FAILURE MAP
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FA-13216 / Numerical aggregation / Open access

Maximum subarray block summary: Summaries are merged in sorted block-total order. · case 01

The reduction disagrees with its explicit aggregation oracle.

Verified by executionVariant 1 · 9 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Summaries are merged in sorted block-total order.

THE FAILURE

Summaries are merged in sorted block-total order.

Unsuccessful approach: Reversing blocks also changes permitted contiguous intervals.

Case contract

Merge ordered disjoint integer blocks, preserving concatenation order. Return [total, maximum prefix sum, maximum suffix sum, maximum contiguous subarray sum]; empty subarrays are allowed and have sum 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(blocks):
    total=prefix=suffix=best=0
    for block in sorted(blocks,key=sum):
        t=sum(block)
        p=max([0]+[sum(block[:i]) for i in range(1,len(block)+1)])
        s=max([0]+[sum(block[i:]) for i in range(len(block))])
        b=max([0]+[sum(block[i:j]) for i in range(len(block)) for j in range(i+1,len(block)+1)])
        new_best=max(best,b,suffix+p)
        new_prefix=max(prefix,total+p)
        new_suffix=max(s,suffix+t)
        total=total+t
        prefix,suffix,best=new_prefix,new_suffix,new_best
    return [total,prefix,suffix,best]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([[-100, 5]],)), [-95, 0, 5, 5])
check('regression 2', solve(*([[2, -7, 4], [3, -1]],)), [1, 2, 6, 7])
check('regression 3', solve(*([[-8, -2], [-3]],)), [-13, 0, 0, 0])
check('regression 4', solve(*([],)), [0, 0, 0, 0])
check('regression 5', solve(*([[4, -2], [], [3, -9, 6]],)), [2, 5, 6, 6])
check('regression 6', solve(*([[-2, 7], [-3, 5, -9], [2]],)), [0, 7, 2, 9])
check('regression 7', solve(*([[8, -9], [1, -4, 9]],)), [5, 8, 9, 9])
check('regression 8', solve(*([[1, 2], [3, 4]],)), [10, 10, 10, 10])
check("variable cross block peak",solve([[-N,2*N],[N,-4*N]]),[-2*N,2*N,0,3*N])
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 fixtureActualExpectedOutcome
regression 1[-95, 0, 5, 5][-95, 0, 5, 5]Passed
regression 2[1, 2, 6, 7][1, 2, 6, 7]Passed
regression 3[-13, 0, 0, 0][-13, 0, 0, 0]Passed
regression 4[0, 0, 0, 0][0, 0, 0, 0]Passed
regression 5[2, 4, 8, 10][2, 5, 6, 6]Failed
regression 6[0, 2, 7, 7][0, 7, 2, 9]Failed
regression 7[5, 8, 9, 9][5, 8, 9, 9]Passed
regression 8[10, 10, 10, 10][10, 10, 10, 10]Passed
variable cross block peak[-2, 1, 2, 2][-2, 2, 0, 3]Failed

SHA-256 / ba8dae8487e8c14aec2f30e8433d4c77677a7fe3530dee57b66cad64f7fc8002

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(blocks):
    total=prefix=suffix=best=0
    for block in reversed(blocks):
        t=sum(block)
        p=max([0]+[sum(block[:i]) for i in range(1,len(block)+1)])
        s=max([0]+[sum(block[i:]) for i in range(len(block))])
        b=max([0]+[sum(block[i:j]) for i in range(len(block)) for j in range(i+1,len(block)+1)])
        new_best=max(best,b,suffix+p)
        new_prefix=max(prefix,total+p)
        new_suffix=max(s,suffix+t)
        total=total+t
        prefix,suffix,best=new_prefix,new_suffix,new_best
    return [total,prefix,suffix,best]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([[-100, 5]],)), [-95, 0, 5, 5])
check('regression 2', solve(*([[2, -7, 4], [3, -1]],)), [1, 2, 6, 7])
check('regression 3', solve(*([[-8, -2], [-3]],)), [-13, 0, 0, 0])
check('regression 4', solve(*([],)), [0, 0, 0, 0])
check('regression 5', solve(*([[4, -2], [], [3, -9, 6]],)), [2, 5, 6, 6])
check('regression 6', solve(*([[-2, 7], [-3, 5, -9], [2]],)), [0, 7, 2, 9])
check('regression 7', solve(*([[8, -9], [1, -4, 9]],)), [5, 8, 9, 9])
check('regression 8', solve(*([[1, 2], [3, 4]],)), [10, 10, 10, 10])
check("variable cross block peak",solve([[-N,2*N],[N,-4*N]]),[-2*N,2*N,0,3*N])
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 fixtureActualExpectedOutcome
regression 1[-95, 0, 5, 5][-95, 0, 5, 5]Passed
regression 2[1, 4, 4, 4][1, 2, 6, 7]Failed
regression 3[-13, 0, 0, 0][-13, 0, 0, 0]Passed
regression 4[0, 0, 0, 0][0, 0, 0, 0]Passed
regression 5[2, 4, 8, 10][2, 5, 6, 6]Failed
regression 6[0, 4, 7, 7][0, 7, 2, 9]Failed
regression 7[5, 14, 8, 17][5, 8, 9, 9]Failed
regression 8[10, 10, 10, 10][10, 10, 10, 10]Passed
variable cross block peak[-2, 1, 2, 2][-2, 2, 0, 3]Failed

SHA-256 / 488a980438b4780bed44083158de3fefa55812a6d4d1466278ea6d13dd7226f4

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 9 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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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 2026-09-29T14:39:04.617375+00:00.

Case digest / 102c7b85f8b8bbf053a2ebd6a7f4d6675d5676720860fbdc434bab50b60fccc8