FAILURE MAP
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FA-14691 / Numerics / Open access

Newton divided difference table: difference order bounds · case 01

The exact newton divided difference table result violates the stated contract at difference order bounds.

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

ROOT CAUSE

The difference order bounds step uses range(1,n-1) instead of range(1,n).

THE FAILURE

The difference order bounds step uses range(1,n-1) instead of range(1,n).

Unsuccessful approach: The partial repair range(2,n) still violates the difference order bounds invariant.

Case contract

Input distinct integer nodes and integer values [nodes,values]; return Newton coefficients [p,q] in original node order.

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):
    nodes,values=x
    c=list(map(Fraction,values));n=len(c)
    for order in range(1,n-1):
     for i in range(n-1,order-1,-1):
      den=nodes[i]-nodes[i-order]
      if den==0:return None
      c[i]=(c[i]-c[i-1])/den
    return [[v.numerator,v.denominator] for v in c]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-2, 0, 3], [-1, -1, 0]], [[-1, 1], [0, 1], [1, 15]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [-1, -1, 1]], [[-1, 1], [0, 1], [2, 15]]), ([[-2, 0, 3], [-1, -1, 2]], [[-1, 1], [0, 1], [1, 5]]), ([[-2, 0, 3], [-1, 0, -1]], [[-1, 1], [1, 2], [-1, 6]]), ([[-2, 0, 3], [-1, 0, 0]], [[-1, 1], [1, 2], [-1, 10]]), ([[-2, 0, 3], [-1, 0, 1]], [[-1, 1], [1, 2], [-1, 30]])], [([[-2, 0, 3], [-1, -1, 1]], [[-1, 1], [0, 1], [2, 15]]), ([[-2, 0, 3], [-1, -1, 2]], [[-1, 1], [0, 1], [1, 5]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [0, -1, 0]], [[0, 1], [-1, 2], [1, 6]]), ([[-2, 0, 3], [0, -1, 1]], [[0, 1], [-1, 2], [7, 30]]), ([[-2, 0, 3], [0, -1, 2]], [[0, 1], [-1, 2], [3, 10]]), ([[-2, 0, 3], [0, 0, -1]], [[0, 1], [0, 1], [-1, 15]])], [([[-2, 0, 3], [-1, -1, 2]], [[-1, 1], [0, 1], [1, 5]]), ([[-2, 0, 3], [-1, 0, 1]], [[-1, 1], [1, 2], [-1, 30]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [1, -1, 1]], [[1, 1], [-1, 1], [1, 3]]), ([[-2, 0, 3], [1, -1, 2]], [[1, 1], [-1, 1], [2, 5]]), ([[-2, 0, 3], [1, 0, -1]], [[1, 1], [-1, 2], [1, 30]]), ([[-2, 0, 3], [1, 0, 0]], [[1, 1], [-1, 2], [1, 10]])], [([[-2, 0, 3], [-1, 0, -1]], [[-1, 1], [1, 2], [-1, 6]]), ([[-2, 0, 3], [-1, 1, 0]], [[-1, 1], [1, 1], [-4, 15]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [2, -1, 2]], [[2, 1], [-3, 2], [1, 2]]), ([[-2, 0, 3], [2, 0, -1]], [[2, 1], [-1, 1], [2, 15]]), ([[-2, 0, 3], [2, 0, 0]], [[2, 1], [-1, 1], [1, 5]]), ([[-2, 0, 3], [2, 0, 1]], [[2, 1], [-1, 1], [4, 15]])], [([[-2, 0, 3], [-1, 0, 0]], [[-1, 1], [1, 2], [-1, 10]]), ([[-2, 0, 3], [-1, 2, -1]], [[-1, 1], [3, 2], [-1, 2]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[0, 1, 2], [-1, 0, -1]], [[-1, 1], [1, 1], [-1, 1]]), ([[0, 1, 2], [-1, 0, 0]], [[-1, 1], [1, 1], [-1, 2]]), ([[0, 1, 2], [-1, 0, 1]], [[-1, 1], [1, 1], [0, 1]]), ([[0, 1, 2], [-1, 0, 2]], [[-1, 1], [1, 1], [1, 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], [0, 1], [1, 3]][[-1, 1], [0, 1], [1, 15]]Failed
explicit oracle 1[[-1, 1], [0, 1], [0, 1]][[-1, 1], [0, 1], [0, 1]]Passed
explicit oracle 2[[2, 1], [0, 1], [0, 1], [0, 1]][[2, 1], [0, 1], [0, 1], [0, 1]]Passed
explicit oracle 3[[-1, 1], [0, 1], [2, 3]][[-1, 1], [0, 1], [2, 15]]Failed
explicit oracle 4[[-1, 1], [0, 1], [1, 1]][[-1, 1], [0, 1], [1, 5]]Failed
explicit oracle 5[[-1, 1], [1, 2], [-1, 3]][[-1, 1], [1, 2], [-1, 6]]Failed
explicit oracle 6[[-1, 1], [1, 2], [0, 1]][[-1, 1], [1, 2], [-1, 10]]Failed
explicit oracle 7[[-1, 1], [1, 2], [1, 3]][[-1, 1], [1, 2], [-1, 30]]Failed

SHA-256 / d7ea53966a6fd3c74c943e07c2e35fa74216621205a7b821980cd524614a398d

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):
    nodes,values=x
    c=list(map(Fraction,values));n=len(c)
    for order in range(2,n):
     for i in range(n-1,order-1,-1):
      den=nodes[i]-nodes[i-order]
      if den==0:return None
      c[i]=(c[i]-c[i-1])/den
    return [[v.numerator,v.denominator] for v in c]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-2, 0, 3], [-1, -1, 0]], [[-1, 1], [0, 1], [1, 15]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [-1, -1, 1]], [[-1, 1], [0, 1], [2, 15]]), ([[-2, 0, 3], [-1, -1, 2]], [[-1, 1], [0, 1], [1, 5]]), ([[-2, 0, 3], [-1, 0, -1]], [[-1, 1], [1, 2], [-1, 6]]), ([[-2, 0, 3], [-1, 0, 0]], [[-1, 1], [1, 2], [-1, 10]]), ([[-2, 0, 3], [-1, 0, 1]], [[-1, 1], [1, 2], [-1, 30]])], [([[-2, 0, 3], [-1, -1, 1]], [[-1, 1], [0, 1], [2, 15]]), ([[-2, 0, 3], [-1, -1, 2]], [[-1, 1], [0, 1], [1, 5]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [0, -1, 0]], [[0, 1], [-1, 2], [1, 6]]), ([[-2, 0, 3], [0, -1, 1]], [[0, 1], [-1, 2], [7, 30]]), ([[-2, 0, 3], [0, -1, 2]], [[0, 1], [-1, 2], [3, 10]]), ([[-2, 0, 3], [0, 0, -1]], [[0, 1], [0, 1], [-1, 15]])], [([[-2, 0, 3], [-1, -1, 2]], [[-1, 1], [0, 1], [1, 5]]), ([[-2, 0, 3], [-1, 0, 1]], [[-1, 1], [1, 2], [-1, 30]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [1, -1, 1]], [[1, 1], [-1, 1], [1, 3]]), ([[-2, 0, 3], [1, -1, 2]], [[1, 1], [-1, 1], [2, 5]]), ([[-2, 0, 3], [1, 0, -1]], [[1, 1], [-1, 2], [1, 30]]), ([[-2, 0, 3], [1, 0, 0]], [[1, 1], [-1, 2], [1, 10]])], [([[-2, 0, 3], [-1, 0, -1]], [[-1, 1], [1, 2], [-1, 6]]), ([[-2, 0, 3], [-1, 1, 0]], [[-1, 1], [1, 1], [-4, 15]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[-2, 0, 3], [2, -1, 2]], [[2, 1], [-3, 2], [1, 2]]), ([[-2, 0, 3], [2, 0, -1]], [[2, 1], [-1, 1], [2, 15]]), ([[-2, 0, 3], [2, 0, 0]], [[2, 1], [-1, 1], [1, 5]]), ([[-2, 0, 3], [2, 0, 1]], [[2, 1], [-1, 1], [4, 15]])], [([[-2, 0, 3], [-1, 0, 0]], [[-1, 1], [1, 2], [-1, 10]]), ([[-2, 0, 3], [-1, 2, -1]], [[-1, 1], [3, 2], [-1, 2]]), ([[-2, 0, 3], [-1, -1, -1]], [[-1, 1], [0, 1], [0, 1]]), ([[0, 2, 3, 7], [2, 2, 2, 2]], [[2, 1], [0, 1], [0, 1], [0, 1]]), ([[0, 1, 2], [-1, 0, -1]], [[-1, 1], [1, 1], [-1, 1]]), ([[0, 1, 2], [-1, 0, 0]], [[-1, 1], [1, 1], [-1, 2]]), ([[0, 1, 2], [-1, 0, 1]], [[-1, 1], [1, 1], [0, 1]]), ([[0, 1, 2], [-1, 0, 2]], [[-1, 1], [1, 1], [1, 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], [1, 5]][[-1, 1], [0, 1], [1, 15]]Failed
explicit oracle 1[[-1, 1], [-1, 1], [0, 1]][[-1, 1], [0, 1], [0, 1]]Failed
explicit oracle 2[[2, 1], [2, 1], [0, 1], [0, 1]][[2, 1], [0, 1], [0, 1], [0, 1]]Failed
explicit oracle 3[[-1, 1], [-1, 1], [2, 5]][[-1, 1], [0, 1], [2, 15]]Failed
explicit oracle 4[[-1, 1], [-1, 1], [3, 5]][[-1, 1], [0, 1], [1, 5]]Failed
explicit oracle 5[[-1, 1], [0, 1], [-1, 5]][[-1, 1], [1, 2], [-1, 6]]Failed
explicit oracle 6[[-1, 1], [0, 1], [0, 1]][[-1, 1], [1, 2], [-1, 10]]Failed
explicit oracle 7[[-1, 1], [0, 1], [1, 5]][[-1, 1], [1, 2], [-1, 30]]Failed

SHA-256 / 9c33ca6b16472eb8dfa497bdef226b050f6368eb8ecfe96c5a288fadf4610901

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 8 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

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 2026-09-29T14:39:19.295135+00:00.

Case digest / a4457b3aa437f68487821ca656d4ea6c00fc14877320f21324d0ab4d319e537b