{"abstract":"The reduction disagrees with its explicit aggregation oracle.","category":"Numerical aggregation","checks":12,"contract":"Rows are integer [value, nonnegative frequency]; 0<=p<=q, q>0. Return the smallest supported value whose cumulative positive frequency reaches p/q of total. At p=0 return minimum positive support. No positive mass returns None.","evaluation_group":"s3-na-frequency-left-quantile","failed_approach":"Adding one count also shifts nonintegral rank thresholds.","family":"s3-numerical-aggregation-frequency-left-quantile-threshold-strict","id":"FA-13101","implementations":{"attempt":{"sha256":"8c97b0d8991794a8d23a7c28801ec3e74a1d99e66f2fd6137a42ffa9531eefde","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nfrom collections import Counter, defaultdict\nimport math\nimport itertools\nN = 1\nobservations = []\ndef solve(rows, p, q):\n    rows=sorted((x,w) for x,w in rows if w>0)\n    if not rows: return None\n    total=sum(w for x,w in rows)\n    threshold=Fraction(p,q)*total\n    running=0\n    for x,w in rows:\n        running+=w\n        if running>=threshold+1: return x\n    return rows[-1][0]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(-8, 1), (-2, 1), (3, 1)], 2, 5)), -2)\ncheck('regression 2', solve(*([(1, 2), (8, 5)], 1, 2)), 8)\ncheck('regression 3', solve(*([(1, 1), (8, 2)], 2, 5)), 8)\ncheck('regression 4', solve(*([(9, 1), (1, 3), (5, 2)], 1, 2)), 1)\ncheck('regression 5', solve(*([(8, 0), (2, 1)], 0, 1)), 2)\ncheck('regression 6', solve(*([], 1, 2)), None)\ncheck('regression 7', solve(*([(1, 0), (5, 0)], 1, 2)), None)\ncheck('regression 8', solve(*([(1, 1), (4, 2), (8, 1)], 3, 4)), 4)\ncheck('regression 9', solve(*([(9, 4), (2, 1)], 1, 1)), 9)\ncheck('regression 10', solve(*([(3, 2), (3, 1), (1, 1)], 1, 2)), 3)\ncheck('regression 11', solve(*([(2, 2), (7, 2)], 1, 2)), 2)\ncheck(\"variable rank\",solve([(N,2),(N+4,1)],1,2),N)\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":"f4441b9a1f3250e6e21b8a8027c75308821dcf7fe79b8125b62e28ba39ae7dfc","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nfrom collections import Counter, defaultdict\nimport math\nimport itertools\nN = 1\nobservations = []\ndef solve(rows, p, q):\n    rows=sorted((x,w) for x,w in rows if w>0)\n    if not rows: return None\n    total=sum(w for x,w in rows)\n    threshold=Fraction(p,q)*total\n    running=0\n    for x,w in rows:\n        running+=w\n        if running>threshold: return x\n    return rows[-1][0]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(-8, 1), (-2, 1), (3, 1)], 2, 5)), -2)\ncheck('regression 2', solve(*([(1, 2), (8, 5)], 1, 2)), 8)\ncheck('regression 3', solve(*([(1, 1), (8, 2)], 2, 5)), 8)\ncheck('regression 4', solve(*([(9, 1), (1, 3), (5, 2)], 1, 2)), 1)\ncheck('regression 5', solve(*([(8, 0), (2, 1)], 0, 1)), 2)\ncheck('regression 6', solve(*([], 1, 2)), None)\ncheck('regression 7', solve(*([(1, 0), (5, 0)], 1, 2)), None)\ncheck('regression 8', solve(*([(1, 1), (4, 2), (8, 1)], 3, 4)), 4)\ncheck('regression 9', solve(*([(9, 4), (2, 1)], 1, 1)), 9)\ncheck('regression 10', solve(*([(3, 2), (3, 1), (1, 1)], 1, 2)), 3)\ncheck('regression 11', solve(*([(2, 2), (7, 2)], 1, 2)), 2)\ncheck(\"variable rank\",solve([(N,2),(N+4,1)],1,2),N)\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":"551bcdb223f6329978a6042ae9bafef33f637bb02ab4034980f21b6095458761","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nfrom collections import Counter, defaultdict\nimport math\nimport itertools\nN = 1\nobservations = []\ndef solve(rows, p, q):\n    rows=sorted((x,w) for x,w in rows if w>0)\n    if not rows: return None\n    total=sum(w for x,w in rows)\n    threshold=Fraction(p,q)*total\n    running=0\n    for x,w in rows:\n        running+=w\n        if running>=threshold: return x\n    return rows[-1][0]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(-8, 1), (-2, 1), (3, 1)], 2, 5)), -2)\ncheck('regression 2', solve(*([(1, 2), (8, 5)], 1, 2)), 8)\ncheck('regression 3', solve(*([(1, 1), (8, 2)], 2, 5)), 8)\ncheck('regression 4', solve(*([(9, 1), (1, 3), (5, 2)], 1, 2)), 1)\ncheck('regression 5', solve(*([(8, 0), (2, 1)], 0, 1)), 2)\ncheck('regression 6', solve(*([], 1, 2)), None)\ncheck('regression 7', solve(*([(1, 0), (5, 0)], 1, 2)), None)\ncheck('regression 8', solve(*([(1, 1), (4, 2), (8, 1)], 3, 4)), 4)\ncheck('regression 9', solve(*([(9, 4), (2, 1)], 1, 1)), 9)\ncheck('regression 10', solve(*([(3, 2), (3, 1), (1, 1)], 1, 2)), 3)\ncheck('regression 11', solve(*([(2, 2), (7, 2)], 1, 2)), 2)\ncheck(\"variable rank\",solve([(N,2),(N+4,1)],1,2),N)\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":"Small offline integer/rational inputs only; no performance, statistical inference, or production-library conformance claim. 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-numerical-aggregation-frequency-left-quantile-threshold-strict","generated_at":"2026-09-29T14:39:03.583429+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Exact bounded examples isolate a reduction defect without floating-point or external-service effects.","repair":"Preserve the frequency left quantile contract at the identified reduction decision.","root_cause":"An exact cumulative rank boundary advances to the next support value.","sha256":"1a57bcb0ffe7b240f37d01de3ad99dbf24eb2c467e7510be233db2a3aaeed109","title":"Frequency left quantile: An exact cumulative rank boundary advances to the next support value. · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":46.205,"exit_code":1,"observations":[{"actual":3,"check":"regression 1","expected":-2,"passed":false},{"actual":8,"check":"regression 2","expected":8,"passed":true},{"actual":8,"check":"regression 3","expected":8,"passed":true},{"actual":5,"check":"regression 4","expected":1,"passed":false},{"actual":2,"check":"regression 5","expected":2,"passed":true},{"actual":null,"check":"regression 6","expected":null,"passed":true},{"actual":null,"check":"regression 7","expected":null,"passed":true},{"actual":8,"check":"regression 8","expected":4,"passed":false},{"actual":9,"check":"regression 9","expected":9,"passed":true},{"actual":3,"check":"regression 10","expected":3,"passed":true},{"actual":7,"check":"regression 11","expected":2,"passed":false},{"actual":5,"check":"variable rank","expected":1,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": 3, \"expected\": -2, \"passed\": false}, {\"check\": \"regression 2\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"regression 3\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"regression 4\", \"actual\": 5, \"expected\": 1, \"passed\": false}, {\"check\": \"regression 5\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 7\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 8\", \"actual\": 8, \"expected\": 4, \"passed\": false}, {\"check\": \"regression 9\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"regression 10\", \"actual\": 3, \"expected\": 3, \"passed\": true}, {\"check\": \"regression 11\", \"actual\": 7, \"expected\": 2, \"passed\": false}, {\"check\": \"variable rank\", \"actual\": 5, \"expected\": 1, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":53.101,"exit_code":1,"observations":[{"actual":-2,"check":"regression 1","expected":-2,"passed":true},{"actual":8,"check":"regression 2","expected":8,"passed":true},{"actual":8,"check":"regression 3","expected":8,"passed":true},{"actual":5,"check":"regression 4","expected":1,"passed":false},{"actual":2,"check":"regression 5","expected":2,"passed":true},{"actual":null,"check":"regression 6","expected":null,"passed":true},{"actual":null,"check":"regression 7","expected":null,"passed":true},{"actual":8,"check":"regression 8","expected":4,"passed":false},{"actual":9,"check":"regression 9","expected":9,"passed":true},{"actual":3,"check":"regression 10","expected":3,"passed":true},{"actual":7,"check":"regression 11","expected":2,"passed":false},{"actual":1,"check":"variable rank","expected":1,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": -2, \"expected\": -2, \"passed\": true}, {\"check\": \"regression 2\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"regression 3\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"regression 4\", \"actual\": 5, \"expected\": 1, \"passed\": false}, {\"check\": \"regression 5\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 7\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 8\", \"actual\": 8, \"expected\": 4, \"passed\": false}, {\"check\": \"regression 9\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"regression 10\", \"actual\": 3, \"expected\": 3, \"passed\": true}, {\"check\": \"regression 11\", \"actual\": 7, \"expected\": 2, \"passed\": false}, {\"check\": \"variable rank\", \"actual\": 1, \"expected\": 1, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.077,"exit_code":0,"observations":[{"actual":-2,"check":"regression 1","expected":-2,"passed":true},{"actual":8,"check":"regression 2","expected":8,"passed":true},{"actual":8,"check":"regression 3","expected":8,"passed":true},{"actual":1,"check":"regression 4","expected":1,"passed":true},{"actual":2,"check":"regression 5","expected":2,"passed":true},{"actual":null,"check":"regression 6","expected":null,"passed":true},{"actual":null,"check":"regression 7","expected":null,"passed":true},{"actual":4,"check":"regression 8","expected":4,"passed":true},{"actual":9,"check":"regression 9","expected":9,"passed":true},{"actual":3,"check":"regression 10","expected":3,"passed":true},{"actual":2,"check":"regression 11","expected":2,"passed":true},{"actual":1,"check":"variable rank","expected":1,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": -2, \"expected\": -2, \"passed\": true}, {\"check\": \"regression 2\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"regression 3\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"regression 4\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"regression 5\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 7\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 8\", \"actual\": 4, \"expected\": 4, \"passed\": true}, {\"check\": \"regression 9\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"regression 10\", \"actual\": 3, \"expected\": 3, \"passed\": true}, {\"check\": \"regression 11\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"variable rank\", \"actual\": 1, \"expected\": 1, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}