{"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.","contract_signature":"rows, k","evaluation_group":"s3-na-frequency-lower-tail-sum","failed_approach":"Forcing at least one observation also violates empty-tail semantics.","family":"s3-numerical-aggregation-frequency-lower-tail-sum-zero-budget-inclusive","id":"FA-13296","implementations":{"attempt":{"sha256":"df4b359221decc34fb581464f49a5cd28388830b177747faeb044d88e7cc1309","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(max(1,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"},"broken":{"sha256":"0f6da74ba5c61011824b0709bc3d242630863164c4292facf7bfb5bd3669f4eb","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+1,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-zero-budget-inclusive","generated_at":"2026-09-29T14:39:05.472954+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.","root_cause":"A zero remaining budget still consumes one observation.","sha256":"4bfbcf796c817ce2b910f877cb3124a5ed9829e8834ad44eb9b1e44a8848b0cd","title":"Frequency lower tail sum: A zero remaining budget still consumes one observation. · 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":45.241,"exit_code":1,"observations":[{"actual":13,"check":"regression 1","expected":13,"passed":true},{"actual":15,"check":"regression 2","expected":7,"passed":false},{"actual":2,"check":"regression 3","expected":0,"passed":false},{"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":13,"check":"regression 7","expected":4,"passed":false},{"actual":15,"check":"regression 8","expected":15,"passed":true},{"actual":9,"check":"variable tail mass","expected":9,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": 13, \"expected\": 13, \"passed\": true}, {\"check\": \"regression 2\", \"actual\": 15, \"expected\": 7, \"passed\": false}, {\"check\": \"regression 3\", \"actual\": 2, \"expected\": 0, \"passed\": false}, {\"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\": 13, \"expected\": 4, \"passed\": false}, {\"check\": \"regression 8\", \"actual\": 15, \"expected\": 15, \"passed\": true}, {\"check\": \"variable tail mass\", \"actual\": 9, \"expected\": 9, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":45.522,"exit_code":1,"observations":[{"actual":22,"check":"regression 1","expected":13,"passed":false},{"actual":11,"check":"regression 2","expected":7,"passed":false},{"actual":2,"check":"regression 3","expected":0,"passed":false},{"actual":0,"check":"regression 4","expected":0,"passed":true},{"actual":2,"check":"regression 5","expected":2,"passed":true},{"actual":-4,"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\": 22, \"expected\": 13, \"passed\": false}, {\"check\": \"regression 2\", \"actual\": 11, \"expected\": 7, \"passed\": false}, {\"check\": \"regression 3\", \"actual\": 2, \"expected\": 0, \"passed\": false}, {\"check\": \"regression 4\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"regression 5\", \"actual\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": -4, \"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"}},"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."}}