FA-14886 / Numerics / Open access

Extended euclidean certificate: gcd sign normalization · case 01

The exact extended euclidean certificate result violates the stated contract at gcd sign normalization.

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

ROOT CAUSE

The gcd sign normalization step uses -r0,s0,t0 instead of -r0,-s0,-t0.

THE FAILURE

The gcd sign normalization step uses -r0,s0,t0 instead of -r0,-s0,-t0.

Unsuccessful approach: The partial repair r0,-s0,-t0 still violates the gcd sign normalization invariant.

Case contract

Input [a,b], integers not both zero; return [g,u,v] with g>=0 and a*u+b*v=g using standard Euclidean quotient recurrence. Bounds: |a|,|b|<=1000000.

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,b=x
    r0,r1=a,b
    s0,s1=1,0
    t0,t1=0,1
    for _ in range(60):
     if r1==0:break
     q=r0//r1
     r0,r1=r1,r0-q*r1
     s0,s1=s1,s0-q*s1
     t0,t1=t1,t0-q*t1
    if r0<0:r0,s0,t0=-r0,s0,t0
    return [r0,s0,t0]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -19], [1, -1, 1]), ([-20, -18], [2, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -16], [4, -1, 1]), ([-20, -15], [5, -1, 1]), ([-20, -14], [2, 2, -3])], [([-20, -19], [1, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -3], [1, 1, -7]), ([-20, -2], [2, 0, -1]), ([-20, -1], [1, 0, -1]), ([-20, 0], [20, -1, 0])], [([-20, -18], [2, -1, 1]), ([-20, -14], [2, 2, -3]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, 14], [2, 2, 3]), ([-20, 15], [5, -1, -1]), ([-20, 16], [4, -1, -1]), ([-20, 17], [1, -6, -7])], [([-20, -17], [1, -6, 7]), ([-20, -11], [1, -5, 9]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, -10], [1, 1, -2]), ([-19, -9], [1, -1, 2]), ([-19, -8], [1, -3, 7]), ([-19, -7], [1, -3, 8])], [([-20, -16], [4, -1, 1]), ([-20, -8], [4, -1, 2]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, 7], [1, -3, -8]), ([-19, 8], [1, -3, -7]), ([-19, 9], [1, -1, -2]), ([-19, 10], [1, 1, 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 fixture	Actual	Expected	Outcome
explicit oracle 0	`[20, 0, 1]`	`[20, 0, -1]`	Failed
explicit oracle 1	`[20, 0, 1]`	`[20, 0, 1]`	Passed
explicit oracle 2	`[1, 1, -1]`	`[1, -1, 1]`	Failed
explicit oracle 3	`[2, 1, -1]`	`[2, -1, 1]`	Failed
explicit oracle 4	`[1, 6, -7]`	`[1, -6, 7]`	Failed
explicit oracle 5	`[4, 1, -1]`	`[4, -1, 1]`	Failed
explicit oracle 6	`[5, 1, -1]`	`[5, -1, 1]`	Failed
explicit oracle 7	`[2, -2, 3]`	`[2, 2, -3]`	Failed

SHA-256 / c90821c74914e1517cfafcfc550bfe5c2242a315c05a68f54a02b7863e85af28

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,b=x
    r0,r1=a,b
    s0,s1=1,0
    t0,t1=0,1
    for _ in range(60):
     if r1==0:break
     q=r0//r1
     r0,r1=r1,r0-q*r1
     s0,s1=s1,s0-q*s1
     t0,t1=t1,t0-q*t1
    if r0<0:r0,s0,t0=r0,-s0,-t0
    return [r0,s0,t0]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -19], [1, -1, 1]), ([-20, -18], [2, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -16], [4, -1, 1]), ([-20, -15], [5, -1, 1]), ([-20, -14], [2, 2, -3])], [([-20, -19], [1, -1, 1]), ([-20, -17], [1, -6, 7]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, -3], [1, 1, -7]), ([-20, -2], [2, 0, -1]), ([-20, -1], [1, 0, -1]), ([-20, 0], [20, -1, 0])], [([-20, -18], [2, -1, 1]), ([-20, -14], [2, 2, -3]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-20, 14], [2, 2, 3]), ([-20, 15], [5, -1, -1]), ([-20, 16], [4, -1, -1]), ([-20, 17], [1, -6, -7])], [([-20, -17], [1, -6, 7]), ([-20, -11], [1, -5, 9]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, -10], [1, 1, -2]), ([-19, -9], [1, -1, 2]), ([-19, -8], [1, -3, 7]), ([-19, -7], [1, -3, 8])], [([-20, -16], [4, -1, 1]), ([-20, -8], [4, -1, 2]), ([-20, -20], [20, 0, -1]), ([20, 20], [20, 0, 1]), ([-19, 7], [1, -3, -8]), ([-19, 8], [1, -3, -7]), ([-19, 9], [1, -1, -2]), ([-19, 10], [1, 1, 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 fixture	Actual	Expected	Outcome
explicit oracle 0	`[-20, 0, -1]`	`[20, 0, -1]`	Failed
explicit oracle 1	`[20, 0, 1]`	`[20, 0, 1]`	Passed
explicit oracle 2	`[-1, -1, 1]`	`[1, -1, 1]`	Failed
explicit oracle 3	`[-2, -1, 1]`	`[2, -1, 1]`	Failed
explicit oracle 4	`[-1, -6, 7]`	`[1, -6, 7]`	Failed
explicit oracle 5	`[-4, -1, 1]`	`[4, -1, 1]`	Failed
explicit oracle 6	`[-5, -1, 1]`	`[5, -1, 1]`	Failed
explicit oracle 7	`[-2, 2, -3]`	`[2, 2, -3]`	Failed

SHA-256 / b4d5b62eebffb543f15fa97f35d780d6c5eb7a0bc1b3b96ccfd591e5ed435b56

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:39:21.245980+00:00.

Case digest / 19721c0366d29d2290838af6c0df3977ff5efa8f9db0aac3d8316576dc655203