FA-6066 / Linear algebra / Open access

Two by two inverse rational · case 01

The adjugate is not divided by the determinant.

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ROOT CAUSE

The adjugate is not divided by the determinant.

VERIFIED REPAIR

Apply the specified mathematical contract directly, preserving all terms and boundary cases: return [[str(Fraction(x,m[0][0]*m[1][1]-m[0][1]*m[1][0])) for x in row] for row in [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]] if m[0][0]*m[1][1]!=m[0][1]*m[1][0] else None

Unsuccessful approach: Elementwise reciprocals do not invert a linear map.

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. Input is 2 by 2; return None for a singular matrix and otherwise a 2 by 2 list of reduced Fraction strings. Exact operational definition: [[str(Fraction(x,m[0][0]*m[1][1]-m[0][1]*m[1][0])) for x in row] for row in [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]] if m[0][0]*m[1][1]!=m[0][1]*m[1][0] else None

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 [[str(x) for x in row] for row in [[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: ([[2, 0], [0, 4]],)', solve(*([[2, 0], [0, 4]],)), [['1/2', '0'], ['0', '1/4']])
check('fixture 2: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [['-2', '1'], ['3/2', '-1/2']])
check('fixture 3: ([[1, 2], [2, 4]],)', solve(*([[1, 2], [2, 4]],)), None)
check('fixture 4: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [['1', '0'], ['0', '1']])
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
fixture 1: ([[2, 0], [0, 4]],)	`[['4', '0'], ['0', '2']]`	`[['1/2', '0'], ['0', '1/4']]`	Failed
fixture 2: ([[1, 2], [3, 4]],)	`[['4', '-2'], ['-3', '1']]`	`[['-2', '1'], ['3/2', '-1/2']]`	Failed
fixture 3: ([[1, 2], [2, 4]],)	`[['4', '-2'], ['-2', '1']]`	`None`	Failed
fixture 4: ([[1, 0], [0, 1]],)	`[['1', '0'], ['0', '1']]`	`[['1', '0'], ['0', '1']]`	Passed

SHA-256 / 30f0264bd4e8d655e6bc3f21b9f75f6738e01bcc87a5055b7ac11319025d9b56

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 [[str(Fraction(1,x)) if x else "0" for x in row] for row in m]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([[2, 0], [0, 4]],)', solve(*([[2, 0], [0, 4]],)), [['1/2', '0'], ['0', '1/4']])
check('fixture 2: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [['-2', '1'], ['3/2', '-1/2']])
check('fixture 3: ([[1, 2], [2, 4]],)', solve(*([[1, 2], [2, 4]],)), None)
check('fixture 4: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [['1', '0'], ['0', '1']])
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
fixture 1: ([[2, 0], [0, 4]],)	`[['1/2', '0'], ['0', '1/4']]`	`[['1/2', '0'], ['0', '1/4']]`	Passed
fixture 2: ([[1, 2], [3, 4]],)	`[['1', '1/2'], ['1/3', '1/4']]`	`[['-2', '1'], ['3/2', '-1/2']]`	Failed
fixture 3: ([[1, 2], [2, 4]],)	`[['1', '1/2'], ['1/2', '1/4']]`	`None`	Failed
fixture 4: ([[1, 0], [0, 1]],)	`[['1', '0'], ['0', '1']]`	`[['1', '0'], ['0', '1']]`	Passed

SHA-256 / 86fb3df6a436b71b0e35f1a283b2598594d726f149bce28ad3cfe9d5007a82a6

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 [[str(Fraction(x,m[0][0]*m[1][1]-m[0][1]*m[1][0])) for x in row] for row in [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]] if m[0][0]*m[1][1]!=m[0][1]*m[1][0] else None
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([[2, 0], [0, 4]],)', solve(*([[2, 0], [0, 4]],)), [['1/2', '0'], ['0', '1/4']])
check('fixture 2: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [['-2', '1'], ['3/2', '-1/2']])
check('fixture 3: ([[1, 2], [2, 4]],)', solve(*([[1, 2], [2, 4]],)), None)
check('fixture 4: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [['1', '0'], ['0', '1']])
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
fixture 1: ([[2, 0], [0, 4]],)	`[['1/2', '0'], ['0', '1/4']]`	`[['1/2', '0'], ['0', '1/4']]`	Passed
fixture 2: ([[1, 2], [3, 4]],)	`[['-2', '1'], ['3/2', '-1/2']]`	`[['-2', '1'], ['3/2', '-1/2']]`	Passed
fixture 3: ([[1, 2], [2, 4]],)	`None`	`None`	Passed
fixture 4: ([[1, 0], [0, 1]],)	`[['1', '0'], ['0', '1']]`	`[['1', '0'], ['0', '1']]`	Passed

SHA-256 / e624fd0139a84f1a0fb2e51105dc99150c52dc2c217dc04ddc33b762d170a51e

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 2026-09-29T14:37:56.999706+00:00.

Case digest / b0bf0ebce921f4f6b8f4b694b581a1acad3b944ef05ca6f69395fc48f906e557