FA-72256 / Error-correcting codes / Open access

RS erasure recovery evaluates data at zero-based points · case 01

Recovered data is shifted by one position and includes a value at x = 0.

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

ROOT CAUSE

The data symbols are evaluated at x = 0..k-1 instead of 1..k.

THE FAILURE

The data symbols are evaluated at x = 0..k-1 instead of 1..k.

Unsuccessful approach: Stopping at x = k - 1 drops the last data symbol.

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(k):
        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 [[[1, 191], [5, 97], [3, 23]], 3]', [[[1, 191], [5, 97], [3, 23]], 3], [191, 141, 23]], ['regression [[[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], ['control [[[4, 14], [5, 158], [7, 75], [8, 90], [4, 15]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [4, 15]], 4], None], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5], None]], [['regression [[[1, 224], [5, 68], [4, 8]], 3]', [[[1, 224], [5, 68], [4, 8]], 3], [224, 86, 14]], ['regression [[[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], ['control [[[1, 88], [7, 101], [1, 88]], 3]', [[[1, 88], [7, 101], [1, 88]], 3], None], ['control [[[1, 88], [7, 101], [3, 242], [1, 89]], 3]', [[[1, 88], [7, 101], [3, 242], [1, 89]], 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]], ['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], ['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]], [['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]], ['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], ['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]]]
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]	`[227, 195]`	`[195, 163]`	Failed
regression [[[1, 195], [2, 163], [6, 35]], 2]	`[227, 195]`	`[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 / 17e8706b17d635dc096376b9615db68a4d9e9928972aacfd91f4919a1d73561f

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):
        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 [[[1, 191], [5, 97], [3, 23]], 3]', [[[1, 191], [5, 97], [3, 23]], 3], [191, 141, 23]], ['regression [[[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], ['control [[[4, 14], [5, 158], [7, 75], [8, 90], [4, 15]], 4]', [[[4, 14], [5, 158], [7, 75], [8, 90], [4, 15]], 4], None], ['control [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5]', [[[4, 122], [1, 54], [2, 149], [5, 153], [4, 122]], 5], None]], [['regression [[[1, 224], [5, 68], [4, 8]], 3]', [[[1, 224], [5, 68], [4, 8]], 3], [224, 86, 14]], ['regression [[[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], ['control [[[1, 88], [7, 101], [1, 88]], 3]', [[[1, 88], [7, 101], [1, 88]], 3], None], ['control [[[1, 88], [7, 101], [3, 242], [1, 89]], 3]', [[[1, 88], [7, 101], [3, 242], [1, 89]], 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]], ['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], ['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]], [['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]], ['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], ['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]]]
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]`	`[195, 163]`	Failed
regression [[[1, 195], [2, 163], [6, 35]], 2]	`[195]`	`[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 / d4300502afa417fb8136840c049e42443a9ac4f30918f2963a843cad2f98b55c

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:48:36.847515+00:00.

Case digest / 75641f14f9e6f9ec01395c0adfcae0fbb8e436c4b2c12809164d88bdb5872583