{"abstract":"The exact pell positive solutions result violates the stated contract at coordinate pair representation.","category":"Numerics","checks":8,"contract":"Input [D,u,v,k], known positive fundamental solution u^2-D*v^2=1, k>=0; return first k positive solutions.","contract_signature":"x","evaluation_group":"s3-numerics-pell-positive-solutions","failed_approach":"The partial repair [a,-b] still violates the coordinate pair representation invariant.","family":"s3-numerics-pell-positive-solutions-coordinate-pair-representation","id":"FA-14936","implementations":{"attempt":{"sha256":"e865704f520a4fd3db6b3e7ec7513a71819335ff4a84c39bc1e76d61c600f80a","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    D,u,v,k=x\n    a,b=1,0\n    out=[]\n    for _ in range(k):\n     a,b=a*u+D*b*v,a*v+b*u\n     out.append([a,-b])\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 3, 2, 1], [[3, 2]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([2, 3, 2, 2], [[3, 2], [17, 12]]), ([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([2, 3, 2, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]])], [([2, 3, 2, 2], [[3, 2], [17, 12]]), ([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([3, 2, 1, 8], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911], [18817, 10864]]), ([5, 9, 4, 0], []), ([5, 9, 4, 1], [[9, 4]]), ([5, 9, 4, 2], [[9, 4], [161, 72]])], [([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 7], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860], [114243, 80782]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([6, 5, 2, 7], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198]]), ([6, 5, 2, 8], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198], [46099201, 18819920]]), ([7, 8, 3, 0], []), ([7, 8, 3, 1], [[8, 3]])], [([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([3, 2, 1, 2], [[2, 1], [7, 4]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([10, 19, 6, 6], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764]]), ([10, 19, 6, 7], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202]]), ([10, 19, 6, 8], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202], [2161873163521, 683644320912]]), ([11, 10, 3, 0], [])], [([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([3, 2, 1, 5], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([12, 7, 2, 5], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658]]), ([12, 7, 2, 6], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780]]), ([12, 7, 2, 7], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262]]), ([12, 7, 2, 8], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262], [708158977, 204427888]])]]\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":"868c6fa2be1cf68fc3b324b23b802f7c9616be6cbd73455351e9e1f4119c59de","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    D,u,v,k=x\n    a,b=1,0\n    out=[]\n    for _ in range(k):\n     a,b=a*u+D*b*v,a*v+b*u\n     out.append([b,a])\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 3, 2, 1], [[3, 2]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([2, 3, 2, 2], [[3, 2], [17, 12]]), ([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([2, 3, 2, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]])], [([2, 3, 2, 2], [[3, 2], [17, 12]]), ([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([3, 2, 1, 8], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911], [18817, 10864]]), ([5, 9, 4, 0], []), ([5, 9, 4, 1], [[9, 4]]), ([5, 9, 4, 2], [[9, 4], [161, 72]])], [([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 7], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860], [114243, 80782]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([6, 5, 2, 7], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198]]), ([6, 5, 2, 8], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198], [46099201, 18819920]]), ([7, 8, 3, 0], []), ([7, 8, 3, 1], [[8, 3]])], [([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([3, 2, 1, 2], [[2, 1], [7, 4]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([10, 19, 6, 6], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764]]), ([10, 19, 6, 7], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202]]), ([10, 19, 6, 8], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202], [2161873163521, 683644320912]]), ([11, 10, 3, 0], [])], [([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([3, 2, 1, 5], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([12, 7, 2, 5], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658]]), ([12, 7, 2, 6], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780]]), ([12, 7, 2, 7], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262]]), ([12, 7, 2, 8], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262], [708158977, 204427888]])]]\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-pell-positive-solutions-coordinate-pair-representation","generated_at":"2026-09-29T14:39:21.835368+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 coordinate pair representation step uses [b,a] instead of [a,b].","sha256":"230f7b3dd419ad049ff634171f81a85d3cc24649f1378609ee11984577595637","title":"Pell positive solutions: coordinate pair representation · 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":42.421,"exit_code":1,"observations":[{"actual":[[3,-2]],"check":"explicit oracle 0","expected":[[3,2]],"passed":false},{"actual":[],"check":"explicit oracle 1","expected":[],"passed":true},{"actual":[[649,-180],[842401,-233640],[1093435849,-303264540],[1419278889601,-393637139280],[1842222905266249,-510940703520900],[2391203911756701601,-663200639532988920],[3103780835237293411849,-860833919173116097260],[4028705132934095091878401,-1117361763886065161254560]],"check":"explicit oracle 2","expected":[[649,180],[842401,233640],[1093435849,303264540],[1419278889601,393637139280],[1842222905266249,510940703520900],[2391203911756701601,663200639532988920],[3103780835237293411849,860833919173116097260],[4028705132934095091878401,1117361763886065161254560]],"passed":false},{"actual":[[3,-2],[17,-12]],"check":"explicit oracle 3","expected":[[3,2],[17,12]],"passed":false},{"actual":[[3,-2],[17,-12],[99,-70]],"check":"explicit oracle 4","expected":[[3,2],[17,12],[99,70]],"passed":false},{"actual":[[3,-2],[17,-12],[99,-70],[577,-408]],"check":"explicit oracle 5","expected":[[3,2],[17,12],[99,70],[577,408]],"passed":false},{"actual":[[3,-2],[17,-12],[99,-70],[577,-408],[3363,-2378]],"check":"explicit oracle 6","expected":[[3,2],[17,12],[99,70],[577,408],[3363,2378]],"passed":false},{"actual":[[3,-2],[17,-12],[99,-70],[577,-408],[3363,-2378],[19601,-13860]],"check":"explicit oracle 7","expected":[[3,2],[17,12],[99,70],[577,408],[3363,2378],[19601,13860]],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[3, -2]], \"expected\": [[3, 2]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[649, -180], [842401, -233640], [1093435849, -303264540], [1419278889601, -393637139280], [1842222905266249, -510940703520900], [2391203911756701601, -663200639532988920], [3103780835237293411849, -860833919173116097260], [4028705132934095091878401, -1117361763886065161254560]], \"expected\": [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [[3, -2], [17, -12]], \"expected\": [[3, 2], [17, 12]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[3, -2], [17, -12], [99, -70]], \"expected\": [[3, 2], [17, 12], [99, 70]], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [[3, -2], [17, -12], [99, -70], [577, -408]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408]], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [[3, -2], [17, -12], [99, -70], [577, -408], [3363, -2378]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [[3, -2], [17, -12], [99, -70], [577, -408], [3363, -2378], [19601, -13860]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.631,"exit_code":1,"observations":[{"actual":[[2,3]],"check":"explicit oracle 0","expected":[[3,2]],"passed":false},{"actual":[],"check":"explicit oracle 1","expected":[],"passed":true},{"actual":[[180,649],[233640,842401],[303264540,1093435849],[393637139280,1419278889601],[510940703520900,1842222905266249],[663200639532988920,2391203911756701601],[860833919173116097260,3103780835237293411849],[1117361763886065161254560,4028705132934095091878401]],"check":"explicit oracle 2","expected":[[649,180],[842401,233640],[1093435849,303264540],[1419278889601,393637139280],[1842222905266249,510940703520900],[2391203911756701601,663200639532988920],[3103780835237293411849,860833919173116097260],[4028705132934095091878401,1117361763886065161254560]],"passed":false},{"actual":[[2,3],[12,17]],"check":"explicit oracle 3","expected":[[3,2],[17,12]],"passed":false},{"actual":[[2,3],[12,17],[70,99]],"check":"explicit oracle 4","expected":[[3,2],[17,12],[99,70]],"passed":false},{"actual":[[2,3],[12,17],[70,99],[408,577]],"check":"explicit oracle 5","expected":[[3,2],[17,12],[99,70],[577,408]],"passed":false},{"actual":[[2,3],[12,17],[70,99],[408,577],[2378,3363]],"check":"explicit oracle 6","expected":[[3,2],[17,12],[99,70],[577,408],[3363,2378]],"passed":false},{"actual":[[2,3],[12,17],[70,99],[408,577],[2378,3363],[13860,19601]],"check":"explicit oracle 7","expected":[[3,2],[17,12],[99,70],[577,408],[3363,2378],[19601,13860]],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[2, 3]], \"expected\": [[3, 2]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[180, 649], [233640, 842401], [303264540, 1093435849], [393637139280, 1419278889601], [510940703520900, 1842222905266249], [663200639532988920, 2391203911756701601], [860833919173116097260, 3103780835237293411849], [1117361763886065161254560, 4028705132934095091878401]], \"expected\": [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [[2, 3], [12, 17]], \"expected\": [[3, 2], [17, 12]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[2, 3], [12, 17], [70, 99]], \"expected\": [[3, 2], [17, 12], [99, 70]], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [[2, 3], [12, 17], [70, 99], [408, 577]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408]], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [[2, 3], [12, 17], [70, 99], [408, 577], [2378, 3363]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [[2, 3], [12, 17], [70, 99], [408, 577], [2378, 3363], [13860, 19601]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]], \"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."}}