FAILURE MAP
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FA-40796 / Heap invariants / Open access

Interval tail removal drops an array node only when its last endpoint is removed · case 01

The bounded interval pop end certificate reports an incorrect nodes.

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

ROOT CAUSE

Interval tail removal drops an array node only when its last endpoint is removed.

THE FAILURE

Interval tail removal drops an array node only when its last endpoint is removed.

Unsuccessful approach: The local patch uses len(a) 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': v,
    'remaining': out,
    'size': sum(map(len,out)),
    'singleton': bool(out) and len(out[-1])==1,
    'survivor': rest[0] if rest else None,
    'nodes': max(0,len(a)-1)}
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': 0, 'remaining': [[9]], 'replacement': 1, 'singleton': True, 'size': 1, 'survivor': 9}{'nodes': 1, 'remaining': [[9]], 'replacement': 1, 'singleton': True, 'size': 1, 'survivor': 9}Failed
regression certificate 4{'nodes': 1, '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}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': 2, '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}Failed
variant-dependent certificate{'nodes': 2, '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}Failed

SHA-256 / ca5e8c78739bfa41c45366410400571f28c144acc6bb31dd053bc2eec5246ab6

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': v,
    'remaining': out,
    'size': sum(map(len,out)),
    'singleton': bool(out) and len(out[-1])==1,
    'survivor': rest[0] if rest else None,
    'nodes': len(a)}
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': 1, 'remaining': [], 'replacement': 4, 'singleton': False, 'size': 0, 'survivor': None}{'nodes': 0, 'remaining': [], 'replacement': 4, 'singleton': False, 'size': 0, 'survivor': None}Failed
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': 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': 2, 'remaining': [[1, 9]], 'replacement': 3, 'singleton': False, 'size': 2, 'survivor': None}{'nodes': 1, 'remaining': [[1, 9]], 'replacement': 3, 'singleton': False, 'size': 2, 'survivor': None}Failed
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 / 1996ed201998d808eb1d9a4edc0f3238b0c1285761e30c0185aed6c5376c28a4

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 / 328fd36d7f6981988d5c474aa5163b005aa87fd8f00b5ae5ac6adab6b2360a08