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FA-14651 / Numerics / Open access

Polynomial taylor shift: coefficient source · case 01

The exact polynomial taylor shift result violates the stated contract at coefficient source.

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

ROOT CAUSE

The coefficient source step uses A[j] instead of A[i].

VERIFIED REPAIR

Use A[i] at the coefficient source step.

Unsuccessful approach: The partial repair A[n-1-i] still violates the coefficient source invariant.

Case contract

Input [A,h] ascending integer polynomial, integer h; return coefficients of A(t+h), preserving array length.

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):
    A,h=x
    n=len(A);r=[0]*n
    for i in range(n):
     for j in range(i+1):
      r[j]+=(A[j]) * (math.comb(i,j) if j<=i else 0) * (h**(i-j) if j<=i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, -1], -1], [0, -2, 2, -1]), ([[-1, -1, -1, -1], 0], [-1, -1, -1, -1]), ([[-1, -1, -1, -1], 1], [-4, -6, -4, -1]), ([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, 0], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, 0], -1], [-1, 1, -1, 0]), ([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, 2], 1], [-1, 3, 5, 2]), ([[-1, -1, -1, 2], 2], [9, 19, 11, 2]), ([[-1, -1, 0, -1], -2], [9, -13, 6, -1])], [([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-1, -1, -1, 1], -1], [-2, 4, -4, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 0, 1], 2], [5, 11, 6, 1]), ([[-1, -1, 0, 2], -2], [-15, 23, -12, 2]), ([[-1, -1, 0, 2], -1], [-2, 5, -6, 2]), ([[-1, -1, 0, 2], 0], [-1, -1, 0, 2])], [([[-1, -1, -1, 0], 2], [-7, -5, -1, 0]), ([[-1, -1, -1, 1], 2], [1, 7, 5, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 1, 1], -1], [0, 0, -2, 1]), ([[-1, -1, 1, 1], 0], [-1, -1, 1, 1]), ([[-1, -1, 1, 1], 1], [0, 4, 4, 1]), ([[-1, -1, 1, 1], 2], [9, 15, 7, 1])], [([[-1, -1, -1, 1], -2], [-11, 15, -7, 1]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 2, 0], 1], [0, 3, 2, 0]), ([[-1, -1, 2, 0], 2], [5, 7, 2, 0]), ([[-1, -1, 2, 1], -2], [1, 3, -4, 1]), ([[-1, -1, 2, 1], -1], [1, -2, -1, 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[5, -9, 5, 0][-3, 3, -1, 0]Failed
explicit oracle 1[5, -9, 5, -1][5, -9, 5, -1]Passed
explicit oracle 2[30, 34, 14, 2][30, 34, 14, 2]Passed
explicit oracle 3[0, -2, 2, -1][0, -2, 2, -1]Passed
explicit oracle 4[-1, -1, -1, -1][-1, -1, -1, -1]Passed
explicit oracle 5[-4, -6, -4, -1][-4, -6, -4, -1]Passed
explicit oracle 6[-15, -17, -7, -1][-15, -17, -7, -1]Passed
explicit oracle 7[0, -2, 2, 0][-1, 1, -1, 0]Failed

SHA-256 / bb9c3aea771b343f1f3cc713d4263c37adf0741404edf8f9176a12dea84d3602

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):
    A,h=x
    n=len(A);r=[0]*n
    for i in range(n):
     for j in range(i+1):
      r[j]+=(A[n-1-i]) * (math.comb(i,j) if j<=i else 0) * (h**(i-j) if j<=i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, -1], -1], [0, -2, 2, -1]), ([[-1, -1, -1, -1], 0], [-1, -1, -1, -1]), ([[-1, -1, -1, -1], 1], [-4, -6, -4, -1]), ([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, 0], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, 0], -1], [-1, 1, -1, 0]), ([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, 2], 1], [-1, 3, 5, 2]), ([[-1, -1, -1, 2], 2], [9, 19, 11, 2]), ([[-1, -1, 0, -1], -2], [9, -13, 6, -1])], [([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-1, -1, -1, 1], -1], [-2, 4, -4, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 0, 1], 2], [5, 11, 6, 1]), ([[-1, -1, 0, 2], -2], [-15, 23, -12, 2]), ([[-1, -1, 0, 2], -1], [-2, 5, -6, 2]), ([[-1, -1, 0, 2], 0], [-1, -1, 0, 2])], [([[-1, -1, -1, 0], 2], [-7, -5, -1, 0]), ([[-1, -1, -1, 1], 2], [1, 7, 5, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 1, 1], -1], [0, 0, -2, 1]), ([[-1, -1, 1, 1], 0], [-1, -1, 1, 1]), ([[-1, -1, 1, 1], 1], [0, 4, 4, 1]), ([[-1, -1, 1, 1], 2], [9, 15, 7, 1])], [([[-1, -1, -1, 1], -2], [-11, 15, -7, 1]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 2, 0], 1], [0, 3, 2, 0]), ([[-1, -1, 2, 0], 2], [5, 7, 2, 0]), ([[-1, -1, 2, 1], -2], [1, 3, -4, 1]), ([[-1, -1, 2, 1], -1], [1, -2, -1, 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, -9, 5, -1][-3, 3, -1, 0]Failed
explicit oracle 1[5, -9, 5, -1][5, -9, 5, -1]Passed
explicit oracle 2[30, 34, 14, 2][30, 34, 14, 2]Passed
explicit oracle 3[0, -2, 2, -1][0, -2, 2, -1]Passed
explicit oracle 4[-1, -1, -1, -1][-1, -1, -1, -1]Passed
explicit oracle 5[-4, -6, -4, -1][-4, -6, -4, -1]Passed
explicit oracle 6[-15, -17, -7, -1][-15, -17, -7, -1]Passed
explicit oracle 7[1, -2, 2, -1][-1, 1, -1, 0]Failed

SHA-256 / 7c48208ae7a8b5594855c02e3436c07ca53ee574169fca87de714000f80b6b9f

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):
    A,h=x
    n=len(A);r=[0]*n
    for i in range(n):
     for j in range(i+1):
      r[j]+=(A[i]) * (math.comb(i,j) if j<=i else 0) * (h**(i-j) if j<=i else 0)
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1, -1, -1, 0], -2], [-3, 3, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, -1], -1], [0, -2, 2, -1]), ([[-1, -1, -1, -1], 0], [-1, -1, -1, -1]), ([[-1, -1, -1, -1], 1], [-4, -6, -4, -1]), ([[-1, -1, -1, -1], 2], [-15, -17, -7, -1]), ([[-1, -1, -1, 0], -1], [-1, 1, -1, 0])], [([[-1, -1, -1, 0], -1], [-1, 1, -1, 0]), ([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, 2], 1], [-1, 3, 5, 2]), ([[-1, -1, -1, 2], 2], [9, 19, 11, 2]), ([[-1, -1, 0, -1], -2], [9, -13, 6, -1])], [([[-1, -1, -1, 0], 1], [-3, -3, -1, 0]), ([[-1, -1, -1, 1], -1], [-2, 4, -4, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 0, 1], 2], [5, 11, 6, 1]), ([[-1, -1, 0, 2], -2], [-15, 23, -12, 2]), ([[-1, -1, 0, 2], -1], [-2, 5, -6, 2]), ([[-1, -1, 0, 2], 0], [-1, -1, 0, 2])], [([[-1, -1, -1, 0], 2], [-7, -5, -1, 0]), ([[-1, -1, -1, 1], 2], [1, 7, 5, 1]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 1, 1], -1], [0, 0, -2, 1]), ([[-1, -1, 1, 1], 0], [-1, -1, 1, 1]), ([[-1, -1, 1, 1], 1], [0, 4, 4, 1]), ([[-1, -1, 1, 1], 2], [9, 15, 7, 1])], [([[-1, -1, -1, 1], -2], [-11, 15, -7, 1]), ([[-1, -1, -1, 2], 0], [-1, -1, -1, 2]), ([[-1, -1, -1, -1], -2], [5, -9, 5, -1]), ([[2, 2, 2, 2], 2], [30, 34, 14, 2]), ([[-1, -1, 2, 0], 1], [0, 3, 2, 0]), ([[-1, -1, 2, 0], 2], [5, 7, 2, 0]), ([[-1, -1, 2, 1], -2], [1, 3, -4, 1]), ([[-1, -1, 2, 1], -1], [1, -2, -1, 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[-3, 3, -1, 0][-3, 3, -1, 0]Passed
explicit oracle 1[5, -9, 5, -1][5, -9, 5, -1]Passed
explicit oracle 2[30, 34, 14, 2][30, 34, 14, 2]Passed
explicit oracle 3[0, -2, 2, -1][0, -2, 2, -1]Passed
explicit oracle 4[-1, -1, -1, -1][-1, -1, -1, -1]Passed
explicit oracle 5[-4, -6, -4, -1][-4, -6, -4, -1]Passed
explicit oracle 6[-15, -17, -7, -1][-15, -17, -7, -1]Passed
explicit oracle 7[-1, 1, -1, 0][-1, 1, -1, 0]Passed

SHA-256 / b2b2344e5bd2db4efba9d1add499a7f0b4632e887f9062f273f831e5f0258a4e

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 2026-09-29T14:39:19.106785+00:00.

Case digest / 426ca19fe86c2bfd655e50ef8c3a02f2580dcebc32932acb85323560bb253f20