FA-6456 / Discrete calculus / Open access
Newton divided difference two · case 01
Uniform-grid second differences are applied to nonuniform nodes.
ROOT CAUSE
Uniform-grid second differences are applied to nonuniform nodes.
VERIFIED REPAIR
Apply the specified mathematical contract directly, preserving all terms and boundary cases: return str((Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))/Fraction(x2-x0))
Unsuccessful approach: The difference of first slopes is not divided by the full node span.
Case contract
Integer sample values and compatible sample-array lengths. Rational results are reduced Fraction strings; spacing is positive unless explicitly stated otherwise. x0,x1,x2 are pairwise distinct integer nodes with integer observations; return the second divided difference. Exact operational definition: str((Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))/Fraction(x2-x0))
Why this case matters
Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Discrete calculus 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(x0, y0, x1, y1, x2, y2):
return str(Fraction(y2-2*y1+y0,x2-x0))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (0, 0, 1, 1, 3, 9)', solve(*(0, 0, 1, 1, 3, 9)), '1')
check('fixture 2: (0, 1, 2, 5, 4, 17)', solve(*(0, 1, 2, 5, 4, 17)), '1')
check('fixture 3: (0, 3, 1, 5, 2, 7)', solve(*(0, 3, 1, 5, 2, 7)), '0')
check('fixture 4: (1, 1, 2, 4, 4, 16)', solve(*(1, 1, 2, 4, 4, 16)), '1')
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: (0, 0, 1, 1, 3, 9) | 7/3 | 1 | Failed |
| fixture 2: (0, 1, 2, 5, 4, 17) | 2 | 1 | Failed |
| fixture 3: (0, 3, 1, 5, 2, 7) | 0 | 0 | Passed |
| fixture 4: (1, 1, 2, 4, 4, 16) | 3 | 1 | Failed |
SHA-256 / 157714a418001485499e8a52fa975a3c4970d249b8a1ddd4ed9357a13ae081a5
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(x0, y0, x1, y1, x2, y2):
return str(Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (0, 0, 1, 1, 3, 9)', solve(*(0, 0, 1, 1, 3, 9)), '1')
check('fixture 2: (0, 1, 2, 5, 4, 17)', solve(*(0, 1, 2, 5, 4, 17)), '1')
check('fixture 3: (0, 3, 1, 5, 2, 7)', solve(*(0, 3, 1, 5, 2, 7)), '0')
check('fixture 4: (1, 1, 2, 4, 4, 16)', solve(*(1, 1, 2, 4, 4, 16)), '1')
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: (0, 0, 1, 1, 3, 9) | 3 | 1 | Failed |
| fixture 2: (0, 1, 2, 5, 4, 17) | 4 | 1 | Failed |
| fixture 3: (0, 3, 1, 5, 2, 7) | 0 | 0 | Passed |
| fixture 4: (1, 1, 2, 4, 4, 16) | 3 | 1 | Failed |
SHA-256 / eb60c83f240bd1bc025591de8f76913a063f81806a7cb39f0684985c877ac1cf
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(x0, y0, x1, y1, x2, y2):
return str((Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))/Fraction(x2-x0))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (0, 0, 1, 1, 3, 9)', solve(*(0, 0, 1, 1, 3, 9)), '1')
check('fixture 2: (0, 1, 2, 5, 4, 17)', solve(*(0, 1, 2, 5, 4, 17)), '1')
check('fixture 3: (0, 3, 1, 5, 2, 7)', solve(*(0, 3, 1, 5, 2, 7)), '0')
check('fixture 4: (1, 1, 2, 4, 4, 16)', solve(*(1, 1, 2, 4, 4, 16)), '1')
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: (0, 0, 1, 1, 3, 9) | 1 | 1 | Passed |
| fixture 2: (0, 1, 2, 5, 4, 17) | 1 | 1 | Passed |
| fixture 3: (0, 3, 1, 5, 2, 7) | 0 | 0 | Passed |
| fixture 4: (1, 1, 2, 4, 4, 16) | 1 | 1 | Passed |
SHA-256 / 8629132a3ab9cadca6d3b1d08cbb583fbd061cb10f15c29ad40724469c87b65c
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 2026-09-29T14:38:02.205490+00:00.
Case digest / 35c138b2bcb97b2e91ca9f43227a451e1f472165ce9ccbe3dcbcc039d4e7e53d