FA-40456 / Heap invariants / Open access

Interval insertion containment includes both parent endpoints · case 01

The bounded interval insert certificate reports an incorrect contained.

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

ROOT CAUSE

Interval insertion containment includes both parent endpoints.

VERIFIED REPAIR

Derive contained using lo<=x<=hi under the stated bounded certificate contract.

Unsuccessful approach: The local patch uses x>=lo or x<=hi and still violates the stated relation.

Case contract

Insert key x into an interval-heap last-node decision. For a singleton last interval s, complete it as sorted(s,x). For a new singleton with parent [lo,hi], x below lo routes to the lower heap, above hi to the upper heap, otherwise stays. Return pair completion, lower route, upper route, contained route, parent endpoint displaced on crossing, and logical-size increment.

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):
    x=d['x']; s=d['singleton']; lo,hi=d['parent']
    return {'completed': sorted([s,x]),
    'lower': x<lo,
    'upper': x>hi,
    'contained': lo<x<hi,
    'displaced': lo if x<lo else hi if x>hi else None,
    'new_size': d["size"]+1}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 7, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 8, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 9, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 10, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 10], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 11, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 11], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})]][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 fixture	Actual	Expected	Outcome
regression certificate 1	`{'completed': [1, 6], 'contained': False, 'displaced': 2, 'lower': True, 'new_size': 4, 'upper': False}`	`{'completed': [1, 6], 'contained': False, 'displaced': 2, 'lower': True, 'new_size': 4, 'upper': False}`	Passed
regression certificate 2	`{'completed': [4, 9], 'contained': False, 'displaced': 8, 'lower': False, 'new_size': 6, 'upper': True}`	`{'completed': [4, 9], 'contained': False, 'displaced': 8, 'lower': False, 'new_size': 6, 'upper': True}`	Passed
regression certificate 3	`{'completed': [5, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 8, 'upper': False}`	`{'completed': [5, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 8, 'upper': False}`	Passed
regression certificate 4	`{'completed': [2, 3], 'contained': False, 'displaced': None, 'lower': False, 'new_size': 10, 'upper': False}`	`{'completed': [2, 3], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 10, 'upper': False}`	Failed
regression certificate 5	`{'completed': [1, 8], 'contained': False, 'displaced': None, 'lower': False, 'new_size': 12, 'upper': False}`	`{'completed': [1, 8], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 12, 'upper': False}`	Failed
regression certificate 6	`{'completed': [6, 6], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	`{'completed': [6, 6], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	Passed
variant-dependent certificate	`{'completed': [6, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	`{'completed': [6, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	Passed

SHA-256 / ce9c7a64294682ffab7f274425f61c432416c7ed4a5beeff61f31d9548b9a1a0

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(d):
    x=d['x']; s=d['singleton']; lo,hi=d['parent']
    return {'completed': sorted([s,x]),
    'lower': x<lo,
    'upper': x>hi,
    'contained': x>=lo or x<=hi,
    'displaced': lo if x<lo else hi if x>hi else None,
    'new_size': d["size"]+1}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 7, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 8, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 9, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 10, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 10], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 11, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 11], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})]][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 fixture	Actual	Expected	Outcome
regression certificate 1	`{'completed': [1, 6], 'contained': True, 'displaced': 2, 'lower': True, 'new_size': 4, 'upper': False}`	`{'completed': [1, 6], 'contained': False, 'displaced': 2, 'lower': True, 'new_size': 4, 'upper': False}`	Failed
regression certificate 2	`{'completed': [4, 9], 'contained': True, 'displaced': 8, 'lower': False, 'new_size': 6, 'upper': True}`	`{'completed': [4, 9], 'contained': False, 'displaced': 8, 'lower': False, 'new_size': 6, 'upper': True}`	Failed
regression certificate 3	`{'completed': [5, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 8, 'upper': False}`	`{'completed': [5, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 8, 'upper': False}`	Passed
regression certificate 4	`{'completed': [2, 3], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 10, 'upper': False}`	`{'completed': [2, 3], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 10, 'upper': False}`	Passed
regression certificate 5	`{'completed': [1, 8], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 12, 'upper': False}`	`{'completed': [1, 8], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 12, 'upper': False}`	Passed
regression certificate 6	`{'completed': [6, 6], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	`{'completed': [6, 6], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	Passed
variant-dependent certificate	`{'completed': [6, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	`{'completed': [6, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	Passed

SHA-256 / 299f399f1e49aa96acb5b87ece24f79504768616146bd3104c1e564db234fb9a

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(d):
    x=d['x']; s=d['singleton']; lo,hi=d['parent']
    return {'completed': sorted([s,x]),
    'lower': x<lo,
    'upper': x>hi,
    'contained': lo<=x<=hi,
    'displaced': lo if x<lo else hi if x>hi else None,
    'new_size': d["size"]+1}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 7, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 8, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 9, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 10, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 10], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})], [({'x': 1, 'singleton': 6, 'parent': [2, 8], 'size': 3}, {'completed': [1, 6], 'lower': True, 'upper': False, 'contained': False, 'displaced': 2, 'new_size': 4}), ({'x': 9, 'singleton': 4, 'parent': [2, 8], 'size': 5}, {'completed': [4, 9], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 6}), ({'x': 5, 'singleton': 7, 'parent': [2, 8], 'size': 7}, {'completed': [5, 7], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 8}), ({'x': 2, 'singleton': 3, 'parent': [2, 8], 'size': 9}, {'completed': [2, 3], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 10}), ({'x': 8, 'singleton': 1, 'parent': [2, 8], 'size': 11}, {'completed': [1, 8], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 12}), ({'x': 6, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 6], 'lower': False, 'upper': False, 'contained': True, 'displaced': None, 'new_size': 2}), ({'x': 11, 'singleton': 6, 'parent': [2, 8], 'size': 1}, {'completed': [6, 11], 'lower': False, 'upper': True, 'contained': False, 'displaced': 8, 'new_size': 2})]][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 fixture	Actual	Expected	Outcome
regression certificate 1	`{'completed': [1, 6], 'contained': False, 'displaced': 2, 'lower': True, 'new_size': 4, 'upper': False}`	`{'completed': [1, 6], 'contained': False, 'displaced': 2, 'lower': True, 'new_size': 4, 'upper': False}`	Passed
regression certificate 2	`{'completed': [4, 9], 'contained': False, 'displaced': 8, 'lower': False, 'new_size': 6, 'upper': True}`	`{'completed': [4, 9], 'contained': False, 'displaced': 8, 'lower': False, 'new_size': 6, 'upper': True}`	Passed
regression certificate 3	`{'completed': [5, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 8, 'upper': False}`	`{'completed': [5, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 8, 'upper': False}`	Passed
regression certificate 4	`{'completed': [2, 3], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 10, 'upper': False}`	`{'completed': [2, 3], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 10, 'upper': False}`	Passed
regression certificate 5	`{'completed': [1, 8], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 12, 'upper': False}`	`{'completed': [1, 8], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 12, 'upper': False}`	Passed
regression certificate 6	`{'completed': [6, 6], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	`{'completed': [6, 6], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	Passed
variant-dependent certificate	`{'completed': [6, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	`{'completed': [6, 7], 'contained': True, 'displaced': None, 'lower': False, 'new_size': 2, 'upper': False}`	Passed

SHA-256 / e85404eca133ad047294489268f657c2f47ea2cafb8a191ccd019a1202626077

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 2026-09-29T14:43:31.326036+00:00.

Case digest / 174405ace8da8a160a6c5f44ec71dbef4f3c2c1e1b462e9cdc0ea601736f33fd