FA-14526 / Numerics / Open access

Stern brocot path: left residual · case 01

The exact stern brocot path result violates the stated contract at left residual.

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

ROOT CAUSE

The left residual step uses q%p instead of q-p.

VERIFIED REPAIR

Use q-p at the left residual step.

Unsuccessful approach: The partial repair q-p-1 still violates the left residual invariant.

Case contract

Input reduced positive [p,q]; return run-expanded L/R path from 1/1. Bounds: 1<=p,q<=24.

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(100):
     if p==q: break
     if p<q:
      out.append('L')
      q=q%p
     else:
      out.append('R')
      p=p-q
    return ''.join(out)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2], 'L'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([1, 3], 'LL'), ([1, 4], 'LLL'), ([1, 5], 'LLLL'), ([1, 6], 'LLLLL'), ([1, 7], 'LLLLLL')], [([1, 3], 'LL'), ([1, 5], 'LLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([1, 18], 'LLLLLLLLLLLLLLLLL'), ([1, 19], 'LLLLLLLLLLLLLLLLLL'), ([1, 20], 'LLLLLLLLLLLLLLLLLLL'), ([1, 21], 'LLLLLLLLLLLLLLLLLLLL')], [([1, 4], 'LLL'), ([1, 8], 'LLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([2, 21], 'LLLLLLLLLLR'), ([2, 23], 'LLLLLLLLLLLR'), ([3, 1], 'RR'), ([3, 2], 'RL')], [([1, 5], 'LLLL'), ([1, 11], 'LLLLLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([3, 23], 'LLLLLLLRL'), ([4, 1], 'RRR'), ([4, 3], 'RLL'), ([4, 5], 'LRRR')], [([1, 6], 'LLLLL'), ([1, 14], 'LLLLLLLLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([5, 6], 'LRRRR'), ([5, 7], 'LRRL'), ([5, 8], 'LRLR'), ([5, 9], 'LRLLL')]]
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	`LRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`L`	Failed
explicit oracle 1			Passed
explicit oracle 2	`RLRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`RLLLLLLLLLLLLLLLLLLLLLL`	Failed
explicit oracle 3	`LRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`LL`	Failed
explicit oracle 4	`LRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`LLL`	Failed
explicit oracle 5	`LRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`LLLL`	Failed
explicit oracle 6	`LRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`LLLLL`	Failed
explicit oracle 7	`LRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`LLLLLL`	Failed

SHA-256 / 07b5732c7dfbeea114a62179b3835faaa3bcf282175f24d23b3b8d5f8944fe9f

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(100):
     if p==q: break
     if p<q:
      out.append('L')
      q=q-p-1
     else:
      out.append('R')
      p=p-q
    return ''.join(out)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2], 'L'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([1, 3], 'LL'), ([1, 4], 'LLL'), ([1, 5], 'LLLL'), ([1, 6], 'LLLLL'), ([1, 7], 'LLLLLL')], [([1, 3], 'LL'), ([1, 5], 'LLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([1, 18], 'LLLLLLLLLLLLLLLLL'), ([1, 19], 'LLLLLLLLLLLLLLLLLL'), ([1, 20], 'LLLLLLLLLLLLLLLLLLL'), ([1, 21], 'LLLLLLLLLLLLLLLLLLLL')], [([1, 4], 'LLL'), ([1, 8], 'LLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([2, 21], 'LLLLLLLLLLR'), ([2, 23], 'LLLLLLLLLLLR'), ([3, 1], 'RR'), ([3, 2], 'RL')], [([1, 5], 'LLLL'), ([1, 11], 'LLLLLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([3, 23], 'LLLLLLLRL'), ([4, 1], 'RRR'), ([4, 3], 'RLL'), ([4, 5], 'LRRR')], [([1, 6], 'LLLLL'), ([1, 14], 'LLLLLLLLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([5, 6], 'LRRRR'), ([5, 7], 'LRRL'), ([5, 8], 'LRLR'), ([5, 9], 'LRLLL')]]
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	`LRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`L`	Failed
explicit oracle 1			Passed
explicit oracle 2	`RLLLLLLLLLLL`	`RLLLLLLLLLLLLLLLLLLLLLL`	Failed
explicit oracle 3	`L`	`LL`	Failed
explicit oracle 4	`LLRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`LLL`	Failed
explicit oracle 5	`LL`	`LLLL`	Failed
explicit oracle 6	`LLLRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR`	`LLLLL`	Failed
explicit oracle 7	`LLL`	`LLLLLL`	Failed

SHA-256 / bd8af4ecf61b9a948de16cc71117803d9e5da27cf8c803668b3be68546b798ea

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(100):
     if p==q: break
     if p<q:
      out.append('L')
      q=q-p
     else:
      out.append('R')
      p=p-q
    return ''.join(out)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2], 'L'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([1, 3], 'LL'), ([1, 4], 'LLL'), ([1, 5], 'LLLL'), ([1, 6], 'LLLLL'), ([1, 7], 'LLLLLL')], [([1, 3], 'LL'), ([1, 5], 'LLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([1, 18], 'LLLLLLLLLLLLLLLLL'), ([1, 19], 'LLLLLLLLLLLLLLLLLL'), ([1, 20], 'LLLLLLLLLLLLLLLLLLL'), ([1, 21], 'LLLLLLLLLLLLLLLLLLLL')], [([1, 4], 'LLL'), ([1, 8], 'LLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([2, 21], 'LLLLLLLLLLR'), ([2, 23], 'LLLLLLLLLLLR'), ([3, 1], 'RR'), ([3, 2], 'RL')], [([1, 5], 'LLLL'), ([1, 11], 'LLLLLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([3, 23], 'LLLLLLLRL'), ([4, 1], 'RRR'), ([4, 3], 'RLL'), ([4, 5], 'LRRR')], [([1, 6], 'LLLLL'), ([1, 14], 'LLLLLLLLLLLLL'), ([1, 1], ''), ([24, 23], 'RLLLLLLLLLLLLLLLLLLLLLL'), ([5, 6], 'LRRRR'), ([5, 7], 'LRRL'), ([5, 8], 'LRLR'), ([5, 9], 'LRLLL')]]
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	`L`	`L`	Passed
explicit oracle 1			Passed
explicit oracle 2	`RLLLLLLLLLLLLLLLLLLLLLL`	`RLLLLLLLLLLLLLLLLLLLLLL`	Passed
explicit oracle 3	`LL`	`LL`	Passed
explicit oracle 4	`LLL`	`LLL`	Passed
explicit oracle 5	`LLLL`	`LLLL`	Passed
explicit oracle 6	`LLLLL`	`LLLLL`	Passed
explicit oracle 7	`LLLLLL`	`LLLLLL`	Passed

SHA-256 / 046af283613ae17e6dee39cec8bd528a022cb4eb3129b56c3b7720e18f3353e5

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 2026-09-29T14:39:17.815557+00:00.

Case digest / fa44c770da4ca108e3fffef05e642c47bb43186ed384195126a56cdc60928545