FA-14416 / Numerics / Open access

Rational continued fraction: floor partial quotient · case 01

The exact rational continued fraction result violates the stated contract at floor partial quotient.

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

ROOT CAUSE

The floor partial quotient step uses int(Fraction(p,q)) instead of p//q.

VERIFIED REPAIR

Use p//q at the floor partial quotient step.

Unsuccessful approach: The partial repair abs(p)//q still violates the floor partial quotient invariant.

Case contract

Input [p,q], q>0; canonical simple continued fraction of p/q with last term>1 unless singleton. Bounds: -25<=p<=25 and 1<=q<=17.

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):
    p,q=x
    out=[]
    for _ in range(30):
     if q==0: break
     a=int(Fraction(p,q))
     out.append(a)
     r=p-a*q
     p,q=q,r
    if len(out)>1 and out[-1]==1:
     out[-2]+=1
     out.pop()
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-25, 2], [-13, 2]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-25, 3], [-9, 1, 2]), ([-25, 4], [-7, 1, 3]), ([-25, 5], [-5]), ([-25, 6], [-5, 1, 5]), ([-25, 7], [-4, 2, 3])], [([-25, 3], [-9, 1, 2]), ([-25, 4], [-7, 1, 3]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-24, 1], [-24]), ([-24, 2], [-12]), ([-24, 3], [-8]), ([-24, 4], [-6])], [([-25, 4], [-7, 1, 3]), ([-25, 7], [-4, 2, 3]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-23, 1], [-23]), ([-23, 2], [-12, 2]), ([-23, 3], [-8, 3]), ([-23, 4], [-6, 4])], [([-25, 6], [-5, 1, 5]), ([-25, 10], [-3, 2]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-22, 1], [-22]), ([-22, 2], [-11]), ([-22, 3], [-8, 1, 2]), ([-22, 4], [-6, 2])], [([-25, 7], [-4, 2, 3]), ([-25, 13], [-2, 13]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-21, 1], [-21]), ([-21, 2], [-11, 2]), ([-21, 3], [-7]), ([-21, 4], [-6, 1, 3])]]
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	`[-12, -2]`	`[-13, 2]`	Failed
explicit oracle 1	`[-25]`	`[-25]`	Passed
explicit oracle 2	`[1, 2, 8]`	`[1, 2, 8]`	Passed
explicit oracle 3	`[-8, -3]`	`[-9, 1, 2]`	Failed
explicit oracle 4	`[-6, -4]`	`[-7, 1, 3]`	Failed
explicit oracle 5	`[-5]`	`[-5]`	Passed
explicit oracle 6	`[-4, -6]`	`[-5, 1, 5]`	Failed
explicit oracle 7	`[-3, -1, -1, -3]`	`[-4, 2, 3]`	Failed

SHA-256 / e5671fac4351e4f71cfc0c2e0591b0ec32795cde35e57fe6ea117ed5e8713629

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):
    p,q=x
    out=[]
    for _ in range(30):
     if q==0: break
     a=abs(p)//q
     out.append(a)
     r=p-a*q
     p,q=q,r
    if len(out)>1 and out[-1]==1:
     out[-2]+=1
     out.pop()
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-25, 2], [-13, 2]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-25, 3], [-9, 1, 2]), ([-25, 4], [-7, 1, 3]), ([-25, 5], [-5]), ([-25, 6], [-5, 1, 5]), ([-25, 7], [-4, 2, 3])], [([-25, 3], [-9, 1, 2]), ([-25, 4], [-7, 1, 3]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-24, 1], [-24]), ([-24, 2], [-12]), ([-24, 3], [-8]), ([-24, 4], [-6])], [([-25, 4], [-7, 1, 3]), ([-25, 7], [-4, 2, 3]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-23, 1], [-23]), ([-23, 2], [-12, 2]), ([-23, 3], [-8, 3]), ([-23, 4], [-6, 4])], [([-25, 6], [-5, 1, 5]), ([-25, 10], [-3, 2]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-22, 1], [-22]), ([-22, 2], [-11]), ([-22, 3], [-8, 1, 2]), ([-22, 4], [-6, 2])], [([-25, 7], [-4, 2, 3]), ([-25, 13], [-2, 13]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-21, 1], [-21]), ([-21, 2], [-11, 2]), ([-21, 3], [-7]), ([-21, 4], [-6, 1, 3])]]
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	`[12, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]`	`[-13, 2]`	Failed
explicit oracle 1	`[25, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]`	`[-25]`	Failed
explicit oracle 2	`[1, 2, 8]`	`[1, 2, 8]`	Passed
explicit oracle 3	`[8, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]`	`[-9, 1, 2]`	Failed
explicit oracle 4	`[6, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]`	`[-7, 1, 3]`	Failed
explicit oracle 5	`[5, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]`	`[-5]`	Failed
explicit oracle 6	`[4, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]`	`[-5, 1, 5]`	Failed
explicit oracle 7	`[3, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]`	`[-4, 2, 3]`	Failed

SHA-256 / e12813f2513bfe71c75445edfecc86ddabefa7c8bb906a5ce2c5eec5403653a7

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):
    p,q=x
    out=[]
    for _ in range(30):
     if q==0: break
     a=p//q
     out.append(a)
     r=p-a*q
     p,q=q,r
    if len(out)>1 and out[-1]==1:
     out[-2]+=1
     out.pop()
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-25, 2], [-13, 2]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-25, 3], [-9, 1, 2]), ([-25, 4], [-7, 1, 3]), ([-25, 5], [-5]), ([-25, 6], [-5, 1, 5]), ([-25, 7], [-4, 2, 3])], [([-25, 3], [-9, 1, 2]), ([-25, 4], [-7, 1, 3]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-24, 1], [-24]), ([-24, 2], [-12]), ([-24, 3], [-8]), ([-24, 4], [-6])], [([-25, 4], [-7, 1, 3]), ([-25, 7], [-4, 2, 3]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-23, 1], [-23]), ([-23, 2], [-12, 2]), ([-23, 3], [-8, 3]), ([-23, 4], [-6, 4])], [([-25, 6], [-5, 1, 5]), ([-25, 10], [-3, 2]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-22, 1], [-22]), ([-22, 2], [-11]), ([-22, 3], [-8, 1, 2]), ([-22, 4], [-6, 2])], [([-25, 7], [-4, 2, 3]), ([-25, 13], [-2, 13]), ([-25, 1], [-25]), ([25, 17], [1, 2, 8]), ([-21, 1], [-21]), ([-21, 2], [-11, 2]), ([-21, 3], [-7]), ([-21, 4], [-6, 1, 3])]]
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	`[-13, 2]`	`[-13, 2]`	Passed
explicit oracle 1	`[-25]`	`[-25]`	Passed
explicit oracle 2	`[1, 2, 8]`	`[1, 2, 8]`	Passed
explicit oracle 3	`[-9, 1, 2]`	`[-9, 1, 2]`	Passed
explicit oracle 4	`[-7, 1, 3]`	`[-7, 1, 3]`	Passed
explicit oracle 5	`[-5]`	`[-5]`	Passed
explicit oracle 6	`[-5, 1, 5]`	`[-5, 1, 5]`	Passed
explicit oracle 7	`[-4, 2, 3]`	`[-4, 2, 3]`	Passed

SHA-256 / eba07eb91b7129c24ed3b0000bde9b1eec8794b17030856b3a0dd0919aff71ec

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 2026-09-29T14:39:16.579020+00:00.

Case digest / 04d1de85ad91d9863fd708c17e67da40e193ebddb0b1a4567f465a353d89bca6