FA-73031 / Probabilistic sketches / Open access
HyperLogLog merge across precisions: empty registers acquire phantom ranks · case 01
Registers that never saw a hash contribute non-zero ranks after folding, inflating the estimate.
ROOT CAUSE
Empty registers are not skipped during folding, so displaced index bits alone produce a rank.
THE FAILURE
Empty registers are not skipped during folding, so displaced index bits alone produce a rank.
Unsuccessful approach: Skipping only empty registers with zero displaced bits still invents ranks for the others.
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']):
pass
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, 3, 3, 6, 4, 3]] | [3, [3, 6, 3, 2, 1, 6, 4, 2]] | Failed |
| empty sketches | [2, [1, 1, 1, 1]] | [2, [0, 0, 0, 0]] | Failed |
SHA-256 / 0f44c37b96b220314d72f3ee1323e4eb520d78c4de70656da618adcdd5fa77d8
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 and (j & ((1 << d) - 1)) == 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, 3, 3, 6, 4, 3]] | [3, [3, 6, 3, 2, 1, 6, 4, 2]] | Failed |
| empty sketches | [2, [1, 1, 1, 1]] | [2, [0, 0, 0, 0]] | Failed |
SHA-256 / f0a73f4b596b20e8d9e03a341de733d2cc60709006768b14d08d24f4e2a00023
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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Sign in to the archive ↗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 2026-09-29T14:48:43.824942+00:00.
Case digest / 1f96cbc03ea6235d48b54b48192a04a0555135ef5c8ccc27dbdcebe08fa88c63