FA-14861 / Numerics / Open access
Jacobi symbol: common factor terminal · case 01
The exact jacobi symbol result violates the stated contract at common factor terminal.
ROOT CAUSE
The common factor terminal step uses sign instead of sign if n==1 else 0.
VERIFIED REPAIR
Use sign if n==1 else 0 at the common factor terminal step.
Unsuccessful approach: The partial repair sign if n>1 else 0 still violates the common factor terminal invariant.
Case contract
Input [a,n], positive odd n; return Jacobi symbol in {-1,0,1}. Bounds: |a|<=100 and odd n<=99.
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):
a,n=x
a=a%n;sign=1
for _ in range(100):
if a==0:break
while a%2==0:
a//=2
if n%8 in (3,5):sign=-sign
if a%4==3 and n%4==3:sign=-sign
a,n=n%a,a
return sign
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-30, 3], 0), ([-30, 1], 1), ([50, 69], -1), ([-29, 1], 1), ([-28, 1], 1), ([-27, 1], 1), ([-26, 1], 1), ([-25, 1], 1)], [([-27, 3], 0), ([-27, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([-13, 1], 1), ([-12, 1], 1), ([-11, 1], 1), ([-10, 1], 1)], [([-24, 3], 0), ([-24, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([4, 1], 1), ([5, 1], 1), ([6, 1], 1), ([7, 1], 1)], [([-21, 3], 0), ([-21, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([21, 1], 1), ([22, 1], 1), ([23, 1], 1), ([24, 1], 1)], [([-18, 3], 0), ([-18, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([38, 1], 1), ([39, 1], 1), ([40, 1], 1), ([41, 1], 1)]]
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 | 0 | Failed |
| explicit oracle 1 | 1 | 1 | Passed |
| explicit oracle 2 | -1 | -1 | Passed |
| explicit oracle 3 | 1 | 1 | 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 / 43cac1b119754c438b4c76f8a02aa45e5c90935caee3f9eccd1d441a66a6bb17
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):
a,n=x
a=a%n;sign=1
for _ in range(100):
if a==0:break
while a%2==0:
a//=2
if n%8 in (3,5):sign=-sign
if a%4==3 and n%4==3:sign=-sign
a,n=n%a,a
return sign if n>1 else 0
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-30, 3], 0), ([-30, 1], 1), ([50, 69], -1), ([-29, 1], 1), ([-28, 1], 1), ([-27, 1], 1), ([-26, 1], 1), ([-25, 1], 1)], [([-27, 3], 0), ([-27, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([-13, 1], 1), ([-12, 1], 1), ([-11, 1], 1), ([-10, 1], 1)], [([-24, 3], 0), ([-24, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([4, 1], 1), ([5, 1], 1), ([6, 1], 1), ([7, 1], 1)], [([-21, 3], 0), ([-21, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([21, 1], 1), ([22, 1], 1), ([23, 1], 1), ([24, 1], 1)], [([-18, 3], 0), ([-18, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([38, 1], 1), ([39, 1], 1), ([40, 1], 1), ([41, 1], 1)]]
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 | 0 | Failed |
| explicit oracle 1 | 0 | 1 | Failed |
| explicit oracle 2 | 0 | -1 | Failed |
| explicit oracle 3 | 0 | 1 | Failed |
| explicit oracle 4 | 0 | 1 | Failed |
| explicit oracle 5 | 0 | 1 | Failed |
| explicit oracle 6 | 0 | 1 | Failed |
| explicit oracle 7 | 0 | 1 | Failed |
SHA-256 / b49c5852ed383fe09cf74ec2c49edfe81cbf3d023e3748f09fb35e4e47f88f8e
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):
a,n=x
a=a%n;sign=1
for _ in range(100):
if a==0:break
while a%2==0:
a//=2
if n%8 in (3,5):sign=-sign
if a%4==3 and n%4==3:sign=-sign
a,n=n%a,a
return sign if n==1 else 0
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-30, 3], 0), ([-30, 1], 1), ([50, 69], -1), ([-29, 1], 1), ([-28, 1], 1), ([-27, 1], 1), ([-26, 1], 1), ([-25, 1], 1)], [([-27, 3], 0), ([-27, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([-13, 1], 1), ([-12, 1], 1), ([-11, 1], 1), ([-10, 1], 1)], [([-24, 3], 0), ([-24, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([4, 1], 1), ([5, 1], 1), ([6, 1], 1), ([7, 1], 1)], [([-21, 3], 0), ([-21, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([21, 1], 1), ([22, 1], 1), ([23, 1], 1), ([24, 1], 1)], [([-18, 3], 0), ([-18, 1], 1), ([-30, 1], 1), ([50, 69], -1), ([38, 1], 1), ([39, 1], 1), ([40, 1], 1), ([41, 1], 1)]]
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 | 0 | Passed |
| explicit oracle 1 | 1 | 1 | Passed |
| explicit oracle 2 | -1 | -1 | Passed |
| explicit oracle 3 | 1 | 1 | 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 / ed1bde85bcddc4290797150393bd719e9b7c8c71d9376d0f97ddc2e8d8327a7d
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 2026-09-29T14:39:21.151750+00:00.
Case digest / 9cf0269d1ba0edd9c5f8160b33bbaaf37ae6d35a30d485916b461870ea202cce