FA-72281 / Error-correcting codes / Open access

Chase combining decides zero LLRs as bit 0 · case 01

Positions with no information are reported as confident zeros.

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

ROOT CAUSE

The hard decision maps a zero combined LLR to 0 instead of undecided.

VERIFIED REPAIR

Report None for a zero combined LLR.

Unsuccessful approach: Mapping zero to bit 1 is the same guess with the other sign.

Case contract

Hybrid-ARQ chase combining. txs are equal-length LLR vectors from successive transmissions (None where a position was not received). Combined LLR = sum of received values, saturated once at the end to [-clip, clip]. Hard decision: positive -> 0, negative -> 1, zero -> None (undecided). Return [combined, hard]; empty or ragged input returns None.

Why this case matters

LTE/5G receivers soft-combine retransmissions before decoding; saturating too early discards information.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import functools
N = 1
observations = []
def solve(txs, clip):
    if not txs or any(len(t) != len(txs[0]) for t in txs):
        return None
    comb = []
    for k in range(len(txs[0])):
        s = sum(t[k] for t in txs if t[k] is not None)
        comb.append(max(-clip, min(clip, s)))
    hard = [1 if c < 0 else 0 for c in comb]
    return [comb, hard]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[2, null], [null, -3], [5, 8]], 10]', [[[2, None], [None, -3], [5, 8]], 10], [[7, 5], [0, 0]]], ['control [[[5, -9]], 15]', [[[5, -9]], 15], [[5, -9], [0, 1]]], ['control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]', [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10], [[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]], ['control [[[-3, -6], [-6, 5]], 15]', [[[-3, -6], [-6, 5]], 15], [[-9, -1], [1, 1]]], ['control [[[null, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, null, -9...', [[[None, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, None, -9, 5]], 7], [[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]], ['control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]', [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15], [[-9, -2, 5, -1], [1, 1, 0, 1]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]', [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15], [[-9, -2, 5, -1], [1, 1, 0, 1]]], ['control [[[null, -1, -9, null, -3], [8, 8, -9, 8, -6]], 15]', [[[None, -1, -9, None, -3], [8, 8, -9, 8, -6]], 15], [[8, 7, -15, 8, -9], [0, 0, 1, 0, 1]]], ['control [[[0, -1, -3, -1], [2, null, 11, -3], [0, null, -6, 0]], 15]', [[[0, -1, -3, -1], [2, None, 11, -3], [0, None, -6, 0]], 15], [[2, -1, 2, -4], [0, 1, 0, 1]]], ['control [[[-1, 0, 0], [-9, 8, 0], [11, -9, -6]], 7]', [[[-1, 0, 0], [-9, 8, 0], [11, -9, -6]], 7], [[1, -1, -6], [0, 1, 1]]], ['control [[[8, -6]], 10]', [[[8, -6]], 10], [[8, -6], [0, 1]]], ['control [[[-6, -3, null], [0, -6, 8], [-9, 2, 0]], 15]', [[[-6, -3, None], [0, -6, 8], [-9, 2, 0]], 15], [[-15, -7, 8], [1, 1, 0]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[-6, -3, null], [0, -6, 8], [-9, 2, 0]], 15]', [[[-6, -3, None], [0, -6, 8], [-9, 2, 0]], 15], [[-15, -7, 8], [1, 1, 0]]], ['control [[[11, -3, -1, null, -6, 8], [-3, -9, 11, null, -6, null], [null, 5...', [[[11, -3, -1, None, -6, 8], [-3, -9, 11, None, -6, None], [None, 5, -6, -6, 2, 5]], 15], [[8, -7, 4, -6, -10, 13], [0, 1, 0, 1, 1, 0]]], ['control [[[-3, 11], [-3, 11], [-6, 5]], 10]', [[[-3, 11], [-3, 11], [-6, 5]], 10], [[-10, 10], [1, 0]]], ['control [[[8, -9], [2, null]], 7]', [[[8, -9], [2, None]], 7], [[7, -7], [0, 1]]], ['control [[[0, 2, -9, -3, -1], [11, -1, -6, null, 0]], 15]', [[[0, 2, -9, -3, -1], [11, -1, -6, None, 0]], 15], [[11, 1, -15, -3, -1], [0, 0, 1, 1, 1]]], ['control [[[11, 8, null], [2, -6, -1], [8, null, -3]], 10]', [[[11, 8, None], [2, -6, -1], [8, None, -3]], 10], [[10, 2, -4], [0, 0, 1]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[11, 8, null], [2, -6, -1], [8, null, -3]], 10]', [[[11, 8, None], [2, -6, -1], [8, None, -3]], 10], [[10, 2, -4], [0, 0, 1]]], ['control [[[-6, null, 8, 2, 8], [-9, 11, null, -3, -1]], 15]', [[[-6, None, 8, 2, 8], [-9, 11, None, -3, -1]], 15], [[-15, 11, 8, -1, 7], [1, 0, 0, 1, 0]]], ['control [[[-1, -6, null, -3], [null, 8, 2, 0]], 7]', [[[-1, -6, None, -3], [None, 8, 2, 0]], 7], [[-1, 2, 2, -3], [1, 0, 0, 1]]], ['control [[[12, -12], [-5, 5]], 10]', [[[12, -12], [-5, 5]], 10], [[7, -7], [0, 1]]], ['control [[[20], [20], [-30]], 16]', [[[20], [20], [-30]], 16], [[10], [0]]], ['control [[], 5]', [[], 5], None]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[], 5]', [[], 5], None], ['control [[[1, 2], [3]], 5]', [[[1, 2], [3]], 5], None], ['control [[[2, null], [null, -3], [5, 8]], 10]', [[[2, None], [None, -3], [5, 8]], 10], [[7, 5], [0, 0]]], ['control [[[5, -9]], 15]', [[[5, -9]], 15], [[5, -9], [0, 1]]], ['control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]', [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10], [[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]], ['control [[[-3, -6], [-6, 5]], 15]', [[[-3, -6], [-6, 5]], 15], [[-9, -1], [1, 1]]]]]
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 [[[5, null]], 7]	`[[5, 0], [0, 0]]`	`[[5, 0], [0, None]]`	Failed
regression [[[5, -6, 0]], 10]	`[[5, -6, 0], [0, 1, 0]]`	`[[5, -6, 0], [0, 1, None]]`	Failed
control [[[2, null], [null, -3], [5, 8]], 10]	`[[7, 5], [0, 0]]`	`[[7, 5], [0, 0]]`	Passed
control [[[5, -9]], 15]	`[[5, -9], [0, 1]]`	`[[5, -9], [0, 1]]`	Passed
control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]	`[[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]`	`[[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]`	Passed
control [[[-3, -6], [-6, 5]], 15]	`[[-9, -1], [1, 1]]`	`[[-9, -1], [1, 1]]`	Passed
control [[[null, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, null, -9...	`[[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]`	`[[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]`	Passed
control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]	`[[-9, -2, 5, -1], [1, 1, 0, 1]]`	`[[-9, -2, 5, -1], [1, 1, 0, 1]]`	Passed

SHA-256 / cc505744d942ae81eedc3cf8c89b74838510206405ac163f7295e688a2d4e3c5

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import functools
N = 1
observations = []
def solve(txs, clip):
    if not txs or any(len(t) != len(txs[0]) for t in txs):
        return None
    comb = []
    for k in range(len(txs[0])):
        s = sum(t[k] for t in txs if t[k] is not None)
        comb.append(max(-clip, min(clip, s)))
    hard = [1 if c <= 0 else 0 for c in comb]
    return [comb, hard]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[2, null], [null, -3], [5, 8]], 10]', [[[2, None], [None, -3], [5, 8]], 10], [[7, 5], [0, 0]]], ['control [[[5, -9]], 15]', [[[5, -9]], 15], [[5, -9], [0, 1]]], ['control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]', [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10], [[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]], ['control [[[-3, -6], [-6, 5]], 15]', [[[-3, -6], [-6, 5]], 15], [[-9, -1], [1, 1]]], ['control [[[null, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, null, -9...', [[[None, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, None, -9, 5]], 7], [[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]], ['control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]', [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15], [[-9, -2, 5, -1], [1, 1, 0, 1]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]', [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15], [[-9, -2, 5, -1], [1, 1, 0, 1]]], ['control [[[null, -1, -9, null, -3], [8, 8, -9, 8, -6]], 15]', [[[None, -1, -9, None, -3], [8, 8, -9, 8, -6]], 15], [[8, 7, -15, 8, -9], [0, 0, 1, 0, 1]]], ['control [[[0, -1, -3, -1], [2, null, 11, -3], [0, null, -6, 0]], 15]', [[[0, -1, -3, -1], [2, None, 11, -3], [0, None, -6, 0]], 15], [[2, -1, 2, -4], [0, 1, 0, 1]]], ['control [[[-1, 0, 0], [-9, 8, 0], [11, -9, -6]], 7]', [[[-1, 0, 0], [-9, 8, 0], [11, -9, -6]], 7], [[1, -1, -6], [0, 1, 1]]], ['control [[[8, -6]], 10]', [[[8, -6]], 10], [[8, -6], [0, 1]]], ['control [[[-6, -3, null], [0, -6, 8], [-9, 2, 0]], 15]', [[[-6, -3, None], [0, -6, 8], [-9, 2, 0]], 15], [[-15, -7, 8], [1, 1, 0]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[-6, -3, null], [0, -6, 8], [-9, 2, 0]], 15]', [[[-6, -3, None], [0, -6, 8], [-9, 2, 0]], 15], [[-15, -7, 8], [1, 1, 0]]], ['control [[[11, -3, -1, null, -6, 8], [-3, -9, 11, null, -6, null], [null, 5...', [[[11, -3, -1, None, -6, 8], [-3, -9, 11, None, -6, None], [None, 5, -6, -6, 2, 5]], 15], [[8, -7, 4, -6, -10, 13], [0, 1, 0, 1, 1, 0]]], ['control [[[-3, 11], [-3, 11], [-6, 5]], 10]', [[[-3, 11], [-3, 11], [-6, 5]], 10], [[-10, 10], [1, 0]]], ['control [[[8, -9], [2, null]], 7]', [[[8, -9], [2, None]], 7], [[7, -7], [0, 1]]], ['control [[[0, 2, -9, -3, -1], [11, -1, -6, null, 0]], 15]', [[[0, 2, -9, -3, -1], [11, -1, -6, None, 0]], 15], [[11, 1, -15, -3, -1], [0, 0, 1, 1, 1]]], ['control [[[11, 8, null], [2, -6, -1], [8, null, -3]], 10]', [[[11, 8, None], [2, -6, -1], [8, None, -3]], 10], [[10, 2, -4], [0, 0, 1]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[11, 8, null], [2, -6, -1], [8, null, -3]], 10]', [[[11, 8, None], [2, -6, -1], [8, None, -3]], 10], [[10, 2, -4], [0, 0, 1]]], ['control [[[-6, null, 8, 2, 8], [-9, 11, null, -3, -1]], 15]', [[[-6, None, 8, 2, 8], [-9, 11, None, -3, -1]], 15], [[-15, 11, 8, -1, 7], [1, 0, 0, 1, 0]]], ['control [[[-1, -6, null, -3], [null, 8, 2, 0]], 7]', [[[-1, -6, None, -3], [None, 8, 2, 0]], 7], [[-1, 2, 2, -3], [1, 0, 0, 1]]], ['control [[[12, -12], [-5, 5]], 10]', [[[12, -12], [-5, 5]], 10], [[7, -7], [0, 1]]], ['control [[[20], [20], [-30]], 16]', [[[20], [20], [-30]], 16], [[10], [0]]], ['control [[], 5]', [[], 5], None]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[], 5]', [[], 5], None], ['control [[[1, 2], [3]], 5]', [[[1, 2], [3]], 5], None], ['control [[[2, null], [null, -3], [5, 8]], 10]', [[[2, None], [None, -3], [5, 8]], 10], [[7, 5], [0, 0]]], ['control [[[5, -9]], 15]', [[[5, -9]], 15], [[5, -9], [0, 1]]], ['control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]', [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10], [[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]], ['control [[[-3, -6], [-6, 5]], 15]', [[[-3, -6], [-6, 5]], 15], [[-9, -1], [1, 1]]]]]
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 [[[5, null]], 7]	`[[5, 0], [0, 1]]`	`[[5, 0], [0, None]]`	Failed
regression [[[5, -6, 0]], 10]	`[[5, -6, 0], [0, 1, 1]]`	`[[5, -6, 0], [0, 1, None]]`	Failed
control [[[2, null], [null, -3], [5, 8]], 10]	`[[7, 5], [0, 0]]`	`[[7, 5], [0, 0]]`	Passed
control [[[5, -9]], 15]	`[[5, -9], [0, 1]]`	`[[5, -9], [0, 1]]`	Passed
control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]	`[[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]`	`[[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]`	Passed
control [[[-3, -6], [-6, 5]], 15]	`[[-9, -1], [1, 1]]`	`[[-9, -1], [1, 1]]`	Passed
control [[[null, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, null, -9...	`[[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]`	`[[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]`	Passed
control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]	`[[-9, -2, 5, -1], [1, 1, 0, 1]]`	`[[-9, -2, 5, -1], [1, 1, 0, 1]]`	Passed

SHA-256 / ac300ece41b75952a64ef9e65adfb39c9e12b306e686423055b6c08e2f950220

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import functools
N = 1
observations = []
def solve(txs, clip):
    if not txs or any(len(t) != len(txs[0]) for t in txs):
        return None
    comb = []
    for k in range(len(txs[0])):
        s = sum(t[k] for t in txs if t[k] is not None)
        comb.append(max(-clip, min(clip, s)))
    hard = [None if c == 0 else (1 if c < 0 else 0) for c in comb]
    return [comb, hard]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[2, null], [null, -3], [5, 8]], 10]', [[[2, None], [None, -3], [5, 8]], 10], [[7, 5], [0, 0]]], ['control [[[5, -9]], 15]', [[[5, -9]], 15], [[5, -9], [0, 1]]], ['control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]', [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10], [[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]], ['control [[[-3, -6], [-6, 5]], 15]', [[[-3, -6], [-6, 5]], 15], [[-9, -1], [1, 1]]], ['control [[[null, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, null, -9...', [[[None, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, None, -9, 5]], 7], [[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]], ['control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]', [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15], [[-9, -2, 5, -1], [1, 1, 0, 1]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]', [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15], [[-9, -2, 5, -1], [1, 1, 0, 1]]], ['control [[[null, -1, -9, null, -3], [8, 8, -9, 8, -6]], 15]', [[[None, -1, -9, None, -3], [8, 8, -9, 8, -6]], 15], [[8, 7, -15, 8, -9], [0, 0, 1, 0, 1]]], ['control [[[0, -1, -3, -1], [2, null, 11, -3], [0, null, -6, 0]], 15]', [[[0, -1, -3, -1], [2, None, 11, -3], [0, None, -6, 0]], 15], [[2, -1, 2, -4], [0, 1, 0, 1]]], ['control [[[-1, 0, 0], [-9, 8, 0], [11, -9, -6]], 7]', [[[-1, 0, 0], [-9, 8, 0], [11, -9, -6]], 7], [[1, -1, -6], [0, 1, 1]]], ['control [[[8, -6]], 10]', [[[8, -6]], 10], [[8, -6], [0, 1]]], ['control [[[-6, -3, null], [0, -6, 8], [-9, 2, 0]], 15]', [[[-6, -3, None], [0, -6, 8], [-9, 2, 0]], 15], [[-15, -7, 8], [1, 1, 0]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[-6, -3, null], [0, -6, 8], [-9, 2, 0]], 15]', [[[-6, -3, None], [0, -6, 8], [-9, 2, 0]], 15], [[-15, -7, 8], [1, 1, 0]]], ['control [[[11, -3, -1, null, -6, 8], [-3, -9, 11, null, -6, null], [null, 5...', [[[11, -3, -1, None, -6, 8], [-3, -9, 11, None, -6, None], [None, 5, -6, -6, 2, 5]], 15], [[8, -7, 4, -6, -10, 13], [0, 1, 0, 1, 1, 0]]], ['control [[[-3, 11], [-3, 11], [-6, 5]], 10]', [[[-3, 11], [-3, 11], [-6, 5]], 10], [[-10, 10], [1, 0]]], ['control [[[8, -9], [2, null]], 7]', [[[8, -9], [2, None]], 7], [[7, -7], [0, 1]]], ['control [[[0, 2, -9, -3, -1], [11, -1, -6, null, 0]], 15]', [[[0, 2, -9, -3, -1], [11, -1, -6, None, 0]], 15], [[11, 1, -15, -3, -1], [0, 0, 1, 1, 1]]], ['control [[[11, 8, null], [2, -6, -1], [8, null, -3]], 10]', [[[11, 8, None], [2, -6, -1], [8, None, -3]], 10], [[10, 2, -4], [0, 0, 1]]]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[[11, 8, null], [2, -6, -1], [8, null, -3]], 10]', [[[11, 8, None], [2, -6, -1], [8, None, -3]], 10], [[10, 2, -4], [0, 0, 1]]], ['control [[[-6, null, 8, 2, 8], [-9, 11, null, -3, -1]], 15]', [[[-6, None, 8, 2, 8], [-9, 11, None, -3, -1]], 15], [[-15, 11, 8, -1, 7], [1, 0, 0, 1, 0]]], ['control [[[-1, -6, null, -3], [null, 8, 2, 0]], 7]', [[[-1, -6, None, -3], [None, 8, 2, 0]], 7], [[-1, 2, 2, -3], [1, 0, 0, 1]]], ['control [[[12, -12], [-5, 5]], 10]', [[[12, -12], [-5, 5]], 10], [[7, -7], [0, 1]]], ['control [[[20], [20], [-30]], 16]', [[[20], [20], [-30]], 16], [[10], [0]]], ['control [[], 5]', [[], 5], None]], [['regression [[[5, null]], 7]', [[[5, None]], 7], [[5, 0], [0, None]]], ['regression [[[5, -6, 0]], 10]', [[[5, -6, 0]], 10], [[5, -6, 0], [0, 1, None]]], ['control [[], 5]', [[], 5], None], ['control [[[1, 2], [3]], 5]', [[[1, 2], [3]], 5], None], ['control [[[2, null], [null, -3], [5, 8]], 10]', [[[2, None], [None, -3], [5, 8]], 10], [[7, 5], [0, 0]]], ['control [[[5, -9]], 15]', [[[5, -9]], 15], [[5, -9], [0, 1]]], ['control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]', [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10], [[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]], ['control [[[-3, -6], [-6, 5]], 15]', [[[-3, -6], [-6, 5]], 15], [[-9, -1], [1, 1]]]]]
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 [[[5, null]], 7]	`[[5, 0], [0, None]]`	`[[5, 0], [0, None]]`	Passed
regression [[[5, -6, 0]], 10]	`[[5, -6, 0], [0, 1, None]]`	`[[5, -6, 0], [0, 1, None]]`	Passed
control [[[2, null], [null, -3], [5, 8]], 10]	`[[7, 5], [0, 0]]`	`[[7, 5], [0, 0]]`	Passed
control [[[5, -9]], 15]	`[[5, -9], [0, 1]]`	`[[5, -9], [0, 1]]`	Passed
control [[[-9, -3, 0, -6, 0], [11, 5, -3, 2, 5]], 10]	`[[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]`	`[[2, 2, -3, -4, 5], [0, 0, 1, 1, 0]]`	Passed
control [[[-3, -6], [-6, 5]], 15]	`[[-9, -1], [1, 1]]`	`[[-9, -1], [1, 1]]`	Passed
control [[[null, 5, 8, 8, 0, 0], [-6, 2, -1, 2, 2, 11], [2, 0, -9, null, -9...	`[[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]`	`[[-4, 7, -2, 7, -7, 7], [1, 0, 1, 0, 1, 0]]`	Passed
control [[[0, -1, 0, 0], [-9, -1, 5, -1]], 15]	`[[-9, -2, 5, -1], [1, 1, 0, 1]]`	`[[-9, -2, 5, -1], [1, 1, 0, 1]]`	Passed

SHA-256 / ee860cc0a4f3a732bdbf0a0fd15116d5af854fd1b5789b0ee4cd0ae39ff8acc2

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 2026-09-29T14:48:37.194135+00:00.

Case digest / 91e1e36e28239a35ef5415ab4656d369784d18766fa1ded480b4e8bdef52672d