{"abstract":"The exact negative radix encoding result violates the stated contract at negative base zero representation.","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 [b] still violates the negative base zero representation invariant.","family":"s3-numerics-negative-radix-encoding-negative-base-zero-representation","id":"FA-15041","implementations":{"attempt":{"sha256":"d3e0dd100efac3fccd419fa0c84d6de3ebe8f2fce25e85977a9dbdbd4c2ddc7c","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 [b]\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 = [[([0, 2], [0]), ([-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])], [([0, 3], [0]), ([0, 5], [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])], [([0, 4], [0]), ([0, 8], [0]), ([-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])], [([0, 5], [0]), ([0, 4], [0]), ([-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])], [([0, 6], [0]), ([0, 7], [0]), ([-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":"c27e4fe758d26998687649bcc653f789d709dc3385629d862ffed7ca1946e54f","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 []\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 = [[([0, 2], [0]), ([-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])], [([0, 3], [0]), ([0, 5], [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])], [([0, 4], [0]), ([0, 8], [0]), ([-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])], [([0, 5], [0]), ([0, 4], [0]), ([-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])], [([0, 6], [0]), ([0, 7], [0]), ([-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-zero-representation","generated_at":"2026-09-29T14:39:22.696993+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 zero representation step uses [] instead of [0].","sha256":"f9308330a14ea98687abe51cb75c20ac36648f4f044aadecb1bb5ef09a772118","title":"Negative radix encoding: negative base zero representation · 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":41.795,"exit_code":1,"observations":[{"actual":[2],"check":"explicit oracle 0","expected":[0],"passed":false},{"actual":[1,1,1,0,1,1,0,0],"check":"explicit oracle 1","expected":[1,1,1,0,1,1,0,0],"passed":true},{"actual":[2,4,4],"check":"explicit oracle 2","expected":[2,4,4],"passed":true},{"actual":[1,2,1,1,1,2],"check":"explicit oracle 3","expected":[1,2,1,1,1,2],"passed":true},{"actual":[2,2,1,0],"check":"explicit oracle 4","expected":[2,2,1,0],"passed":true},{"actual":[1,1,0,0],"check":"explicit oracle 5","expected":[1,1,0,0],"passed":true},{"actual":[1,4,5,2],"check":"explicit oracle 6","expected":[1,4,5,2],"passed":true},{"actual":[1,5,1,5],"check":"explicit oracle 7","expected":[1,5,1,5],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [2], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 1, 1, 0, 1, 1, 0, 0], \"expected\": [1, 1, 1, 0, 1, 1, 0, 0], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [2, 4, 4], \"expected\": [2, 4, 4], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [1, 2, 1, 1, 1, 2], \"expected\": [1, 2, 1, 1, 1, 2], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [2, 2, 1, 0], \"expected\": [2, 2, 1, 0], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [1, 1, 0, 0], \"expected\": [1, 1, 0, 0], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 4, 5, 2], \"expected\": [1, 4, 5, 2], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [1, 5, 1, 5], \"expected\": [1, 5, 1, 5], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.013,"exit_code":1,"observations":[{"actual":[],"check":"explicit oracle 0","expected":[0],"passed":false},{"actual":[1,1,1,0,1,1,0,0],"check":"explicit oracle 1","expected":[1,1,1,0,1,1,0,0],"passed":true},{"actual":[2,4,4],"check":"explicit oracle 2","expected":[2,4,4],"passed":true},{"actual":[1,2,1,1,1,2],"check":"explicit oracle 3","expected":[1,2,1,1,1,2],"passed":true},{"actual":[2,2,1,0],"check":"explicit oracle 4","expected":[2,2,1,0],"passed":true},{"actual":[1,1,0,0],"check":"explicit oracle 5","expected":[1,1,0,0],"passed":true},{"actual":[1,4,5,2],"check":"explicit oracle 6","expected":[1,4,5,2],"passed":true},{"actual":[1,5,1,5],"check":"explicit oracle 7","expected":[1,5,1,5],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [], \"expected\": [0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 1, 1, 0, 1, 1, 0, 0], \"expected\": [1, 1, 1, 0, 1, 1, 0, 0], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [2, 4, 4], \"expected\": [2, 4, 4], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [1, 2, 1, 1, 1, 2], \"expected\": [1, 2, 1, 1, 1, 2], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [2, 2, 1, 0], \"expected\": [2, 2, 1, 0], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [1, 1, 0, 0], \"expected\": [1, 1, 0, 0], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 4, 5, 2], \"expected\": [1, 4, 5, 2], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [1, 5, 1, 5], \"expected\": [1, 5, 1, 5], \"passed\": true}], \"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."}}