FA-72241 / Error-correcting codes / Open access

RS erasure recovery flips the Lagrange denominator · case 01

Recovered symbols are wrong whenever k - 1 is odd.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The basis denominator uses (xj - xi) instead of (xi - xj).

VERIFIED REPAIR

Use prod over j != i of (xi - xj) for basis polynomial i.

Unsuccessful approach: Taking the absolute difference drops the sign that GF(p) arithmetic needs.

Case contract

Systematic Reed-Solomon erasure code over GF(257): the k data symbols are the values of a degree < k polynomial at x = 1..k, and any symbol is its value at another x. Given surviving shares [x, y] (duplicates allowed if they agree; conflicting duplicates return None; y is reduced mod 257), recover the k data symbols by Lagrange interpolation over the k shares with the smallest x. Fewer than k distinct x values returns None.

Why this case matters

Distributed storage rebuilds lost data fragments from any k surviving fragments.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(shares, k):
    p = 257
    pts = {}
    for x, y in shares:
        if x in pts and pts[x] != y % p:
            return None
        pts[x] = y % p
    if len(pts) < k:
        return None
    use = sorted(pts.items())[:k]
    out = []
    for t in range(1, k + 1):
        total = 0
        for i, (xi, yi) in enumerate(use):
            num, den = 1, 1
            for j, (xj, _) in enumerate(use):
                if j != i:
                    num = num * (t - xj) % p
                    den = den * (xj - xi) % p
            total = (total + yi * num * pow(den, p - 2, p)) % p
        out.append(total)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[[1, 195], [2, 163]], 2]', [[[1, 195], [2, 163]], 2], [195, 163]], ['regression [[[1, 195], [2, 163], [6, 35]], 2]', [[[1, 195], [2, 163], [6, 35]], 2], [195, 163]], ['control [[[1, 195], [1, 195]], 2]', [[[1, 195], [1, 195]], 2], None], ['control [[[1, 195], [2, 163], [1, 196]], 2]', [[[1, 195], [2, 163], [1, 196]], 2], None], ['control [[[1, 191], [5, 97], [1, 191]], 3]', [[[1, 191], [5, 97], [1, 191]], 3], None], ['control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]', [[[1, 191], [5, 97], [3, 23], [1, 192]], 3], None], ['control [[[1, 224], [5, 68], [1, 224]], 3]', [[[1, 224], [5, 68], [1, 224]], 3], None], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None]], [['regression [[[8, 25], [4, 191], [3, 238], [1, 28]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28]], 4], [28, 103, 238, 191]], ['regression [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4], [28, 103, 238, 191]], ['partial-repair [[[1, 191], [5, 97], [3, 23]], 3]', [[[1, 191], [5, 97], [3, 23]], 3], [191, 141, 23]], ['partial-repair [[[1, 191], [5, 97], [3, 23], [4, 94]], 3]', [[[1, 191], [5, 97], [3, 23], [4, 94]], 3], [191, 141, 23]], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None], ['control [[[8, 25], [4, 191], [3, 238], [8, 25]], 4]', [[[8, 25], [4, 191], [3, 238], [8, 25]], 4], None], ['control [[[8, 25], [4, 191], [3, 238], [1, 28], [8, 26]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [8, 26]], 4], None], ['control [[[4, 14], [5, 158], [7, 75], [4, 14]], 4]', [[[4, 14], [5, 158], [7, 75], [4, 14]], 4], None]], [['regression [[[4, 14], [5, 158], [7, 75], [8, 90]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90]], 4], [215, 35, 99, 14]], ['regression [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4], [215, 35, 99, 14]], ['partial-repair [[[1, 224], [5, 68], [4, 8]], 3]', [[[1, 224], [5, 68], [4, 8]], 3], [224, 86, 14]], ['partial-repair [[[1, 224], [5, 68], [4, 8], [6, 194]], 3]', [[[1, 224], [5, 68], [4, 8], [6, 194]], 3], [224, 86, 14]], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5], None], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [3, 70], [4, 123]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [3, 70], [4, 123]], 5], None], ['control [[[4, 127], [4, 127]], 2]', [[[4, 127], [4, 127]], 2], None], ['control [[[4, 127], [5, 207], [4, 128]], 2]', [[[4, 127], [5, 207], [4, 128]], 2], None]], [['regression [[[4, 127], [5, 207]], 2]', [[[4, 127], [5, 207]], 2], [144, 224]], ['regression [[[4, 127], [5, 207], [1, 144]], 2]', [[[4, 127], [5, 207], [1, 144]], 2], [144, 224]], ['partial-repair [[[8, 25], [4, 191], [3, 238], [1, 28]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28]], 4], [28, 103, 238, 191]], ['partial-repair [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4], [28, 103, 238, 191]], ['control [[[1, 88], [7, 101], [3, 242], [1, 89]], 3]', [[[1, 88], [7, 101], [3, 242], [1, 89]], 3], None], ['control [[[3, 5]], 2]', [[[3, 5]], 2], None], ['control [[[1, 195], [1, 195]], 2]', [[[1, 195], [1, 195]], 2], None], ['control [[[1, 195], [2, 163], [1, 196]], 2]', [[[1, 195], [2, 163], [1, 196]], 2], None]], [['regression [[[1, 10], [2, 20]], 2]', [[[1, 10], [2, 20]], 2], [10, 20]], ['regression [[[1, 7], [1, 7], [2, 9]], 2]', [[[1, 7], [1, 7], [2, 9]], 2], [7, 9]], ['partial-repair [[[4, 14], [5, 158], [7, 75], [8, 90]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90]], 4], [215, 35, 99, 14]], ['partial-repair [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4], [215, 35, 99, 14]], ['control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]', [[[1, 191], [5, 97], [3, 23], [1, 192]], 3], None], ['control [[[1, 224], [5, 68], [1, 224]], 3]', [[[1, 224], [5, 68], [1, 224]], 3], None], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None], ['control [[[8, 25], [4, 191], [3, 238], [8, 25]], 4]', [[[8, 25], [4, 191], [3, 238], [8, 25]], 4], None]]]
for label, args, expected in fixtures[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
regression [[[1, 195], [2, 163]], 2]	`[62, 94]`	`[195, 163]`	Failed
regression [[[1, 195], [2, 163], [6, 35]], 2]	`[62, 94]`	`[195, 163]`	Failed
control [[[1, 195], [1, 195]], 2]	`None`	`None`	Passed
control [[[1, 195], [2, 163], [1, 196]], 2]	`None`	`None`	Passed
control [[[1, 191], [5, 97], [1, 191]], 3]	`None`	`None`	Passed
control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]	`None`	`None`	Passed
control [[[1, 224], [5, 68], [1, 224]], 3]	`None`	`None`	Passed
control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]	`None`	`None`	Passed

SHA-256 / df2adaebd6145a6b7251f3e5c7f285fa2b6d4c1ef1116a32adaaeeb06d2346ca

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(shares, k):
    p = 257
    pts = {}
    for x, y in shares:
        if x in pts and pts[x] != y % p:
            return None
        pts[x] = y % p
    if len(pts) < k:
        return None
    use = sorted(pts.items())[:k]
    out = []
    for t in range(1, k + 1):
        total = 0
        for i, (xi, yi) in enumerate(use):
            num, den = 1, 1
            for j, (xj, _) in enumerate(use):
                if j != i:
                    num = num * (t - xj) % p
                    den = den * abs(xi - xj) % p
            total = (total + yi * num * pow(den, p - 2, p)) % p
        out.append(total)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[[1, 195], [2, 163]], 2]', [[[1, 195], [2, 163]], 2], [195, 163]], ['regression [[[1, 195], [2, 163], [6, 35]], 2]', [[[1, 195], [2, 163], [6, 35]], 2], [195, 163]], ['control [[[1, 195], [1, 195]], 2]', [[[1, 195], [1, 195]], 2], None], ['control [[[1, 195], [2, 163], [1, 196]], 2]', [[[1, 195], [2, 163], [1, 196]], 2], None], ['control [[[1, 191], [5, 97], [1, 191]], 3]', [[[1, 191], [5, 97], [1, 191]], 3], None], ['control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]', [[[1, 191], [5, 97], [3, 23], [1, 192]], 3], None], ['control [[[1, 224], [5, 68], [1, 224]], 3]', [[[1, 224], [5, 68], [1, 224]], 3], None], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None]], [['regression [[[8, 25], [4, 191], [3, 238], [1, 28]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28]], 4], [28, 103, 238, 191]], ['regression [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4], [28, 103, 238, 191]], ['partial-repair [[[1, 191], [5, 97], [3, 23]], 3]', [[[1, 191], [5, 97], [3, 23]], 3], [191, 141, 23]], ['partial-repair [[[1, 191], [5, 97], [3, 23], [4, 94]], 3]', [[[1, 191], [5, 97], [3, 23], [4, 94]], 3], [191, 141, 23]], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None], ['control [[[8, 25], [4, 191], [3, 238], [8, 25]], 4]', [[[8, 25], [4, 191], [3, 238], [8, 25]], 4], None], ['control [[[8, 25], [4, 191], [3, 238], [1, 28], [8, 26]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [8, 26]], 4], None], ['control [[[4, 14], [5, 158], [7, 75], [4, 14]], 4]', [[[4, 14], [5, 158], [7, 75], [4, 14]], 4], None]], [['regression [[[4, 14], [5, 158], [7, 75], [8, 90]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90]], 4], [215, 35, 99, 14]], ['regression [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4], [215, 35, 99, 14]], ['partial-repair [[[1, 224], [5, 68], [4, 8]], 3]', [[[1, 224], [5, 68], [4, 8]], 3], [224, 86, 14]], ['partial-repair [[[1, 224], [5, 68], [4, 8], [6, 194]], 3]', [[[1, 224], [5, 68], [4, 8], [6, 194]], 3], [224, 86, 14]], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5], None], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [3, 70], [4, 123]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [3, 70], [4, 123]], 5], None], ['control [[[4, 127], [4, 127]], 2]', [[[4, 127], [4, 127]], 2], None], ['control [[[4, 127], [5, 207], [4, 128]], 2]', [[[4, 127], [5, 207], [4, 128]], 2], None]], [['regression [[[4, 127], [5, 207]], 2]', [[[4, 127], [5, 207]], 2], [144, 224]], ['regression [[[4, 127], [5, 207], [1, 144]], 2]', [[[4, 127], [5, 207], [1, 144]], 2], [144, 224]], ['partial-repair [[[8, 25], [4, 191], [3, 238], [1, 28]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28]], 4], [28, 103, 238, 191]], ['partial-repair [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4], [28, 103, 238, 191]], ['control [[[1, 88], [7, 101], [3, 242], [1, 89]], 3]', [[[1, 88], [7, 101], [3, 242], [1, 89]], 3], None], ['control [[[3, 5]], 2]', [[[3, 5]], 2], None], ['control [[[1, 195], [1, 195]], 2]', [[[1, 195], [1, 195]], 2], None], ['control [[[1, 195], [2, 163], [1, 196]], 2]', [[[1, 195], [2, 163], [1, 196]], 2], None]], [['regression [[[1, 10], [2, 20]], 2]', [[[1, 10], [2, 20]], 2], [10, 20]], ['regression [[[1, 7], [1, 7], [2, 9]], 2]', [[[1, 7], [1, 7], [2, 9]], 2], [7, 9]], ['partial-repair [[[4, 14], [5, 158], [7, 75], [8, 90]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90]], 4], [215, 35, 99, 14]], ['partial-repair [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4], [215, 35, 99, 14]], ['control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]', [[[1, 191], [5, 97], [3, 23], [1, 192]], 3], None], ['control [[[1, 224], [5, 68], [1, 224]], 3]', [[[1, 224], [5, 68], [1, 224]], 3], None], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None], ['control [[[8, 25], [4, 191], [3, 238], [8, 25]], 4]', [[[8, 25], [4, 191], [3, 238], [8, 25]], 4], None]]]
for label, args, expected in fixtures[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
regression [[[1, 195], [2, 163]], 2]	`[62, 163]`	`[195, 163]`	Failed
regression [[[1, 195], [2, 163], [6, 35]], 2]	`[62, 163]`	`[195, 163]`	Failed
control [[[1, 195], [1, 195]], 2]	`None`	`None`	Passed
control [[[1, 195], [2, 163], [1, 196]], 2]	`None`	`None`	Passed
control [[[1, 191], [5, 97], [1, 191]], 3]	`None`	`None`	Passed
control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]	`None`	`None`	Passed
control [[[1, 224], [5, 68], [1, 224]], 3]	`None`	`None`	Passed
control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]	`None`	`None`	Passed

SHA-256 / a1a0859c34d814a606f56a3ef9308535c098cdbf247571b142d17f2920e2aac2

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(shares, k):
    p = 257
    pts = {}
    for x, y in shares:
        if x in pts and pts[x] != y % p:
            return None
        pts[x] = y % p
    if len(pts) < k:
        return None
    use = sorted(pts.items())[:k]
    out = []
    for t in range(1, k + 1):
        total = 0
        for i, (xi, yi) in enumerate(use):
            num, den = 1, 1
            for j, (xj, _) in enumerate(use):
                if j != i:
                    num = num * (t - xj) % p
                    den = den * (xi - xj) % p
            total = (total + yi * num * pow(den, p - 2, p)) % p
        out.append(total)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[[1, 195], [2, 163]], 2]', [[[1, 195], [2, 163]], 2], [195, 163]], ['regression [[[1, 195], [2, 163], [6, 35]], 2]', [[[1, 195], [2, 163], [6, 35]], 2], [195, 163]], ['control [[[1, 195], [1, 195]], 2]', [[[1, 195], [1, 195]], 2], None], ['control [[[1, 195], [2, 163], [1, 196]], 2]', [[[1, 195], [2, 163], [1, 196]], 2], None], ['control [[[1, 191], [5, 97], [1, 191]], 3]', [[[1, 191], [5, 97], [1, 191]], 3], None], ['control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]', [[[1, 191], [5, 97], [3, 23], [1, 192]], 3], None], ['control [[[1, 224], [5, 68], [1, 224]], 3]', [[[1, 224], [5, 68], [1, 224]], 3], None], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None]], [['regression [[[8, 25], [4, 191], [3, 238], [1, 28]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28]], 4], [28, 103, 238, 191]], ['regression [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4], [28, 103, 238, 191]], ['partial-repair [[[1, 191], [5, 97], [3, 23]], 3]', [[[1, 191], [5, 97], [3, 23]], 3], [191, 141, 23]], ['partial-repair [[[1, 191], [5, 97], [3, 23], [4, 94]], 3]', [[[1, 191], [5, 97], [3, 23], [4, 94]], 3], [191, 141, 23]], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None], ['control [[[8, 25], [4, 191], [3, 238], [8, 25]], 4]', [[[8, 25], [4, 191], [3, 238], [8, 25]], 4], None], ['control [[[8, 25], [4, 191], [3, 238], [1, 28], [8, 26]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [8, 26]], 4], None], ['control [[[4, 14], [5, 158], [7, 75], [4, 14]], 4]', [[[4, 14], [5, 158], [7, 75], [4, 14]], 4], None]], [['regression [[[4, 14], [5, 158], [7, 75], [8, 90]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90]], 4], [215, 35, 99, 14]], ['regression [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4], [215, 35, 99, 14]], ['partial-repair [[[1, 224], [5, 68], [4, 8]], 3]', [[[1, 224], [5, 68], [4, 8]], 3], [224, 86, 14]], ['partial-repair [[[1, 224], [5, 68], [4, 8], [6, 194]], 3]', [[[1, 224], [5, 68], [4, 8], [6, 194]], 3], [224, 86, 14]], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5], None], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [3, 70], [4, 123]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [3, 70], [4, 123]], 5], None], ['control [[[4, 127], [4, 127]], 2]', [[[4, 127], [4, 127]], 2], None], ['control [[[4, 127], [5, 207], [4, 128]], 2]', [[[4, 127], [5, 207], [4, 128]], 2], None]], [['regression [[[4, 127], [5, 207]], 2]', [[[4, 127], [5, 207]], 2], [144, 224]], ['regression [[[4, 127], [5, 207], [1, 144]], 2]', [[[4, 127], [5, 207], [1, 144]], 2], [144, 224]], ['partial-repair [[[8, 25], [4, 191], [3, 238], [1, 28]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28]], 4], [28, 103, 238, 191]], ['partial-repair [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4]', [[[8, 25], [4, 191], [3, 238], [1, 28], [7, 136]], 4], [28, 103, 238, 191]], ['control [[[1, 88], [7, 101], [3, 242], [1, 89]], 3]', [[[1, 88], [7, 101], [3, 242], [1, 89]], 3], None], ['control [[[3, 5]], 2]', [[[3, 5]], 2], None], ['control [[[1, 195], [1, 195]], 2]', [[[1, 195], [1, 195]], 2], None], ['control [[[1, 195], [2, 163], [1, 196]], 2]', [[[1, 195], [2, 163], [1, 196]], 2], None]], [['regression [[[1, 10], [2, 20]], 2]', [[[1, 10], [2, 20]], 2], [10, 20]], ['regression [[[1, 7], [1, 7], [2, 9]], 2]', [[[1, 7], [1, 7], [2, 9]], 2], [7, 9]], ['partial-repair [[[4, 14], [5, 158], [7, 75], [8, 90]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90]], 4], [215, 35, 99, 14]], ['partial-repair [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [1, 215]], 4], [215, 35, 99, 14]], ['control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]', [[[1, 191], [5, 97], [3, 23], [1, 192]], 3], None], ['control [[[1, 224], [5, 68], [1, 224]], 3]', [[[1, 224], [5, 68], [1, 224]], 3], None], ['control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]', [[[1, 224], [5, 68], [4, 8], [1, 225]], 3], None], ['control [[[8, 25], [4, 191], [3, 238], [8, 25]], 4]', [[[8, 25], [4, 191], [3, 238], [8, 25]], 4], None]]]
for label, args, expected in fixtures[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
regression [[[1, 195], [2, 163]], 2]	`[195, 163]`	`[195, 163]`	Passed
regression [[[1, 195], [2, 163], [6, 35]], 2]	`[195, 163]`	`[195, 163]`	Passed
control [[[1, 195], [1, 195]], 2]	`None`	`None`	Passed
control [[[1, 195], [2, 163], [1, 196]], 2]	`None`	`None`	Passed
control [[[1, 191], [5, 97], [1, 191]], 3]	`None`	`None`	Passed
control [[[1, 191], [5, 97], [3, 23], [1, 192]], 3]	`None`	`None`	Passed
control [[[1, 224], [5, 68], [1, 224]], 3]	`None`	`None`	Passed
control [[[1, 224], [5, 68], [4, 8], [1, 225]], 3]	`None`	`None`	Passed

SHA-256 / 7184d30d52052bfa2d21fbc99b429422b4483114e5676add2ecc4797a3f44402

Verification & scope

A deterministic, bounded teaching model of the named code under the stated contract; not a production codec. 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 2026-09-29T14:48:36.678821+00:00.

Case digest / 75c29f03f10acf309c720355e184c0fec56658ac99943e1e1320067ef5924990