FAILURE MAP
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FA-6061 / Linear algebra / Open access

Two by two adjugate · case 01

Off-diagonal cofactors lack negative signs.

Verified by executionVariant 1 · 4 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Off-diagonal cofactors lack negative signs.

VERIFIED REPAIR

Apply the specified mathematical contract directly, preserving all terms and boundary cases: return [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]

Unsuccessful approach: The untransposed cofactor matrix is returned.

Case contract

Integer matrix/vector entries, rectangular rows, compatible multiplication dimensions, and square matrices for determinant, trace, inverse, symmetry and characteristic-polynomial operations. Rational matrix outputs are reduced Fraction strings. Two by two adjugate. Exact operational definition: [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]

Why this case matters

Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Linear algebra results depend on the stated convention.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(m):
    return [[m[1][1],m[0][1]],[m[1][0],m[0][0]]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [[4, -2], [-3, 1]])
check('fixture 2: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [[1, 0], [0, 1]])
check('fixture 3: ([[0, 1], [2, 0]],)', solve(*([[0, 1], [2, 0]],)), [[0, -1], [-2, 0]])
check('fixture 4: ([[0, 0], [0, 0]],)', solve(*([[0, 0], [0, 0]],)), [[0, 0], [0, 0]])
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 fixtureActualExpectedOutcome
fixture 1: ([[1, 2], [3, 4]],)[[4, 2], [3, 1]][[4, -2], [-3, 1]]Failed
fixture 2: ([[1, 0], [0, 1]],)[[1, 0], [0, 1]][[1, 0], [0, 1]]Passed
fixture 3: ([[0, 1], [2, 0]],)[[0, 1], [2, 0]][[0, -1], [-2, 0]]Failed
fixture 4: ([[0, 0], [0, 0]],)[[0, 0], [0, 0]][[0, 0], [0, 0]]Passed

SHA-256 / 11904c7d2105387ad3ef43c68d366b8dbf56ce161c82b03ddd47aff38f45ee1c

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(m):
    return [[m[1][1],-m[1][0]],[-m[0][1],m[0][0]]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [[4, -2], [-3, 1]])
check('fixture 2: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [[1, 0], [0, 1]])
check('fixture 3: ([[0, 1], [2, 0]],)', solve(*([[0, 1], [2, 0]],)), [[0, -1], [-2, 0]])
check('fixture 4: ([[0, 0], [0, 0]],)', solve(*([[0, 0], [0, 0]],)), [[0, 0], [0, 0]])
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 fixtureActualExpectedOutcome
fixture 1: ([[1, 2], [3, 4]],)[[4, -3], [-2, 1]][[4, -2], [-3, 1]]Failed
fixture 2: ([[1, 0], [0, 1]],)[[1, 0], [0, 1]][[1, 0], [0, 1]]Passed
fixture 3: ([[0, 1], [2, 0]],)[[0, -2], [-1, 0]][[0, -1], [-2, 0]]Failed
fixture 4: ([[0, 0], [0, 0]],)[[0, 0], [0, 0]][[0, 0], [0, 0]]Passed

SHA-256 / fe44250288353bd65b4d9245b7a0cb03ebd82bf0fd55b5805cbdff0f3fdd6e3d

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(m):
    return [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [[4, -2], [-3, 1]])
check('fixture 2: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [[1, 0], [0, 1]])
check('fixture 3: ([[0, 1], [2, 0]],)', solve(*([[0, 1], [2, 0]],)), [[0, -1], [-2, 0]])
check('fixture 4: ([[0, 0], [0, 0]],)', solve(*([[0, 0], [0, 0]],)), [[0, 0], [0, 0]])
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 fixtureActualExpectedOutcome
fixture 1: ([[1, 2], [3, 4]],)[[4, -2], [-3, 1]][[4, -2], [-3, 1]]Passed
fixture 2: ([[1, 0], [0, 1]],)[[1, 0], [0, 1]][[1, 0], [0, 1]]Passed
fixture 3: ([[0, 1], [2, 0]],)[[0, -1], [-2, 0]][[0, -1], [-2, 0]]Passed
fixture 4: ([[0, 0], [0, 0]],)[[0, 0], [0, 0]][[0, 0], [0, 0]]Passed

SHA-256 / d4a554b366a04eec0242d4fe5a64d1c11e3b32fcf818d84ab700fc993e668837

Verification & scope

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 2026-09-29T14:37:57.000754+00:00.

Case digest / 201f21b6f0aa004b877e54e93286f7acf549fa5008f81c192a6488782065d3a1