FAILURE MAP
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FA-14741 / Numerics / Open access

Bernstein to power basis: control point coverage · case 01

The exact bernstein to power basis result violates the stated contract at control point coverage.

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

ROOT CAUSE

The control point coverage step uses range(n) instead of range(n+1).

VERIFIED REPAIR

Use range(n+1) at the control point coverage step.

Unsuccessful approach: The partial repair range(1,n+1) still violates the control point coverage invariant.

Case contract

Input Bernstein coefficients b0..bn as integers; return exact ascending power coefficients of sum bi*choose(n,i)*t^i*(1-t)^(n-i).

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):
    b=x;n=len(b)-1;r=[0]*(n+1)
    for i in range(n):
     for j in range(n-i+1):
      k=i+j
      if k<0 or k>n:return None
      r[k]+=b[i]*(math.comb(n,i)) * (math.comb(n-i,j)*(-1)**j if j<=n-i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([2], [2]), ([-1, -1], [-1, 0]), ([-1, 0], [-1, 1]), ([-1, 1], [-1, 2])], [([1], [1]), ([-1, -1], [-1, 0]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, 0], [2, -2]), ([2, 1], [2, -1]), ([2, 2], [2, 0]), ([-1, -1, -1], [-1, 0, 0])], [([2], [2]), ([-1, 2], [-1, 3]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([-1, 2, 1], [-1, 6, -4]), ([-1, 2, 2], [-1, 6, -3]), ([0, -1, -1], [0, -2, 1]), ([0, -1, 0], [0, -2, 2])], [([-1, -1], [-1, 0]), ([1, 1], [1, 0]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0, 2, 2], [0, 4, -2]), ([1, -1, -1], [1, -4, 2]), ([1, -1, 0], [1, -4, 3]), ([1, -1, 1], [1, -4, 4])], [([-1, 1], [-1, 2]), ([2, 0], [2, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, -1, -1], [2, -6, 3]), ([2, -1, 0], [2, -6, 4]), ([2, -1, 1], [2, -6, 5]), ([2, -1, 2], [2, -6, 6])]]
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]Failed
explicit oracle 1[2, 0, 0, 0, -2][2, 0, 0, 0, 0]Failed
explicit oracle 2[0][0]Passed
explicit oracle 3[0][1]Failed
explicit oracle 4[0][2]Failed
explicit oracle 5[-1, 1][-1, 0]Failed
explicit oracle 6[-1, 1][-1, 1]Passed
explicit oracle 7[-1, 1][-1, 2]Failed

SHA-256 / 2353200a1f72b8581683b588e2aa1385b62630aa1f6e7befdbc0adfd8baa3971

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):
    b=x;n=len(b)-1;r=[0]*(n+1)
    for i in range(1,n+1):
     for j in range(n-i+1):
      k=i+j
      if k<0 or k>n:return None
      r[k]+=b[i]*(math.comb(n,i)) * (math.comb(n-i,j)*(-1)**j if j<=n-i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([2], [2]), ([-1, -1], [-1, 0]), ([-1, 0], [-1, 1]), ([-1, 1], [-1, 2])], [([1], [1]), ([-1, -1], [-1, 0]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, 0], [2, -2]), ([2, 1], [2, -1]), ([2, 2], [2, 0]), ([-1, -1, -1], [-1, 0, 0])], [([2], [2]), ([-1, 2], [-1, 3]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([-1, 2, 1], [-1, 6, -4]), ([-1, 2, 2], [-1, 6, -3]), ([0, -1, -1], [0, -2, 1]), ([0, -1, 0], [0, -2, 2])], [([-1, -1], [-1, 0]), ([1, 1], [1, 0]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0, 2, 2], [0, 4, -2]), ([1, -1, -1], [1, -4, 2]), ([1, -1, 0], [1, -4, 3]), ([1, -1, 1], [1, -4, 4])], [([-1, 1], [-1, 2]), ([2, 0], [2, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, -1, -1], [2, -6, 3]), ([2, -1, 0], [2, -6, 4]), ([2, -1, 1], [2, -6, 5]), ([2, -1, 2], [2, -6, 6])]]
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]Failed
explicit oracle 1[0, 8, -12, 8, -2][2, 0, 0, 0, 0]Failed
explicit oracle 2[0][0]Passed
explicit oracle 3[0][1]Failed
explicit oracle 4[0][2]Failed
explicit oracle 5[0, -1][-1, 0]Failed
explicit oracle 6[0, 0][-1, 1]Failed
explicit oracle 7[0, 1][-1, 2]Failed

SHA-256 / 1f6af64c4439779f25e1bcb09480714335f9a48a281f881ba53593d4df85bd75

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):
    b=x;n=len(b)-1;r=[0]*(n+1)
    for i in range(n+1):
     for j in range(n-i+1):
      k=i+j
      if k<0 or k>n:return None
      r[k]+=b[i]*(math.comb(n,i)) * (math.comb(n-i,j)*(-1)**j if j<=n-i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([2], [2]), ([-1, -1], [-1, 0]), ([-1, 0], [-1, 1]), ([-1, 1], [-1, 2])], [([1], [1]), ([-1, -1], [-1, 0]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, 0], [2, -2]), ([2, 1], [2, -1]), ([2, 2], [2, 0]), ([-1, -1, -1], [-1, 0, 0])], [([2], [2]), ([-1, 2], [-1, 3]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([-1, 2, 1], [-1, 6, -4]), ([-1, 2, 2], [-1, 6, -3]), ([0, -1, -1], [0, -2, 1]), ([0, -1, 0], [0, -2, 2])], [([-1, -1], [-1, 0]), ([1, 1], [1, 0]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([0, 2, 2], [0, 4, -2]), ([1, -1, -1], [1, -4, 2]), ([1, -1, 0], [1, -4, 3]), ([1, -1, 1], [1, -4, 4])], [([-1, 1], [-1, 2]), ([2, 0], [2, -2]), ([-1], [-1]), ([2, 2, 2, 2, 2], [2, 0, 0, 0, 0]), ([2, -1, -1], [2, -6, 3]), ([2, -1, 0], [2, -6, 4]), ([2, -1, 1], [2, -6, 5]), ([2, -1, 2], [2, -6, 6])]]
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]Passed
explicit oracle 1[2, 0, 0, 0, 0][2, 0, 0, 0, 0]Passed
explicit oracle 2[0][0]Passed
explicit oracle 3[1][1]Passed
explicit oracle 4[2][2]Passed
explicit oracle 5[-1, 0][-1, 0]Passed
explicit oracle 6[-1, 1][-1, 1]Passed
explicit oracle 7[-1, 2][-1, 2]Passed

SHA-256 / 6261987effb3e0b5ad327be523782ab5dcb7fe320640a3c09109e97b8f9156b4

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 2026-09-29T14:39:19.872477+00:00.

Case digest / 08493cba349ab3e75e04d40c20c98c423bb66a0ce72df678277aafaca6add9fa