{"abstract":"The exact pell positive solutions result violates the stated contract at quadratic unit real product.","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*u-D*b*v still violates the quadratic unit real product invariant.","family":"s3-numerics-pell-positive-solutions-quadratic-unit-real-product","id":"FA-14926","implementations":{"attempt":{"sha256":"7465f0b3c1ac4ba2357e80daa22a14db1653b6d8fc511b4ea2f4d234261fd42d","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, 2], [[3, 2], [17, 12]]), ([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, 1], [[3, 2]]), ([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, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([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, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 8], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860], [114243, 80782], [665857, 470832]]), ([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, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([3, 2, 1, 4], [[2, 1], [7, 4], [26, 15], [97, 56]]), ([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, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]]), ([3, 2, 1, 7], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911]]), ([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":"1047fd6e2358e7714fd1b23d0a3acad59071510258cf4aab9cee87d7619f9094","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+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, 2], [[3, 2], [17, 12]]), ([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, 1], [[3, 2]]), ([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, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([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, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 8], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860], [114243, 80782], [665857, 470832]]), ([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, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([3, 2, 1, 4], [[2, 1], [7, 4], [26, 15], [97, 56]]), ([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, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]]), ([3, 2, 1, 7], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911]]), ([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-quadratic-unit-real-product","generated_at":"2026-09-29T14:39:21.795919+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 quadratic unit real product step uses a*u+b*v instead of a*u+D*b*v.","sha256":"9383d6e2e2a7b34ff3281bcfa1ffcdcb2504bfb0a018c8ea1038a046b0ff7c10","title":"Pell positive solutions: quadratic unit real product · 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":45.508,"exit_code":1,"observations":[{"actual":[[3,2],[1,12]],"check":"explicit oracle 0","expected":[[3,2],[17,12]],"passed":false},{"actual":[],"check":"explicit oracle 1","expected":[],"passed":true},{"actual":[[649,180],[1,233640],[-546716951,151632540],[-709639444799,467280],[-460557093109751,-127734796799100],[-2128918334399,-165800159882371080],[387970992456749302249,-107604686968959022740],[503588141611794410371201,-663200639531353440]],"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]],"check":"explicit oracle 3","expected":[[3,2]],"passed":true},{"actual":[[3,2],[1,12],[-45,38]],"check":"explicit oracle 4","expected":[[3,2],[17,12],[99,70]],"passed":false},{"actual":[[3,2],[1,12],[-45,38],[-287,24]],"check":"explicit oracle 5","expected":[[3,2],[17,12],[99,70],[577,408]],"passed":false},{"actual":[[3,2],[1,12],[-45,38],[-287,24],[-957,-502]],"check":"explicit oracle 6","expected":[[3,2],[17,12],[99,70],[577,408],[3363,2378]],"passed":false},{"actual":[[3,2],[1,12],[-45,38],[-287,24],[-957,-502],[-863,-3420]],"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], [1, 12]], \"expected\": [[3, 2], [17, 12]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[649, 180], [1, 233640], [-546716951, 151632540], [-709639444799, 467280], [-460557093109751, -127734796799100], [-2128918334399, -165800159882371080], [387970992456749302249, -107604686968959022740], [503588141611794410371201, -663200639531353440]], \"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]], \"expected\": [[3, 2]], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [[3, 2], [1, 12], [-45, 38]], \"expected\": [[3, 2], [17, 12], [99, 70]], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [[3, 2], [1, 12], [-45, 38], [-287, 24]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408]], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [[3, 2], [1, 12], [-45, 38], [-287, 24], [-957, -502]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [[3, 2], [1, 12], [-45, 38], [-287, 24], [-957, -502], [-863, -3420]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.998,"exit_code":1,"observations":[{"actual":[[3,2],[13,12]],"check":"explicit oracle 0","expected":[[3,2],[17,12]],"passed":false},{"actual":[],"check":"explicit oracle 1","expected":[],"passed":true},{"actual":[[649,180],[453601,233640],[336442249,233280540],[260341516801,211958675280],[207114205954249,184422653280900],[167613197254869601,156970859051068920],[137035719647602776649,132044463030020257260],[112704185396697848352001,110363286043051646758560]],"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]],"check":"explicit oracle 3","expected":[[3,2]],"passed":true},{"actual":[[3,2],[13,12],[63,62]],"check":"explicit oracle 4","expected":[[3,2],[17,12],[99,70]],"passed":false},{"actual":[[3,2],[13,12],[63,62],[313,312]],"check":"explicit oracle 5","expected":[[3,2],[17,12],[99,70],[577,408]],"passed":false},{"actual":[[3,2],[13,12],[63,62],[313,312],[1563,1562]],"check":"explicit oracle 6","expected":[[3,2],[17,12],[99,70],[577,408],[3363,2378]],"passed":false},{"actual":[[3,2],[13,12],[63,62],[313,312],[1563,1562],[7813,7812]],"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], [13, 12]], \"expected\": [[3, 2], [17, 12]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[649, 180], [453601, 233640], [336442249, 233280540], [260341516801, 211958675280], [207114205954249, 184422653280900], [167613197254869601, 156970859051068920], [137035719647602776649, 132044463030020257260], [112704185396697848352001, 110363286043051646758560]], \"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]], \"expected\": [[3, 2]], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [[3, 2], [13, 12], [63, 62]], \"expected\": [[3, 2], [17, 12], [99, 70]], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [[3, 2], [13, 12], [63, 62], [313, 312]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408]], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [[3, 2], [13, 12], [63, 62], [313, 312], [1563, 1562]], \"expected\": [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [[3, 2], [13, 12], [63, 62], [313, 312], [1563, 1562], [7813, 7812]], \"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."}}