{"abstract":"The exact polynomial taylor shift result violates the stated contract at translation exponent.","category":"Numerics","checks":8,"contract":"Input [A,h] ascending integer polynomial, integer h; return coefficients of A(t+h), preserving array length.","contract_signature":"x","evaluation_group":"s3-numerics-polynomial-taylor-shift","failed_approach":"The partial repair h**i still violates the translation exponent invariant.","family":"s3-numerics-polynomial-taylor-shift-translation-exponent","id":"FA-14661","implementations":{"attempt":{"sha256":"404e3731179cb6c8ff1c08828c33fa70ca5254655076b661d166a12cf6133d1e","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    A,h=x\n    n=len(A);r=[0]*n\n    for i in range(n):\n     for j in range(i+1):\n      r[j]+=(A[i]) * (math.comb(i,j) if j<=i else 0) * (h**i)\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, -1], -1], [0, -2, 2, -1]), ([[-1, -1, -1, -1], 0], [-1, -1, -1, -1]), ([[-1, -1, -1, -1], 1], [-4, -6, -4, -1]), ([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-1, -1, -1, 0], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, -1], -1], [0, -2, 2, -1]), ([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, 2], 1], [-1, 3, 5, 2]), ([[-1, -1, -1, 2], 2], [9, 19, 11, 2]), ([[-1, -1, 0, -1], -2], [9, -13, 6, -1])], [([[-1, -1, -1, -1], 0], [-1, -1, -1, -1]), ([[-1, -1, -1, 0], 0], [-1, -1, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 0, 1], 2], [5, 11, 6, 1]), ([[-1, -1, 0, 2], -2], [-15, 23, -12, 2]), ([[-1, -1, 0, 2], -1], [-2, 5, -6, 2]), ([[-1, -1, 0, 2], 0], [-1, -1, 0, 2])], [([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, 1], -1], [-2, 4, -4, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 1, 1], -1], [0, 0, -2, 1]), ([[-1, -1, 1, 1], 0], [-1, -1, 1, 1]), ([[-1, -1, 1, 1], 1], [0, 4, 4, 1]), ([[-1, -1, 1, 1], 2], [9, 15, 7, 1])], [([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-1, -1, -1, 2], -2], [-19, 27, -13, 2]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 2, 0], 1], [0, 3, 2, 0]), ([[-1, -1, 2, 0], 2], [5, 7, 2, 0]), ([[-1, -1, 2, 1], -2], [1, 3, -4, 1]), ([[-1, -1, 2, 1], -1], [1, -2, -1, 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":"805f43b78d38fea43411229a0d6a15fe112fafcea49801407e077da6bef6c088","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    A,h=x\n    n=len(A);r=[0]*n\n    for i in range(n):\n     for j in range(i+1):\n      r[j]+=(A[i]) * (math.comb(i,j) if j<=i else 0) * (h**j)\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, -1], -1], [0, -2, 2, -1]), ([[-1, -1, -1, -1], 0], [-1, -1, -1, -1]), ([[-1, -1, -1, -1], 1], [-4, -6, -4, -1]), ([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-1, -1, -1, 0], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, -1], -1], [0, -2, 2, -1]), ([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, 2], 1], [-1, 3, 5, 2]), ([[-1, -1, -1, 2], 2], [9, 19, 11, 2]), ([[-1, -1, 0, -1], -2], [9, -13, 6, -1])], [([[-1, -1, -1, -1], 0], [-1, -1, -1, -1]), ([[-1, -1, -1, 0], 0], [-1, -1, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 0, 1], 2], [5, 11, 6, 1]), ([[-1, -1, 0, 2], -2], [-15, 23, -12, 2]), ([[-1, -1, 0, 2], -1], [-2, 5, -6, 2]), ([[-1, -1, 0, 2], 0], [-1, -1, 0, 2])], [([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, 1], -1], [-2, 4, -4, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 1, 1], -1], [0, 0, -2, 1]), ([[-1, -1, 1, 1], 0], [-1, -1, 1, 1]), ([[-1, -1, 1, 1], 1], [0, 4, 4, 1]), ([[-1, -1, 1, 1], 2], [9, 15, 7, 1])], [([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-1, -1, -1, 2], -2], [-19, 27, -13, 2]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 2, 0], 1], [0, 3, 2, 0]), ([[-1, -1, 2, 0], 2], [5, 7, 2, 0]), ([[-1, -1, 2, 1], -2], [1, 3, -4, 1]), ([[-1, -1, 2, 1], -1], [1, -2, -1, 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-polynomial-taylor-shift-translation-exponent","generated_at":"2026-09-29T14:39:19.161184+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 translation exponent step uses h**j instead of h**(i-j) if j<=i else 0.","sha256":"a7fdbcd30a1517cce82b53d876da0e55e1a1b4388a3e541c7abc5f820d1bed16","title":"Polynomial taylor shift: translation exponent · 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":44.284,"exit_code":1,"observations":[{"actual":[5,18,20,8],"check":"explicit oracle 0","expected":[5,-9,5,-1],"passed":false},{"actual":[30,68,56,16],"check":"explicit oracle 1","expected":[30,34,14,2],"passed":false},{"actual":[0,2,2,1],"check":"explicit oracle 2","expected":[0,-2,2,-1],"passed":false},{"actual":[-1,0,0,0],"check":"explicit oracle 3","expected":[-1,-1,-1,-1],"passed":false},{"actual":[-4,-6,-4,-1],"check":"explicit oracle 4","expected":[-4,-6,-4,-1],"passed":true},{"actual":[-15,-34,-28,-8],"check":"explicit oracle 5","expected":[-15,-17,-7,-1],"passed":false},{"actual":[-3,-6,-4,0],"check":"explicit oracle 6","expected":[-3,3,-1,0],"passed":false},{"actual":[-1,-1,-1,0],"check":"explicit oracle 7","expected":[-1,1,-1,0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [5, 18, 20, 8], \"expected\": [5, -9, 5, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [30, 68, 56, 16], \"expected\": [30, 34, 14, 2], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 2, 2, 1], \"expected\": [0, -2, 2, -1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [-1, 0, 0, 0], \"expected\": [-1, -1, -1, -1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [-4, -6, -4, -1], \"expected\": [-4, -6, -4, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-15, -34, -28, -8], \"expected\": [-15, -17, -7, -1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [-3, -6, -4, 0], \"expected\": [-3, 3, -1, 0], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, -1, -1, 0], \"expected\": [-1, 1, -1, 0], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.989,"exit_code":1,"observations":[{"actual":[-4,12,-16,8],"check":"explicit oracle 0","expected":[5,-9,5,-1],"passed":false},{"actual":[8,24,32,16],"check":"explicit oracle 1","expected":[30,34,14,2],"passed":false},{"actual":[-4,6,-4,1],"check":"explicit oracle 2","expected":[0,-2,2,-1],"passed":false},{"actual":[-4,0,0,0],"check":"explicit oracle 3","expected":[-1,-1,-1,-1],"passed":false},{"actual":[-4,-6,-4,-1],"check":"explicit oracle 4","expected":[-4,-6,-4,-1],"passed":true},{"actual":[-4,-12,-16,-8],"check":"explicit oracle 5","expected":[-15,-17,-7,-1],"passed":false},{"actual":[-3,6,-4,0],"check":"explicit oracle 6","expected":[-3,3,-1,0],"passed":false},{"actual":[-3,3,-1,0],"check":"explicit oracle 7","expected":[-1,1,-1,0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [-4, 12, -16, 8], \"expected\": [5, -9, 5, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [8, 24, 32, 16], \"expected\": [30, 34, 14, 2], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [-4, 6, -4, 1], \"expected\": [0, -2, 2, -1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [-4, 0, 0, 0], \"expected\": [-1, -1, -1, -1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [-4, -6, -4, -1], \"expected\": [-4, -6, -4, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-4, -12, -16, -8], \"expected\": [-15, -17, -7, -1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [-3, 6, -4, 0], \"expected\": [-3, 3, -1, 0], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [-3, 3, -1, 0], \"expected\": [-1, 1, -1, 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."}}