{"abstract":"The exact rational generating recurrence result violates the stated contract at numerator coefficient alignment.","category":"Numerics","checks":8,"contract":"Input [P,Q,k] integer ascending arrays with Q[0]=1 and k>=0; return first k coefficients of formal series P/Q.","contract_signature":"x","evaluation_group":"s3-numerics-rational-generating-recurrence","failed_approach":"The partial repair P[0] if P else 0 still violates the numerator coefficient alignment invariant.","family":"s3-numerics-rational-generating-recurrence-numerator-coefficient-alignment","id":"FA-14771","implementations":{"attempt":{"sha256":"61cb934e0681308933149c6c19dd959274c16aa205ed5bdb5585f361198a6d48","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    P,Q,k=x\n    r=[]\n    for n in range(k):\n     v=P[0] if P else 0\n     for j in range(1,min(n,len(Q)-1)+1):\n      v=v-Q[j]*r[n-j]\n     r.append(v)\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[-1, -1], [1, -2, -2], 1], [-1]), ([[-1, -1], [1, -2, -2], 3], [-1, -3, -8]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, -2], 2], [-1, -3]), ([[-1, -1], [1, -2, -2], 4], [-1, -3, -8, -22]), ([[-1, -1], [1, -2, -2], 5], [-1, -3, -8, -22, -60]), ([[-1, -1], [1, -2, -2], 6], [-1, -3, -8, -22, -60, -164]), ([[-1, -1], [1, -2, -2], 7], [-1, -3, -8, -22, -60, -164, -448])], [([[-1, -1], [1, -2, -2], 2], [-1, -3]), ([[-1, -1], [1, -2, -2], 6], [-1, -3, -8, -22, -60, -164]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, 0], 2], [-1, -3]), ([[-1, -1], [1, -2, 0], 3], [-1, -3, -6]), ([[-1, -1], [1, -2, 0], 4], [-1, -3, -6, -12]), ([[-1, -1], [1, -2, 0], 5], [-1, -3, -6, -12, -24])], [([[-1, -1], [1, -2, -2], 3], [-1, -3, -8]), ([[-1, -1], [1, -2, -1], 3], [-1, -3, -7]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, 2], 3], [-1, -3, -4]), ([[-1, -1], [1, -2, 2], 4], [-1, -3, -4, -2]), ([[-1, -1], [1, -2, 2], 5], [-1, -3, -4, -2, 4]), ([[-1, -1], [1, -2, 2], 6], [-1, -3, -4, -2, 4, 12])], [([[-1, -1], [1, -2, -2], 4], [-1, -3, -8, -22]), ([[-1, -1], [1, -2, -1], 6], [-1, -3, -7, -17, -41, -99]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -1, -1], 4], [-1, -2, -3, -5]), ([[-1, -1], [1, -1, -1], 5], [-1, -2, -3, -5, -8]), ([[-1, -1], [1, -1, -1], 6], [-1, -2, -3, -5, -8, -13]), ([[-1, -1], [1, -1, -1], 7], [-1, -2, -3, -5, -8, -13, -21])], [([[-1, -1], [1, -2, -2], 5], [-1, -3, -8, -22, -60]), ([[-1, -1], [1, -2, 0], 3], [-1, -3, -6]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -1, 1], 5], [-1, -2, -1, 1, 2]), ([[-1, -1], [1, -1, 1], 6], [-1, -2, -1, 1, 2, 1]), ([[-1, -1], [1, -1, 1], 7], [-1, -2, -1, 1, 2, 1, -1]), ([[-1, -1], [1, -1, 1], 8], [-1, -2, -1, 1, 2, 1, -1, -2])]]\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":"5d20cf217e8a90eae192ad03fbafcca78834a05b61fa08ef529340911a8ccb7d","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    P,Q,k=x\n    r=[]\n    for n in range(k):\n     v=P[n-1] if 0<n<=len(P) else 0\n     for j in range(1,min(n,len(Q)-1)+1):\n      v=v-Q[j]*r[n-j]\n     r.append(v)\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[-1, -1], [1, -2, -2], 1], [-1]), ([[-1, -1], [1, -2, -2], 3], [-1, -3, -8]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, -2], 2], [-1, -3]), ([[-1, -1], [1, -2, -2], 4], [-1, -3, -8, -22]), ([[-1, -1], [1, -2, -2], 5], [-1, -3, -8, -22, -60]), ([[-1, -1], [1, -2, -2], 6], [-1, -3, -8, -22, -60, -164]), ([[-1, -1], [1, -2, -2], 7], [-1, -3, -8, -22, -60, -164, -448])], [([[-1, -1], [1, -2, -2], 2], [-1, -3]), ([[-1, -1], [1, -2, -2], 6], [-1, -3, -8, -22, -60, -164]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, 0], 2], [-1, -3]), ([[-1, -1], [1, -2, 0], 3], [-1, -3, -6]), ([[-1, -1], [1, -2, 0], 4], [-1, -3, -6, -12]), ([[-1, -1], [1, -2, 0], 5], [-1, -3, -6, -12, -24])], [([[-1, -1], [1, -2, -2], 3], [-1, -3, -8]), ([[-1, -1], [1, -2, -1], 3], [-1, -3, -7]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, 2], 3], [-1, -3, -4]), ([[-1, -1], [1, -2, 2], 4], [-1, -3, -4, -2]), ([[-1, -1], [1, -2, 2], 5], [-1, -3, -4, -2, 4]), ([[-1, -1], [1, -2, 2], 6], [-1, -3, -4, -2, 4, 12])], [([[-1, -1], [1, -2, -2], 4], [-1, -3, -8, -22]), ([[-1, -1], [1, -2, -1], 6], [-1, -3, -7, -17, -41, -99]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -1, -1], 4], [-1, -2, -3, -5]), ([[-1, -1], [1, -1, -1], 5], [-1, -2, -3, -5, -8]), ([[-1, -1], [1, -1, -1], 6], [-1, -2, -3, -5, -8, -13]), ([[-1, -1], [1, -1, -1], 7], [-1, -2, -3, -5, -8, -13, -21])], [([[-1, -1], [1, -2, -2], 5], [-1, -3, -8, -22, -60]), ([[-1, -1], [1, -2, 0], 3], [-1, -3, -6]), ([[-1, -1], [1, -2, -2], 1], [-1]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -1, 1], 5], [-1, -2, -1, 1, 2]), ([[-1, -1], [1, -1, 1], 6], [-1, -2, -1, 1, 2, 1]), ([[-1, -1], [1, -1, 1], 7], [-1, -2, -1, 1, 2, 1, -1]), ([[-1, -1], [1, -1, 1], 8], [-1, -2, -1, 1, 2, 1, -1, -2])]]\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-rational-generating-recurrence-numerator-coefficient-alignment","generated_at":"2026-09-29T14:39:20.238396+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 numerator coefficient alignment step uses P[n-1] if 0<n<=len(P) else 0 instead of P[n] if n<len(P) else 0.","sha256":"1d51c5717888781964f4e2acc56f833cc4565ee7a1bc65c2f861173a3f0928b5","title":"Rational generating recurrence: numerator coefficient alignment · 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":41.147,"exit_code":1,"observations":[{"actual":[-1],"check":"explicit oracle 0","expected":[-1],"passed":true},{"actual":[-1,-3,-9],"check":"explicit oracle 1","expected":[-1,-3,-8],"passed":false},{"actual":[1,-1,1,1,-3,5,-3,-3],"check":"explicit oracle 2","expected":[1,-1,0,2,-4,4,0,-8],"passed":false},{"actual":[-1,-3],"check":"explicit oracle 3","expected":[-1,-3],"passed":true},{"actual":[-1,-3,-9,-25],"check":"explicit oracle 4","expected":[-1,-3,-8,-22],"passed":false},{"actual":[-1,-3,-9,-25,-69],"check":"explicit oracle 5","expected":[-1,-3,-8,-22,-60],"passed":false},{"actual":[-1,-3,-9,-25,-69,-189],"check":"explicit oracle 6","expected":[-1,-3,-8,-22,-60,-164],"passed":false},{"actual":[-1,-3,-9,-25,-69,-189,-517],"check":"explicit oracle 7","expected":[-1,-3,-8,-22,-60,-164,-448],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [-1], \"expected\": [-1], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [-1, -3, -9], \"expected\": [-1, -3, -8], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1, -1, 1, 1, -3, 5, -3, -3], \"expected\": [1, -1, 0, 2, -4, 4, 0, -8], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [-1, -3], \"expected\": [-1, -3], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -3, -9, -25], \"expected\": [-1, -3, -8, -22], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [-1, -3, -9, -25, -69], \"expected\": [-1, -3, -8, -22, -60], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [-1, -3, -9, -25, -69, -189], \"expected\": [-1, -3, -8, -22, -60, -164], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, -3, -9, -25, -69, -189, -517], \"expected\": [-1, -3, -8, -22, -60, -164, -448], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.849,"exit_code":1,"observations":[{"actual":[0],"check":"explicit oracle 0","expected":[-1],"passed":false},{"actual":[0,-1,-3],"check":"explicit oracle 1","expected":[-1,-3,-8],"passed":false},{"actual":[0,1,-1,0,2,-4,4,0],"check":"explicit oracle 2","expected":[1,-1,0,2,-4,4,0,-8],"passed":false},{"actual":[0,-1],"check":"explicit oracle 3","expected":[-1,-3],"passed":false},{"actual":[0,-1,-3,-8],"check":"explicit oracle 4","expected":[-1,-3,-8,-22],"passed":false},{"actual":[0,-1,-3,-8,-22],"check":"explicit oracle 5","expected":[-1,-3,-8,-22,-60],"passed":false},{"actual":[0,-1,-3,-8,-22,-60],"check":"explicit oracle 6","expected":[-1,-3,-8,-22,-60,-164],"passed":false},{"actual":[0,-1,-3,-8,-22,-60,-164],"check":"explicit oracle 7","expected":[-1,-3,-8,-22,-60,-164,-448],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [0], \"expected\": [-1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [0, -1, -3], \"expected\": [-1, -3, -8], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 1, -1, 0, 2, -4, 4, 0], \"expected\": [1, -1, 0, 2, -4, 4, 0, -8], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [0, -1], \"expected\": [-1, -3], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [0, -1, -3, -8], \"expected\": [-1, -3, -8, -22], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [0, -1, -3, -8, -22], \"expected\": [-1, -3, -8, -22, -60], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [0, -1, -3, -8, -22, -60], \"expected\": [-1, -3, -8, -22, -60, -164], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [0, -1, -3, -8, -22, -60, -164], \"expected\": [-1, -3, -8, -22, -60, -164, -448], \"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."}}