FA-341 / Integer arithmetic / Open access
A positive-only tie rule rounds negative halves toward zero · case 01
Exact midpoint values receive even rounding or asymmetric rounding instead of the required away-from-zero result.
ROOT CAUSE
A default tie convention or the positive-only floor(x+0.5) identity is used for signed rational values.
VERIFIED REPAIR
Compare twice the absolute remainder with the denominator, then restore the original sign after rounding the magnitude.
Unsuccessful approach: Adding one half before floor fixes positive ties but turns a negative tie toward zero.
Case contract
Round numerator/denominator to the nearest integer, with exact ties away from zero. Inputs are integers and denominator must be positive; invalid denominators return None. No decimal parsing or floating-point conversion occurs.
Why this case matters
Quantizers and fixed-point protocols need an explicit signed tie rule. The failure here is a rounding-policy mismatch, separate from binary representation error when parsing decimals.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
import math
N = 1
observations = []
def solve(numerator, denominator):
return round(Fraction(numerator, denominator)) if denominator > 0 else None
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive half after even integer', solve(4 * N + 1, 2), 2 * N + 1)
check('negative half after even magnitude', solve(-4 * N - 1, 2), -2 * N - 1)
check('just below midpoint', solve(200 * N + 49, 100), 2 * N)
check('just above midpoint', solve(200 * N + 51, 100), 2 * N + 1)
check('negative value below midpoint magnitude', solve(-200 * N - 49, 100), -2 * N)
check('exact negative integer', solve(-N * 7, 7), -N)
check('zero', solve(0, N), 0)
check('zero denominator rejected', solve(N, 0), None)
check('negative denominator rejected', solve(N, -1), None)
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 |
|---|---|---|---|
| positive half after even integer | 2 | 3 | Failed |
| negative half after even magnitude | -2 | -3 | Failed |
| just below midpoint | 2 | 2 | Passed |
| just above midpoint | 3 | 3 | Passed |
| negative value below midpoint magnitude | -2 | -2 | Passed |
| exact negative integer | -1 | -1 | Passed |
| zero | 0 | 0 | Passed |
| zero denominator rejected | None | None | Passed |
| negative denominator rejected | None | None | Passed |
SHA-256 / 9035bd73f15b979682fe58b316db3ded346faf095f3d148eff1adc555d607277
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
import math
N = 1
observations = []
def solve(numerator, denominator):
return math.floor(Fraction(numerator, denominator) + Fraction(1, 2)) if denominator > 0 else None
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive half after even integer', solve(4 * N + 1, 2), 2 * N + 1)
check('negative half after even magnitude', solve(-4 * N - 1, 2), -2 * N - 1)
check('just below midpoint', solve(200 * N + 49, 100), 2 * N)
check('just above midpoint', solve(200 * N + 51, 100), 2 * N + 1)
check('negative value below midpoint magnitude', solve(-200 * N - 49, 100), -2 * N)
check('exact negative integer', solve(-N * 7, 7), -N)
check('zero', solve(0, N), 0)
check('zero denominator rejected', solve(N, 0), None)
check('negative denominator rejected', solve(N, -1), None)
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 |
|---|---|---|---|
| positive half after even integer | 3 | 3 | Passed |
| negative half after even magnitude | -2 | -3 | Failed |
| just below midpoint | 2 | 2 | Passed |
| just above midpoint | 3 | 3 | Passed |
| negative value below midpoint magnitude | -2 | -2 | Passed |
| exact negative integer | -1 | -1 | Passed |
| zero | 0 | 0 | Passed |
| zero denominator rejected | None | None | Passed |
| negative denominator rejected | None | None | Passed |
SHA-256 / 38dc0407a0c46da9affd3ab399283fc5da2cfefe8761609ad30a7f856f1f816a
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
import math
N = 1
observations = []
def solve(numerator, denominator):
if denominator <= 0:
return None
whole, remainder = divmod(abs(numerator), denominator)
magnitude = whole + (2 * remainder >= denominator)
return -magnitude if numerator < 0 else magnitude
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive half after even integer', solve(4 * N + 1, 2), 2 * N + 1)
check('negative half after even magnitude', solve(-4 * N - 1, 2), -2 * N - 1)
check('just below midpoint', solve(200 * N + 49, 100), 2 * N)
check('just above midpoint', solve(200 * N + 51, 100), 2 * N + 1)
check('negative value below midpoint magnitude', solve(-200 * N - 49, 100), -2 * N)
check('exact negative integer', solve(-N * 7, 7), -N)
check('zero', solve(0, N), 0)
check('zero denominator rejected', solve(N, 0), None)
check('negative denominator rejected', solve(N, -1), None)
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 |
|---|---|---|---|
| positive half after even integer | 3 | 3 | Passed |
| negative half after even magnitude | -3 | -3 | Passed |
| just below midpoint | 2 | 2 | Passed |
| just above midpoint | 3 | 3 | Passed |
| negative value below midpoint magnitude | -2 | -2 | Passed |
| exact negative integer | -1 | -1 | Passed |
| zero | 0 | 0 | Passed |
| zero denominator rejected | None | None | Passed |
| negative denominator rejected | None | None | Passed |
SHA-256 / 4953ff45b056e8da09c2ee3d267e5ee697916f7aa6530504339d804c2858f8c1
Verification & scope
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 2026-09-29T14:36:52.120851+00:00.
Case digest / 5ea61b6fc79dcbe340d5720d98a94fdefbf0901f30bacdf1c80de4faeaaef738