FA-15581 / Numerics / Open access
Prime power unit factorial: all prime factor stripping · case 01
The exact prime power unit factorial result violates the stated contract at all prime factor stripping.
ROOT CAUSE
The all prime factor stripping step uses v<p instead of v%p!=0.
VERIFIED REPAIR
Use v%p!=0 at the all prime factor stripping step.
Unsuccessful approach: The partial repair v==p still violates the all prime factor stripping invariant.
Case contract
Input [n,p,e], n>=0 p prime e>=1; return product of positive integers<=n after removing ALL p factors from each, modulo p^e. Bounds: n<=64.
Why this case matters
Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
n,p,e=x;mod=p**e;r=1
for v in range(1,n+1):
for _ in range(30):
if v<p:break
v//=p
r=r*v%mod
return r
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([3, 2, 2], 3), ([1, 2, 1], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([0, 2, 2], 1), ([0, 2, 3], 1), ([0, 2, 4], 1), ([1, 2, 2], 1)], [([3, 2, 3], 3), ([1, 2, 4], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([4, 2, 2], 3), ([4, 2, 3], 3), ([4, 2, 4], 3), ([5, 2, 1], 1)], [([3, 2, 4], 3), ([2, 2, 3], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([8, 2, 3], 3), ([8, 2, 4], 11), ([9, 2, 1], 1), ([9, 2, 2], 3)], [([4, 2, 2], 3), ([3, 2, 2], 3), ([0, 2, 1], 1), ([64, 7, 4], 282), ([12, 2, 4], 15), ([13, 2, 1], 1), ([13, 2, 2], 3), ([13, 2, 3], 3)], [([4, 2, 3], 3), ([4, 2, 1], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([17, 2, 1], 1), ([17, 2, 2], 3), ([17, 2, 3], 3), ([17, 2, 4], 11)]]
for i, (args, expected) in enumerate(fixtures[N-1]):
check("explicit oracle %d" % i, solve(args), expected)
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 |
|---|---|---|---|
| explicit oracle 0 | 1 | 3 | Failed |
| explicit oracle 1 | 1 | 1 | Passed |
| explicit oracle 2 | 1 | 1 | Passed |
| explicit oracle 3 | 2122 | 282 | Failed |
| explicit oracle 4 | 1 | 1 | Passed |
| explicit oracle 5 | 1 | 1 | Passed |
| explicit oracle 6 | 1 | 1 | Passed |
| explicit oracle 7 | 1 | 1 | Passed |
SHA-256 / bbcb1443b2c344ae989d2b2159b4fab15b9955faebddb877e7d8d431fb49e906
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
n,p,e=x;mod=p**e;r=1
for v in range(1,n+1):
for _ in range(30):
if v==p:break
v//=p
r=r*v%mod
return r
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([3, 2, 2], 3), ([1, 2, 1], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([0, 2, 2], 1), ([0, 2, 3], 1), ([0, 2, 4], 1), ([1, 2, 2], 1)], [([3, 2, 3], 3), ([1, 2, 4], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([4, 2, 2], 3), ([4, 2, 3], 3), ([4, 2, 4], 3), ([5, 2, 1], 1)], [([3, 2, 4], 3), ([2, 2, 3], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([8, 2, 3], 3), ([8, 2, 4], 11), ([9, 2, 1], 1), ([9, 2, 2], 3)], [([4, 2, 2], 3), ([3, 2, 2], 3), ([0, 2, 1], 1), ([64, 7, 4], 282), ([12, 2, 4], 15), ([13, 2, 1], 1), ([13, 2, 2], 3), ([13, 2, 3], 3)], [([4, 2, 3], 3), ([4, 2, 1], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([17, 2, 1], 1), ([17, 2, 2], 3), ([17, 2, 3], 3), ([17, 2, 4], 11)]]
for i, (args, expected) in enumerate(fixtures[N-1]):
check("explicit oracle %d" % i, solve(args), expected)
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 |
|---|---|---|---|
| explicit oracle 0 | 0 | 3 | Failed |
| explicit oracle 1 | 0 | 1 | Failed |
| explicit oracle 2 | 1 | 1 | Passed |
| explicit oracle 3 | 0 | 282 | Failed |
| explicit oracle 4 | 1 | 1 | Passed |
| explicit oracle 5 | 1 | 1 | Passed |
| explicit oracle 6 | 1 | 1 | Passed |
| explicit oracle 7 | 0 | 1 | Failed |
SHA-256 / 78bae66f12acae36595d66d9f8da188373baba5d5efc787facaa6bfddc989e58
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
n,p,e=x;mod=p**e;r=1
for v in range(1,n+1):
for _ in range(30):
if v%p!=0:break
v//=p
r=r*v%mod
return r
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([3, 2, 2], 3), ([1, 2, 1], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([0, 2, 2], 1), ([0, 2, 3], 1), ([0, 2, 4], 1), ([1, 2, 2], 1)], [([3, 2, 3], 3), ([1, 2, 4], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([4, 2, 2], 3), ([4, 2, 3], 3), ([4, 2, 4], 3), ([5, 2, 1], 1)], [([3, 2, 4], 3), ([2, 2, 3], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([8, 2, 3], 3), ([8, 2, 4], 11), ([9, 2, 1], 1), ([9, 2, 2], 3)], [([4, 2, 2], 3), ([3, 2, 2], 3), ([0, 2, 1], 1), ([64, 7, 4], 282), ([12, 2, 4], 15), ([13, 2, 1], 1), ([13, 2, 2], 3), ([13, 2, 3], 3)], [([4, 2, 3], 3), ([4, 2, 1], 1), ([0, 2, 1], 1), ([64, 7, 4], 282), ([17, 2, 1], 1), ([17, 2, 2], 3), ([17, 2, 3], 3), ([17, 2, 4], 11)]]
for i, (args, expected) in enumerate(fixtures[N-1]):
check("explicit oracle %d" % i, solve(args), expected)
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 |
|---|---|---|---|
| explicit oracle 0 | 3 | 3 | Passed |
| explicit oracle 1 | 1 | 1 | Passed |
| explicit oracle 2 | 1 | 1 | Passed |
| explicit oracle 3 | 282 | 282 | Passed |
| explicit oracle 4 | 1 | 1 | Passed |
| explicit oracle 5 | 1 | 1 | Passed |
| explicit oracle 6 | 1 | 1 | Passed |
| explicit oracle 7 | 1 | 1 | Passed |
SHA-256 / f8f8f0fed887120e7244453ca652d990e972828b247b2d0dc63588c96c7515ff
Verification & scope
A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. 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 2026-09-29T14:39:28.097097+00:00.
Case digest / 786c9813b323f0ba1d55e99e6c782613bc0471176f9b43126ea176ce03e76e0c