0,0],"passed":false},{"actual":[7,1,3,4],"check":"explicit oracle 1","expected":[2,4,4],"passed":false},{"actual":[2,0,1,0,2],"check":"explicit oracle 2","expected":[1,2,1,1,1,2],"passed":false},{"actual":[3,1,1,1,0],"check":"explicit oracle 3","expected":[2,2,1,0],"passed":false},{"actual":[1,0,0],"check":"explicit oracle 4","expected":[1,1,0,0],"passed":false},{"actual":[3,4,2],"check":"explicit oracle 5","expected":[1,4,5,2],"passed":false},{"actual":[5,0,5],"check":"explicit oracle 6","expected":[1,5,1,5],"passed":false},{"actual":[6,4,4],"check":"explicit oracle 7","expected":[1,7,5,4],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [1, 1, 0, 1, 0, 0, 1, 0, 0], \"expected\": [1, 1, 1, 0, 1, 1, 0, 0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [7, 1, 3, 4], \"expected\": [2, 4, 4], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [2, 0, 1, 0, 2], \"expected\": [1, 2, 1, 1, 1, 2], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [3, 1, 1, 1, 0], \"expected\": [2, 2, 1, 0], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1, 0, 0], \"expected\": [1, 1, 0, 0], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [3, 4, 2], \"expected\": [1, 4, 5, 2], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [5, 0, 5], \"expected\": [1, 5, 1, 5], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [6, 4, 4], \"expected\": [1, 7, 5, 4], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.082,"exit_code":1,"observations":[{"actual":[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,0,0],"check":"explicit oracle 0","expected":[1,1,1,0,1,1,0,0],"passed":false},{"actual":[1,4,4],"check":"explicit oracle 1","expected":[2,4,4],"passed":false},{"actual":[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,0,2,2],"check":"explicit oracle 2","expected":[1,2,1,1,1,2],"passed":false},{"actual":[3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,1,3,0],"check":"explicit oracle 3","expected":[2,2,1,0],"passed":false},{"actual":[4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,1,0,0],"check":"explicit oracle 4","expected":[1,1,0,0],"passed":false},{"actual":[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,3,1,2],"check":"explicit oracle 5","expected":[1,4,5,2],"passed":false},{"actual":[6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,4,6,5],"check":"explicit oracle 6","expected":[1,5,1,5],"passed":false},{"actual":[7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,3,4],"check":"explicit oracle 7","expected":[1,7,5,4],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0], \"expected\": [1, 1, 1, 0, 1, 1, 0, 0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 4, 4], \"expected\": [2, 4, 4], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [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, 1, 2, 0, 2, 2], \"expected\": [1, 2, 1, 1, 1, 2], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 1, 3, 0], \"expected\": [2, 2, 1, 0], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 0, 0], \"expected\": [1, 1, 0, 0], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 1, 2], \"expected\": [1, 4, 5, 2], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 6, 5], \"expected\": [1, 5, 1, 5], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 3, 4], \"expected\": [1, 7, 5, 4], \"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."}}