{"abstract":"The reduction disagrees with its explicit aggregation oracle.","category":"Numerical aggregation","checks":9,"contract":"Rows [integer value, nonnegative frequency] expand to a multiset. Sum its smallest min(k,total frequency) observations for nonnegative integer k; zero/empty tail sums to zero.","evaluation_group":"s3-na-frequency-lower-tail-sum","failed_approach":"Using the full frequency still overcharges a partially selected bin.","family":"s3-numerical-aggregation-frequency-lower-tail-sum-value-without-multiplicity","id":"FA-13276","implementations":{"attempt":{"sha256":"676a12d9d550ebb7bcff75af13b978afb665776a0ba5f66faf42ce3faa92eda2","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, k):\n    remaining=k\n    out=0\n    for x,w in sorted(rows):\n        take=min(remaining,w)\n        out+=x*w if take else 0\n        remaining-=take\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(2, 1), (2, 1), (9, 3)], 3)), 13)\ncheck('regression 2', solve(*([(8, 2), (1, 3), (4, 2)], 4)), 7)\ncheck('regression 3', solve(*([(2, 3)], 0)), 0)\ncheck('regression 4', solve(*([], 3)), 0)\ncheck('regression 5', solve(*([(9, 0), (2, 1)], 8)), 2)\ncheck('regression 6', solve(*([(-5, 2), (3, 4)], 3)), -7)\ncheck('regression 7', solve(*([(2, 1), (2, 2), (9, 1)], 2)), 4)\ncheck('regression 8', solve(*([(1, 8), (7, 2)], 9)), 15)\ncheck(\"variable tail mass\",solve([(N,3),(N+5,2)],4),4*N+5)\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":"c0e41af73e9349665a07ab36f15c73fe0620af0f5b697003e79f71258802c4f3","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, k):\n    remaining=k\n    out=0\n    for x,w in sorted(rows):\n        take=min(remaining,w)\n        out+=x if take else 0\n        remaining-=take\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(2, 1), (2, 1), (9, 3)], 3)), 13)\ncheck('regression 2', solve(*([(8, 2), (1, 3), (4, 2)], 4)), 7)\ncheck('regression 3', solve(*([(2, 3)], 0)), 0)\ncheck('regression 4', solve(*([], 3)), 0)\ncheck('regression 5', solve(*([(9, 0), (2, 1)], 8)), 2)\ncheck('regression 6', solve(*([(-5, 2), (3, 4)], 3)), -7)\ncheck('regression 7', solve(*([(2, 1), (2, 2), (9, 1)], 2)), 4)\ncheck('regression 8', solve(*([(1, 8), (7, 2)], 9)), 15)\ncheck(\"variable tail mass\",solve([(N,3),(N+5,2)],4),4*N+5)\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":"417ea1a368ee3e5d21cca4194af0275ebd8f4420b8ab532e12385aebec4c9142","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, k):\n    remaining=k\n    out=0\n    for x,w in sorted(rows):\n        take=min(remaining,w)\n        out+=x*take\n        remaining-=take\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(2, 1), (2, 1), (9, 3)], 3)), 13)\ncheck('regression 2', solve(*([(8, 2), (1, 3), (4, 2)], 4)), 7)\ncheck('regression 3', solve(*([(2, 3)], 0)), 0)\ncheck('regression 4', solve(*([], 3)), 0)\ncheck('regression 5', solve(*([(9, 0), (2, 1)], 8)), 2)\ncheck('regression 6', solve(*([(-5, 2), (3, 4)], 3)), -7)\ncheck('regression 7', solve(*([(2, 1), (2, 2), (9, 1)], 2)), 4)\ncheck('regression 8', solve(*([(1, 8), (7, 2)], 9)), 15)\ncheck(\"variable tail mass\",solve([(N,3),(N+5,2)],4),4*N+5)\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-lower-tail-sum-value-without-multiplicity","generated_at":"2026-09-29T14:39:05.323414+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 lower tail sum contract at the identified reduction decision.","root_cause":"The selected bin contributes its value once.","sha256":"d8980378402af0d4b2ee54041493ef68fa82ba204e6ed2081d262ee89f147514","title":"Frequency lower tail sum: The selected bin contributes its value once. · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":43.361,"exit_code":1,"observations":[{"actual":31,"check":"regression 1","expected":13,"passed":false},{"actual":11,"check":"regression 2","expected":7,"passed":false},{"actual":0,"check":"regression 3","expected":0,"passed":true},{"actual":0,"check":"regression 4","expected":0,"passed":true},{"actual":2,"check":"regression 5","expected":2,"passed":true},{"actual":2,"check":"regression 6","expected":-7,"passed":false},{"actual":6,"check":"regression 7","expected":4,"passed":false},{"actual":22,"check":"regression 8","expected":15,"passed":false},{"actual":15,"check":"variable tail mass","expected":9,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": 31, \"expected\": 13, \"passed\": false}, {\"check\": \"regression 2\", \"actual\": 11, \"expected\": 7, \"passed\": false}, {\"check\": \"regression 3\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"regression 4\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"regression 5\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": 2, \"expected\": -7, \"passed\": false}, {\"check\": \"regression 7\", \"actual\": 6, \"expected\": 4, \"passed\": false}, {\"check\": \"regression 8\", \"actual\": 22, \"expected\": 15, \"passed\": false}, {\"check\": \"variable tail mass\", \"actual\": 15, \"expected\": 9, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":45.682,"exit_code":1,"observations":[{"actual":13,"check":"regression 1","expected":13,"passed":true},{"actual":5,"check":"regression 2","expected":7,"passed":false},{"actual":0,"check":"regression 3","expected":0,"passed":true},{"actual":0,"check":"regression 4","expected":0,"passed":true},{"actual":2,"check":"regression 5","expected":2,"passed":true},{"actual":-2,"check":"regression 6","expected":-7,"passed":false},{"actual":4,"check":"regression 7","expected":4,"passed":true},{"actual":8,"check":"regression 8","expected":15,"passed":false},{"actual":7,"check":"variable tail mass","expected":9,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": 13, \"expected\": 13, \"passed\": true}, {\"check\": \"regression 2\", \"actual\": 5, \"expected\": 7, \"passed\": false}, {\"check\": \"regression 3\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"regression 4\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"regression 5\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": -2, \"expected\": -7, \"passed\": false}, {\"check\": \"regression 7\", \"actual\": 4, \"expected\": 4, \"passed\": true}, {\"check\": \"regression 8\", \"actual\": 8, \"expected\": 15, \"passed\": false}, {\"check\": \"variable tail mass\", \"actual\": 7, \"expected\": 9, \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":46.848,"exit_code":0,"observations":[{"actual":13,"check":"regression 1","expected":13,"passed":true},{"actual":7,"check":"regression 2","expected":7,"passed":true},{"actual":0,"check":"regression 3","expected":0,"passed":true},{"actual":0,"check":"regression 4","expected":0,"passed":true},{"actual":2,"check":"regression 5","expected":2,"passed":true},{"actual":-7,"check":"regression 6","expected":-7,"passed":true},{"actual":4,"check":"regression 7","expected":4,"passed":true},{"actual":15,"check":"regression 8","expected":15,"passed":true},{"actual":9,"check":"variable tail mass","expected":9,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": 13, \"expected\": 13, \"passed\": true}, {\"check\": \"regression 2\", \"actual\": 7, \"expected\": 7, \"passed\": true}, {\"check\": \"regression 3\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"regression 4\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"regression 5\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": -7, \"expected\": -7, \"passed\": true}, {\"check\": \"regression 7\", \"actual\": 4, \"expected\": 4, \"passed\": true}, {\"check\": \"regression 8\", \"actual\": 15, \"expected\": 15, \"passed\": true}, {\"check\": \"variable tail mass\", \"actual\": 9, \"expected\": 9, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}