FAILURE MAP
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FA-40431 / Heap invariants / Open access

Interval heap maximum is the root upper endpoint even with a singleton tail · case 01

The bounded interval endpoints certificate reports an incorrect maximum.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Interval heap maximum is the root upper endpoint even with a singleton tail.

THE FAILURE

Interval heap maximum is the root upper endpoint even with a singleton tail.

Unsuccessful approach: The local patch uses highs[-1] if a else None and still violates the stated relation.

Case contract

An interval heap stores ordered pairs [lo,hi] except the last node may be singleton [x]. Report reversed intervals, low-end heap violations, high-end heap violations treating singleton as both endpoints, global minimum, global maximum, and member count.

Why this case matters

This isolates an internal heap representation or priority-structure invariant using deterministic finite records.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(d):
    a=d['intervals']; n=len(a); lows=[x[0] for x in a]; highs=[x[-1] for x in a]
    return {'reversed': [i for i,x in enumerate(a) if x[0]>x[-1]],
    'low_order': [i for i in range(1,n) if lows[i]<lows[(i-1)//2]],
    'high_order': [i for i in range(1,n) if highs[i]>highs[(i-1)//2]],
    'minimum': lows[0] if a else None,
    'maximum': max(lows) if a else None,
    'members': sum(len(x) for x in a)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 10], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 10, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 11], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 11, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 12], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 12, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 13], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 13, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 14], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 14, 'members': 8})]][N-1]
check('regression certificate 1', solve(cases[0][0]), cases[0][1])
check('regression certificate 2', solve(cases[1][0]), cases[1][1])
check('regression certificate 3', solve(cases[2][0]), cases[2][1])
check('regression certificate 4', solve(cases[3][0]), cases[3][1])
check('regression certificate 5', solve(cases[4][0]), cases[4][1])
check('regression certificate 6', solve(cases[5][0]), cases[5][1])
check('variant-dependent certificate', solve(cases[6][0]), cases[6][1])
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 certificate 1{'high_order': [], 'low_order': [], 'maximum': None, 'members': 0, 'minimum': None, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': None, 'members': 0, 'minimum': None, 'reversed': []}Passed
regression certificate 2{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 1, 'minimum': 4, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 1, 'minimum': 4, 'reversed': []}Passed
regression certificate 3{'high_order': [], 'low_order': [], 'maximum': 3, 'members': 5, 'minimum': 1, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 9, 'members': 5, 'minimum': 1, 'reversed': []}Failed
regression certificate 4{'high_order': [2], 'low_order': [], 'maximum': 9, 'members': 5, 'minimum': 1, 'reversed': []}{'high_order': [2], 'low_order': [], 'maximum': 8, 'members': 5, 'minimum': 1, 'reversed': []}Failed
regression certificate 5{'high_order': [1, 2], 'low_order': [1, 2], 'maximum': 5, 'members': 6, 'minimum': 5, 'reversed': [0]}{'high_order': [1, 2], 'low_order': [1, 2], 'maximum': 3, 'members': 6, 'minimum': 5, 'reversed': [0]}Failed
regression certificate 6{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 8, 'minimum': 1, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 9, 'members': 8, 'minimum': 1, 'reversed': []}Failed
variant-dependent certificate{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 8, 'minimum': 1, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 10, 'members': 8, 'minimum': 1, 'reversed': []}Failed

SHA-256 / c31e3282429b57bdecfc73cadf817571d5f8901a77c6a1d8602d2116dba8929f

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(d):
    a=d['intervals']; n=len(a); lows=[x[0] for x in a]; highs=[x[-1] for x in a]
    return {'reversed': [i for i,x in enumerate(a) if x[0]>x[-1]],
    'low_order': [i for i in range(1,n) if lows[i]<lows[(i-1)//2]],
    'high_order': [i for i in range(1,n) if highs[i]>highs[(i-1)//2]],
    'minimum': lows[0] if a else None,
    'maximum': highs[-1] if a else None,
    'members': sum(len(x) for x in a)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 10], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 10, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 11], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 11, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 12], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 12, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 13], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 13, 'members': 8})], [({'intervals': []}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': None, 'maximum': None, 'members': 0}), ({'intervals': [[4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 4, 'maximum': 4, 'members': 1}), ({'intervals': [[1, 9], [2, 7], [3]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 5}), ({'intervals': [[1, 8], [2, 7], [9]]}, {'reversed': [], 'low_order': [], 'high_order': [2], 'minimum': 1, 'maximum': 8, 'members': 5}), ({'intervals': [[5, 3], [1, 6], [4, 8]]}, {'reversed': [0], 'low_order': [1, 2], 'high_order': [1, 2], 'minimum': 5, 'maximum': 3, 'members': 6}), ({'intervals': [[1, 9], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 9, 'members': 8}), ({'intervals': [[1, 14], [2, 8], [3, 7], [4, 4]]}, {'reversed': [], 'low_order': [], 'high_order': [], 'minimum': 1, 'maximum': 14, 'members': 8})]][N-1]
check('regression certificate 1', solve(cases[0][0]), cases[0][1])
check('regression certificate 2', solve(cases[1][0]), cases[1][1])
check('regression certificate 3', solve(cases[2][0]), cases[2][1])
check('regression certificate 4', solve(cases[3][0]), cases[3][1])
check('regression certificate 5', solve(cases[4][0]), cases[4][1])
check('regression certificate 6', solve(cases[5][0]), cases[5][1])
check('variant-dependent certificate', solve(cases[6][0]), cases[6][1])
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 certificate 1{'high_order': [], 'low_order': [], 'maximum': None, 'members': 0, 'minimum': None, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': None, 'members': 0, 'minimum': None, 'reversed': []}Passed
regression certificate 2{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 1, 'minimum': 4, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 1, 'minimum': 4, 'reversed': []}Passed
regression certificate 3{'high_order': [], 'low_order': [], 'maximum': 3, 'members': 5, 'minimum': 1, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 9, 'members': 5, 'minimum': 1, 'reversed': []}Failed
regression certificate 4{'high_order': [2], 'low_order': [], 'maximum': 9, 'members': 5, 'minimum': 1, 'reversed': []}{'high_order': [2], 'low_order': [], 'maximum': 8, 'members': 5, 'minimum': 1, 'reversed': []}Failed
regression certificate 5{'high_order': [1, 2], 'low_order': [1, 2], 'maximum': 8, 'members': 6, 'minimum': 5, 'reversed': [0]}{'high_order': [1, 2], 'low_order': [1, 2], 'maximum': 3, 'members': 6, 'minimum': 5, 'reversed': [0]}Failed
regression certificate 6{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 8, 'minimum': 1, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 9, 'members': 8, 'minimum': 1, 'reversed': []}Failed
variant-dependent certificate{'high_order': [], 'low_order': [], 'maximum': 4, 'members': 8, 'minimum': 1, 'reversed': []}{'high_order': [], 'low_order': [], 'maximum': 10, 'members': 8, 'minimum': 1, 'reversed': []}Failed

SHA-256 / 8315de4f14978e0f9c2cc5ac127c42c60c3f682910bfdda0c883732494da6fc3

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 7 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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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Verification & scope

A stipulated offline diagnostic model; it does not implement a production allocator, concurrency protocol, or complete heap library. 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 2026-09-29T14:43:31.142837+00:00.

Case digest / 1e854ee7f44e0817c83af9e4af1b9d5b9982d01294626181bca5ab14772c625b