FA-6111 / Polynomial arithmetic / Open access

Formal polynomial integral zero constant · case 01

Antiderivative coefficients are shifted without degree division.

Verified by executionVariant 1 · 4 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Antiderivative coefficients are shifted without degree division.

VERIFIED REPAIR

Apply the specified mathematical contract directly, preserving all terms and boundary cases: return ["0"]+[str(Fraction(c,i+1)) for i,c in enumerate(coeff)]

Unsuccessful approach: The zero integration constant is omitted from ascending order.

Case contract

Integer coefficients in ascending power order and integer evaluation bounds; missing high-degree coefficients are zero. Rational outputs are reduced Fraction strings; the zero-polynomial degree is -1. Formal polynomial integral zero constant. Exact operational definition: ["0"]+[str(Fraction(c,i+1)) for i,c in enumerate(coeff)]

Why this case matters

Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Polynomial arithmetic results depend on the stated convention.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(coeff):
    return ["0"]+[str(c) for c in coeff]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([2, 3],)', solve(*([2, 3],)), ['0', '2', '3/2'])
check('fixture 2: ([0, 0, 3],)', solve(*([0, 0, 3],)), ['0', '0', '0', '1'])
check('fixture 3: ([],)', solve(*([],)), ['0'])
check('fixture 4: ([5],)', solve(*([5],)), ['0', '5'])
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
fixture 1: ([2, 3],)	`['0', '2', '3']`	`['0', '2', '3/2']`	Failed
fixture 2: ([0, 0, 3],)	`['0', '0', '0', '3']`	`['0', '0', '0', '1']`	Failed
fixture 3: ([],)	`['0']`	`['0']`	Passed
fixture 4: ([5],)	`['0', '5']`	`['0', '5']`	Passed

SHA-256 / 6c0cc5b779f3fbe77aa9cf3be453122aea86326c104458b8d745268a5b07ed3e

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(coeff):
    return [str(Fraction(c,i+1)) for i,c in enumerate(coeff)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([2, 3],)', solve(*([2, 3],)), ['0', '2', '3/2'])
check('fixture 2: ([0, 0, 3],)', solve(*([0, 0, 3],)), ['0', '0', '0', '1'])
check('fixture 3: ([],)', solve(*([],)), ['0'])
check('fixture 4: ([5],)', solve(*([5],)), ['0', '5'])
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
fixture 1: ([2, 3],)	`['2', '3/2']`	`['0', '2', '3/2']`	Failed
fixture 2: ([0, 0, 3],)	`['0', '0', '1']`	`['0', '0', '0', '1']`	Failed
fixture 3: ([],)	`[]`	`['0']`	Failed
fixture 4: ([5],)	`['5']`	`['0', '5']`	Failed

SHA-256 / e306ace5c4c5534db2d417ad121c6f06c28e943847dc853787b6900bd6375814

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(coeff):
    return ["0"]+[str(Fraction(c,i+1)) for i,c in enumerate(coeff)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([2, 3],)', solve(*([2, 3],)), ['0', '2', '3/2'])
check('fixture 2: ([0, 0, 3],)', solve(*([0, 0, 3],)), ['0', '0', '0', '1'])
check('fixture 3: ([],)', solve(*([],)), ['0'])
check('fixture 4: ([5],)', solve(*([5],)), ['0', '5'])
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
fixture 1: ([2, 3],)	`['0', '2', '3/2']`	`['0', '2', '3/2']`	Passed
fixture 2: ([0, 0, 3],)	`['0', '0', '0', '1']`	`['0', '0', '0', '1']`	Passed
fixture 3: ([],)	`['0']`	`['0']`	Passed
fixture 4: ([5],)	`['0', '5']`	`['0', '5']`	Passed

SHA-256 / 493834cb51b3344ca70dd5b4a0e16764650714e4b00732e4aaea358d36b49c26

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 2026-09-29T14:37:57.757694+00:00.

Case digest / 97cb1f375c5e9839a0bc77453226c16cb8c9946a5c3cab0a362dbc49fe579311