FA-13266 / Numerical aggregation / Open access
Frequency lower tail sum: A terminal frequency bin contributes its whole mass. · case 01
The reduction disagrees with its explicit aggregation oracle.
ROOT CAUSE
A terminal frequency bin contributes its whole mass.
VERIFIED REPAIR
Preserve the frequency lower tail sum contract at the identified reduction decision.
Unsuccessful approach: Assigning all remaining demand ignores available bin frequency.
Case 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.
Why this case matters
Exact bounded examples isolate a reduction defect without floating-point or external-service effects.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(rows, k):
remaining=k
out=0
for x,w in sorted(rows):
take=w if remaining>0 else 0
out+=x*take
remaining-=take
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([(2, 1), (2, 1), (9, 3)], 3)), 13)
check('regression 2', solve(*([(8, 2), (1, 3), (4, 2)], 4)), 7)
check('regression 3', solve(*([(2, 3)], 0)), 0)
check('regression 4', solve(*([], 3)), 0)
check('regression 5', solve(*([(9, 0), (2, 1)], 8)), 2)
check('regression 6', solve(*([(-5, 2), (3, 4)], 3)), -7)
check('regression 7', solve(*([(2, 1), (2, 2), (9, 1)], 2)), 4)
check('regression 8', solve(*([(1, 8), (7, 2)], 9)), 15)
check("variable tail mass",solve([(N,3),(N+5,2)],4),4*N+5)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 1 | 31 | 13 | Failed |
| regression 2 | 11 | 7 | Failed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 2 | 2 | Passed |
| regression 6 | 2 | -7 | Failed |
| regression 7 | 6 | 4 | Failed |
| regression 8 | 22 | 15 | Failed |
| variable tail mass | 15 | 9 | Failed |
SHA-256 / a9ff846de7fcdab2b449cb2aee6803ca810cc5b12af34f14ccd222822028a67c
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(rows, k):
remaining=k
out=0
for x,w in sorted(rows):
take=remaining if remaining>0 else 0
out+=x*take
remaining-=take
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([(2, 1), (2, 1), (9, 3)], 3)), 13)
check('regression 2', solve(*([(8, 2), (1, 3), (4, 2)], 4)), 7)
check('regression 3', solve(*([(2, 3)], 0)), 0)
check('regression 4', solve(*([], 3)), 0)
check('regression 5', solve(*([(9, 0), (2, 1)], 8)), 2)
check('regression 6', solve(*([(-5, 2), (3, 4)], 3)), -7)
check('regression 7', solve(*([(2, 1), (2, 2), (9, 1)], 2)), 4)
check('regression 8', solve(*([(1, 8), (7, 2)], 9)), 15)
check("variable tail mass",solve([(N,3),(N+5,2)],4),4*N+5)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 1 | 6 | 13 | Failed |
| regression 2 | 4 | 7 | Failed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 16 | 2 | Failed |
| regression 6 | -15 | -7 | Failed |
| regression 7 | 4 | 4 | Passed |
| regression 8 | 9 | 15 | Failed |
| variable tail mass | 4 | 9 | Failed |
SHA-256 / e78cbef39d302e96c9fd10ad60248117d966bdf752f3247abfb6346b9519d0fc
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(rows, k):
remaining=k
out=0
for x,w in sorted(rows):
take=min(remaining,w)
out+=x*take
remaining-=take
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([(2, 1), (2, 1), (9, 3)], 3)), 13)
check('regression 2', solve(*([(8, 2), (1, 3), (4, 2)], 4)), 7)
check('regression 3', solve(*([(2, 3)], 0)), 0)
check('regression 4', solve(*([], 3)), 0)
check('regression 5', solve(*([(9, 0), (2, 1)], 8)), 2)
check('regression 6', solve(*([(-5, 2), (3, 4)], 3)), -7)
check('regression 7', solve(*([(2, 1), (2, 2), (9, 1)], 2)), 4)
check('regression 8', solve(*([(1, 8), (7, 2)], 9)), 15)
check("variable tail mass",solve([(N,3),(N+5,2)],4),4*N+5)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 1 | 13 | 13 | Passed |
| regression 2 | 7 | 7 | Passed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 2 | 2 | Passed |
| regression 6 | -7 | -7 | Passed |
| regression 7 | 4 | 4 | Passed |
| regression 8 | 15 | 15 | Passed |
| variable tail mass | 9 | 9 | Passed |
SHA-256 / 417ea1a368ee3e5d21cca4194af0275ebd8f4420b8ab532e12385aebec4c9142
Verification & scope
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.
Observations recorded using Python 3.12.14 at 2026-09-29T14:39:05.322948+00:00.
Case digest / 83182bf764dd24b156551c154b2206982ff1fe4e223cdf1399d3754f1cad4e50