FAILURE MAP
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FA-14676 / Numerics / Open access

Lagrange interpolation value: basis denominator orientation · case 01

The exact lagrange interpolation value result violates the stated contract at basis denominator orientation.

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

ROOT CAUSE

The basis denominator orientation step uses xj-xi instead of xi-xj.

THE FAILURE

The basis denominator orientation step uses xj-xi instead of xi-xj.

Unsuccessful approach: The partial repair abs(xi-xj) still violates the basis denominator orientation invariant.

Case contract

Input [points,t] distinct integer abscissae and integer ordinates; exact rational interpolation value [p,q].

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):
    points,t=x
    ans=Fraction(0)
    for i,(xi,yi) in enumerate(points):
     term=Fraction(yi)
     for j,(xj,yj) in enumerate(points):
      if i==j:continue
      den=xj-xi
      if den==0:return None
      term*=Fraction(t-xj,den)
     ans=ans+term
    return [ans.numerator,ans.denominator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[[0, -2], [2, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1])], [([[[0, -2], [2, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 2], [10, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 3], [18, 1])], [([[[0, -2], [2, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 2]], 3], [38, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -3], [-5, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -1], [-1, 1])], [([[[0, -2], [2, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], 3], [8, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 0]], -1], [-1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 1], [1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 2], [2, 1])], [([[[0, -2], [2, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 2], [14, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 3], [23, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -3], [-8, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -2], [-2, 1])]]
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[2, 1][-2, 1]Failed
explicit oracle 1[-2, 1][-2, 1]Passed
explicit oracle 2[-2, 1][2, 1]Failed
explicit oracle 3[-2, 1][-2, 1]Passed
explicit oracle 4[-2, 1][-2, 1]Passed
explicit oracle 5[-2, 1][-2, 1]Passed
explicit oracle 6[-2, 1][-2, 1]Passed
explicit oracle 7[-2, 1][-2, 1]Passed

SHA-256 / 7e2f498dc5fdc67fc028e41a4e798e2c550da0b6783023938695ec2ef7806626

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):
    points,t=x
    ans=Fraction(0)
    for i,(xi,yi) in enumerate(points):
     term=Fraction(yi)
     for j,(xj,yj) in enumerate(points):
      if i==j:continue
      den=abs(xi-xj)
      if den==0:return None
      term*=Fraction(t-xj,den)
     ans=ans+term
    return [ans.numerator,ans.denominator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[[0, -2], [2, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1])], [([[[0, -2], [2, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 2], [10, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 3], [18, 1])], [([[[0, -2], [2, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 2]], 3], [38, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -3], [-5, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -1], [-1, 1])], [([[[0, -2], [2, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], 3], [8, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 0]], -1], [-1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 1], [1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 2], [2, 1])], [([[[0, -2], [2, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 2], [14, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 3], [23, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -3], [-8, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -2], [-2, 1])]]
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[6, 1][-2, 1]Failed
explicit oracle 1[-14, 1][-2, 1]Failed
explicit oracle 2[4, 1][2, 1]Failed
explicit oracle 3[-2, 1][-2, 1]Passed
explicit oracle 4[2, 1][-2, 1]Failed
explicit oracle 5[-2, 1][-2, 1]Passed
explicit oracle 6[-14, 1][-2, 1]Failed
explicit oracle 7[-34, 1][-2, 1]Failed

SHA-256 / eea85b9275c86b82518ef5ee2f8e24f5cfd5d304ca528ed958a5d819db21d0d2

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 2026-09-29T14:39:19.197651+00:00.

Case digest / 5c03f1385ef14202a0921e9e015ac9f503bcf12bd1e7179d4a232ca802acb257