FA-14581 / Numerics / Open access

Rational mediant walk: left bound replacement · case 01

The exact rational mediant walk result violates the stated contract at left bound replacement.

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

ROOT CAUSE

The left bound replacement step uses lo=mid instead of hi=mid.

VERIFIED REPAIR

Use hi=mid at the left bound replacement step.

Unsuccessful approach: The partial repair hi=[mid[1],mid[0]] still violates the left bound replacement invariant.

Case contract

Input path over L/R; return Stern-Brocot interval bounds and its mediant.

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):
    lo=[0,1]
    hi=[1,0]
    for step in x:
     mid=[lo[0]+hi[0],lo[1]+hi[1]]
     if step=='L': lo=mid
     else: lo=mid
    return [lo,hi,[lo[0]+hi[0],lo[1]+hi[1]]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('L', [[0, 1], [1, 1], [1, 2]]), ('LL', [[0, 1], [1, 2], [1, 3]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('R', [[1, 1], [1, 0], [2, 1]]), ('LR', [[1, 2], [1, 1], [2, 3]]), ('RL', [[1, 1], [2, 1], [3, 2]]), ('RR', [[2, 1], [1, 0], [3, 1]])], [('LL', [[0, 1], [1, 2], [1, 3]]), ('LLR', [[1, 3], [1, 2], [2, 5]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLRL', [[1, 3], [2, 5], [3, 8]]), ('LLRR', [[2, 5], [1, 2], [3, 7]]), ('LRLL', [[1, 2], [3, 5], [4, 7]]), ('LRLR', [[3, 5], [2, 3], [5, 8]])], [('LR', [[1, 2], [1, 1], [2, 3]]), ('RRL', [[2, 1], [3, 1], [5, 2]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLLRR', [[2, 7], [1, 3], [3, 10]]), ('LLRLL', [[1, 3], [3, 8], [4, 11]]), ('LLRLR', [[3, 8], [2, 5], [5, 13]]), ('LLRRL', [[2, 5], [3, 7], [5, 12]])], [('RL', [[1, 1], [2, 1], [3, 2]]), ('LLRL', [[1, 3], [2, 5], [3, 8]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('RLRLL', [[3, 2], [8, 5], [11, 7]]), ('RLRLR', [[8, 5], [5, 3], [13, 8]]), ('RLRRL', [[5, 3], [7, 4], [12, 7]]), ('RLRRR', [[7, 4], [2, 1], [9, 5]])], [('LLL', [[0, 1], [1, 3], [1, 4]]), ('LRLR', [[3, 5], [2, 3], [5, 8]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLLRLR', [[3, 11], [2, 7], [5, 18]]), ('LLLRRL', [[2, 7], [3, 10], [5, 17]]), ('LLLRRR', [[3, 10], [1, 3], [4, 13]]), ('LLRLLL', [[1, 3], [4, 11], [5, 14]])]]
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, 1], [1, 0], [2, 1]]`	`[[0, 1], [1, 1], [1, 2]]`	Failed
explicit oracle 1	`[[2, 1], [1, 0], [3, 1]]`	`[[0, 1], [1, 2], [1, 3]]`	Failed
explicit oracle 2	`[[0, 1], [1, 0], [1, 1]]`	`[[0, 1], [1, 0], [1, 1]]`	Passed
explicit oracle 3	`[[6, 1], [1, 0], [7, 1]]`	`[[6, 1], [1, 0], [7, 1]]`	Passed
explicit oracle 4	`[[1, 1], [1, 0], [2, 1]]`	`[[1, 1], [1, 0], [2, 1]]`	Passed
explicit oracle 5	`[[2, 1], [1, 0], [3, 1]]`	`[[1, 2], [1, 1], [2, 3]]`	Failed
explicit oracle 6	`[[2, 1], [1, 0], [3, 1]]`	`[[1, 1], [2, 1], [3, 2]]`	Failed
explicit oracle 7	`[[2, 1], [1, 0], [3, 1]]`	`[[2, 1], [1, 0], [3, 1]]`	Passed

SHA-256 / e5dabbbb7622be7eff06015e2a6afaceee011b5a4be73ca2bcdf8f8fb6185020

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):
    lo=[0,1]
    hi=[1,0]
    for step in x:
     mid=[lo[0]+hi[0],lo[1]+hi[1]]
     if step=='L': hi=[mid[1],mid[0]]
     else: lo=mid
    return [lo,hi,[lo[0]+hi[0],lo[1]+hi[1]]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('L', [[0, 1], [1, 1], [1, 2]]), ('LL', [[0, 1], [1, 2], [1, 3]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('R', [[1, 1], [1, 0], [2, 1]]), ('LR', [[1, 2], [1, 1], [2, 3]]), ('RL', [[1, 1], [2, 1], [3, 2]]), ('RR', [[2, 1], [1, 0], [3, 1]])], [('LL', [[0, 1], [1, 2], [1, 3]]), ('LLR', [[1, 3], [1, 2], [2, 5]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLRL', [[1, 3], [2, 5], [3, 8]]), ('LLRR', [[2, 5], [1, 2], [3, 7]]), ('LRLL', [[1, 2], [3, 5], [4, 7]]), ('LRLR', [[3, 5], [2, 3], [5, 8]])], [('LR', [[1, 2], [1, 1], [2, 3]]), ('RRL', [[2, 1], [3, 1], [5, 2]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLLRR', [[2, 7], [1, 3], [3, 10]]), ('LLRLL', [[1, 3], [3, 8], [4, 11]]), ('LLRLR', [[3, 8], [2, 5], [5, 13]]), ('LLRRL', [[2, 5], [3, 7], [5, 12]])], [('RL', [[1, 1], [2, 1], [3, 2]]), ('LLRL', [[1, 3], [2, 5], [3, 8]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('RLRLL', [[3, 2], [8, 5], [11, 7]]), ('RLRLR', [[8, 5], [5, 3], [13, 8]]), ('RLRRL', [[5, 3], [7, 4], [12, 7]]), ('RLRRR', [[7, 4], [2, 1], [9, 5]])], [('LLL', [[0, 1], [1, 3], [1, 4]]), ('LRLR', [[3, 5], [2, 3], [5, 8]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLLRLR', [[3, 11], [2, 7], [5, 18]]), ('LLLRRL', [[2, 7], [3, 10], [5, 17]]), ('LLLRRR', [[3, 10], [1, 3], [4, 13]]), ('LLRLLL', [[1, 3], [4, 11], [5, 14]])]]
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, 1], [1, 1], [1, 2]]`	`[[0, 1], [1, 1], [1, 2]]`	Passed
explicit oracle 1	`[[0, 1], [2, 1], [2, 2]]`	`[[0, 1], [1, 2], [1, 3]]`	Failed
explicit oracle 2	`[[0, 1], [1, 0], [1, 1]]`	`[[0, 1], [1, 0], [1, 1]]`	Passed
explicit oracle 3	`[[6, 1], [1, 0], [7, 1]]`	`[[6, 1], [1, 0], [7, 1]]`	Passed
explicit oracle 4	`[[1, 1], [1, 0], [2, 1]]`	`[[1, 1], [1, 0], [2, 1]]`	Passed
explicit oracle 5	`[[1, 2], [1, 1], [2, 3]]`	`[[1, 2], [1, 1], [2, 3]]`	Passed
explicit oracle 6	`[[1, 1], [1, 2], [2, 3]]`	`[[1, 1], [2, 1], [3, 2]]`	Failed
explicit oracle 7	`[[2, 1], [1, 0], [3, 1]]`	`[[2, 1], [1, 0], [3, 1]]`	Passed

SHA-256 / f46c1583d622a5cb9b9d6df6ca39f7c862270f57f61e3097024b2a0493611c22

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):
    lo=[0,1]
    hi=[1,0]
    for step in x:
     mid=[lo[0]+hi[0],lo[1]+hi[1]]
     if step=='L': hi=mid
     else: lo=mid
    return [lo,hi,[lo[0]+hi[0],lo[1]+hi[1]]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('L', [[0, 1], [1, 1], [1, 2]]), ('LL', [[0, 1], [1, 2], [1, 3]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('R', [[1, 1], [1, 0], [2, 1]]), ('LR', [[1, 2], [1, 1], [2, 3]]), ('RL', [[1, 1], [2, 1], [3, 2]]), ('RR', [[2, 1], [1, 0], [3, 1]])], [('LL', [[0, 1], [1, 2], [1, 3]]), ('LLR', [[1, 3], [1, 2], [2, 5]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLRL', [[1, 3], [2, 5], [3, 8]]), ('LLRR', [[2, 5], [1, 2], [3, 7]]), ('LRLL', [[1, 2], [3, 5], [4, 7]]), ('LRLR', [[3, 5], [2, 3], [5, 8]])], [('LR', [[1, 2], [1, 1], [2, 3]]), ('RRL', [[2, 1], [3, 1], [5, 2]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLLRR', [[2, 7], [1, 3], [3, 10]]), ('LLRLL', [[1, 3], [3, 8], [4, 11]]), ('LLRLR', [[3, 8], [2, 5], [5, 13]]), ('LLRRL', [[2, 5], [3, 7], [5, 12]])], [('RL', [[1, 1], [2, 1], [3, 2]]), ('LLRL', [[1, 3], [2, 5], [3, 8]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('RLRLL', [[3, 2], [8, 5], [11, 7]]), ('RLRLR', [[8, 5], [5, 3], [13, 8]]), ('RLRRL', [[5, 3], [7, 4], [12, 7]]), ('RLRRR', [[7, 4], [2, 1], [9, 5]])], [('LLL', [[0, 1], [1, 3], [1, 4]]), ('LRLR', [[3, 5], [2, 3], [5, 8]]), ('', [[0, 1], [1, 0], [1, 1]]), ('RRRRRR', [[6, 1], [1, 0], [7, 1]]), ('LLLRLR', [[3, 11], [2, 7], [5, 18]]), ('LLLRRL', [[2, 7], [3, 10], [5, 17]]), ('LLLRRR', [[3, 10], [1, 3], [4, 13]]), ('LLRLLL', [[1, 3], [4, 11], [5, 14]])]]
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, 1], [1, 1], [1, 2]]`	`[[0, 1], [1, 1], [1, 2]]`	Passed
explicit oracle 1	`[[0, 1], [1, 2], [1, 3]]`	`[[0, 1], [1, 2], [1, 3]]`	Passed
explicit oracle 2	`[[0, 1], [1, 0], [1, 1]]`	`[[0, 1], [1, 0], [1, 1]]`	Passed
explicit oracle 3	`[[6, 1], [1, 0], [7, 1]]`	`[[6, 1], [1, 0], [7, 1]]`	Passed
explicit oracle 4	`[[1, 1], [1, 0], [2, 1]]`	`[[1, 1], [1, 0], [2, 1]]`	Passed
explicit oracle 5	`[[1, 2], [1, 1], [2, 3]]`	`[[1, 2], [1, 1], [2, 3]]`	Passed
explicit oracle 6	`[[1, 1], [2, 1], [3, 2]]`	`[[1, 1], [2, 1], [3, 2]]`	Passed
explicit oracle 7	`[[2, 1], [1, 0], [3, 1]]`	`[[2, 1], [1, 0], [3, 1]]`	Passed

SHA-256 / 0909c461d2efa2738c5a5c4148eaca9bed70b1e46641151d7db3adf08dff1af2

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 2026-09-29T14:39:18.397704+00:00.

Case digest / 49053788d56a4caf0e907c2919dde5cd35f1d4f685a7988d58004b542ccc6505