FA-14871 / Numerics / Open access

Extended euclidean certificate: second Bezout seed · case 01

The exact extended euclidean certificate result violates the stated contract at second Bezout seed.

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

ROOT CAUSE

The second Bezout seed step uses 1,0 instead of 0,1.

VERIFIED REPAIR

Use 0,1 at the second Bezout seed step.

Unsuccessful approach: The partial repair 0,0 still violates the second Bezout seed 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=1,0
    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, 0]`	`[20, 0, -1]`	Failed
explicit oracle 1	`[20, 0, 0]`	`[20, 0, 1]`	Failed
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, -6]`	`[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, 2]`	`[2, 2, -3]`	Failed

SHA-256 / 755112fb09a95304f724c34a20768a2fa660cd6c0694b9073ed64d1b7afb5fb5

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=0,0
    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, 0]`	`[20, 0, -1]`	Failed
explicit oracle 1	`[20, 0, 0]`	`[20, 0, 1]`	Failed
explicit oracle 2	`[1, -1, 0]`	`[1, -1, 1]`	Failed
explicit oracle 3	`[2, -1, 0]`	`[2, -1, 1]`	Failed
explicit oracle 4	`[1, -6, 0]`	`[1, -6, 7]`	Failed
explicit oracle 5	`[4, -1, 0]`	`[4, -1, 1]`	Failed
explicit oracle 6	`[5, -1, 0]`	`[5, -1, 1]`	Failed
explicit oracle 7	`[2, 2, 0]`	`[2, 2, -3]`	Failed

SHA-256 / ebbfcadf4576b11f0a23992b28e10394f8c775ba42511fea505c6a69dc8796dd

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,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]`	Passed
explicit oracle 1	`[20, 0, 1]`	`[20, 0, 1]`	Passed
explicit oracle 2	`[1, -1, 1]`	`[1, -1, 1]`	Passed
explicit oracle 3	`[2, -1, 1]`	`[2, -1, 1]`	Passed
explicit oracle 4	`[1, -6, 7]`	`[1, -6, 7]`	Passed
explicit oracle 5	`[4, -1, 1]`	`[4, -1, 1]`	Passed
explicit oracle 6	`[5, -1, 1]`	`[5, -1, 1]`	Passed
explicit oracle 7	`[2, 2, -3]`	`[2, 2, -3]`	Passed

SHA-256 / 106e76ad395b8e5b6838ec1fa5b2dbe3e99d1b812b3d70ad55fc4f07cc7b6e8b

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 2026-09-29T14:39:21.197394+00:00.

Case digest / fdafbff4d0bec74ed27e8c3ab24ccda26da9aa9d50391e17924f2ad88dd7a7a7