FA-73216 / Probabilistic sketches / Open access
MinHash LSH banding: band slice uses the band count as width · case 01
Band boundaries shift whenever b differs from r, so candidate pairs change.
ROOT CAUSE
Bands are sliced with width b instead of rows-per-band r.
THE FAILURE
Bands are sliced with width b instead of rows-per-band r.
Unsuccessful approach: Striding by b interleaves components instead of forming contiguous bands.
Case contract
Input {b, r, sigs} with sigs a list of [id, signature]. Every signature must have exactly b*r components, otherwise "invalid". Band i covers components i*r .. i*r+r-1 and hashes to the key (i, band tuple). Ids sharing any band key form a candidate pair [smaller, larger], reported once, sorted. Also return the similarity threshold (1/b)^(1/r) rounded to 3 decimals.
Why this case matters
Locality-sensitive hashing prunes all-pairs similarity search; banding mistakes either flood the verifier with false candidates or silently miss near-duplicates.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
b = x['b']
r = x['r']
sigs = x['sigs']
if any(len(sig) != b * r for ident, sig in sigs):
return 'invalid'
buckets = {}
for ident, sig in sigs:
for i in range(b):
key = (i, tuple(sig[i * b:(i + 1) * b]))
buckets.setdefault(key, []).append(ident)
pairs = set()
for members in buckets.values():
for u in members:
for v in members:
if u < v:
pairs.add((u, v))
return [[list(p) for p in sorted(pairs)], round((1 / b) ** (1 / r), 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [4, 2, 3, 1, 0, 4]],
[2, [2, 1, 3, 1, 1, 4]],
[3, [2, 4, 3, 1, 2, 4]],
[4, [2, 1, 3, 1, 1, 4]],
[5, [2, 1, 1, 1, 2, 2]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4], [4, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [2, 1, 3, 1, 1, 3]],
[2, [2, 1, 3, 1, 4, 3]],
[3, [2, 1, 3, 1, 4, 4]],
[4, [1, 1, 2, 4, 0, 2]],
[5, [2, 1, 3, 1, 4, 4]]]},
[[[1, 2], [1, 3], [1, 5], [2, 3], [2, 5], [3, 5]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [1, 2, 4, 0, 0, 1, 3, 4]],
[2, [0, 4, 4, 0, 1, 1, 3, 4]],
[3, [3, 1, 0, 3, 1, 3, 1, 2]],
[4, [1, 1, 4, 1, 4, 3, 3, 4]],
[5, [1, 4, 4, 3, 1, 3, 1, 3]]]},
[[[1, 2], [1, 4], [2, 4], [3, 5]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [1, 1, 1, 1, 0, 2, 1, 0, 4]],
[2, [3, 1, 1, 1, 0, 0, 3, 3, 3]],
[3, [1, 1, 3, 1, 0, 3, 3, 1, 2]],
[4, [2, 2, 1, 4, 0, 1, 2, 1, 2]],
[5, [0, 2, 1, 1, 0, 2, 3, 1, 2]]]},
[[[1, 5], [3, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 1, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 1]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 1]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [0, 0, 2, 2, 3, 1]],
[2, [4, 2, 4, 2, 3, 1]],
[3, [0, 0, 2, 2, 3, 1]],
[4, [0, 0, 2, 2, 3, 1]],
[5, [1, 0, 2, 4, 4, 1]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 0, 1, 2, 4, 0]],
[2, [3, 1, 2, 2, 2, 0]],
[3, [3, 1, 2, 3, 4, 0]],
[4, [2, 1, 2, 2, 2, 0]],
[5, [3, 3, 3, 3, 3, 0]]]},
[[[2, 3], [2, 4]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [2, 0, 4, 3, 0, 4, 4, 3]],
[2, [2, 0, 2, 3, 0, 3, 0, 3]],
[3, [2, 1, 0, 2, 0, 3, 3, 3]],
[4, [0, 0, 0, 3, 0, 3, 3, 3]],
[5, [3, 3, 2, 1, 0, 3, 0, 4]]]},
[[[1, 2], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [2, 3, 0, 4, 3, 1, 2, 2, 3]],
[2, [2, 3, 3, 4, 0, 1, 4, 0, 3]],
[3, [2, 3, 0, 4, 3, 1, 2, 2, 3]],
[4, [2, 3, 3, 2, 3, 1, 1, 2, 3]],
[5, [2, 3, 0, 4, 3, 1, 2, 2, 3]]]},
[[[1, 3], [1, 5], [2, 4], [3, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 2, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 2]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 2]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [0, 1, 2, 0, 3, 4]],
[2, [4, 0, 2, 0, 3, 3]],
[3, [0, 0, 2, 0, 4, 4]],
[4, [2, 1, 4, 0, 3, 2]],
[5, [0, 0, 3, 3, 4, 2]]]},
[[[1, 2], [1, 3], [2, 3], [3, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [4, 3, 0, 0, 1, 2]],
[2, [4, 3, 0, 0, 1, 2]],
[3, [3, 3, 0, 0, 1, 2]],
[4, [4, 3, 0, 0, 1, 2]],
[5, [4, 0, 4, 1, 2, 2]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [0, 0, 3, 1, 2, 3, 4, 3]],
[2, [0, 0, 0, 4, 2, 0, 3, 4]],
[3, [3, 0, 0, 1, 2, 0, 4, 3]],
[4, [0, 3, 0, 0, 2, 4, 3, 2]],
[5, [1, 0, 3, 1, 1, 4, 0, 4]]]},
[[[1, 2], [1, 3], [1, 5], [2, 3]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [4, 3, 0, 0, 4, 4, 3, 2, 1]],
[2, [4, 1, 3, 0, 0, 0, 1, 4, 1]],
[3, [4, 1, 3, 0, 0, 0, 4, 0, 1]],
[4, [4, 4, 4, 0, 0, 4, 0, 1, 1]],
[5, [1, 2, 3, 0, 0, 0, 1, 1, 1]]]},
[[[2, 3], [2, 5], [3, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 3, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 3]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 3]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [4, 3, 4, 3, 0, 4]],
[2, [4, 2, 3, 3, 0, 1]],
[3, [4, 2, 1, 3, 0, 1]],
[4, [4, 2, 3, 3, 2, 1]],
[5, [4, 4, 3, 3, 0, 0]]]},
[[[2, 3], [2, 4], [2, 5], [3, 4], [4, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [3, 1, 4, 4, 2, 3]],
[2, [1, 4, 0, 2, 2, 3]],
[3, [3, 0, 4, 4, 2, 2]],
[4, [3, 1, 0, 4, 3, 3]],
[5, [4, 1, 4, 1, 3, 3]]]},
[[], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [3, 3, 2, 2, 0, 4, 4, 4]],
[2, [1, 3, 1, 1, 4, 0, 4, 1]],
[3, [3, 3, 1, 1, 0, 2, 1, 4]],
[4, [4, 0, 1, 1, 0, 4, 1, 4]],
[5, [3, 3, 1, 2, 0, 0, 1, 4]]]},
[[[1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [3, 4], [3, 5], [4, 5]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [1, 1, 3, 2, 2, 2, 4, 3, 2]],
[2, [3, 1, 3, 0, 2, 2, 4, 4, 3]],
[3, [1, 1, 4, 1, 2, 2, 4, 3, 2]],
[4, [1, 1, 3, 2, 2, 2, 4, 3, 2]],
[5, [1, 1, 2, 4, 2, 2, 2, 3, 1]]]},
[[[1, 3], [1, 4], [3, 4]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 4, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 4]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 4]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [3, 3, 2, 3, 0, 4]],
[2, [4, 3, 2, 3, 1, 0]],
[3, [1, 2, 2, 3, 3, 0]],
[4, [1, 3, 2, 3, 0, 0]],
[5, [1, 3, 4, 3, 4, 3]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4], [4, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [2, 1, 4, 2, 2, 4]],
[2, [3, 1, 4, 2, 2, 0]],
[3, [2, 1, 4, 2, 2, 4]],
[4, [3, 2, 4, 2, 2, 0]],
[5, [3, 1, 4, 4, 0, 0]]]},
[[[1, 3], [2, 4], [2, 5]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [1, 0, 0, 1, 3, 1, 4, 3]],
[2, [3, 2, 3, 1, 4, 4, 3, 3]],
[3, [1, 0, 0, 1, 4, 1, 4, 3]],
[4, [1, 0, 3, 2, 4, 3, 3, 3]],
[5, [4, 0, 0, 4, 1, 3, 1, 3]]]},
[[[1, 3], [1, 4], [2, 4], [3, 4]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [1, 4, 1, 3, 1, 3, 1, 2, 4]],
[2, [0, 1, 4, 0, 4, 3, 1, 3, 0]],
[3, [1, 4, 1, 0, 4, 1, 3, 4, 3]],
[4, [1, 4, 3, 4, 4, 3, 1, 3, 3]],
[5, [1, 2, 3, 0, 4, 3, 1, 3, 4]]]},
[[[1, 3], [2, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 5, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 5]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 5]]]},
[[], 0.707]]]]
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 |
|---|---|---|---|
| bands 0 b=3 r=2 | [[[1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]], 0.577] | [[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4], [4, 5]], 0.577] | Failed |
| bands 1 b=2 r=3 | [[[1, 2], [1, 3], [1, 5], [2, 3], [2, 5], [3, 5]], 0.794] | [[[1, 2], [1, 3], [1, 5], [2, 3], [2, 5], [3, 5]], 0.794] | Passed |
| bands 2 b=4 r=2 | [[[1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]], 0.5] | [[[1, 2], [1, 4], [2, 4], [3, 5]], 0.5] | Failed |
| bands 3 b=3 r=3 | [[[1, 5], [3, 5]], 0.693] | [[[1, 5], [3, 5]], 0.693] | Passed |
| same values in different bands | [[], 0.707] | [[], 0.707] | Passed |
| overlong signature | invalid | invalid | Passed |
| interleaved agreement | [[[1, 3]], 0.794] | [[[1, 3]], 0.794] | Passed |
| equal sums differ | [[], 0.707] | [[], 0.707] | Passed |
SHA-256 / 8a76299d03fcd2c7f1dda0402d0bf661bd94a07e1dbd39f7173839b20aaa6472
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
b = x['b']
r = x['r']
sigs = x['sigs']
if any(len(sig) != b * r for ident, sig in sigs):
return 'invalid'
buckets = {}
for ident, sig in sigs:
for i in range(b):
key = (i, tuple(sig[i::b]))
buckets.setdefault(key, []).append(ident)
pairs = set()
for members in buckets.values():
for u in members:
for v in members:
if u < v:
pairs.add((u, v))
return [[list(p) for p in sorted(pairs)], round((1 / b) ** (1 / r), 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [4, 2, 3, 1, 0, 4]],
[2, [2, 1, 3, 1, 1, 4]],
[3, [2, 4, 3, 1, 2, 4]],
[4, [2, 1, 3, 1, 1, 4]],
[5, [2, 1, 1, 1, 2, 2]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4], [4, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [2, 1, 3, 1, 1, 3]],
[2, [2, 1, 3, 1, 4, 3]],
[3, [2, 1, 3, 1, 4, 4]],
[4, [1, 1, 2, 4, 0, 2]],
[5, [2, 1, 3, 1, 4, 4]]]},
[[[1, 2], [1, 3], [1, 5], [2, 3], [2, 5], [3, 5]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [1, 2, 4, 0, 0, 1, 3, 4]],
[2, [0, 4, 4, 0, 1, 1, 3, 4]],
[3, [3, 1, 0, 3, 1, 3, 1, 2]],
[4, [1, 1, 4, 1, 4, 3, 3, 4]],
[5, [1, 4, 4, 3, 1, 3, 1, 3]]]},
[[[1, 2], [1, 4], [2, 4], [3, 5]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [1, 1, 1, 1, 0, 2, 1, 0, 4]],
[2, [3, 1, 1, 1, 0, 0, 3, 3, 3]],
[3, [1, 1, 3, 1, 0, 3, 3, 1, 2]],
[4, [2, 2, 1, 4, 0, 1, 2, 1, 2]],
[5, [0, 2, 1, 1, 0, 2, 3, 1, 2]]]},
[[[1, 5], [3, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 1, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 1]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 1]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [0, 0, 2, 2, 3, 1]],
[2, [4, 2, 4, 2, 3, 1]],
[3, [0, 0, 2, 2, 3, 1]],
[4, [0, 0, 2, 2, 3, 1]],
[5, [1, 0, 2, 4, 4, 1]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 0, 1, 2, 4, 0]],
[2, [3, 1, 2, 2, 2, 0]],
[3, [3, 1, 2, 3, 4, 0]],
[4, [2, 1, 2, 2, 2, 0]],
[5, [3, 3, 3, 3, 3, 0]]]},
[[[2, 3], [2, 4]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [2, 0, 4, 3, 0, 4, 4, 3]],
[2, [2, 0, 2, 3, 0, 3, 0, 3]],
[3, [2, 1, 0, 2, 0, 3, 3, 3]],
[4, [0, 0, 0, 3, 0, 3, 3, 3]],
[5, [3, 3, 2, 1, 0, 3, 0, 4]]]},
[[[1, 2], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [2, 3, 0, 4, 3, 1, 2, 2, 3]],
[2, [2, 3, 3, 4, 0, 1, 4, 0, 3]],
[3, [2, 3, 0, 4, 3, 1, 2, 2, 3]],
[4, [2, 3, 3, 2, 3, 1, 1, 2, 3]],
[5, [2, 3, 0, 4, 3, 1, 2, 2, 3]]]},
[[[1, 3], [1, 5], [2, 4], [3, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 2, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 2]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 2]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [0, 1, 2, 0, 3, 4]],
[2, [4, 0, 2, 0, 3, 3]],
[3, [0, 0, 2, 0, 4, 4]],
[4, [2, 1, 4, 0, 3, 2]],
[5, [0, 0, 3, 3, 4, 2]]]},
[[[1, 2], [1, 3], [2, 3], [3, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [4, 3, 0, 0, 1, 2]],
[2, [4, 3, 0, 0, 1, 2]],
[3, [3, 3, 0, 0, 1, 2]],
[4, [4, 3, 0, 0, 1, 2]],
[5, [4, 0, 4, 1, 2, 2]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [0, 0, 3, 1, 2, 3, 4, 3]],
[2, [0, 0, 0, 4, 2, 0, 3, 4]],
[3, [3, 0, 0, 1, 2, 0, 4, 3]],
[4, [0, 3, 0, 0, 2, 4, 3, 2]],
[5, [1, 0, 3, 1, 1, 4, 0, 4]]]},
[[[1, 2], [1, 3], [1, 5], [2, 3]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [4, 3, 0, 0, 4, 4, 3, 2, 1]],
[2, [4, 1, 3, 0, 0, 0, 1, 4, 1]],
[3, [4, 1, 3, 0, 0, 0, 4, 0, 1]],
[4, [4, 4, 4, 0, 0, 4, 0, 1, 1]],
[5, [1, 2, 3, 0, 0, 0, 1, 1, 1]]]},
[[[2, 3], [2, 5], [3, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 3, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 3]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 3]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [4, 3, 4, 3, 0, 4]],
[2, [4, 2, 3, 3, 0, 1]],
[3, [4, 2, 1, 3, 0, 1]],
[4, [4, 2, 3, 3, 2, 1]],
[5, [4, 4, 3, 3, 0, 0]]]},
[[[2, 3], [2, 4], [2, 5], [3, 4], [4, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [3, 1, 4, 4, 2, 3]],
[2, [1, 4, 0, 2, 2, 3]],
[3, [3, 0, 4, 4, 2, 2]],
[4, [3, 1, 0, 4, 3, 3]],
[5, [4, 1, 4, 1, 3, 3]]]},
[[], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [3, 3, 2, 2, 0, 4, 4, 4]],
[2, [1, 3, 1, 1, 4, 0, 4, 1]],
[3, [3, 3, 1, 1, 0, 2, 1, 4]],
[4, [4, 0, 1, 1, 0, 4, 1, 4]],
[5, [3, 3, 1, 2, 0, 0, 1, 4]]]},
[[[1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [3, 4], [3, 5], [4, 5]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [1, 1, 3, 2, 2, 2, 4, 3, 2]],
[2, [3, 1, 3, 0, 2, 2, 4, 4, 3]],
[3, [1, 1, 4, 1, 2, 2, 4, 3, 2]],
[4, [1, 1, 3, 2, 2, 2, 4, 3, 2]],
[5, [1, 1, 2, 4, 2, 2, 2, 3, 1]]]},
[[[1, 3], [1, 4], [3, 4]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 4, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 4]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 4]]]},
[[], 0.707]]],
[['bands 0 b=3 r=2',
{'b': 3,
'r': 2,
'sigs': [[1, [3, 3, 2, 3, 0, 4]],
[2, [4, 3, 2, 3, 1, 0]],
[3, [1, 2, 2, 3, 3, 0]],
[4, [1, 3, 2, 3, 0, 0]],
[5, [1, 3, 4, 3, 4, 3]]]},
[[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4], [4, 5]], 0.577]],
['bands 1 b=2 r=3',
{'b': 2,
'r': 3,
'sigs': [[1, [2, 1, 4, 2, 2, 4]],
[2, [3, 1, 4, 2, 2, 0]],
[3, [2, 1, 4, 2, 2, 4]],
[4, [3, 2, 4, 2, 2, 0]],
[5, [3, 1, 4, 4, 0, 0]]]},
[[[1, 3], [2, 4], [2, 5]], 0.794]],
['bands 2 b=4 r=2',
{'b': 4,
'r': 2,
'sigs': [[1, [1, 0, 0, 1, 3, 1, 4, 3]],
[2, [3, 2, 3, 1, 4, 4, 3, 3]],
[3, [1, 0, 0, 1, 4, 1, 4, 3]],
[4, [1, 0, 3, 2, 4, 3, 3, 3]],
[5, [4, 0, 0, 4, 1, 3, 1, 3]]]},
[[[1, 3], [1, 4], [2, 4], [3, 4]], 0.5]],
['bands 3 b=3 r=3',
{'b': 3,
'r': 3,
'sigs': [[1, [1, 4, 1, 3, 1, 3, 1, 2, 4]],
[2, [0, 1, 4, 0, 4, 3, 1, 3, 0]],
[3, [1, 4, 1, 0, 4, 1, 3, 4, 3]],
[4, [1, 4, 3, 4, 4, 3, 1, 3, 3]],
[5, [1, 2, 3, 0, 4, 3, 1, 3, 4]]]},
[[[1, 3], [2, 5]], 0.693]],
['same values in different bands',
{'b': 2, 'r': 2, 'sigs': [[1, [7, 7, 1, 2]], [2, [3, 4, 7, 7]], [3, [5, 5, 6, 6]]]},
[[], 0.707]],
['overlong signature',
{'b': 2, 'r': 2, 'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 2, 3, 4]]]},
'invalid'],
['interleaved agreement',
{'b': 2,
'r': 3,
'sigs': [[1, [1, 2, 3, 4, 5, 6]], [2, [1, 9, 3, 9, 5, 9]], [3, [1, 2, 3, 0, 0, 5]]]},
[[[1, 3]], 0.794]],
['equal sums differ',
{'b': 2, 'r': 2, 'sigs': [[4, [1, 3, 8, 8]], [9, [2, 2, 9, 9]], [5, [3, 1, 0, 5]]]},
[[], 0.707]]]]
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 |
|---|---|---|---|
| bands 0 b=3 r=2 | [[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]], 0.577] | [[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4], [4, 5]], 0.577] | Failed |
| bands 1 b=2 r=3 | [[[1, 2], [2, 3], [2, 5], [3, 5]], 0.794] | [[[1, 2], [1, 3], [1, 5], [2, 3], [2, 5], [3, 5]], 0.794] | Failed |
| bands 2 b=4 r=2 | [[[1, 2], [1, 4], [2, 4], [3, 4]], 0.5] | [[[1, 2], [1, 4], [2, 4], [3, 5]], 0.5] | Failed |
| bands 3 b=3 r=3 | [[[4, 5]], 0.693] | [[[1, 5], [3, 5]], 0.693] | Failed |
| same values in different bands | [[], 0.707] | [[], 0.707] | Passed |
| overlong signature | invalid | invalid | Passed |
| interleaved agreement | [[[1, 2]], 0.794] | [[[1, 3]], 0.794] | Failed |
| equal sums differ | [[], 0.707] | [[], 0.707] | Passed |
SHA-256 / 2d685a887c5bf87d5d7f04bf7932048462b2d0506351eee0a5cf5a6e7f56180e
HELD IN THE MEMBER ARCHIVE
The verified repair and its recorded checks are member-only.
This mechanism has 8 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 2026-09-29T14:48:45.580469+00:00.
Case digest / 2ca472c24fd68f81659b14be72df6acad9c0964e07135a2c7d21a1e1d9d4c913