FA-14521 / Numerics / Open access

Stern brocot path: left edge label · case 01

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

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

ROOT CAUSE

The left edge label step uses 'R' instead of 'L'.

VERIFIED REPAIR

Use 'L' at the left edge label step.

Unsuccessful approach: The partial repair '' still violates the left edge label 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('R')
      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	`R`	`L`	Failed
explicit oracle 1			Passed
explicit oracle 2	`RRRRRRRRRRRRRRRRRRRRRRR`	`RLLLLLLLLLLLLLLLLLLLLLL`	Failed
explicit oracle 3	`RR`	`LL`	Failed
explicit oracle 4	`RRR`	`LLL`	Failed
explicit oracle 5	`RRRR`	`LLLL`	Failed
explicit oracle 6	`RRRRR`	`LLLLL`	Failed
explicit oracle 7	`RRRRRR`	`LLLLLL`	Failed

SHA-256 / 9cce2cd0004f228a4ac73ef9b2f35ab2ec6a63a1d3c88dd8ff487b8c62917f85

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('')
      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`	Failed
explicit oracle 1			Passed
explicit oracle 2	`R`	`RLLLLLLLLLLLLLLLLLLLLLL`	Failed
explicit oracle 3		`LL`	Failed
explicit oracle 4		`LLL`	Failed
explicit oracle 5		`LLLL`	Failed
explicit oracle 6		`LLLLL`	Failed
explicit oracle 7		`LLLLLL`	Failed

SHA-256 / b59e925106e592e3b54dbb0acf70026a2ab2beb5fbe918bacc1c3ae70b1d62a7

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 2026-09-29T14:39:17.778544+00:00.

Case digest / d63315755662c26b99a33fa6bdfa978c77356d53cb7fa094a7988a7155977c6b