{"abstract":"The exact bounded diophantine result violates the stated contract at second coordinate quotient.","category":"Numerics","checks":8,"contract":"Input [a,b,c,L], positive a,b and nonnegative L; lexicographically ordered nonnegative solutions a*u+b*v=c with u,v<=L.","evaluation_group":"s3-numerics-bounded-diophantine","failed_approach":"The partial repair abs(t)//b still violates the second coordinate quotient invariant.","family":"s3-numerics-bounded-diophantine-second-coordinate-quotient","id":"FA-14406","implementations":{"attempt":{"sha256":"d2f90e1aad23de560598865ddc875e7ed5ebe65cc440415b44f4a90b2090b63d","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,c,L=x\n    out=[]\n    for u in range(L+1):\n     t=c-a*u\n     if t%b != 0: continue\n     v=abs(t)//b\n     if 0<=v<=L: out.append([u,v])\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 2, 2, 1], [[0, 1]]), ([1, 1, -2, 2], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, -2, 1], []), ([1, 1, -2, 3], []), ([1, 1, -2, 4], []), ([1, 1, -2, 5], [])], [([1, 2, 2, 2], [[0, 1], [2, 0]]), ([1, 1, -2, 5], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 0, 5], [[0, 0]]), ([1, 1, 1, 0], []), ([1, 1, 1, 1], [[0, 1], [1, 0]]), ([1, 1, 1, 2], [[0, 1], [1, 0]])], [([1, 2, 2, 3], [[0, 1], [2, 0]]), ([1, 1, -1, 3], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 3, 4], [[0, 3], [1, 2], [2, 1], [3, 0]]), ([1, 1, 3, 5], [[0, 3], [1, 2], [2, 1], [3, 0]]), ([1, 1, 4, 0], []), ([1, 1, 4, 1], [])], [([1, 2, 2, 4], [[0, 1], [2, 0]]), ([1, 1, 0, 1], [[0, 0]]), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 6, 3], [[3, 3]]), ([1, 1, 6, 4], [[2, 4], [3, 3], [4, 2]]), ([1, 1, 6, 5], [[1, 5], [2, 4], [3, 3], [4, 2], [5, 1]]), ([1, 1, 7, 0], [])], [([1, 2, 2, 5], [[0, 1], [2, 0]]), ([1, 1, 0, 4], [[0, 0]]), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 9, 2], []), ([1, 1, 9, 3], []), ([1, 1, 9, 4], []), ([1, 1, 9, 5], [[4, 5], [5, 4]])]]\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":"e85eb4b6a8cfe2ded012210d2f617ec0223ba4673589d5623a4dee58228c2539","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,c,L=x\n    out=[]\n    for u in range(L+1):\n     t=c-a*u\n     if t%b != 0: continue\n     v=t\n     if 0<=v<=L: out.append([u,v])\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 2, 2, 1], [[0, 1]]), ([1, 1, -2, 2], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, -2, 1], []), ([1, 1, -2, 3], []), ([1, 1, -2, 4], []), ([1, 1, -2, 5], [])], [([1, 2, 2, 2], [[0, 1], [2, 0]]), ([1, 1, -2, 5], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 0, 5], [[0, 0]]), ([1, 1, 1, 0], []), ([1, 1, 1, 1], [[0, 1], [1, 0]]), ([1, 1, 1, 2], [[0, 1], [1, 0]])], [([1, 2, 2, 3], [[0, 1], [2, 0]]), ([1, 1, -1, 3], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 3, 4], [[0, 3], [1, 2], [2, 1], [3, 0]]), ([1, 1, 3, 5], [[0, 3], [1, 2], [2, 1], [3, 0]]), ([1, 1, 4, 0], []), ([1, 1, 4, 1], [])], [([1, 2, 2, 4], [[0, 1], [2, 0]]), ([1, 1, 0, 1], [[0, 0]]), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 6, 3], [[3, 3]]), ([1, 1, 6, 4], [[2, 4], [3, 3], [4, 2]]), ([1, 1, 6, 5], [[1, 5], [2, 4], [3, 3], [4, 2], [5, 1]]), ([1, 1, 7, 0], [])], [([1, 2, 2, 5], [[0, 1], [2, 0]]), ([1, 1, 0, 4], [[0, 0]]), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 9, 2], []), ([1, 1, 9, 3], []), ([1, 1, 9, 4], []), ([1, 1, 9, 5], [[4, 5], [5, 4]])]]\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":"20ac2f935cb7d1a68336db9481a6d82144953b0edaeafcf4d164217b59adf8b5","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,c,L=x\n    out=[]\n    for u in range(L+1):\n     t=c-a*u\n     if t%b != 0: continue\n     v=t//b\n     if 0<=v<=L: out.append([u,v])\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 2, 2, 1], [[0, 1]]), ([1, 1, -2, 2], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, -2, 1], []), ([1, 1, -2, 3], []), ([1, 1, -2, 4], []), ([1, 1, -2, 5], [])], [([1, 2, 2, 2], [[0, 1], [2, 0]]), ([1, 1, -2, 5], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 0, 5], [[0, 0]]), ([1, 1, 1, 0], []), ([1, 1, 1, 1], [[0, 1], [1, 0]]), ([1, 1, 1, 2], [[0, 1], [1, 0]])], [([1, 2, 2, 3], [[0, 1], [2, 0]]), ([1, 1, -1, 3], []), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 3, 4], [[0, 3], [1, 2], [2, 1], [3, 0]]), ([1, 1, 3, 5], [[0, 3], [1, 2], [2, 1], [3, 0]]), ([1, 1, 4, 0], []), ([1, 1, 4, 1], [])], [([1, 2, 2, 4], [[0, 1], [2, 0]]), ([1, 1, 0, 1], [[0, 0]]), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 6, 3], [[3, 3]]), ([1, 1, 6, 4], [[2, 4], [3, 3], [4, 2]]), ([1, 1, 6, 5], [[1, 5], [2, 4], [3, 3], [4, 2], [5, 1]]), ([1, 1, 7, 0], [])], [([1, 2, 2, 5], [[0, 1], [2, 0]]), ([1, 1, 0, 4], [[0, 0]]), ([1, 1, -2, 0], []), ([4, 4, 14, 5], []), ([1, 1, 9, 2], []), ([1, 1, 9, 3], []), ([1, 1, 9, 4], []), ([1, 1, 9, 5], [[4, 5], [5, 4]])]]\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-bounded-diophantine-second-coordinate-quotient","generated_at":"2026-09-29T14:39:16.490452+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 t//b at the second coordinate quotient step.","root_cause":"The second coordinate quotient step uses t instead of t//b.","sha256":"d960c93dfa21e3d37bfe1732108ad5dd51c8b9072ebe485c9b799329677685d3","title":"Bounded diophantine: second coordinate quotient · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":45.293,"exit_code":1,"observations":[{"actual":[[0,1]],"check":"explicit oracle 0","expected":[[0,1]],"passed":true},{"actual":[[0,2]],"check":"explicit oracle 1","expected":[],"passed":false},{"actual":[],"check":"explicit oracle 2","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 3","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 4","expected":[],"passed":true},{"actual":[[0,2],[1,3]],"check":"explicit oracle 5","expected":[],"passed":false},{"actual":[[0,2],[1,3],[2,4]],"check":"explicit oracle 6","expected":[],"passed":false},{"actual":[[0,2],[1,3],[2,4],[3,5]],"check":"explicit oracle 7","expected":[],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[0, 1]], \"expected\": [[0, 1]], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [[0, 2]], \"expected\": [], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[0, 2], [1, 3]], \"expected\": [], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [[0, 2], [1, 3], [2, 4]], \"expected\": [], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [[0, 2], [1, 3], [2, 4], [3, 5]], \"expected\": [], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.312,"exit_code":1,"observations":[{"actual":[],"check":"explicit oracle 0","expected":[[0,1]],"passed":false},{"actual":[],"check":"explicit oracle 1","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 2","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 3","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 4","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 5","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 6","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 7","expected":[],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [], \"expected\": [[0, 1]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [], \"expected\": [], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.638,"exit_code":0,"observations":[{"actual":[[0,1]],"check":"explicit oracle 0","expected":[[0,1]],"passed":true},{"actual":[],"check":"explicit oracle 1","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 2","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 3","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 4","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 5","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 6","expected":[],"passed":true},{"actual":[],"check":"explicit oracle 7","expected":[],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[0, 1]], \"expected\": [[0, 1]], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [], \"expected\": [], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}