FA-5626 / Combinatorics / Open access
Prime length necklace colorings · case 01
Dividing all words by the length mishandles constant words with smaller rotation orbits.
ROOT CAUSE
Dividing all words by the length mishandles constant words with smaller rotation orbits.
VERIFIED REPAIR
Apply the specified mathematical contract directly, preserving all terms and boundary cases: return (colors**length+(length-1)*colors)//length
Unsuccessful approach: Rotationally equivalent colorings are counted separately.
Case contract
Nonnegative integer counts. Permutation and combination counts are zero when a requested selection exceeds the available objects. Shape sizes satisfy the preconditions stated below. length is prime and colors>=1; count colorings of a labelled-position cycle up to rotations, retaining reflection distinctions. Exact operational definition: (colors**length+(length-1)*colors)//length
Why this case matters
Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Combinatorics 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(length, colors):
return colors**length//length
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (3, 2)', solve(*(3, 2)), 4)
check('fixture 2: (5, 2)', solve(*(5, 2)), 8)
check('fixture 3: (2, 3)', solve(*(2, 3)), 6)
check('fixture 4: (3, 1)', solve(*(3, 1)), 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: (3, 2) | 2 | 4 | Failed |
| fixture 2: (5, 2) | 6 | 8 | Failed |
| fixture 3: (2, 3) | 4 | 6 | Failed |
| fixture 4: (3, 1) | 0 | 1 | Failed |
SHA-256 / 919cf139f81d820ecb217792c5f2560cb81782f581051d15f77b7b51bef591a3
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(length, colors):
return colors**length
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (3, 2)', solve(*(3, 2)), 4)
check('fixture 2: (5, 2)', solve(*(5, 2)), 8)
check('fixture 3: (2, 3)', solve(*(2, 3)), 6)
check('fixture 4: (3, 1)', solve(*(3, 1)), 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: (3, 2) | 8 | 4 | Failed |
| fixture 2: (5, 2) | 32 | 8 | Failed |
| fixture 3: (2, 3) | 9 | 6 | Failed |
| fixture 4: (3, 1) | 1 | 1 | Passed |
SHA-256 / 822e3ef8d431bcb2e2e665d1f900b4a5fe5dd32003dd6b59309b8be5a656345a
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(length, colors):
return (colors**length+(length-1)*colors)//length
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (3, 2)', solve(*(3, 2)), 4)
check('fixture 2: (5, 2)', solve(*(5, 2)), 8)
check('fixture 3: (2, 3)', solve(*(2, 3)), 6)
check('fixture 4: (3, 1)', solve(*(3, 1)), 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: (3, 2) | 4 | 4 | Passed |
| fixture 2: (5, 2) | 8 | 8 | Passed |
| fixture 3: (2, 3) | 6 | 6 | Passed |
| fixture 4: (3, 1) | 1 | 1 | Passed |
SHA-256 / e4438b6ff9841ed4b6e237f5741db96f036fa670ded63f35243ef01a71f4ef1a
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 2026-09-29T14:37:51.774206+00:00.
Case digest / 42bfd64065ec920f07ca300c37bef99a83462090e045c2c941fbf470357d472e