{"abstract":"The exact extended euclidean certificate result violates the stated contract at second coefficient recurrence.","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 t0,t0-q*t1 still violates the second coefficient recurrence invariant.","family":"s3-numerics-extended-euclidean-certificate-second-coefficient-recurrence","id":"FA-14881","implementations":{"attempt":{"sha256":"be5a0062c1560800fd718c358c61c70783b440fef54368b7517eb98929f6e17c","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=t0,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, -19], [1, -1, 1]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 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, -18], [2, -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, -17], [1, -6, 7]), ([-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, -16], [4, -1, 1]), ([-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, -15], [5, -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":"a79b6b081bd915813e6048f634ad0cdadb17eac6901ebe350fbfcc010655d8aa","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, -19], [1, -1, 1]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 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, -18], [2, -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, -17], [1, -6, 7]), ([-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, -16], [4, -1, 1]), ([-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, -15], [5, -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":"a4fc93ab0f1b0dd512ad1283a06b7c26eb7ac4fdcb0900fa0aa83979154e6f3a","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, -19], [1, -1, 1]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 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, -18], [2, -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, -17], [1, -6, 7]), ([-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, -16], [4, -1, 1]), ([-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, -15], [5, -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-coefficient-recurrence","generated_at":"2026-09-29T14:39:21.250515+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 t1,t0-q*t1 at the second coefficient recurrence step.","root_cause":"The second coefficient recurrence step uses t1,t0+q*t1 instead of t1,t0-q*t1.","sha256":"93dbb21d4fb9b9e19536c0f537e983d761415697e16bf7011c33774bb5aeb7c6","title":"Extended euclidean certificate: second coefficient recurrence · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":44.49,"exit_code":1,"observations":[{"actual":[1,-1,0],"check":"explicit oracle 0","expected":[1,-1,1],"passed":false},{"actual":[20,0,0],"check":"explicit oracle 1","expected":[20,0,-1],"passed":false},{"actual":[20,0,0],"check":"explicit oracle 2","expected":[20,0,1],"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\": [1, -1, 0], \"expected\": [1, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 0, 0], \"expected\": [20, 0, -1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [20, 0, 0], \"expected\": [20, 0, 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":43.853,"exit_code":1,"observations":[{"actual":[1,-1,-1],"check":"explicit oracle 0","expected":[1,-1,1],"passed":false},{"actual":[20,0,-1],"check":"explicit oracle 1","expected":[20,0,-1],"passed":true},{"actual":[20,0,1],"check":"explicit oracle 2","expected":[20,0,1],"passed":true},{"actual":[2,-1,-1],"check":"explicit oracle 3","expected":[2,-1,1],"passed":false},{"actual":[1,-6,-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":[2,2,-3],"check":"explicit oracle 7","expected":[2,2,-3],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [1, -1, -1], \"expected\": [1, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 0, -1], \"expected\": [20, 0, -1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [20, 0, 1], \"expected\": [20, 0, 1], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [2, -1, -1], \"expected\": [2, -1, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1, -6, -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, 2, -3], \"expected\": [2, 2, -3], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.282,"exit_code":0,"observations":[{"actual":[1,-1,1],"check":"explicit oracle 0","expected":[1,-1,1],"passed":true},{"actual":[20,0,-1],"check":"explicit oracle 1","expected":[20,0,-1],"passed":true},{"actual":[20,0,1],"check":"explicit oracle 2","expected":[20,0,1],"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\": [1, -1, 1], \"expected\": [1, -1, 1], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [20, 0, -1], \"expected\": [20, 0, -1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [20, 0, 1], \"expected\": [20, 0, 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"}