FA-6096 / Linear algebra / Open access

Three dimensional scalar triple product · case 01

Coordinatewise triple multiplication omits determinant permutations.

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

ROOT CAUSE

Coordinatewise triple multiplication omits determinant permutations.

THE FAILURE

Coordinatewise triple multiplication omits determinant permutations.

Unsuccessful approach: Taking absolute volume erases orientation.

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 scalar triple product. Exact operational definition: a[0]*(b[1]*c[2]-b[2]*c[1])+a[1]*(b[2]*c[0]-b[0]*c[2])+a[2]*(b[0]*c[1]-b[1]*c[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, c):
    return sum(x*y*z for x,y,z in zip(a,b,c))
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], [0, 0, 1])', solve(*([1, 0, 0], [0, 1, 0], [0, 0, 1])), 1)
check('fixture 2: ([0, 1, 0], [1, 0, 0], [0, 0, 1])', solve(*([0, 1, 0], [1, 0, 0], [0, 0, 1])), -1)
check('fixture 3: ([1, 2, 3], [1, 2, 3], [2, 1, 0])', solve(*([1, 2, 3], [1, 2, 3], [2, 1, 0])), 0)
check('fixture 4: ([2, 0, 0], [0, 3, 0], [0, 0, 4])', solve(*([2, 0, 0], [0, 3, 0], [0, 0, 4])), 24)
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`	`1`	Failed
fixture 2: ([0, 1, 0], [1, 0, 0], [0, 0, 1])	`0`	`-1`	Failed
fixture 3: ([1, 2, 3], [1, 2, 3], [2, 1, 0])	`6`	`0`	Failed
fixture 4: ([2, 0, 0], [0, 3, 0], [0, 0, 4])	`0`	`24`	Failed

SHA-256 / ccc26def3beac0d0df3c4807b93d37b887fc5e5fe872d020280b4a9e7d125bf9

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, c):
    return abs(a[0]*(b[1]*c[2]-b[2]*c[1])+a[1]*(b[2]*c[0]-b[0]*c[2])+a[2]*(b[0]*c[1]-b[1]*c[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], [0, 0, 1])', solve(*([1, 0, 0], [0, 1, 0], [0, 0, 1])), 1)
check('fixture 2: ([0, 1, 0], [1, 0, 0], [0, 0, 1])', solve(*([0, 1, 0], [1, 0, 0], [0, 0, 1])), -1)
check('fixture 3: ([1, 2, 3], [1, 2, 3], [2, 1, 0])', solve(*([1, 2, 3], [1, 2, 3], [2, 1, 0])), 0)
check('fixture 4: ([2, 0, 0], [0, 3, 0], [0, 0, 4])', solve(*([2, 0, 0], [0, 3, 0], [0, 0, 4])), 24)
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])	`1`	`1`	Passed
fixture 2: ([0, 1, 0], [1, 0, 0], [0, 0, 1])	`1`	`-1`	Failed
fixture 3: ([1, 2, 3], [1, 2, 3], [2, 1, 0])	`0`	`0`	Passed
fixture 4: ([2, 0, 0], [0, 3, 0], [0, 0, 4])	`24`	`24`	Passed

SHA-256 / b5347880f85509f34b390820e455bf06338440bf9905d09e973168ab96b9325d

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 2026-09-29T14:37:57.409610+00:00.

Case digest / 103a901608c67e7c3f13b0a4a0c90803db9bc93a790a62356dba40d5a5ea065e