FA-73026 / Probabilistic sketches / Open access
HyperLogLog merge across precisions: displaced-bit rank counts from the wrong end · case 01
Folded ranks grow with the value of displaced bits rather than with leading zeros.
ROOT CAUSE
The rank from displaced bits uses their bit length instead of their leading-zero count.
VERIFIED REPAIR
Use d - bitlen(low) + 1.
Unsuccessful approach: Dropping the +1 counts leading zeros without the terminating one bit.
Case contract
Input {a, b}, each {p, regs}. Both sketches are folded to q = min precision. Folding from p to q with d = p - q maps register j to j >> d; if the low d bits of j are non-zero they become the leading bits of the new remainder, giving rank d - bitlen(low) + 1, otherwise the rank is r + d. Empty registers contribute nothing. The merged sketch is the element-wise maximum. Return [q, regs].
Why this case matters
Merging distinct-count sketches produced with different precision settings requires re-deriving ranks, not just re-indexing registers.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
def fold(sk, q):
d = sk['p'] - q
out = [0] * (1 << q)
for j, r in enumerate(sk['regs']):
if r == 0:
continue
nj = j >> d
low = j & ((1 << d) - 1)
if low != 0:
nr = low.bit_length()
else:
nr = r + d
out[nj] = max(out[nj], nr)
return out
q = min(x['a']['p'], x['b']['p'])
fa = fold(x['a'], q)
fb = fold(x['b'], q)
return [q, [max(u, v) for u, v in zip(fa, fb)]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['fold p5 into p3',
{'a': {'p': 5,
'regs': [3,
2,
4,
3,
0,
4,
5,
1,
1,
1,
0,
0,
0,
5,
3,
0,
4,
2,
5,
3,
0,
3,
0,
0,
3,
0,
0,
5,
4,
3,
1,
0]},
'b': {'p': 3, 'regs': [2, 1, 0, 4, 3, 0, 1, 3]}},
[3, [5, 2, 3, 4, 6, 2, 5, 6]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 3, 0, 2]},
'b': {'p': 4, 'regs': [5, 0, 5, 3, 3, 2, 2, 2, 4, 4, 0, 3, 4, 0, 4, 4]}},
[2, [7, 5, 6, 6]]],
['same precision',
{'a': {'p': 3, 'regs': [0, 0, 4, 0, 1, 5, 0, 0]},
'b': {'p': 3, 'regs': [0, 2, 1, 4, 0, 0, 0, 0]}},
[3, [0, 2, 4, 4, 1, 5, 0, 0]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [4, 4, 4, 4]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 1]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 5, 0, 0, 0, 0, 4, 2]},
'b': {'p': 6,
'regs': [0,
5,
0,
0,
0,
0,
0,
0,
3,
2,
0,
0,
0,
0,
0,
0,
0,
3,
0,
0,
5,
0,
0,
1,
0,
0,
4,
0,
4,
0,
0,
0,
0,
0,
0,
0,
0,
0,
4,
0,
3,
0,
0,
0,
0,
0,
5,
0,
0,
0,
0,
3,
0,
0,
0,
0,
0,
0,
2,
1,
3,
0,
0,
0]}},
[3, [3, 6, 3, 2, 1, 6, 4, 2]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
0,
1,
2,
0,
2,
0,
3,
0,
4,
5,
0,
3,
0,
0,
3,
3,
1,
2,
0,
5,
0,
3,
0,
4,
1,
0,
1,
1,
4,
3,
4]},
'b': {'p': 3, 'regs': [3, 0, 3, 3, 0, 1, 1, 2]}},
[3, [3, 2, 3, 5, 5, 7, 6, 3]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 0, 0, 0]},
'b': {'p': 4, 'regs': [1, 2, 5, 0, 1, 0, 2, 0, 3, 2, 4, 1, 4, 0, 3, 1]}},
[2, [3, 3, 5, 6]]],
['same precision',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 1, 2, 5, 4]},
'b': {'p': 3, 'regs': [2, 1, 0, 0, 0, 0, 0, 0]}},
[3, [2, 1, 0, 0, 1, 2, 5, 4]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [5, 5, 5, 5]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 2]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 0, 2, 4, 5, 0, 5, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
0,
1,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1,
2,
5,
0,
0,
0,
0,
0,
1,
0,
0,
1,
4,
2,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
4,
0,
1,
0,
0,
0,
2,
0,
0,
0,
4,
3,
0,
0,
2,
5,
0,
0,
0]}},
[3, [1, 0, 3, 4, 5, 4, 5, 6]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [1,
2,
2,
1,
3,
3,
3,
0,
1,
0,
4,
5,
0,
0,
0,
0,
0,
2,
4,
0,
2,
4,
0,
4,
0,
3,
1,
2,
5,
5,
0,
0]},
'b': {'p': 3, 'regs': [4, 0, 0, 1, 4, 4, 0, 0]}},
[3, [4, 5, 3, 1, 4, 4, 2, 7]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 2, 0, 0]},
'b': {'p': 4, 'regs': [2, 5, 1, 1, 1, 2, 4, 5, 1, 3, 3, 3, 0, 3, 1, 3]}},
[2, [4, 3, 3, 2]]],
['same precision',
{'a': {'p': 3, 'regs': [5, 0, 0, 4, 0, 0, 2, 5]},
'b': {'p': 3, 'regs': [3, 0, 1, 1, 5, 3, 1, 0]}},
[3, [5, 0, 1, 4, 5, 3, 2, 5]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [6, 6, 6, 6]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [1, 0, 0, 3, 0, 0, 0, 5]},
'b': {'p': 6,
'regs': [0,
2,
2,
0,
0,
0,
0,
0,
0,
5,
0,
0,
0,
5,
3,
0,
3,
4,
0,
4,
0,
4,
0,
0,
0,
0,
1,
0,
0,
0,
0,
4,
0,
0,
0,
0,
3,
0,
3,
0,
0,
0,
0,
0,
0,
1,
0,
0,
2,
0,
0,
0,
0,
0,
0,
0,
4,
0,
0,
5,
0,
0,
0,
0]}},
[3, [3, 3, 6, 3, 1, 1, 5, 7]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
0,
4,
0,
0,
0,
0,
5,
1,
4,
4,
0,
2,
3,
5,
1,
0,
0,
1,
1,
5,
3,
0,
0,
0,
5,
4,
3,
4,
5,
3,
2]},
'b': {'p': 3, 'regs': [5, 3, 1, 2, 0, 3, 3, 3]}},
[3, [5, 3, 3, 4, 1, 7, 3, 6]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 0, 0, 0]},
'b': {'p': 4, 'regs': [1, 0, 3, 0, 4, 4, 2, 1, 4, 4, 2, 0, 0, 1, 2, 1]}},
[2, [3, 6, 6, 2]]],
['same precision',
{'a': {'p': 3, 'regs': [3, 3, 2, 1, 5, 3, 0, 0]},
'b': {'p': 3, 'regs': [0, 1, 0, 0, 0, 2, 2, 5]}},
[3, [3, 3, 2, 1, 5, 3, 2, 5]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [3, 3, 3, 3]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 1]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [4, 2, 0, 0, 4, 0, 2, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
2,
0,
0,
5,
0,
0,
5,
0,
4,
2,
0,
0,
1,
0,
0,
0,
2,
1,
0,
0,
0,
0,
5,
0,
0,
0,
5,
3,
4,
5,
0,
5,
0,
0,
0,
0,
0,
3,
3,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
2,
0,
5,
0,
1,
0,
0,
0,
0,
0,
0]}},
[3, [4, 3, 2, 3, 8, 6, 2, 3]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
5,
1,
2,
0,
5,
0,
1,
1,
0,
5,
1,
0,
1,
0,
2,
3,
1,
0,
3,
4,
0,
0,
5,
0,
5,
1,
0,
0,
0,
0,
0]},
'b': {'p': 3, 'regs': [5, 1, 1, 4, 2, 3, 5, 0]}},
[3, [5, 2, 3, 4, 5, 6, 5, 0]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 3, 3, 0]},
'b': {'p': 4, 'regs': [1, 3, 2, 0, 0, 5, 2, 2, 3, 2, 4, 2, 3, 3, 1, 3]}},
[2, [3, 3, 5, 5]]],
['same precision',
{'a': {'p': 3, 'regs': [4, 0, 0, 3, 0, 4, 1, 1]},
'b': {'p': 3, 'regs': [0, 0, 0, 0, 3, 0, 4, 0]}},
[3, [4, 0, 0, 3, 3, 4, 4, 1]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [4, 4, 4, 4]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 2]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 0, 4, 4, 0, 5, 3, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
0,
2,
0,
0,
0,
1,
0,
0,
3,
0,
0,
0,
0,
0,
0,
0,
2,
0,
5,
0,
0,
4,
0,
0,
0,
0,
1,
0,
0,
0,
1,
0,
0,
4,
0,
0,
0,
5,
0,
0,
0,
0,
0,
0,
0,
0,
3,
1,
3,
1,
5,
0,
0,
0,
0,
2,
0,
3,
0,
0,
4]}},
[3, [1, 4, 4, 7, 3, 8, 3, 2]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]]]
for label, args, expected in cases[N - 1]:
check(label, solve(args), expected)
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 |
|---|---|---|---|
| fold p5 into p3 | [3, [5, 2, 3, 4, 6, 1, 5, 6]] | [3, [5, 2, 3, 4, 6, 2, 5, 6]] | Failed |
| fold p4 into p2 | [2, [7, 5, 6, 6]] | [2, [7, 5, 6, 6]] | Passed |
| same precision | [3, [0, 2, 4, 4, 1, 5, 0, 0]] | [3, [0, 2, 4, 4, 1, 5, 0, 0]] | Passed |
| only low-zero slots | [2, [4, 4, 4, 4]] | [2, [4, 4, 4, 4]] | Passed |
| only odd slots | [2, [2, 2, 2, 2]] | [2, [2, 2, 2, 2]] | Passed |
| fold p6 into p3 | [3, [1, 6, 3, 3, 3, 6, 4, 3]] | [3, [3, 6, 3, 2, 1, 6, 4, 2]] | Failed |
| empty sketches | [2, [0, 0, 0, 0]] | [2, [0, 0, 0, 0]] | Passed |
SHA-256 / 3cb0b99aeaccf750cd8f92a13e39e1248f4fc9b9f56be28e3b070f2b83cdae70
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
def fold(sk, q):
d = sk['p'] - q
out = [0] * (1 << q)
for j, r in enumerate(sk['regs']):
if r == 0:
continue
nj = j >> d
low = j & ((1 << d) - 1)
if low != 0:
nr = d - low.bit_length()
else:
nr = r + d
out[nj] = max(out[nj], nr)
return out
q = min(x['a']['p'], x['b']['p'])
fa = fold(x['a'], q)
fb = fold(x['b'], q)
return [q, [max(u, v) for u, v in zip(fa, fb)]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['fold p5 into p3',
{'a': {'p': 5,
'regs': [3,
2,
4,
3,
0,
4,
5,
1,
1,
1,
0,
0,
0,
5,
3,
0,
4,
2,
5,
3,
0,
3,
0,
0,
3,
0,
0,
5,
4,
3,
1,
0]},
'b': {'p': 3, 'regs': [2, 1, 0, 4, 3, 0, 1, 3]}},
[3, [5, 2, 3, 4, 6, 2, 5, 6]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 3, 0, 2]},
'b': {'p': 4, 'regs': [5, 0, 5, 3, 3, 2, 2, 2, 4, 4, 0, 3, 4, 0, 4, 4]}},
[2, [7, 5, 6, 6]]],
['same precision',
{'a': {'p': 3, 'regs': [0, 0, 4, 0, 1, 5, 0, 0]},
'b': {'p': 3, 'regs': [0, 2, 1, 4, 0, 0, 0, 0]}},
[3, [0, 2, 4, 4, 1, 5, 0, 0]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [4, 4, 4, 4]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 1]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 5, 0, 0, 0, 0, 4, 2]},
'b': {'p': 6,
'regs': [0,
5,
0,
0,
0,
0,
0,
0,
3,
2,
0,
0,
0,
0,
0,
0,
0,
3,
0,
0,
5,
0,
0,
1,
0,
0,
4,
0,
4,
0,
0,
0,
0,
0,
0,
0,
0,
0,
4,
0,
3,
0,
0,
0,
0,
0,
5,
0,
0,
0,
0,
3,
0,
0,
0,
0,
0,
0,
2,
1,
3,
0,
0,
0]}},
[3, [3, 6, 3, 2, 1, 6, 4, 2]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
0,
1,
2,
0,
2,
0,
3,
0,
4,
5,
0,
3,
0,
0,
3,
3,
1,
2,
0,
5,
0,
3,
0,
4,
1,
0,
1,
1,
4,
3,
4]},
'b': {'p': 3, 'regs': [3, 0, 3, 3, 0, 1, 1, 2]}},
[3, [3, 2, 3, 5, 5, 7, 6, 3]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 0, 0, 0]},
'b': {'p': 4, 'regs': [1, 2, 5, 0, 1, 0, 2, 0, 3, 2, 4, 1, 4, 0, 3, 1]}},
[2, [3, 3, 5, 6]]],
['same precision',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 1, 2, 5, 4]},
'b': {'p': 3, 'regs': [2, 1, 0, 0, 0, 0, 0, 0]}},
[3, [2, 1, 0, 0, 1, 2, 5, 4]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [5, 5, 5, 5]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 2]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 0, 2, 4, 5, 0, 5, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
0,
1,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1,
2,
5,
0,
0,
0,
0,
0,
1,
0,
0,
1,
4,
2,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
4,
0,
1,
0,
0,
0,
2,
0,
0,
0,
4,
3,
0,
0,
2,
5,
0,
0,
0]}},
[3, [1, 0, 3, 4, 5, 4, 5, 6]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [1,
2,
2,
1,
3,
3,
3,
0,
1,
0,
4,
5,
0,
0,
0,
0,
0,
2,
4,
0,
2,
4,
0,
4,
0,
3,
1,
2,
5,
5,
0,
0]},
'b': {'p': 3, 'regs': [4, 0, 0, 1, 4, 4, 0, 0]}},
[3, [4, 5, 3, 1, 4, 4, 2, 7]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 2, 0, 0]},
'b': {'p': 4, 'regs': [2, 5, 1, 1, 1, 2, 4, 5, 1, 3, 3, 3, 0, 3, 1, 3]}},
[2, [4, 3, 3, 2]]],
['same precision',
{'a': {'p': 3, 'regs': [5, 0, 0, 4, 0, 0, 2, 5]},
'b': {'p': 3, 'regs': [3, 0, 1, 1, 5, 3, 1, 0]}},
[3, [5, 0, 1, 4, 5, 3, 2, 5]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [6, 6, 6, 6]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [1, 0, 0, 3, 0, 0, 0, 5]},
'b': {'p': 6,
'regs': [0,
2,
2,
0,
0,
0,
0,
0,
0,
5,
0,
0,
0,
5,
3,
0,
3,
4,
0,
4,
0,
4,
0,
0,
0,
0,
1,
0,
0,
0,
0,
4,
0,
0,
0,
0,
3,
0,
3,
0,
0,
0,
0,
0,
0,
1,
0,
0,
2,
0,
0,
0,
0,
0,
0,
0,
4,
0,
0,
5,
0,
0,
0,
0]}},
[3, [3, 3, 6, 3, 1, 1, 5, 7]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
0,
4,
0,
0,
0,
0,
5,
1,
4,
4,
0,
2,
3,
5,
1,
0,
0,
1,
1,
5,
3,
0,
0,
0,
5,
4,
3,
4,
5,
3,
2]},
'b': {'p': 3, 'regs': [5, 3, 1, 2, 0, 3, 3, 3]}},
[3, [5, 3, 3, 4, 1, 7, 3, 6]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 0, 0, 0]},
'b': {'p': 4, 'regs': [1, 0, 3, 0, 4, 4, 2, 1, 4, 4, 2, 0, 0, 1, 2, 1]}},
[2, [3, 6, 6, 2]]],
['same precision',
{'a': {'p': 3, 'regs': [3, 3, 2, 1, 5, 3, 0, 0]},
'b': {'p': 3, 'regs': [0, 1, 0, 0, 0, 2, 2, 5]}},
[3, [3, 3, 2, 1, 5, 3, 2, 5]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [3, 3, 3, 3]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 1]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [4, 2, 0, 0, 4, 0, 2, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
2,
0,
0,
5,
0,
0,
5,
0,
4,
2,
0,
0,
1,
0,
0,
0,
2,
1,
0,
0,
0,
0,
5,
0,
0,
0,
5,
3,
4,
5,
0,
5,
0,
0,
0,
0,
0,
3,
3,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
2,
0,
5,
0,
1,
0,
0,
0,
0,
0,
0]}},
[3, [4, 3, 2, 3, 8, 6, 2, 3]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
5,
1,
2,
0,
5,
0,
1,
1,
0,
5,
1,
0,
1,
0,
2,
3,
1,
0,
3,
4,
0,
0,
5,
0,
5,
1,
0,
0,
0,
0,
0]},
'b': {'p': 3, 'regs': [5, 1, 1, 4, 2, 3, 5, 0]}},
[3, [5, 2, 3, 4, 5, 6, 5, 0]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 3, 3, 0]},
'b': {'p': 4, 'regs': [1, 3, 2, 0, 0, 5, 2, 2, 3, 2, 4, 2, 3, 3, 1, 3]}},
[2, [3, 3, 5, 5]]],
['same precision',
{'a': {'p': 3, 'regs': [4, 0, 0, 3, 0, 4, 1, 1]},
'b': {'p': 3, 'regs': [0, 0, 0, 0, 3, 0, 4, 0]}},
[3, [4, 0, 0, 3, 3, 4, 4, 1]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [4, 4, 4, 4]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 2]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 0, 4, 4, 0, 5, 3, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
0,
2,
0,
0,
0,
1,
0,
0,
3,
0,
0,
0,
0,
0,
0,
0,
2,
0,
5,
0,
0,
4,
0,
0,
0,
0,
1,
0,
0,
0,
1,
0,
0,
4,
0,
0,
0,
5,
0,
0,
0,
0,
0,
0,
0,
0,
3,
1,
3,
1,
5,
0,
0,
0,
0,
2,
0,
3,
0,
0,
4]}},
[3, [1, 4, 4, 7, 3, 8, 3, 2]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]]]
for label, args, expected in cases[N - 1]:
check(label, solve(args), expected)
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 |
|---|---|---|---|
| fold p5 into p3 | [3, [5, 1, 3, 4, 6, 1, 5, 6]] | [3, [5, 2, 3, 4, 6, 2, 5, 6]] | Failed |
| fold p4 into p2 | [2, [7, 5, 6, 6]] | [2, [7, 5, 6, 6]] | Passed |
| same precision | [3, [0, 2, 4, 4, 1, 5, 0, 0]] | [3, [0, 2, 4, 4, 1, 5, 0, 0]] | Passed |
| only low-zero slots | [2, [4, 4, 4, 4]] | [2, [4, 4, 4, 4]] | Passed |
| only odd slots | [2, [1, 1, 1, 1]] | [2, [2, 2, 2, 2]] | Failed |
| fold p6 into p3 | [3, [2, 6, 2, 1, 0, 6, 4, 2]] | [3, [3, 6, 3, 2, 1, 6, 4, 2]] | Failed |
| empty sketches | [2, [0, 0, 0, 0]] | [2, [0, 0, 0, 0]] | Passed |
SHA-256 / 14532bbdae98bfce1185cdeb5a0b3d523f05c90ee622b3161f9feae491983d55
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
def fold(sk, q):
d = sk['p'] - q
out = [0] * (1 << q)
for j, r in enumerate(sk['regs']):
if r == 0:
continue
nj = j >> d
low = j & ((1 << d) - 1)
if low != 0:
nr = d - low.bit_length() + 1
else:
nr = r + d
out[nj] = max(out[nj], nr)
return out
q = min(x['a']['p'], x['b']['p'])
fa = fold(x['a'], q)
fb = fold(x['b'], q)
return [q, [max(u, v) for u, v in zip(fa, fb)]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['fold p5 into p3',
{'a': {'p': 5,
'regs': [3,
2,
4,
3,
0,
4,
5,
1,
1,
1,
0,
0,
0,
5,
3,
0,
4,
2,
5,
3,
0,
3,
0,
0,
3,
0,
0,
5,
4,
3,
1,
0]},
'b': {'p': 3, 'regs': [2, 1, 0, 4, 3, 0, 1, 3]}},
[3, [5, 2, 3, 4, 6, 2, 5, 6]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 3, 0, 2]},
'b': {'p': 4, 'regs': [5, 0, 5, 3, 3, 2, 2, 2, 4, 4, 0, 3, 4, 0, 4, 4]}},
[2, [7, 5, 6, 6]]],
['same precision',
{'a': {'p': 3, 'regs': [0, 0, 4, 0, 1, 5, 0, 0]},
'b': {'p': 3, 'regs': [0, 2, 1, 4, 0, 0, 0, 0]}},
[3, [0, 2, 4, 4, 1, 5, 0, 0]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [4, 4, 4, 4]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 1]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 5, 0, 0, 0, 0, 4, 2]},
'b': {'p': 6,
'regs': [0,
5,
0,
0,
0,
0,
0,
0,
3,
2,
0,
0,
0,
0,
0,
0,
0,
3,
0,
0,
5,
0,
0,
1,
0,
0,
4,
0,
4,
0,
0,
0,
0,
0,
0,
0,
0,
0,
4,
0,
3,
0,
0,
0,
0,
0,
5,
0,
0,
0,
0,
3,
0,
0,
0,
0,
0,
0,
2,
1,
3,
0,
0,
0]}},
[3, [3, 6, 3, 2, 1, 6, 4, 2]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
0,
1,
2,
0,
2,
0,
3,
0,
4,
5,
0,
3,
0,
0,
3,
3,
1,
2,
0,
5,
0,
3,
0,
4,
1,
0,
1,
1,
4,
3,
4]},
'b': {'p': 3, 'regs': [3, 0, 3, 3, 0, 1, 1, 2]}},
[3, [3, 2, 3, 5, 5, 7, 6, 3]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 0, 0, 0]},
'b': {'p': 4, 'regs': [1, 2, 5, 0, 1, 0, 2, 0, 3, 2, 4, 1, 4, 0, 3, 1]}},
[2, [3, 3, 5, 6]]],
['same precision',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 1, 2, 5, 4]},
'b': {'p': 3, 'regs': [2, 1, 0, 0, 0, 0, 0, 0]}},
[3, [2, 1, 0, 0, 1, 2, 5, 4]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [5, 5, 5, 5]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 2]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 0, 2, 4, 5, 0, 5, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
0,
1,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1,
2,
5,
0,
0,
0,
0,
0,
1,
0,
0,
1,
4,
2,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
4,
0,
1,
0,
0,
0,
2,
0,
0,
0,
4,
3,
0,
0,
2,
5,
0,
0,
0]}},
[3, [1, 0, 3, 4, 5, 4, 5, 6]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [1,
2,
2,
1,
3,
3,
3,
0,
1,
0,
4,
5,
0,
0,
0,
0,
0,
2,
4,
0,
2,
4,
0,
4,
0,
3,
1,
2,
5,
5,
0,
0]},
'b': {'p': 3, 'regs': [4, 0, 0, 1, 4, 4, 0, 0]}},
[3, [4, 5, 3, 1, 4, 4, 2, 7]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 2, 0, 0]},
'b': {'p': 4, 'regs': [2, 5, 1, 1, 1, 2, 4, 5, 1, 3, 3, 3, 0, 3, 1, 3]}},
[2, [4, 3, 3, 2]]],
['same precision',
{'a': {'p': 3, 'regs': [5, 0, 0, 4, 0, 0, 2, 5]},
'b': {'p': 3, 'regs': [3, 0, 1, 1, 5, 3, 1, 0]}},
[3, [5, 0, 1, 4, 5, 3, 2, 5]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [6, 6, 6, 6]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [1, 0, 0, 3, 0, 0, 0, 5]},
'b': {'p': 6,
'regs': [0,
2,
2,
0,
0,
0,
0,
0,
0,
5,
0,
0,
0,
5,
3,
0,
3,
4,
0,
4,
0,
4,
0,
0,
0,
0,
1,
0,
0,
0,
0,
4,
0,
0,
0,
0,
3,
0,
3,
0,
0,
0,
0,
0,
0,
1,
0,
0,
2,
0,
0,
0,
0,
0,
0,
0,
4,
0,
0,
5,
0,
0,
0,
0]}},
[3, [3, 3, 6, 3, 1, 1, 5, 7]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
0,
4,
0,
0,
0,
0,
5,
1,
4,
4,
0,
2,
3,
5,
1,
0,
0,
1,
1,
5,
3,
0,
0,
0,
5,
4,
3,
4,
5,
3,
2]},
'b': {'p': 3, 'regs': [5, 3, 1, 2, 0, 3, 3, 3]}},
[3, [5, 3, 3, 4, 1, 7, 3, 6]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 0, 0, 0]},
'b': {'p': 4, 'regs': [1, 0, 3, 0, 4, 4, 2, 1, 4, 4, 2, 0, 0, 1, 2, 1]}},
[2, [3, 6, 6, 2]]],
['same precision',
{'a': {'p': 3, 'regs': [3, 3, 2, 1, 5, 3, 0, 0]},
'b': {'p': 3, 'regs': [0, 1, 0, 0, 0, 2, 2, 5]}},
[3, [3, 3, 2, 1, 5, 3, 2, 5]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [3, 3, 3, 3]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 1]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [4, 2, 0, 0, 4, 0, 2, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
2,
0,
0,
5,
0,
0,
5,
0,
4,
2,
0,
0,
1,
0,
0,
0,
2,
1,
0,
0,
0,
0,
5,
0,
0,
0,
5,
3,
4,
5,
0,
5,
0,
0,
0,
0,
0,
3,
3,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
2,
0,
5,
0,
1,
0,
0,
0,
0,
0,
0]}},
[3, [4, 3, 2, 3, 8, 6, 2, 3]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]],
[['fold p5 into p3',
{'a': {'p': 5,
'regs': [0,
5,
1,
2,
0,
5,
0,
1,
1,
0,
5,
1,
0,
1,
0,
2,
3,
1,
0,
3,
4,
0,
0,
5,
0,
5,
1,
0,
0,
0,
0,
0]},
'b': {'p': 3, 'regs': [5, 1, 1, 4, 2, 3, 5, 0]}},
[3, [5, 2, 3, 4, 5, 6, 5, 0]]],
['fold p4 into p2',
{'a': {'p': 2, 'regs': [0, 3, 3, 0]},
'b': {'p': 4, 'regs': [1, 3, 2, 0, 0, 5, 2, 2, 3, 2, 4, 2, 3, 3, 1, 3]}},
[2, [3, 3, 5, 5]]],
['same precision',
{'a': {'p': 3, 'regs': [4, 0, 0, 3, 0, 4, 1, 1]},
'b': {'p': 3, 'regs': [0, 0, 0, 0, 3, 0, 4, 0]}},
[3, [4, 0, 0, 3, 3, 4, 4, 1]]],
['only low-zero slots',
{'a': {'p': 4, 'regs': [2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0]},
'b': {'p': 2, 'regs': [0, 1, 0, 0]}},
[2, [4, 4, 4, 4]]],
['only odd slots',
{'a': {'p': 4, 'regs': [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2]},
'b': {'p': 2, 'regs': [0, 0, 0, 2]}},
[2, [2, 2, 2, 2]]],
['fold p6 into p3',
{'a': {'p': 3, 'regs': [0, 0, 4, 4, 0, 5, 3, 0]},
'b': {'p': 6,
'regs': [0,
0,
0,
0,
2,
0,
0,
0,
1,
0,
0,
3,
0,
0,
0,
0,
0,
0,
0,
2,
0,
5,
0,
0,
4,
0,
0,
0,
0,
1,
0,
0,
0,
1,
0,
0,
4,
0,
0,
0,
5,
0,
0,
0,
0,
0,
0,
0,
0,
3,
1,
3,
1,
5,
0,
0,
0,
0,
2,
0,
3,
0,
0,
4]}},
[3, [1, 4, 4, 7, 3, 8, 3, 2]]],
['empty sketches',
{'a': {'p': 3, 'regs': [0, 0, 0, 0, 0, 0, 0, 0]}, 'b': {'p': 2, 'regs': [0, 0, 0, 0]}},
[2, [0, 0, 0, 0]]]]]
for label, args, expected in cases[N - 1]:
check(label, solve(args), expected)
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 |
|---|---|---|---|
| fold p5 into p3 | [3, [5, 2, 3, 4, 6, 2, 5, 6]] | [3, [5, 2, 3, 4, 6, 2, 5, 6]] | Passed |
| fold p4 into p2 | [2, [7, 5, 6, 6]] | [2, [7, 5, 6, 6]] | Passed |
| same precision | [3, [0, 2, 4, 4, 1, 5, 0, 0]] | [3, [0, 2, 4, 4, 1, 5, 0, 0]] | Passed |
| only low-zero slots | [2, [4, 4, 4, 4]] | [2, [4, 4, 4, 4]] | Passed |
| only odd slots | [2, [2, 2, 2, 2]] | [2, [2, 2, 2, 2]] | Passed |
| fold p6 into p3 | [3, [3, 6, 3, 2, 1, 6, 4, 2]] | [3, [3, 6, 3, 2, 1, 6, 4, 2]] | Passed |
| empty sketches | [2, [0, 0, 0, 0]] | [2, [0, 0, 0, 0]] | Passed |
SHA-256 / 90487cbbe4bc3f7e2e4b4b7a1387c0680be67d996a0fe4dd526d15f86395efaa
Verification & scope
A deterministic, bounded teaching model with stipulated constants and pre-hashed or explicitly hashed inputs; it is not a production implementation and makes no claim of conformance to any library or paper beyond the stated contract. 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 2026-09-29T14:48:43.658748+00:00.
Case digest / ae28a1a0cad489d614ce257c1ff2b745a46fce4e680d3babb54b7822d6b28b57