{"abstract":"The exact extended euclidean certificate result violates the stated contract at second 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.","contract_signature":"x","evaluation_group":"s3-numerics-extended-euclidean-certificate","failed_approach":"The partial repair 0,0 still violates the second Bezout seed invariant.","family":"s3-numerics-extended-euclidean-certificate-second-Bezout-seed","id":"FA-14871","implementations":{"attempt":{"sha256":"ebbfcadf4576b11f0a23992b28e10394f8c775ba42511fea505c6a69dc8796dd","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=0,0\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":"755112fb09a95304f724c34a20768a2fa660cd6c0694b9073ed64d1b7afb5fb5","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=1,0\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-second-Bezout-seed","generated_at":"2026-09-29T14:39:21.245980+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 second Bezout seed step uses 1,0 instead of 0,1.","sha256":"68cd60a781c1eadcf7a0e36d802c9323f6765719d401ef49d5280475ed0fb304","title":"Extended euclidean certificate: second Bezout seed · 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":39.728,"exit_code":1,"observations":[{"actual":[20,0,0],"check":"explicit oracle 0","expected":[20,0,-1],"passed":false},{"actual":[20,0,0],"check":"explicit oracle 1","expected":[20,0,1],"passed":false},{"actual":[1,-1,0],"check":"explicit oracle 2","expected":[1,-1,1],"passed":false},{"actual":[2,-1,0],"check":"explicit oracle 3","expected":[2,-1,1],"passed":false},{"actual":[1,-6,0],"check":"explicit oracle 4","expected":[1,-6,7],"passed":false},{"actual":[4,-1,0],"check":"explicit oracle 5","expected":[4,-1,1],"passed":false},{"actual":[5,-1,0],"check":"explicit oracle 6","expected":[5,-1,1],"passed":false},{"actual":[2,2,0],"check":"explicit oracle 7","expected":[2,2,-3],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [20, 0, 0], \"expected\": [20, 0, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 0, 0], \"expected\": [20, 0, 1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1, -1, 0], \"expected\": [1, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [2, -1, 0], \"expected\": [2, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1, -6, 0], \"expected\": [1, -6, 7], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [4, -1, 0], \"expected\": [4, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [5, -1, 0], \"expected\": [5, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [2, 2, 0], \"expected\": [2, 2, -3], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.433,"exit_code":1,"observations":[{"actual":[20,0,0],"check":"explicit oracle 0","expected":[20,0,-1],"passed":false},{"actual":[20,0,0],"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,-6,-6],"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":[2,2,2],"check":"explicit oracle 7","expected":[2,2,-3],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [20, 0, 0], \"expected\": [20, 0, -1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 0, 0], \"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, -6, -6], \"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, 2, 2], \"expected\": [2, 2, -3], \"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."}}