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FA-40541 / Heap invariants / Open access

Weak heap ancestor cost counts only orientation-driven climbs · case 01

The bounded weak ancestor certificate reports an incorrect cost.

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

ROOT CAUSE

Weak heap ancestor cost counts only orientation-driven climbs.

THE FAILURE

Weak heap ancestor cost counts only orientation-driven climbs.

Unsuccessful approach: The local patch uses len(path)+1 and still violates the stated relation.

Case contract

A weak heap stores reverse bits r in a zero-based array. For nonroot node j, climb while j is a left child under its parent reverse bit: (j mod 2)==r[j//2]. Its distinguished ancestor is j//2 after climbing; zero has none. Report ancestor, climb path, climb count, stopping child, orientation bit, and whether direct parent suffices.

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):
    r=d['reverse']; start=d['node']; j=start; path=[]
    while j>0 and j%2==r[j//2]:
        path.append(j); j//=2
    ancestor=None if start==0 else j//2
    return {'ancestor': ancestor,
    'path': path,
    'cost': 0 if start==0 else 1,
    'stop': j,
    'orientation': None if start==0 else r[j//2],
    'direct': start!=0 and ancestor==start//2}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 2}, {'ancestor': 1, 'path': [], 'cost': 0, 'stop': 2, 'orientation': 1, 'direct': True})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 3}, {'ancestor': 0, 'path': [3], 'cost': 1, 'stop': 1, 'orientation': 0, 'direct': False})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 4}, {'ancestor': 2, 'path': [], 'cost': 0, 'stop': 4, 'orientation': 1, 'direct': True})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 5}, {'ancestor': 1, 'path': [5], 'cost': 1, 'stop': 2, 'orientation': 1, 'direct': False})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True})]][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{'ancestor': None, 'cost': 0, 'direct': False, 'orientation': None, 'path': [], 'stop': 0}{'ancestor': None, 'cost': 0, 'direct': False, 'orientation': None, 'path': [], 'stop': 0}Passed
regression certificate 2{'ancestor': 0, 'cost': 1, 'direct': True, 'orientation': 0, 'path': [], 'stop': 1}{'ancestor': 0, 'cost': 0, 'direct': True, 'orientation': 0, 'path': [], 'stop': 1}Failed
regression certificate 3{'ancestor': 0, 'cost': 1, 'direct': False, 'orientation': 0, 'path': [4, 2], 'stop': 1}{'ancestor': 0, 'cost': 2, 'direct': False, 'orientation': 0, 'path': [4, 2], 'stop': 1}Failed
regression certificate 4{'ancestor': 0, 'cost': 1, 'direct': False, 'orientation': 0, 'path': [7, 3], 'stop': 1}{'ancestor': 0, 'cost': 2, 'direct': False, 'orientation': 0, 'path': [7, 3], 'stop': 1}Failed
regression certificate 5{'ancestor': 0, 'cost': 1, 'direct': False, 'orientation': 0, 'path': [5, 2], 'stop': 1}{'ancestor': 0, 'cost': 2, 'direct': False, 'orientation': 0, 'path': [5, 2], 'stop': 1}Failed
regression certificate 6{'ancestor': 3, 'cost': 1, 'direct': True, 'orientation': 1, 'path': [], 'stop': 6}{'ancestor': 3, 'cost': 0, 'direct': True, 'orientation': 1, 'path': [], 'stop': 6}Failed
variant-dependent certificate{'ancestor': 1, 'cost': 1, 'direct': True, 'orientation': 1, 'path': [], 'stop': 2}{'ancestor': 1, 'cost': 0, 'direct': True, 'orientation': 1, 'path': [], 'stop': 2}Failed

SHA-256 / 70ab7fa569e7643a60ffa3d61331d09b13a9c550490ecdb3b3902f0369a8faa2

2 / The unsuccessful fix

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

N = 1
observations = []
def solve(d):
    r=d['reverse']; start=d['node']; j=start; path=[]
    while j>0 and j%2==r[j//2]:
        path.append(j); j//=2
    ancestor=None if start==0 else j//2
    return {'ancestor': ancestor,
    'path': path,
    'cost': len(path)+1,
    'stop': j,
    'orientation': None if start==0 else r[j//2],
    'direct': start!=0 and ancestor==start//2}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 2}, {'ancestor': 1, 'path': [], 'cost': 0, 'stop': 2, 'orientation': 1, 'direct': True})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 3}, {'ancestor': 0, 'path': [3], 'cost': 1, 'stop': 1, 'orientation': 0, 'direct': False})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 4}, {'ancestor': 2, 'path': [], 'cost': 0, 'stop': 4, 'orientation': 1, 'direct': True})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 5}, {'ancestor': 1, 'path': [5], 'cost': 1, 'stop': 2, 'orientation': 1, 'direct': False})], [({'reverse': [0], 'node': 0}, {'ancestor': None, 'path': [], 'cost': 0, 'stop': 0, 'orientation': None, 'direct': False}), ({'reverse': [0, 0], 'node': 1}, {'ancestor': 0, 'path': [], 'cost': 0, 'stop': 1, 'orientation': 0, 'direct': True}), ({'reverse': [0, 0, 0, 0, 0, 0, 0, 0], 'node': 4}, {'ancestor': 0, 'path': [4, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 0, 1, 0, 0, 0, 0], 'node': 7}, {'ancestor': 0, 'path': [7, 3], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 0, 1, 0, 0, 0, 0, 0], 'node': 5}, {'ancestor': 0, 'path': [5, 2], 'cost': 2, 'stop': 1, 'orientation': 0, 'direct': False}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True}), ({'reverse': [0, 1, 1, 1, 0, 0, 0, 0], 'node': 6}, {'ancestor': 3, 'path': [], 'cost': 0, 'stop': 6, 'orientation': 1, 'direct': True})]][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{'ancestor': None, 'cost': 1, 'direct': False, 'orientation': None, 'path': [], 'stop': 0}{'ancestor': None, 'cost': 0, 'direct': False, 'orientation': None, 'path': [], 'stop': 0}Failed
regression certificate 2{'ancestor': 0, 'cost': 1, 'direct': True, 'orientation': 0, 'path': [], 'stop': 1}{'ancestor': 0, 'cost': 0, 'direct': True, 'orientation': 0, 'path': [], 'stop': 1}Failed
regression certificate 3{'ancestor': 0, 'cost': 3, 'direct': False, 'orientation': 0, 'path': [4, 2], 'stop': 1}{'ancestor': 0, 'cost': 2, 'direct': False, 'orientation': 0, 'path': [4, 2], 'stop': 1}Failed
regression certificate 4{'ancestor': 0, 'cost': 3, 'direct': False, 'orientation': 0, 'path': [7, 3], 'stop': 1}{'ancestor': 0, 'cost': 2, 'direct': False, 'orientation': 0, 'path': [7, 3], 'stop': 1}Failed
regression certificate 5{'ancestor': 0, 'cost': 3, 'direct': False, 'orientation': 0, 'path': [5, 2], 'stop': 1}{'ancestor': 0, 'cost': 2, 'direct': False, 'orientation': 0, 'path': [5, 2], 'stop': 1}Failed
regression certificate 6{'ancestor': 3, 'cost': 1, 'direct': True, 'orientation': 1, 'path': [], 'stop': 6}{'ancestor': 3, 'cost': 0, 'direct': True, 'orientation': 1, 'path': [], 'stop': 6}Failed
variant-dependent certificate{'ancestor': 1, 'cost': 1, 'direct': True, 'orientation': 1, 'path': [], 'stop': 2}{'ancestor': 1, 'cost': 0, 'direct': True, 'orientation': 1, 'path': [], 'stop': 2}Failed

SHA-256 / 480588c47dda87b796a1e99710c4c6fa1381b09acbc04889e5a6c238a0d7ab68

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 2026-09-29T14:43:32.179336+00:00.

Case digest / 0b98dea5d2388d4d5705872e619bfd8b07131539b4c1cbfe9c3c69e70e477973