FA-6391 / Discrete calculus / Open access

Central second derivative samples · case 01

A second derivative is divided by one step instead of squared step.

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

ROOT CAUSE

A second derivative is divided by one step instead of squared step.

VERIFIED REPAIR

Apply the specified mathematical contract directly, preserving all terms and boundary cases: return str(Fraction(left-2*center+right,h*h))

Unsuccessful approach: The central sample coefficient is one instead of two.

Case contract

Integer sample values and compatible sample-array lengths. Rational results are reduced Fraction strings; spacing is positive unless explicitly stated otherwise. Central second derivative samples. Exact operational definition: str(Fraction(left-2*center+right,h*h))

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(left, center, right, h):
    return str(Fraction(left-2*center+right,h))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (0, 4, 16, 2)', solve(*(0, 4, 16, 2)), '2')
check('fixture 2: (3, 3, 3, 1)', solve(*(3, 3, 3, 1)), '0')
check('fixture 3: (1, 0, 1, 1)', solve(*(1, 0, 1, 1)), '2')
check('fixture 4: (0, 1, 0, 2)', solve(*(0, 1, 0, 2)), '-1/2')
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, 4, 16, 2)	`4`	`2`	Failed
fixture 2: (3, 3, 3, 1)	`0`	`0`	Passed
fixture 3: (1, 0, 1, 1)	`2`	`2`	Passed
fixture 4: (0, 1, 0, 2)	`-1`	`-1/2`	Failed

SHA-256 / e72a886cd8493285fca8cdc49ce579703a79d2145644adfc9ba7259f383efd2b

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(left, center, right, h):
    return str(Fraction(left-center+right,h*h))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (0, 4, 16, 2)', solve(*(0, 4, 16, 2)), '2')
check('fixture 2: (3, 3, 3, 1)', solve(*(3, 3, 3, 1)), '0')
check('fixture 3: (1, 0, 1, 1)', solve(*(1, 0, 1, 1)), '2')
check('fixture 4: (0, 1, 0, 2)', solve(*(0, 1, 0, 2)), '-1/2')
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, 4, 16, 2)	`3`	`2`	Failed
fixture 2: (3, 3, 3, 1)	`3`	`0`	Failed
fixture 3: (1, 0, 1, 1)	`2`	`2`	Passed
fixture 4: (0, 1, 0, 2)	`-1/4`	`-1/2`	Failed

SHA-256 / 993dd4ec6811c6781948af4b3eff587024807936ff780b446a180edbfb7dd467

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(left, center, right, h):
    return str(Fraction(left-2*center+right,h*h))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (0, 4, 16, 2)', solve(*(0, 4, 16, 2)), '2')
check('fixture 2: (3, 3, 3, 1)', solve(*(3, 3, 3, 1)), '0')
check('fixture 3: (1, 0, 1, 1)', solve(*(1, 0, 1, 1)), '2')
check('fixture 4: (0, 1, 0, 2)', solve(*(0, 1, 0, 2)), '-1/2')
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, 4, 16, 2)	`2`	`2`	Passed
fixture 2: (3, 3, 3, 1)	`0`	`0`	Passed
fixture 3: (1, 0, 1, 1)	`2`	`2`	Passed
fixture 4: (0, 1, 0, 2)	`-1/2`	`-1/2`	Passed

SHA-256 / 11e9d342a492e31f4bc6512ccbf4dd914cddcbea4be5f350f4a5fc1d2089d96c

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 2026-09-29T14:38:01.394936+00:00.

Case digest / 07d3ee1b294e5f0093df5be3358d98f73838fc98e7ea6b9212e63f3cd96cd738