{"abstract":"The exact rational generating recurrence result violates the stated contract at coefficient retention.","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 v if n==0 else r[-1] still violates the coefficient retention invariant.","family":"s3-numerics-rational-generating-recurrence-coefficient-retention","id":"FA-14786","implementations":{"attempt":{"sha256":"c78a4cd7bb6b3915e214d0b2b9649f576c251b8c7b3482d88eda292b719df527","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] if 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 if n==0 else r[-1])\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], 2], [-1, -3]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, -2], 3], [-1, -3, -8]), ([[-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], 5], [-1, -3, -8, -22, -60]), ([[-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, -2], 8], [-1, -3, -8, -22, -60, -164, -448, -1224]), ([[-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], 4], [-1, -3, -7, -17]), ([[-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, -1], 7], [-1, -3, -7, -17, -41, -99, -239]), ([[-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":"f193dc7cf476d17f9a3b92ee7ecaf86866f1fb0f2c4fc9c869226c9224f6099b","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] if 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(abs(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], 2], [-1, -3]), ([[1, 1], [1, 2, 2], 8], [1, -1, 0, 2, -4, 4, 0, -8]), ([[-1, -1], [1, -2, -2], 3], [-1, -3, -8]), ([[-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], 5], [-1, -3, -8, -22, -60]), ([[-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, -2], 8], [-1, -3, -8, -22, -60, -164, -448, -1224]), ([[-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], 4], [-1, -3, -7, -17]), ([[-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, -1], 7], [-1, -3, -7, -17, -41, -99, -239]), ([[-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-coefficient-retention","generated_at":"2026-09-29T14:39:20.420762+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 coefficient retention step uses abs(v) instead of v.","sha256":"633d5ef5dbb5b145cadf14e755d70abf9b28ba5da6bf2ad6153704927dca19a0","title":"Rational generating recurrence: coefficient retention · 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":42.203,"exit_code":1,"observations":[{"actual":[-1],"check":"explicit oracle 0","expected":[-1],"passed":true},{"actual":[-1,-1],"check":"explicit oracle 1","expected":[-1,-3],"passed":false},{"actual":[1,1,1,1,1,1,1,1],"check":"explicit oracle 2","expected":[1,-1,0,2,-4,4,0,-8],"passed":false},{"actual":[-1,-1,-1],"check":"explicit oracle 3","expected":[-1,-3,-8],"passed":false},{"actual":[-1,-1,-1,-1],"check":"explicit oracle 4","expected":[-1,-3,-8,-22],"passed":false},{"actual":[-1,-1,-1,-1,-1],"check":"explicit oracle 5","expected":[-1,-3,-8,-22,-60],"passed":false},{"actual":[-1,-1,-1,-1,-1,-1],"check":"explicit oracle 6","expected":[-1,-3,-8,-22,-60,-164],"passed":false},{"actual":[-1,-1,-1,-1,-1,-1,-1],"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, -1], \"expected\": [-1, -3], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 1, 1, 1, 1, 1, 1, 1], \"expected\": [1, -1, 0, 2, -4, 4, 0, -8], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [-1, -1, -1], \"expected\": [-1, -3, -8], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [-1, -1, -1, -1], \"expected\": [-1, -3, -8, -22], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [-1, -1, -1, -1, -1], \"expected\": [-1, -3, -8, -22, -60], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [-1, -1, -1, -1, -1, -1], \"expected\": [-1, -3, -8, -22, -60, -164], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, -1, -1, -1, -1, -1, -1], \"expected\": [-1, -3, -8, -22, -60, -164, -448], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.288,"exit_code":1,"observations":[{"actual":[1],"check":"explicit oracle 0","expected":[-1],"passed":false},{"actual":[1,1],"check":"explicit oracle 1","expected":[-1,-3],"passed":false},{"actual":[1,1,4,10,28,76,208,568],"check":"explicit oracle 2","expected":[1,-1,0,2,-4,4,0,-8],"passed":false},{"actual":[1,1,4],"check":"explicit oracle 3","expected":[-1,-3,-8],"passed":false},{"actual":[1,1,4,10],"check":"explicit oracle 4","expected":[-1,-3,-8,-22],"passed":false},{"actual":[1,1,4,10,28],"check":"explicit oracle 5","expected":[-1,-3,-8,-22,-60],"passed":false},{"actual":[1,1,4,10,28,76],"check":"explicit oracle 6","expected":[-1,-3,-8,-22,-60,-164],"passed":false},{"actual":[1,1,4,10,28,76,208],"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\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 1], \"expected\": [-1, -3], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 1, 4, 10, 28, 76, 208, 568], \"expected\": [1, -1, 0, 2, -4, 4, 0, -8], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [1, 1, 4], \"expected\": [-1, -3, -8], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1, 1, 4, 10], \"expected\": [-1, -3, -8, -22], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [1, 1, 4, 10, 28], \"expected\": [-1, -3, -8, -22, -60], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 1, 4, 10, 28, 76], \"expected\": [-1, -3, -8, -22, -60, -164], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [1, 1, 4, 10, 28, 76, 208], \"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."}}