FA-5566 / Number theory / Open access
Multiplicative order · case 01
Starting at exponent zero returns the trivial identity.
ROOT CAUSE
Starting at exponent zero returns the trivial identity.
VERIFIED REPAIR
Apply the specified mathematical contract directly, preserving all terms and boundary cases: return next((k for k in range(1,m+1) if pow(a,k,m)==1),None)
Unsuccessful approach: The group size is an upper bound rather than every element order.
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. Multiplicative order. Exact operational definition: next((k for k in range(1,m+1) if pow(a,k,m)==1),None)
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, m):
return next((k for k in range(m+1) if pow(a,k,m)==1),None)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (2, 7)', solve(*(2, 7)), 3)
check('fixture 2: (3, 7)', solve(*(3, 7)), 6)
check('fixture 3: (2, 4)', solve(*(2, 4)), None)
check('fixture 4: (1, 9)', solve(*(1, 9)), 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, 7) | 0 | 3 | Failed |
| fixture 2: (3, 7) | 0 | 6 | Failed |
| fixture 3: (2, 4) | 0 | None | Failed |
| fixture 4: (1, 9) | 0 | 1 | Failed |
SHA-256 / 5a7abdc0d1ba383e776d1628b455461341c3b3526e96f4dfa56b14ae6272893d
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, m):
return m-1 if math.gcd(a,m)==1 else None
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (2, 7)', solve(*(2, 7)), 3)
check('fixture 2: (3, 7)', solve(*(3, 7)), 6)
check('fixture 3: (2, 4)', solve(*(2, 4)), None)
check('fixture 4: (1, 9)', solve(*(1, 9)), 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, 7) | 6 | 3 | Failed |
| fixture 2: (3, 7) | 6 | 6 | Passed |
| fixture 3: (2, 4) | None | None | Passed |
| fixture 4: (1, 9) | 8 | 1 | Failed |
SHA-256 / 46e39b607887318b8cb2158e2e1b5cbc6a189a995b16ea25b97c5d369d8f1ac2
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, m):
return next((k for k in range(1,m+1) if pow(a,k,m)==1),None)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (2, 7)', solve(*(2, 7)), 3)
check('fixture 2: (3, 7)', solve(*(3, 7)), 6)
check('fixture 3: (2, 4)', solve(*(2, 4)), None)
check('fixture 4: (1, 9)', solve(*(1, 9)), 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, 7) | 3 | 3 | Passed |
| fixture 2: (3, 7) | 6 | 6 | Passed |
| fixture 3: (2, 4) | None | None | Passed |
| fixture 4: (1, 9) | 1 | 1 | Passed |
SHA-256 / 3eadcafc315806222fbe09b222cbf0d6ace61547b015ec9d474941fce0c5d5ca
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 2026-09-29T14:37:50.979653+00:00.
Case digest / b8645307ee84c6680591df9c6e49df85a21c9753928d1580e32961ec53d1c70b