FAILURE MAP
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FA-40771 / Heap invariants / Open access

Interval delete chooses the requested tail endpoint while respecting singleton tails · case 01

The bounded interval pop end certificate reports an incorrect replacement.

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

ROOT CAUSE

Interval delete chooses the requested tail endpoint while respecting singleton tails.

THE FAILURE

Interval delete chooses the requested tail endpoint while respecting singleton tails.

Unsuccessful approach: The local patch uses tail[0] if tail else None and still violates the stated relation.

Case contract

A bounded interval-tail removal receives intervals, and side low or high. Remove the requested endpoint from the final interval (singleton removed wholly), returning extracted tail replacement, remaining interval list, new logical size, singleton status, surviving endpoint, and array-node count. This isolates tail extraction before global bubbling.

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']; side=d['side']; tail=a[-1] if a else []; k=0 if side=='low' else -1; v=tail[k] if tail else None; rest=tail[1:] if side=='low' else tail[:-1]; out=a[:-1]+([rest] if rest else []) if a else []
    return {'replacement': tail[-1] if tail else None,
    'remaining': out,
    'size': sum(map(len,out)),
    'singleton': bool(out) and len(out[-1])==1,
    'survivor': rest[0] if rest else None,
    'nodes': len(out)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 7]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [7]], 'size': 5, 'singleton': True, 'survivor': 7, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 8]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [8]], 'size': 5, 'singleton': True, 'survivor': 8, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 9]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [9]], 'size': 5, 'singleton': True, 'survivor': 9, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 10]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [10]], 'size': 5, 'singleton': True, 'survivor': 10, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 11]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [11]], 'size': 5, 'singleton': True, 'survivor': 11, 'nodes': 3})]][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{'nodes': 0, 'remaining': [], 'replacement': None, 'singleton': False, 'size': 0, 'survivor': None}{'nodes': 0, 'remaining': [], 'replacement': None, 'singleton': False, 'size': 0, 'survivor': None}Passed
regression certificate 2{'nodes': 0, 'remaining': [], 'replacement': 4, 'singleton': False, 'size': 0, 'survivor': None}{'nodes': 0, 'remaining': [], 'replacement': 4, 'singleton': False, 'size': 0, 'survivor': None}Passed
regression certificate 3{'nodes': 1, 'remaining': [[9]], 'replacement': 9, 'singleton': True, 'size': 1, 'survivor': 9}{'nodes': 1, 'remaining': [[9]], 'replacement': 1, 'singleton': True, 'size': 1, 'survivor': 9}Failed
regression certificate 4{'nodes': 2, 'remaining': [[1, 9], [3]], 'replacement': 7, 'singleton': True, 'size': 3, 'survivor': 3}{'nodes': 2, 'remaining': [[1, 9], [3]], 'replacement': 7, 'singleton': True, 'size': 3, 'survivor': 3}Passed
regression certificate 5{'nodes': 1, 'remaining': [[1, 9]], 'replacement': 3, 'singleton': False, 'size': 2, 'survivor': None}{'nodes': 1, 'remaining': [[1, 9]], 'replacement': 3, 'singleton': False, 'size': 2, 'survivor': None}Passed
regression certificate 6{'nodes': 3, 'remaining': [[1, 9], [2, 8], [6]], 'replacement': 6, 'singleton': True, 'size': 5, 'survivor': 6}{'nodes': 3, 'remaining': [[1, 9], [2, 8], [6]], 'replacement': 4, 'singleton': True, 'size': 5, 'survivor': 6}Failed
variant-dependent certificate{'nodes': 3, 'remaining': [[1, 9], [2, 8], [7]], 'replacement': 7, 'singleton': True, 'size': 5, 'survivor': 7}{'nodes': 3, 'remaining': [[1, 9], [2, 8], [7]], 'replacement': 4, 'singleton': True, 'size': 5, 'survivor': 7}Failed

SHA-256 / 7c2acb5849f263e94d83f73222291f756bcdc66f19e4c5b371500dd868c80aa0

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']; side=d['side']; tail=a[-1] if a else []; k=0 if side=='low' else -1; v=tail[k] if tail else None; rest=tail[1:] if side=='low' else tail[:-1]; out=a[:-1]+([rest] if rest else []) if a else []
    return {'replacement': tail[0] if tail else None,
    'remaining': out,
    'size': sum(map(len,out)),
    'singleton': bool(out) and len(out[-1])==1,
    'survivor': rest[0] if rest else None,
    'nodes': len(out)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 7]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [7]], 'size': 5, 'singleton': True, 'survivor': 7, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 8]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [8]], 'size': 5, 'singleton': True, 'survivor': 8, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 9]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [9]], 'size': 5, 'singleton': True, 'survivor': 9, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 10]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [10]], 'size': 5, 'singleton': True, 'survivor': 10, 'nodes': 3})], [({'intervals': [], 'side': 'low'}, {'replacement': None, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[4]], 'side': 'high'}, {'replacement': 4, 'remaining': [], 'size': 0, 'singleton': False, 'survivor': None, 'nodes': 0}), ({'intervals': [[1, 9]], 'side': 'low'}, {'replacement': 1, 'remaining': [[9]], 'size': 1, 'singleton': True, 'survivor': 9, 'nodes': 1}), ({'intervals': [[1, 9], [3, 7]], 'side': 'high'}, {'replacement': 7, 'remaining': [[1, 9], [3]], 'size': 3, 'singleton': True, 'survivor': 3, 'nodes': 2}), ({'intervals': [[1, 9], [3]], 'side': 'low'}, {'replacement': 3, 'remaining': [[1, 9]], 'size': 2, 'singleton': False, 'survivor': None, 'nodes': 1}), ({'intervals': [[1, 9], [2, 8], [4, 6]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [6]], 'size': 5, 'singleton': True, 'survivor': 6, 'nodes': 3}), ({'intervals': [[1, 9], [2, 8], [4, 11]], 'side': 'low'}, {'replacement': 4, 'remaining': [[1, 9], [2, 8], [11]], 'size': 5, 'singleton': True, 'survivor': 11, 'nodes': 3})]][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{'nodes': 0, 'remaining': [], 'replacement': None, 'singleton': False, 'size': 0, 'survivor': None}{'nodes': 0, 'remaining': [], 'replacement': None, 'singleton': False, 'size': 0, 'survivor': None}Passed
regression certificate 2{'nodes': 0, 'remaining': [], 'replacement': 4, 'singleton': False, 'size': 0, 'survivor': None}{'nodes': 0, 'remaining': [], 'replacement': 4, 'singleton': False, 'size': 0, 'survivor': None}Passed
regression certificate 3{'nodes': 1, 'remaining': [[9]], 'replacement': 1, 'singleton': True, 'size': 1, 'survivor': 9}{'nodes': 1, 'remaining': [[9]], 'replacement': 1, 'singleton': True, 'size': 1, 'survivor': 9}Passed
regression certificate 4{'nodes': 2, 'remaining': [[1, 9], [3]], 'replacement': 3, 'singleton': True, 'size': 3, 'survivor': 3}{'nodes': 2, 'remaining': [[1, 9], [3]], 'replacement': 7, 'singleton': True, 'size': 3, 'survivor': 3}Failed
regression certificate 5{'nodes': 1, 'remaining': [[1, 9]], 'replacement': 3, 'singleton': False, 'size': 2, 'survivor': None}{'nodes': 1, 'remaining': [[1, 9]], 'replacement': 3, 'singleton': False, 'size': 2, 'survivor': None}Passed
regression certificate 6{'nodes': 3, 'remaining': [[1, 9], [2, 8], [6]], 'replacement': 4, 'singleton': True, 'size': 5, 'survivor': 6}{'nodes': 3, 'remaining': [[1, 9], [2, 8], [6]], 'replacement': 4, 'singleton': True, 'size': 5, 'survivor': 6}Passed
variant-dependent certificate{'nodes': 3, 'remaining': [[1, 9], [2, 8], [7]], 'replacement': 4, 'singleton': True, 'size': 5, 'survivor': 7}{'nodes': 3, 'remaining': [[1, 9], [2, 8], [7]], 'replacement': 4, 'singleton': True, 'size': 5, 'survivor': 7}Passed

SHA-256 / e6d673dbc7c190d6f2388d029e19532bd17b84c8281127d79cab2cff9c81945d

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.

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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 2026-09-29T14:43:34.517850+00:00.

Case digest / bc3e76f218f799f85d39c2f46e29df65f3d8657fae2f3fddb0c54364ea7d90ec