{"abstract":"The exact polynomial taylor shift result violates the stated contract at coefficient source.","category":"Numerics","checks":8,"contract":"Input [A,h] ascending integer polynomial, integer h; return coefficients of A(t+h), preserving array length.","evaluation_group":"s3-numerics-polynomial-taylor-shift","failed_approach":"The partial repair A[n-1-i] still violates the coefficient source invariant.","family":"s3-numerics-polynomial-taylor-shift-coefficient-source","id":"FA-14651","implementations":{"attempt":{"sha256":"7c48208ae7a8b5594855c02e3436c07ca53ee574169fca87de714000f80b6b9f","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[n-1-i]) * (math.comb(i,j) if j<=i else 0) * (h**(i-j) if j<=i else 0)\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-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], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, 0], -1], [-1, 1, -1, 0]), ([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-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, 0], 1], [-3, -3, -1, 0]), ([[-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, 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, 0], 2], [-7, -5, -1, 0]), ([[-1, -1, -1, 1], 2], [1, 7, 5, 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, 1], -2], [-11, 15, -7, 1]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 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":"bb9c3aea771b343f1f3cc713d4263c37adf0741404edf8f9176a12dea84d3602","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[j]) * (math.comb(i,j) if j<=i else 0) * (h**(i-j) if j<=i else 0)\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-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], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, 0], -1], [-1, 1, -1, 0]), ([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-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, 0], 1], [-3, -3, -1, 0]), ([[-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, 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, 0], 2], [-7, -5, -1, 0]), ([[-1, -1, -1, 1], 2], [1, 7, 5, 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, 1], -2], [-11, 15, -7, 1]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 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"},"fixed":{"sha256":"b2b2344e5bd2db4efba9d1add499a7f0b4632e887f9062f273f831e5f0258a4e","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-j) if j<=i else 0)\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-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], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, 0], -1], [-1, 1, -1, 0]), ([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-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, 0], 1], [-3, -3, -1, 0]), ([[-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, 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, 0], 2], [-7, -5, -1, 0]), ([[-1, -1, -1, 1], 2], [1, 7, 5, 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, 1], -2], [-11, 15, -7, 1]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 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-coefficient-source","generated_at":"2026-09-29T14:39:19.106785+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 A[i] at the coefficient source step.","root_cause":"The coefficient source step uses A[j] instead of A[i].","sha256":"426ca19fe86c2bfd655e50ef8c3a02f2580dcebc32932acb85323560bb253f20","title":"Polynomial taylor shift: coefficient source · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":45.246,"exit_code":1,"observations":[{"actual":[6,-9,5,-1],"check":"explicit oracle 0","expected":[-3,3,-1,0],"passed":false},{"actual":[5,-9,5,-1],"check":"explicit oracle 1","expected":[5,-9,5,-1],"passed":true},{"actual":[30,34,14,2],"check":"explicit oracle 2","expected":[30,34,14,2],"passed":true},{"actual":[0,-2,2,-1],"check":"explicit oracle 3","expected":[0,-2,2,-1],"passed":true},{"actual":[-1,-1,-1,-1],"check":"explicit oracle 4","expected":[-1,-1,-1,-1],"passed":true},{"actual":[-4,-6,-4,-1],"check":"explicit oracle 5","expected":[-4,-6,-4,-1],"passed":true},{"actual":[-15,-17,-7,-1],"check":"explicit oracle 6","expected":[-15,-17,-7,-1],"passed":true},{"actual":[1,-2,2,-1],"check":"explicit oracle 7","expected":[-1,1,-1,0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [6, -9, 5, -1], \"expected\": [-3, 3, -1, 0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [5, -9, 5, -1], \"expected\": [5, -9, 5, -1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [30, 34, 14, 2], \"expected\": [30, 34, 14, 2], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [0, -2, 2, -1], \"expected\": [0, -2, 2, -1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -1, -1, -1], \"expected\": [-1, -1, -1, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-4, -6, -4, -1], \"expected\": [-4, -6, -4, -1], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [-15, -17, -7, -1], \"expected\": [-15, -17, -7, -1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [1, -2, 2, -1], \"expected\": [-1, 1, -1, 0], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.956,"exit_code":1,"observations":[{"actual":[5,-9,5,0],"check":"explicit oracle 0","expected":[-3,3,-1,0],"passed":false},{"actual":[5,-9,5,-1],"check":"explicit oracle 1","expected":[5,-9,5,-1],"passed":true},{"actual":[30,34,14,2],"check":"explicit oracle 2","expected":[30,34,14,2],"passed":true},{"actual":[0,-2,2,-1],"check":"explicit oracle 3","expected":[0,-2,2,-1],"passed":true},{"actual":[-1,-1,-1,-1],"check":"explicit oracle 4","expected":[-1,-1,-1,-1],"passed":true},{"actual":[-4,-6,-4,-1],"check":"explicit oracle 5","expected":[-4,-6,-4,-1],"passed":true},{"actual":[-15,-17,-7,-1],"check":"explicit oracle 6","expected":[-15,-17,-7,-1],"passed":true},{"actual":[0,-2,2,0],"check":"explicit oracle 7","expected":[-1,1,-1,0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [5, -9, 5, 0], \"expected\": [-3, 3, -1, 0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [5, -9, 5, -1], \"expected\": [5, -9, 5, -1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [30, 34, 14, 2], \"expected\": [30, 34, 14, 2], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [0, -2, 2, -1], \"expected\": [0, -2, 2, -1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -1, -1, -1], \"expected\": [-1, -1, -1, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-4, -6, -4, -1], \"expected\": [-4, -6, -4, -1], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [-15, -17, -7, -1], \"expected\": [-15, -17, -7, -1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [0, -2, 2, 0], \"expected\": [-1, 1, -1, 0], \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.318,"exit_code":0,"observations":[{"actual":[-3,3,-1,0],"check":"explicit oracle 0","expected":[-3,3,-1,0],"passed":true},{"actual":[5,-9,5,-1],"check":"explicit oracle 1","expected":[5,-9,5,-1],"passed":true},{"actual":[30,34,14,2],"check":"explicit oracle 2","expected":[30,34,14,2],"passed":true},{"actual":[0,-2,2,-1],"check":"explicit oracle 3","expected":[0,-2,2,-1],"passed":true},{"actual":[-1,-1,-1,-1],"check":"explicit oracle 4","expected":[-1,-1,-1,-1],"passed":true},{"actual":[-4,-6,-4,-1],"check":"explicit oracle 5","expected":[-4,-6,-4,-1],"passed":true},{"actual":[-15,-17,-7,-1],"check":"explicit oracle 6","expected":[-15,-17,-7,-1],"passed":true},{"actual":[-1,1,-1,0],"check":"explicit oracle 7","expected":[-1,1,-1,0],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [-3, 3, -1, 0], \"expected\": [-3, 3, -1, 0], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [5, -9, 5, -1], \"expected\": [5, -9, 5, -1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [30, 34, 14, 2], \"expected\": [30, 34, 14, 2], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [0, -2, 2, -1], \"expected\": [0, -2, 2, -1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -1, -1, -1], \"expected\": [-1, -1, -1, -1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-4, -6, -4, -1], \"expected\": [-4, -6, -4, -1], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [-15, -17, -7, -1], \"expected\": [-15, -17, -7, -1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, 1, -1, 0], \"expected\": [-1, 1, -1, 0], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}