{"abstract":"The exact continued fraction convergents result violates the stated contract at numerator recurrence.","category":"Numerics","checks":8,"contract":"Input nonempty coefficients with positive terms after first; return every convergent as [numerator,positive denominator].","evaluation_group":"s3-numerics-continued-fraction-convergents","failed_approach":"The partial repair c*p1-p0 still violates the numerator recurrence invariant.","family":"s3-numerics-continued-fraction-convergents-numerator-recurrence","id":"FA-14451","implementations":{"attempt":{"sha256":"19e7aef2374f35d6f2d64ca59a19138572136a11d7066e74539c585ac9f67897","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=x\n    p0,p1=0,1\n    q0,q1=1,0\n    out=[]\n    for c in a:\n     p=c*p1-p0\n     q=c*q1+q0\n     out.append([p,q])\n     p0,p1,q0,q1=p1,p,q1,q\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 1], [[2, 1], [3, 1]]), ([1, 1], [[1, 1], [2, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2], [[2, 1]]), ([3], [[3, 1]]), ([4], [[4, 1]]), ([1, 2], [[1, 1], [3, 2]])], [([2, 2], [[2, 1], [5, 2]]), ([1, 4], [[1, 1], [5, 4]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 2], [[4, 1], [9, 2]]), ([4, 3], [[4, 1], [13, 3]]), ([4, 4], [[4, 1], [17, 4]]), ([1, 1, 1], [[1, 1], [2, 1], [3, 2]])], [([2, 3], [[2, 1], [7, 3]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([1, 4, 3], [[1, 1], [5, 4], [16, 13]]), ([1, 4, 4], [[1, 1], [5, 4], [21, 17]]), ([2, 1, 1], [[2, 1], [3, 1], [5, 2]]), ([2, 1, 2], [[2, 1], [3, 1], [8, 3]]), ([2, 1, 3], [[2, 1], [3, 1], [11, 4]])], [([2, 4], [[2, 1], [9, 4]]), ([3, 2], [[3, 1], [7, 2]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2, 4, 4], [[2, 1], [9, 4], [38, 17]]), ([3, 1, 1], [[3, 1], [4, 1], [7, 2]]), ([3, 1, 2], [[3, 1], [4, 1], [11, 3]]), ([3, 1, 3], [[3, 1], [4, 1], [15, 4]])], [([3, 1], [[3, 1], [4, 1]]), ([4, 1], [[4, 1], [5, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 1, 1], [[4, 1], [5, 1], [9, 2]]), ([4, 1, 2], [[4, 1], [5, 1], [14, 3]]), ([4, 1, 3], [[4, 1], [5, 1], [19, 4]]), ([4, 1, 4], [[4, 1], [5, 1], [24, 5]])]]\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":"bf520b58b3dafae09194e07e7c333fd85d6cd9a8ebc192eaaf20063f1f484322","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=x\n    p0,p1=0,1\n    q0,q1=1,0\n    out=[]\n    for c in a:\n     p=c+p0\n     q=c*q1+q0\n     out.append([p,q])\n     p0,p1,q0,q1=p1,p,q1,q\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 1], [[2, 1], [3, 1]]), ([1, 1], [[1, 1], [2, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2], [[2, 1]]), ([3], [[3, 1]]), ([4], [[4, 1]]), ([1, 2], [[1, 1], [3, 2]])], [([2, 2], [[2, 1], [5, 2]]), ([1, 4], [[1, 1], [5, 4]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 2], [[4, 1], [9, 2]]), ([4, 3], [[4, 1], [13, 3]]), ([4, 4], [[4, 1], [17, 4]]), ([1, 1, 1], [[1, 1], [2, 1], [3, 2]])], [([2, 3], [[2, 1], [7, 3]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([1, 4, 3], [[1, 1], [5, 4], [16, 13]]), ([1, 4, 4], [[1, 1], [5, 4], [21, 17]]), ([2, 1, 1], [[2, 1], [3, 1], [5, 2]]), ([2, 1, 2], [[2, 1], [3, 1], [8, 3]]), ([2, 1, 3], [[2, 1], [3, 1], [11, 4]])], [([2, 4], [[2, 1], [9, 4]]), ([3, 2], [[3, 1], [7, 2]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2, 4, 4], [[2, 1], [9, 4], [38, 17]]), ([3, 1, 1], [[3, 1], [4, 1], [7, 2]]), ([3, 1, 2], [[3, 1], [4, 1], [11, 3]]), ([3, 1, 3], [[3, 1], [4, 1], [15, 4]])], [([3, 1], [[3, 1], [4, 1]]), ([4, 1], [[4, 1], [5, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 1, 1], [[4, 1], [5, 1], [9, 2]]), ([4, 1, 2], [[4, 1], [5, 1], [14, 3]]), ([4, 1, 3], [[4, 1], [5, 1], [19, 4]]), ([4, 1, 4], [[4, 1], [5, 1], [24, 5]])]]\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":"9561e7435c078aaa411cf4bab318b08426a7a6887b975223e5693aac39aeb9ba","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=x\n    p0,p1=0,1\n    q0,q1=1,0\n    out=[]\n    for c in a:\n     p=c*p1+p0\n     q=c*q1+q0\n     out.append([p,q])\n     p0,p1,q0,q1=p1,p,q1,q\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 1], [[2, 1], [3, 1]]), ([1, 1], [[1, 1], [2, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2], [[2, 1]]), ([3], [[3, 1]]), ([4], [[4, 1]]), ([1, 2], [[1, 1], [3, 2]])], [([2, 2], [[2, 1], [5, 2]]), ([1, 4], [[1, 1], [5, 4]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 2], [[4, 1], [9, 2]]), ([4, 3], [[4, 1], [13, 3]]), ([4, 4], [[4, 1], [17, 4]]), ([1, 1, 1], [[1, 1], [2, 1], [3, 2]])], [([2, 3], [[2, 1], [7, 3]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([1, 4, 3], [[1, 1], [5, 4], [16, 13]]), ([1, 4, 4], [[1, 1], [5, 4], [21, 17]]), ([2, 1, 1], [[2, 1], [3, 1], [5, 2]]), ([2, 1, 2], [[2, 1], [3, 1], [8, 3]]), ([2, 1, 3], [[2, 1], [3, 1], [11, 4]])], [([2, 4], [[2, 1], [9, 4]]), ([3, 2], [[3, 1], [7, 2]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2, 4, 4], [[2, 1], [9, 4], [38, 17]]), ([3, 1, 1], [[3, 1], [4, 1], [7, 2]]), ([3, 1, 2], [[3, 1], [4, 1], [11, 3]]), ([3, 1, 3], [[3, 1], [4, 1], [15, 4]])], [([3, 1], [[3, 1], [4, 1]]), ([4, 1], [[4, 1], [5, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 1, 1], [[4, 1], [5, 1], [9, 2]]), ([4, 1, 2], [[4, 1], [5, 1], [14, 3]]), ([4, 1, 3], [[4, 1], [5, 1], [19, 4]]), ([4, 1, 4], [[4, 1], [5, 1], [24, 5]])]]\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-continued-fraction-convergents-numerator-recurrence","generated_at":"2026-09-29T14:39:17.087207+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 c*p1+p0 at the numerator recurrence step.","root_cause":"The numerator recurrence step uses c+p0 instead of c*p1+p0.","sha256":"e638cda14948563e908545f0824e4fd168eb2eed15a47c16e8633062ab101bcf","title":"Continued fraction convergents: numerator recurrence · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":44.215,"exit_code":1,"observations":[{"actual":[[2,1],[1,1]],"check":"explicit oracle 0","expected":[[2,1],[3,1]],"passed":false},{"actual":[[1,1],[0,1]],"check":"explicit oracle 1","expected":[[1,1],[2,1]],"passed":false},{"actual":[[1,1]],"check":"explicit oracle 2","expected":[[1,1]],"passed":true},{"actual":[[-2,1],[-7,3],[-26,13]],"check":"explicit oracle 3","expected":[[-2,1],[-5,3],[-22,13]],"passed":false},{"actual":[[2,1]],"check":"explicit oracle 4","expected":[[2,1]],"passed":true},{"actual":[[3,1]],"check":"explicit oracle 5","expected":[[3,1]],"passed":true},{"actual":[[4,1]],"check":"explicit oracle 6","expected":[[4,1]],"passed":true},{"actual":[[1,1],[1,2]],"check":"explicit oracle 7","expected":[[1,1],[3,2]],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[2, 1], [1, 1]], \"expected\": [[2, 1], [3, 1]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[1, 1], [0, 1]], \"expected\": [[1, 1], [2, 1]], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[1, 1]], \"expected\": [[1, 1]], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[-2, 1], [-7, 3], [-26, 13]], \"expected\": [[-2, 1], [-5, 3], [-22, 13]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[2, 1]], \"expected\": [[2, 1]], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[3, 1]], \"expected\": [[3, 1]], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[4, 1]], \"expected\": [[4, 1]], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[1, 1], [1, 2]], \"expected\": [[1, 1], [3, 2]], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.405,"exit_code":1,"observations":[{"actual":[[2,1],[2,1]],"check":"explicit oracle 0","expected":[[2,1],[3,1]],"passed":false},{"actual":[[1,1],[2,1]],"check":"explicit oracle 1","expected":[[1,1],[2,1]],"passed":true},{"actual":[[1,1]],"check":"explicit oracle 2","expected":[[1,1]],"passed":true},{"actual":[[-2,1],[4,3],[2,13]],"check":"explicit oracle 3","expected":[[-2,1],[-5,3],[-22,13]],"passed":false},{"actual":[[2,1]],"check":"explicit oracle 4","expected":[[2,1]],"passed":true},{"actual":[[3,1]],"check":"explicit oracle 5","expected":[[3,1]],"passed":true},{"actual":[[4,1]],"check":"explicit oracle 6","expected":[[4,1]],"passed":true},{"actual":[[1,1],[3,2]],"check":"explicit oracle 7","expected":[[1,1],[3,2]],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[2, 1], [2, 1]], \"expected\": [[2, 1], [3, 1]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[1, 1], [2, 1]], \"expected\": [[1, 1], [2, 1]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[1, 1]], \"expected\": [[1, 1]], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[-2, 1], [4, 3], [2, 13]], \"expected\": [[-2, 1], [-5, 3], [-22, 13]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[2, 1]], \"expected\": [[2, 1]], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[3, 1]], \"expected\": [[3, 1]], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[4, 1]], \"expected\": [[4, 1]], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[1, 1], [3, 2]], \"expected\": [[1, 1], [3, 2]], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":42.079,"exit_code":0,"observations":[{"actual":[[2,1],[3,1]],"check":"explicit oracle 0","expected":[[2,1],[3,1]],"passed":true},{"actual":[[1,1],[2,1]],"check":"explicit oracle 1","expected":[[1,1],[2,1]],"passed":true},{"actual":[[1,1]],"check":"explicit oracle 2","expected":[[1,1]],"passed":true},{"actual":[[-2,1],[-5,3],[-22,13]],"check":"explicit oracle 3","expected":[[-2,1],[-5,3],[-22,13]],"passed":true},{"actual":[[2,1]],"check":"explicit oracle 4","expected":[[2,1]],"passed":true},{"actual":[[3,1]],"check":"explicit oracle 5","expected":[[3,1]],"passed":true},{"actual":[[4,1]],"check":"explicit oracle 6","expected":[[4,1]],"passed":true},{"actual":[[1,1],[3,2]],"check":"explicit oracle 7","expected":[[1,1],[3,2]],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[2, 1], [3, 1]], \"expected\": [[2, 1], [3, 1]], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [[1, 1], [2, 1]], \"expected\": [[1, 1], [2, 1]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[1, 1]], \"expected\": [[1, 1]], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[-2, 1], [-5, 3], [-22, 13]], \"expected\": [[-2, 1], [-5, 3], [-22, 13]], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [[2, 1]], \"expected\": [[2, 1]], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[3, 1]], \"expected\": [[3, 1]], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[4, 1]], \"expected\": [[4, 1]], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[1, 1], [3, 2]], \"expected\": [[1, 1], [3, 2]], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}