FA-5491 / Number theory / Open access

Coprime predicate · case 01

Nondivisibility does not imply absence of a shared factor.

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

ROOT CAUSE

Nondivisibility does not imply absence of a shared factor.

THE FAILURE

Nondivisibility does not imply absence of a shared factor.

Unsuccessful approach: Unequal values may still share factors and equal units are coprime.

Case contract

Integer inputs; n is positive unless a zero or negative fixture explicitly extends that operation. A modulus is greater than one; p is prime; exponent k may be signed when an inverse exists. a and b may be signed or zero; return whether gcd(a,b)=1. Exact operational definition: math.gcd(a,b)==1

Why this case matters

Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Number theory 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%b != 0 if b else False
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (6, 35)', solve(*(6, 35)), True)
check('fixture 2: (6, 9)', solve(*(6, 9)), False)
check('fixture 3: (1, 1)', solve(*(1, 1)), True)
check('fixture 4: (0, 1)', solve(*(0, 1)), True)
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: (6, 35)	`True`	`True`	Passed
fixture 2: (6, 9)	`True`	`False`	Failed
fixture 3: (1, 1)	`False`	`True`	Failed
fixture 4: (0, 1)	`False`	`True`	Failed

SHA-256 / 96fa46fdbd1fde4c3f27fdf393c534e3470344097c2e7f08c697dc9b38bb5614

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!=b
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (6, 35)', solve(*(6, 35)), True)
check('fixture 2: (6, 9)', solve(*(6, 9)), False)
check('fixture 3: (1, 1)', solve(*(1, 1)), True)
check('fixture 4: (0, 1)', solve(*(0, 1)), True)
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: (6, 35)	`True`	`True`	Passed
fixture 2: (6, 9)	`True`	`False`	Failed
fixture 3: (1, 1)	`False`	`True`	Failed
fixture 4: (0, 1)	`True`	`True`	Passed

SHA-256 / 1b8eb4b96542bb70b092f4c7f27475a57a7b44c70c5a9caac4a39ac67b4794cd

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:37:50.172883+00:00.

Case digest / 6c49ea83992ab49cd08b824e84f975fbe7af5c39bdbd1c11e38b7d12367a6770