FA-6011 / Linear algebra / Open access

Three dimensional cross product · case 01

Symmetric sums replace antisymmetric determinant terms.

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ROOT CAUSE

Symmetric sums replace antisymmetric determinant terms.

THE FAILURE

Symmetric sums replace antisymmetric determinant terms.

Unsuccessful approach: The middle cofactor has the wrong sign.

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. Three dimensional cross product. Exact operational definition: [a[1]*b[2]-a[2]*b[1],a[2]*b[0]-a[0]*b[2],a[0]*b[1]-a[1]*b[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(a, b):
    return [a[1]*b[2]+a[2]*b[1],a[2]*b[0]+a[0]*b[2],a[0]*b[1]+a[1]*b[0]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([1, 0, 0], [0, 1, 0])', solve(*([1, 0, 0], [0, 1, 0])), [0, 0, 1])
check('fixture 2: ([1, 0, 0], [0, 0, 1])', solve(*([1, 0, 0], [0, 0, 1])), [0, -1, 0])
check('fixture 3: ([1, 2, 3], [1, 2, 3])', solve(*([1, 2, 3], [1, 2, 3])), [0, 0, 0])
check('fixture 4: ([0, 0, 1], [0, 1, 0])', solve(*([0, 0, 1], [0, 1, 0])), [-1, 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 fixture	Actual	Expected	Outcome
fixture 1: ([1, 0, 0], [0, 1, 0])	`[0, 0, 1]`	`[0, 0, 1]`	Passed
fixture 2: ([1, 0, 0], [0, 0, 1])	`[0, 1, 0]`	`[0, -1, 0]`	Failed
fixture 3: ([1, 2, 3], [1, 2, 3])	`[12, 6, 4]`	`[0, 0, 0]`	Failed
fixture 4: ([0, 0, 1], [0, 1, 0])	`[1, 0, 0]`	`[-1, 0, 0]`	Failed

SHA-256 / 682b9508239ae0c70a28372fa5cd367c4c86a1f440f24805efff75127d1d58a3

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(a, b):
    return [a[1]*b[2]-a[2]*b[1],a[0]*b[2]-a[2]*b[0],a[0]*b[1]-a[1]*b[0]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([1, 0, 0], [0, 1, 0])', solve(*([1, 0, 0], [0, 1, 0])), [0, 0, 1])
check('fixture 2: ([1, 0, 0], [0, 0, 1])', solve(*([1, 0, 0], [0, 0, 1])), [0, -1, 0])
check('fixture 3: ([1, 2, 3], [1, 2, 3])', solve(*([1, 2, 3], [1, 2, 3])), [0, 0, 0])
check('fixture 4: ([0, 0, 1], [0, 1, 0])', solve(*([0, 0, 1], [0, 1, 0])), [-1, 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 fixture	Actual	Expected	Outcome
fixture 1: ([1, 0, 0], [0, 1, 0])	`[0, 0, 1]`	`[0, 0, 1]`	Passed
fixture 2: ([1, 0, 0], [0, 0, 1])	`[0, 1, 0]`	`[0, -1, 0]`	Failed
fixture 3: ([1, 2, 3], [1, 2, 3])	`[0, 0, 0]`	`[0, 0, 0]`	Passed
fixture 4: ([0, 0, 1], [0, 1, 0])	`[-1, 0, 0]`	`[-1, 0, 0]`	Passed

SHA-256 / f5b56113595a1305872a31400e38b46bef35aefc8b7d0a8b3dc62d8845cebb6c

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 4 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.

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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 2026-09-29T14:37:56.285410+00:00.

Case digest / d3b22f91c2e67f8b64fcf5937eccd4860b05886689e33157526899a2b8092a1b