FA-6411 / Discrete calculus / Open access

Quadratic sequence extrapolation · case 01

Linear extrapolation omits second differences.

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

ROOT CAUSE

Linear extrapolation omits second differences.

THE FAILURE

Linear extrapolation omits second differences.

Unsuccessful approach: The middle coefficient in the quadratic stencil is too small.

Case contract

Integer sample values and compatible sample-array lengths. Rational results are reduced Fraction strings; spacing is positive unless explicitly stated otherwise. At least three values are provided; extrapolate using the final three equally spaced observations. Exact operational definition: 3*values[-1]-3*values[-2]+values[-3]

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(values):
    return 2*values[-1]-values[-2]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([1, 4, 9],)', solve(*([1, 4, 9],)), 16)
check('fixture 2: ([0, 1, 4],)', solve(*([0, 1, 4],)), 9)
check('fixture 3: ([3, 3, 3],)', solve(*([3, 3, 3],)), 3)
check('fixture 4: ([1, 2, 3],)', solve(*([1, 2, 3],)), 4)
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: ([1, 4, 9],)	`14`	`16`	Failed
fixture 2: ([0, 1, 4],)	`7`	`9`	Failed
fixture 3: ([3, 3, 3],)	`3`	`3`	Passed
fixture 4: ([1, 2, 3],)	`4`	`4`	Passed

SHA-256 / a83966f8638fd1bd7995a0f1dda575b7ac91306459f98f6f5c81e129a8b5771c

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(values):
    return 3*values[-1]-values[-2]+values[-3]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: ([1, 4, 9],)', solve(*([1, 4, 9],)), 16)
check('fixture 2: ([0, 1, 4],)', solve(*([0, 1, 4],)), 9)
check('fixture 3: ([3, 3, 3],)', solve(*([3, 3, 3],)), 3)
check('fixture 4: ([1, 2, 3],)', solve(*([1, 2, 3],)), 4)
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: ([1, 4, 9],)	`24`	`16`	Failed
fixture 2: ([0, 1, 4],)	`11`	`9`	Failed
fixture 3: ([3, 3, 3],)	`9`	`3`	Failed
fixture 4: ([1, 2, 3],)	`8`	`4`	Failed

SHA-256 / 8b476c972581be61d0cf016b7aa67dd2009ff8347d9a81b65ae4a72b157c9be1

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 4 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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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 2026-09-29T14:38:01.542877+00:00.

Case digest / 312fc1fb93c444122e249313b3d93039a1346bace10cd479755da25c4320af86