FAILURE MAP
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FA-14721 / Numerics / Open access

Forward difference newton basis: left edge extraction · case 01

The exact forward difference newton basis result violates the stated contract at left edge extraction.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The left edge extraction step uses row[-1] instead of row[0].

VERIFIED REPAIR

Use row[0] at the left edge extraction step.

Unsuccessful approach: The partial repair row[len(row)//2] still violates the left edge extraction invariant.

Case contract

Input nonempty integer sequence f(0)..f(n); return binomial-basis coefficients Delta^k f(0).

Why this case matters

Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    row=x[:];out=[]
    for _ in range(len(x)):
     if not row:break
     out.append(row[-1])
     row=[row[i+1]-row[i] for i in range(len(row)-1)]
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1, 0], [-1, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([-1, -1], [-1, 0]), ([-1, 1], [-1, 2]), ([0, -1], [0, -1])], [([-1, 1], [-1, 2]), ([0, 1], [0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 1], [-1, 1, 0]), ([-1, 1, -1], [-1, 2, -4]), ([-1, 1, 0], [-1, 2, -3]), ([-1, 1, 1], [-1, 2, -2])], [([0, -1], [0, -1]), ([-1, -1, 0], [-1, 0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([1, 0, 0], [1, -1, 1]), ([1, 0, 1], [1, -1, 2]), ([1, 1, -1], [1, 0, -2]), ([1, 1, 0], [1, 0, -1])], [([0, 1], [0, 1]), ([-1, 0, 0], [-1, 1, -1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 0, -1], [-1, 1, -1, 0]), ([-1, 0, 0, 0], [-1, 1, -1, 1]), ([-1, 0, 0, 1], [-1, 1, -1, 2]), ([-1, 0, 1, -1], [-1, 1, 0, -3])], [([1, -1], [1, -2]), ([-1, 1, 0], [-1, 2, -3]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0, -1, -1, 1], [0, -1, 1, 1]), ([0, -1, 0, -1], [0, -1, 2, -4]), ([0, -1, 0, 0], [0, -1, 2, -3]), ([0, -1, 0, 1], [0, -1, 2, -2])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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 fixtureActualExpectedOutcome
explicit oracle 0[0, 1][-1, 1]Failed
explicit oracle 1[-1][-1]Passed
explicit oracle 2[1, 0, 0, 0, 0][1, 0, 0, 0, 0]Passed
explicit oracle 3[0][0]Passed
explicit oracle 4[1][1]Passed
explicit oracle 5[-1, 0][-1, 0]Passed
explicit oracle 6[1, 2][-1, 2]Failed
explicit oracle 7[-1, -1][0, -1]Failed

SHA-256 / 3474eb637f9f8b40aa977905c9a0de087afdccaa4aaaeab018f6d2fb05c56db8

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    row=x[:];out=[]
    for _ in range(len(x)):
     if not row:break
     out.append(row[len(row)//2])
     row=[row[i+1]-row[i] for i in range(len(row)-1)]
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1, 0], [-1, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([-1, -1], [-1, 0]), ([-1, 1], [-1, 2]), ([0, -1], [0, -1])], [([-1, 1], [-1, 2]), ([0, 1], [0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 1], [-1, 1, 0]), ([-1, 1, -1], [-1, 2, -4]), ([-1, 1, 0], [-1, 2, -3]), ([-1, 1, 1], [-1, 2, -2])], [([0, -1], [0, -1]), ([-1, -1, 0], [-1, 0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([1, 0, 0], [1, -1, 1]), ([1, 0, 1], [1, -1, 2]), ([1, 1, -1], [1, 0, -2]), ([1, 1, 0], [1, 0, -1])], [([0, 1], [0, 1]), ([-1, 0, 0], [-1, 1, -1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 0, -1], [-1, 1, -1, 0]), ([-1, 0, 0, 0], [-1, 1, -1, 1]), ([-1, 0, 0, 1], [-1, 1, -1, 2]), ([-1, 0, 1, -1], [-1, 1, 0, -3])], [([1, -1], [1, -2]), ([-1, 1, 0], [-1, 2, -3]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0, -1, -1, 1], [0, -1, 1, 1]), ([0, -1, 0, -1], [0, -1, 2, -4]), ([0, -1, 0, 0], [0, -1, 2, -3]), ([0, -1, 0, 1], [0, -1, 2, -2])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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 fixtureActualExpectedOutcome
explicit oracle 0[0, 1][-1, 1]Failed
explicit oracle 1[-1][-1]Passed
explicit oracle 2[1, 0, 0, 0, 0][1, 0, 0, 0, 0]Passed
explicit oracle 3[0][0]Passed
explicit oracle 4[1][1]Passed
explicit oracle 5[-1, 0][-1, 0]Passed
explicit oracle 6[1, 2][-1, 2]Failed
explicit oracle 7[-1, -1][0, -1]Failed

SHA-256 / 2ed284d9b15461d25488ed192dd13a8268a521bd08115cc109e4cc3c0896cd9e

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    row=x[:];out=[]
    for _ in range(len(x)):
     if not row:break
     out.append(row[0])
     row=[row[i+1]-row[i] for i in range(len(row)-1)]
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1, 0], [-1, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([-1, -1], [-1, 0]), ([-1, 1], [-1, 2]), ([0, -1], [0, -1])], [([-1, 1], [-1, 2]), ([0, 1], [0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 1], [-1, 1, 0]), ([-1, 1, -1], [-1, 2, -4]), ([-1, 1, 0], [-1, 2, -3]), ([-1, 1, 1], [-1, 2, -2])], [([0, -1], [0, -1]), ([-1, -1, 0], [-1, 0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([1, 0, 0], [1, -1, 1]), ([1, 0, 1], [1, -1, 2]), ([1, 1, -1], [1, 0, -2]), ([1, 1, 0], [1, 0, -1])], [([0, 1], [0, 1]), ([-1, 0, 0], [-1, 1, -1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 0, -1], [-1, 1, -1, 0]), ([-1, 0, 0, 0], [-1, 1, -1, 1]), ([-1, 0, 0, 1], [-1, 1, -1, 2]), ([-1, 0, 1, -1], [-1, 1, 0, -3])], [([1, -1], [1, -2]), ([-1, 1, 0], [-1, 2, -3]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0, -1, -1, 1], [0, -1, 1, 1]), ([0, -1, 0, -1], [0, -1, 2, -4]), ([0, -1, 0, 0], [0, -1, 2, -3]), ([0, -1, 0, 1], [0, -1, 2, -2])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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 fixtureActualExpectedOutcome
explicit oracle 0[-1, 1][-1, 1]Passed
explicit oracle 1[-1][-1]Passed
explicit oracle 2[1, 0, 0, 0, 0][1, 0, 0, 0, 0]Passed
explicit oracle 3[0][0]Passed
explicit oracle 4[1][1]Passed
explicit oracle 5[-1, 0][-1, 0]Passed
explicit oracle 6[-1, 2][-1, 2]Passed
explicit oracle 7[0, -1][0, -1]Passed

SHA-256 / 43d8370c016e2e179afa436e6beec62646e6264363e14ab2971f5495b2bed9cf

Verification & scope

A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. 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 2026-09-29T14:39:19.781670+00:00.

Case digest / 447a9cddae8a4e3a0129a04be76df567837564c231de5e72ad8c6b08a78370b8