{"abstract":"The exact extended euclidean certificate result violates the stated contract at first Bezout seed.","category":"Numerics","checks":8,"contract":"Input [a,b], integers not both zero; return [g,u,v] with g>=0 and a*u+b*v=g using standard Euclidean quotient recurrence. Bounds: |a|,|b|<=1000000.","evaluation_group":"s3-numerics-extended-euclidean-certificate","failed_approach":"The partial repair 1,1 still violates the first Bezout seed invariant.","family":"s3-numerics-extended-euclidean-certificate-first-Bezout-seed","id":"FA-14866","implementations":{"attempt":{"sha256":"1292d746dff23eca985b44eb357a1c4e7cf197130dbf63242e82102587cf0f1f","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,b=x\n    r0,r1=a,b\n    s0,s1=1,1\n    t0,t1=0,1\n    for _ in range(60):\n     if r1==0:break\n     q=r0//r1\n     r0,r1=r1,r0-q*r1\n     s0,s1=s1,s0-q*s1\n     t0,t1=t1,t0-q*t1\n    if r0<0:r0,s0,t0=-r0,-s0,-t0\n    return [r0,s0,t0]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -19], [1, -1, 1]), ([-20, -18], [2, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -16], [4, -1, 1]), ([-20, -15], [5, -1, 1]), ([-20, -14], [2, 2, -3])], [([-20, -19], [1, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -3], [1, 1, -7]), ([-20, -2], [2, 0, -1]), ([-20, -1], [1, 0, -1]), ([-20, 0], [20, -1, 0])], [([-20, -18], [2, -1, 1]), ([-20, -14], [2, 2, -3]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, 14], [2, 2, 3]), ([-20, 15], [5, -1, -1]), ([-20, 16], [4, -1, -1]), ([-20, 17], [1, -6, -7])], [([-20, -17], [1, -6, 7]), ([-20, -11], [1, -5, 9]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, -10], [1, 1, -2]), ([-19, -9], [1, -1, 2]), ([-19, -8], [1, -3, 7]), ([-19, -7], [1, -3, 8])], [([-20, -16], [4, -1, 1]), ([-20, -8], [4, -1, 2]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, 7], [1, -3, -8]), ([-19, 8], [1, -3, -7]), ([-19, 9], [1, -1, -2]), ([-19, 10], [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":"21dab8624c6714815cb17a955fde95a3c298603c45720b0d39f2a35d95028f10","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,b=x\n    r0,r1=a,b\n    s0,s1=0,1\n    t0,t1=0,1\n    for _ in range(60):\n     if r1==0:break\n     q=r0//r1\n     r0,r1=r1,r0-q*r1\n     s0,s1=s1,s0-q*s1\n     t0,t1=t1,t0-q*t1\n    if r0<0:r0,s0,t0=-r0,-s0,-t0\n    return [r0,s0,t0]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -19], [1, -1, 1]), ([-20, -18], [2, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -16], [4, -1, 1]), ([-20, -15], [5, -1, 1]), ([-20, -14], [2, 2, -3])], [([-20, -19], [1, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -3], [1, 1, -7]), ([-20, -2], [2, 0, -1]), ([-20, -1], [1, 0, -1]), ([-20, 0], [20, -1, 0])], [([-20, -18], [2, -1, 1]), ([-20, -14], [2, 2, -3]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, 14], [2, 2, 3]), ([-20, 15], [5, -1, -1]), ([-20, 16], [4, -1, -1]), ([-20, 17], [1, -6, -7])], [([-20, -17], [1, -6, 7]), ([-20, -11], [1, -5, 9]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, -10], [1, 1, -2]), ([-19, -9], [1, -1, 2]), ([-19, -8], [1, -3, 7]), ([-19, -7], [1, -3, 8])], [([-20, -16], [4, -1, 1]), ([-20, -8], [4, -1, 2]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, 7], [1, -3, -8]), ([-19, 8], [1, -3, -7]), ([-19, 9], [1, -1, -2]), ([-19, 10], [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"},"fixed":{"sha256":"106e76ad395b8e5b6838ec1fa5b2dbe3e99d1b812b3d70ad55fc4f07cc7b6e8b","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,b=x\n    r0,r1=a,b\n    s0,s1=1,0\n    t0,t1=0,1\n    for _ in range(60):\n     if r1==0:break\n     q=r0//r1\n     r0,r1=r1,r0-q*r1\n     s0,s1=s1,s0-q*s1\n     t0,t1=t1,t0-q*t1\n    if r0<0:r0,s0,t0=-r0,-s0,-t0\n    return [r0,s0,t0]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -19], [1, -1, 1]), ([-20, -18], [2, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -16], [4, -1, 1]), ([-20, -15], [5, -1, 1]), ([-20, -14], [2, 2, -3])], [([-20, -19], [1, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -3], [1, 1, -7]), ([-20, -2], [2, 0, -1]), ([-20, -1], [1, 0, -1]), ([-20, 0], [20, -1, 0])], [([-20, -18], [2, -1, 1]), ([-20, -14], [2, 2, -3]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, 14], [2, 2, 3]), ([-20, 15], [5, -1, -1]), ([-20, 16], [4, -1, -1]), ([-20, 17], [1, -6, -7])], [([-20, -17], [1, -6, 7]), ([-20, -11], [1, -5, 9]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, -10], [1, 1, -2]), ([-19, -9], [1, -1, 2]), ([-19, -8], [1, -3, 7]), ([-19, -7], [1, -3, 8])], [([-20, -16], [4, -1, 1]), ([-20, -8], [4, -1, 2]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, 7], [1, -3, -8]), ([-19, 8], [1, -3, -7]), ([-19, 9], [1, -1, -2]), ([-19, 10], [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-extended-euclidean-certificate-first-Bezout-seed","generated_at":"2026-09-29T14:39:21.152947+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 1,0 at the first Bezout seed step.","root_cause":"The first Bezout seed step uses 0,1 instead of 1,0.","sha256":"cc73736fcd4031faff53fde212d6f1d83ed75da4211c4afef77fcfa96c25325e","title":"Extended euclidean certificate: first Bezout seed · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":42.219,"exit_code":1,"observations":[{"actual":[20,-1,-1],"check":"explicit oracle 0","expected":[20,0,-1],"passed":false},{"actual":[20,1,1],"check":"explicit oracle 1","expected":[20,0,1],"passed":false},{"actual":[1,0,1],"check":"explicit oracle 2","expected":[1,-1,1],"passed":false},{"actual":[2,0,1],"check":"explicit oracle 3","expected":[2,-1,1],"passed":false},{"actual":[1,1,7],"check":"explicit oracle 4","expected":[1,-6,7],"passed":false},{"actual":[4,0,1],"check":"explicit oracle 5","expected":[4,-1,1],"passed":false},{"actual":[5,0,1],"check":"explicit oracle 6","expected":[5,-1,1],"passed":false},{"actual":[2,-1,-3],"check":"explicit oracle 7","expected":[2,2,-3],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [20, -1, -1], \"expected\": [20, 0, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 1, 1], \"expected\": [20, 0, 1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 0, 1], \"expected\": [1, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [2, 0, 1], \"expected\": [2, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1, 1, 7], \"expected\": [1, -6, 7], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [4, 0, 1], \"expected\": [4, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [5, 0, 1], \"expected\": [5, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [2, -1, -3], \"expected\": [2, 2, -3], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.725,"exit_code":1,"observations":[{"actual":[20,-1,-1],"check":"explicit oracle 0","expected":[20,0,-1],"passed":false},{"actual":[20,1,1],"check":"explicit oracle 1","expected":[20,0,1],"passed":false},{"actual":[1,1,1],"check":"explicit oracle 2","expected":[1,-1,1],"passed":false},{"actual":[2,1,1],"check":"explicit oracle 3","expected":[2,-1,1],"passed":false},{"actual":[1,7,7],"check":"explicit oracle 4","expected":[1,-6,7],"passed":false},{"actual":[4,1,1],"check":"explicit oracle 5","expected":[4,-1,1],"passed":false},{"actual":[5,1,1],"check":"explicit oracle 6","expected":[5,-1,1],"passed":false},{"actual":[2,-3,-3],"check":"explicit oracle 7","expected":[2,2,-3],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [20, -1, -1], \"expected\": [20, 0, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 1, 1], \"expected\": [20, 0, 1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 1, 1], \"expected\": [1, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [2, 1, 1], \"expected\": [2, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1, 7, 7], \"expected\": [1, -6, 7], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [4, 1, 1], \"expected\": [4, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [5, 1, 1], \"expected\": [5, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [2, -3, -3], \"expected\": [2, 2, -3], \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":39.606,"exit_code":0,"observations":[{"actual":[20,0,-1],"check":"explicit oracle 0","expected":[20,0,-1],"passed":true},{"actual":[20,0,1],"check":"explicit oracle 1","expected":[20,0,1],"passed":true},{"actual":[1,-1,1],"check":"explicit oracle 2","expected":[1,-1,1],"passed":true},{"actual":[2,-1,1],"check":"explicit oracle 3","expected":[2,-1,1],"passed":true},{"actual":[1,-6,7],"check":"explicit oracle 4","expected":[1,-6,7],"passed":true},{"actual":[4,-1,1],"check":"explicit oracle 5","expected":[4,-1,1],"passed":true},{"actual":[5,-1,1],"check":"explicit oracle 6","expected":[5,-1,1],"passed":true},{"actual":[2,2,-3],"check":"explicit oracle 7","expected":[2,2,-3],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [20, 0, -1], \"expected\": [20, 0, -1], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 0, 1], \"expected\": [20, 0, 1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [1, -1, 1], \"expected\": [1, -1, 1], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [2, -1, 1], \"expected\": [2, -1, 1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [1, -6, 7], \"expected\": [1, -6, 7], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [4, -1, 1], \"expected\": [4, -1, 1], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [5, -1, 1], \"expected\": [5, -1, 1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [2, 2, -3], \"expected\": [2, 2, -3], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}