FA-6086 / Linear algebra / Open access

Orthogonality zero dot · case 01

Zero coordinate sum is confused with zero mutual dot product.

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

ROOT CAUSE

Zero coordinate sum is confused with zero mutual dot product.

VERIFIED REPAIR

Apply the specified mathematical contract directly, preserving all terms and boundary cases: return sum(x*y for x,y in zip(a,b))==0

Unsuccessful approach: Unit normalization is confused with orthogonality.

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. Orthogonality zero dot. Exact operational definition: sum(x*y for x,y in zip(a,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 sum(a)==0 or sum(b)==0
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([1, 0], [0, 1])', solve(*([1, 0], [0, 1])), True)
check('fixture 2: ([1, -1], [1, 0])', solve(*([1, -1], [1, 0])), False)
check('fixture 3: ([1, 1], [1, -1])', solve(*([1, 1], [1, -1])), True)
check('fixture 4: ([1, 0], [1, 0])', solve(*([1, 0], [1, 0])), False)
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, 1])	`False`	`True`	Failed
fixture 2: ([1, -1], [1, 0])	`True`	`False`	Failed
fixture 3: ([1, 1], [1, -1])	`True`	`True`	Passed
fixture 4: ([1, 0], [1, 0])	`False`	`False`	Passed

SHA-256 / 8c2c3fca742a2fffee18b5e2a96dc9b06f623c4ec2591650eee9f9adc6d83d79

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 sum(x*y for x,y in zip(a,b))==1
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([1, 0], [0, 1])', solve(*([1, 0], [0, 1])), True)
check('fixture 2: ([1, -1], [1, 0])', solve(*([1, -1], [1, 0])), False)
check('fixture 3: ([1, 1], [1, -1])', solve(*([1, 1], [1, -1])), True)
check('fixture 4: ([1, 0], [1, 0])', solve(*([1, 0], [1, 0])), False)
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, 1])	`False`	`True`	Failed
fixture 2: ([1, -1], [1, 0])	`True`	`False`	Failed
fixture 3: ([1, 1], [1, -1])	`False`	`True`	Failed
fixture 4: ([1, 0], [1, 0])	`True`	`False`	Failed

SHA-256 / 47639dfacbed6bf19bc04f29b13833c925651ba83731d456cd8e020c17d72ca9

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(a, b):
    return sum(x*y for x,y in zip(a,b))==0
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([1, 0], [0, 1])', solve(*([1, 0], [0, 1])), True)
check('fixture 2: ([1, -1], [1, 0])', solve(*([1, -1], [1, 0])), False)
check('fixture 3: ([1, 1], [1, -1])', solve(*([1, 1], [1, -1])), True)
check('fixture 4: ([1, 0], [1, 0])', solve(*([1, 0], [1, 0])), False)
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, 1])	`True`	`True`	Passed
fixture 2: ([1, -1], [1, 0])	`False`	`False`	Passed
fixture 3: ([1, 1], [1, -1])	`True`	`True`	Passed
fixture 4: ([1, 0], [1, 0])	`False`	`False`	Passed

SHA-256 / 677df10dc0b338d0ad540a08539669002257c24c9e13080270ee993b86f83b1c

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 2026-09-29T14:37:57.357762+00:00.

Case digest / 2b9bc2feda1e788bdf5e4abe7021bb62819f4ebb2ae68e3d80fc15af71965a64