FA-40786 / Heap invariants / Open access

Interval delete reports whether the resulting final interval is incomplete · case 01

The bounded interval pop end certificate reports an incorrect singleton.

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

ROOT CAUSE

Interval delete reports whether the resulting final interval is incomplete.

VERIFIED REPAIR

Derive singleton using bool(out) and len(out[-1])==1 under the stated bounded certificate contract.

Unsuccessful approach: The local patch uses bool(out) and len(out[-1])==2 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),
    '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 fixture	Actual	Expected	Outcome
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': 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': True, '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 / 9e9b5fa0634e3a64de525da07dab21a7c64da85b31409fcc28aa57f479f5b1e8

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])==2,
    '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 fixture	Actual	Expected	Outcome
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': False, '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': False, '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': True, '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': False, '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': 4, 'singleton': False, 'size': 5, 'survivor': 7}`	`{'nodes': 3, 'remaining': [[1, 9], [2, 8], [7]], 'replacement': 4, 'singleton': True, 'size': 5, 'survivor': 7}`	Failed

SHA-256 / d5a09cf4d8f914b14ee802c614688b7bbe73fce319453dd4bf3d74ec8e16552a

3 / The verified repair

Exit 0

"""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(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 fixture	Actual	Expected	Outcome
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': 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': 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 / c2d9549ec39fb3315dac6c3ed90f7c75830dbf63ab083f7bf59e92bc6e9aedb4

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 2026-09-29T14:43:34.463880+00:00.

Case digest / 8e98bda9ad9fb3d4bf835ff3782286cc9547cbadd12edd769f8c4795c04b12dc