FA-13296 / Numerical aggregation / Open access
Frequency lower tail sum: A zero remaining budget still consumes one observation. · case 01
The reduction disagrees with its explicit aggregation oracle.
ROOT CAUSE
A zero remaining budget still consumes one observation.
VERIFIED REPAIR
Preserve the frequency lower tail sum contract at the identified reduction decision.
Unsuccessful approach: Forcing at least one observation also violates empty-tail semantics.
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=min(remaining+1,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 | 22 | 13 | Failed |
| regression 2 | 11 | 7 | Failed |
| regression 3 | 2 | 0 | Failed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 2 | 2 | Passed |
| regression 6 | -4 | -7 | Failed |
| regression 7 | 6 | 4 | Failed |
| regression 8 | 22 | 15 | Failed |
| variable tail mass | 15 | 9 | Failed |
SHA-256 / 0f6da74ba5c61011824b0709bc3d242630863164c4292facf7bfb5bd3669f4eb
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=min(max(1,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 | 15 | 7 | Failed |
| regression 3 | 2 | 0 | Failed |
| regression 4 | 0 | 0 | Passed |
| regression 5 | 2 | 2 | Passed |
| regression 6 | -7 | -7 | Passed |
| regression 7 | 13 | 4 | Failed |
| regression 8 | 15 | 15 | Passed |
| variable tail mass | 9 | 9 | Passed |
SHA-256 / df4b359221decc34fb581464f49a5cd28388830b177747faeb044d88e7cc1309
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 2026-09-29T14:39:05.472954+00:00.
Case digest / 4d3382f0007276502501fee7f01c28bcbaebbac5933911997281a0d79c94a40e