FAILURE MAP
← Case archive

FA-14361 / Numerics / Open access

Generalized crt: canonical residue · case 01

The exact generalized crt result violates the stated contract at canonical residue.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The canonical residue step uses r instead of r%period.

VERIFIED REPAIR

Use r%period at the canonical residue step.

Unsuccessful approach: The partial repair abs(r)%period still violates the canonical residue invariant.

Case contract

Input [a,m,b,n], positive moduli; return least nonnegative common residue and lcm modulus or None.

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,m,b,n=x
    g=math.gcd(m,n)
    if (b-a)%g != 0: return None
    q=n//g
    k=((b-a)//g*pow(m//g,-1,q))%q if q>1 else 0
    period=m*(n//g)
    r=a+m*k
    return [r,period]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-3, 1, -3, 1], [0, 1]), ([-3, 1, -2, 3], [1, 3]), ([3, 8, 3, 8], [3, 8]), ([-3, 1, -2, 1], [0, 1]), ([-3, 1, -1, 1], [0, 1]), ([-3, 1, 0, 1], [0, 1]), ([-3, 1, 1, 1], [0, 1]), ([-3, 1, 2, 1], [0, 1])], [([-3, 1, -2, 1], [0, 1]), ([-3, 1, 2, 3], [2, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-1, 1, 0, 1], [0, 1]), ([-1, 1, 1, 1], [0, 1]), ([-1, 1, 2, 1], [0, 1]), ([-1, 1, 3, 1], [0, 1])], [([-3, 1, -1, 1], [0, 1]), ([-2, 1, 1, 3], [1, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([1, 1, 3, 1], [0, 1]), ([2, 1, -3, 1], [0, 1]), ([2, 1, -2, 1], [0, 1]), ([2, 1, -1, 1], [0, 1])], [([-3, 1, 0, 1], [0, 1]), ([-1, 1, 2, 3], [2, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-3, 1, -1, 2], [1, 2]), ([-3, 1, 0, 2], [0, 2]), ([-3, 1, 1, 2], [1, 2]), ([-3, 1, 2, 2], [0, 2])], [([-3, 1, 1, 1], [0, 1]), ([-3, 1, 1, 4], [1, 4]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-1, 1, 2, 2], [0, 2]), ([-1, 1, 3, 2], [1, 2]), ([0, 1, -3, 2], [1, 2]), ([0, 1, -2, 2], [0, 2])]]
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 fixtureActualExpectedOutcome
explicit oracle 0[-3, 1][0, 1]Failed
explicit oracle 1[-2, 3][1, 3]Failed
explicit oracle 2[3, 8][3, 8]Passed
explicit oracle 3[-3, 1][0, 1]Failed
explicit oracle 4[-3, 1][0, 1]Failed
explicit oracle 5[-3, 1][0, 1]Failed
explicit oracle 6[-3, 1][0, 1]Failed
explicit oracle 7[-3, 1][0, 1]Failed

SHA-256 / f55097a2f1309fb06a133cbfeb45f8e76e3f50fb67f168c57a395165547ea5d9

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,m,b,n=x
    g=math.gcd(m,n)
    if (b-a)%g != 0: return None
    q=n//g
    k=((b-a)//g*pow(m//g,-1,q))%q if q>1 else 0
    period=m*(n//g)
    r=a+m*k
    return [abs(r)%period,period]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-3, 1, -3, 1], [0, 1]), ([-3, 1, -2, 3], [1, 3]), ([3, 8, 3, 8], [3, 8]), ([-3, 1, -2, 1], [0, 1]), ([-3, 1, -1, 1], [0, 1]), ([-3, 1, 0, 1], [0, 1]), ([-3, 1, 1, 1], [0, 1]), ([-3, 1, 2, 1], [0, 1])], [([-3, 1, -2, 1], [0, 1]), ([-3, 1, 2, 3], [2, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-1, 1, 0, 1], [0, 1]), ([-1, 1, 1, 1], [0, 1]), ([-1, 1, 2, 1], [0, 1]), ([-1, 1, 3, 1], [0, 1])], [([-3, 1, -1, 1], [0, 1]), ([-2, 1, 1, 3], [1, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([1, 1, 3, 1], [0, 1]), ([2, 1, -3, 1], [0, 1]), ([2, 1, -2, 1], [0, 1]), ([2, 1, -1, 1], [0, 1])], [([-3, 1, 0, 1], [0, 1]), ([-1, 1, 2, 3], [2, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-3, 1, -1, 2], [1, 2]), ([-3, 1, 0, 2], [0, 2]), ([-3, 1, 1, 2], [1, 2]), ([-3, 1, 2, 2], [0, 2])], [([-3, 1, 1, 1], [0, 1]), ([-3, 1, 1, 4], [1, 4]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-1, 1, 2, 2], [0, 2]), ([-1, 1, 3, 2], [1, 2]), ([0, 1, -3, 2], [1, 2]), ([0, 1, -2, 2], [0, 2])]]
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 fixtureActualExpectedOutcome
explicit oracle 0[0, 1][0, 1]Passed
explicit oracle 1[2, 3][1, 3]Failed
explicit oracle 2[3, 8][3, 8]Passed
explicit oracle 3[0, 1][0, 1]Passed
explicit oracle 4[0, 1][0, 1]Passed
explicit oracle 5[0, 1][0, 1]Passed
explicit oracle 6[0, 1][0, 1]Passed
explicit oracle 7[0, 1][0, 1]Passed

SHA-256 / 9fd0df18e595c9629ab4c7b8cddbfa83d0028d7b32ecff9c25dd5c937c0b7322

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,m,b,n=x
    g=math.gcd(m,n)
    if (b-a)%g != 0: return None
    q=n//g
    k=((b-a)//g*pow(m//g,-1,q))%q if q>1 else 0
    period=m*(n//g)
    r=a+m*k
    return [r%period,period]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-3, 1, -3, 1], [0, 1]), ([-3, 1, -2, 3], [1, 3]), ([3, 8, 3, 8], [3, 8]), ([-3, 1, -2, 1], [0, 1]), ([-3, 1, -1, 1], [0, 1]), ([-3, 1, 0, 1], [0, 1]), ([-3, 1, 1, 1], [0, 1]), ([-3, 1, 2, 1], [0, 1])], [([-3, 1, -2, 1], [0, 1]), ([-3, 1, 2, 3], [2, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-1, 1, 0, 1], [0, 1]), ([-1, 1, 1, 1], [0, 1]), ([-1, 1, 2, 1], [0, 1]), ([-1, 1, 3, 1], [0, 1])], [([-3, 1, -1, 1], [0, 1]), ([-2, 1, 1, 3], [1, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([1, 1, 3, 1], [0, 1]), ([2, 1, -3, 1], [0, 1]), ([2, 1, -2, 1], [0, 1]), ([2, 1, -1, 1], [0, 1])], [([-3, 1, 0, 1], [0, 1]), ([-1, 1, 2, 3], [2, 3]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-3, 1, -1, 2], [1, 2]), ([-3, 1, 0, 2], [0, 2]), ([-3, 1, 1, 2], [1, 2]), ([-3, 1, 2, 2], [0, 2])], [([-3, 1, 1, 1], [0, 1]), ([-3, 1, 1, 4], [1, 4]), ([-3, 1, -3, 1], [0, 1]), ([3, 8, 3, 8], [3, 8]), ([-1, 1, 2, 2], [0, 2]), ([-1, 1, 3, 2], [1, 2]), ([0, 1, -3, 2], [1, 2]), ([0, 1, -2, 2], [0, 2])]]
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 fixtureActualExpectedOutcome
explicit oracle 0[0, 1][0, 1]Passed
explicit oracle 1[1, 3][1, 3]Passed
explicit oracle 2[3, 8][3, 8]Passed
explicit oracle 3[0, 1][0, 1]Passed
explicit oracle 4[0, 1][0, 1]Passed
explicit oracle 5[0, 1][0, 1]Passed
explicit oracle 6[0, 1][0, 1]Passed
explicit oracle 7[0, 1][0, 1]Passed

SHA-256 / 502fe444169f17353fd99fab5ce6b7d530a91c178276c6b8380b9568a958f49b

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 2026-09-29T14:39:16.178431+00:00.

Case digest / 51d4399b0497f11ff810b0d5a86c1cbbd490d2b5a13df1d662c9bd78f0db077a