FAILURE MAP
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FA-14626 / Numerics / Open access

Polynomial composition: product allocation · case 01

The exact polynomial composition result violates the stated contract at product allocation.

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

ROOT CAUSE

The product allocation step uses max(1,len(r)+len(B)-2) instead of (len(r)+len(B)-1).

VERIFIED REPAIR

Use (len(r)+len(B)-1) at the product allocation step.

Unsuccessful approach: The partial repair max(len(r),len(B)) still violates the product allocation invariant.

Case contract

Input [A,B] nonempty ascending integer polynomials; return canonical ascending coefficients A(B(t)), zero represented [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):
    A,B=x
    r=[0]
    for a in reversed(A):
     t=[0]*max(1,len(r)+len(B)-2)
     for i,u in enumerate(r):
      for j,v in enumerate(B):
       if i+j>=len(t):return None
       t[i+j]+=u*v
     t[0]+=a
     r=t
    while len(r)>1 and r[-1]==0:r.pop()
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, -1, -1], [-1, 0]], [-1]), ([[-1, -1, -1], [-1, 1]], [-1, 1, -1]), ([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, -1], [0, 0]], [-1]), ([[-1, -1, -1], [0, 1]], [-1, -1, -1]), ([[-1, -1, -1], [1, -1]], [-3, 3, -1])], [([[-1, -1, -1], [-1, 0]], [-1]), ([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, -1, 0], [1, 1]], [-2, -1]), ([[-1, -1, 1], [-1, -1]], [1, 3, 1]), ([[-1, -1, 1], [-1, 0]], [1]), ([[-1, -1, 1], [-1, 1]], [1, -3, 1])], [([[-1, -1, -1], [-1, 1]], [-1, 1, -1]), ([[-1, -1, -1], [1, -1]], [-3, 3, -1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 0, -1], [1, 0]], [-2]), ([[-1, 0, -1], [1, 1]], [-2, -2, -1]), ([[-1, 0, 0], [-1, -1]], [-1]), ([[-1, 0, 0], [-1, 0]], [-1])], [([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, 0], [-1, -1]], [0, 1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 0, 1], [1, -1]], [0, -2, 1]), ([[-1, 0, 1], [1, 0]], [0]), ([[-1, 0, 1], [1, 1]], [0, 2, 1]), ([[-1, 1, -1], [-1, -1]], [-3, -3, -1])], [([[-1, -1, -1], [0, 0]], [-1]), ([[-1, -1, 0], [0, -1]], [-1, 1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 1, 0], [0, 1]], [-1, 1]), ([[-1, 1, 0], [1, -1]], [0, -1]), ([[-1, 1, 0], [1, 0]], [0]), ([[-1, 1, 0], [1, 1]], [0, 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 0None[-1, -1, -1]Failed
explicit oracle 1None[0]Failed
explicit oracle 2None[-1]Failed
explicit oracle 3None[-1, 1, -1]Failed
explicit oracle 4None[-1, 1, -1]Failed
explicit oracle 5None[-1]Failed
explicit oracle 6None[-1, -1, -1]Failed
explicit oracle 7None[-3, 3, -1]Failed

SHA-256 / f0c66d4e137d8281cd6b310fae709c8c5f2306691d48309c4d00527bffc32b21

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,B=x
    r=[0]
    for a in reversed(A):
     t=[0]*max(len(r),len(B))
     for i,u in enumerate(r):
      for j,v in enumerate(B):
       if i+j>=len(t):return None
       t[i+j]+=u*v
     t[0]+=a
     r=t
    while len(r)>1 and r[-1]==0:r.pop()
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, -1, -1], [-1, 0]], [-1]), ([[-1, -1, -1], [-1, 1]], [-1, 1, -1]), ([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, -1], [0, 0]], [-1]), ([[-1, -1, -1], [0, 1]], [-1, -1, -1]), ([[-1, -1, -1], [1, -1]], [-3, 3, -1])], [([[-1, -1, -1], [-1, 0]], [-1]), ([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, -1, 0], [1, 1]], [-2, -1]), ([[-1, -1, 1], [-1, -1]], [1, 3, 1]), ([[-1, -1, 1], [-1, 0]], [1]), ([[-1, -1, 1], [-1, 1]], [1, -3, 1])], [([[-1, -1, -1], [-1, 1]], [-1, 1, -1]), ([[-1, -1, -1], [1, -1]], [-3, 3, -1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 0, -1], [1, 0]], [-2]), ([[-1, 0, -1], [1, 1]], [-2, -2, -1]), ([[-1, 0, 0], [-1, -1]], [-1]), ([[-1, 0, 0], [-1, 0]], [-1])], [([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, 0], [-1, -1]], [0, 1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 0, 1], [1, -1]], [0, -2, 1]), ([[-1, 0, 1], [1, 0]], [0]), ([[-1, 0, 1], [1, 1]], [0, 2, 1]), ([[-1, 1, -1], [-1, -1]], [-3, -3, -1])], [([[-1, -1, -1], [0, 0]], [-1]), ([[-1, -1, 0], [0, -1]], [-1, 1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 1, 0], [0, 1]], [-1, 1]), ([[-1, 1, 0], [1, -1]], [0, -1]), ([[-1, 1, 0], [1, 0]], [0]), ([[-1, 1, 0], [1, 1]], [0, 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 0None[-1, -1, -1]Failed
explicit oracle 1None[0]Failed
explicit oracle 2None[-1]Failed
explicit oracle 3None[-1, 1, -1]Failed
explicit oracle 4None[-1, 1, -1]Failed
explicit oracle 5None[-1]Failed
explicit oracle 6None[-1, -1, -1]Failed
explicit oracle 7None[-3, 3, -1]Failed

SHA-256 / 9bf42b88aa8babadbd18b271edf036c52836944190be03bb29328d3c5cb70a4c

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,B=x
    r=[0]
    for a in reversed(A):
     t=[0]*(len(r)+len(B)-1)
     for i,u in enumerate(r):
      for j,v in enumerate(B):
       if i+j>=len(t):return None
       t[i+j]+=u*v
     t[0]+=a
     r=t
    while len(r)>1 and r[-1]==0:r.pop()
    return r
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, -1, -1], [-1, 0]], [-1]), ([[-1, -1, -1], [-1, 1]], [-1, 1, -1]), ([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, -1], [0, 0]], [-1]), ([[-1, -1, -1], [0, 1]], [-1, -1, -1]), ([[-1, -1, -1], [1, -1]], [-3, 3, -1])], [([[-1, -1, -1], [-1, 0]], [-1]), ([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, -1, 0], [1, 1]], [-2, -1]), ([[-1, -1, 1], [-1, -1]], [1, 3, 1]), ([[-1, -1, 1], [-1, 0]], [1]), ([[-1, -1, 1], [-1, 1]], [1, -3, 1])], [([[-1, -1, -1], [-1, 1]], [-1, 1, -1]), ([[-1, -1, -1], [1, -1]], [-3, 3, -1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 0, -1], [1, 0]], [-2]), ([[-1, 0, -1], [1, 1]], [-2, -2, -1]), ([[-1, 0, 0], [-1, -1]], [-1]), ([[-1, 0, 0], [-1, 0]], [-1])], [([[-1, -1, -1], [0, -1]], [-1, 1, -1]), ([[-1, -1, 0], [-1, -1]], [0, 1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 0, 1], [1, -1]], [0, -2, 1]), ([[-1, 0, 1], [1, 0]], [0]), ([[-1, 0, 1], [1, 1]], [0, 2, 1]), ([[-1, 1, -1], [-1, -1]], [-3, -3, -1])], [([[-1, -1, -1], [0, 0]], [-1]), ([[-1, -1, 0], [0, -1]], [-1, 1]), ([[-1, -1, -1], [-1, -1]], [-1, -1, -1]), ([[0, 0], [1, 1]], [0]), ([[-1, 1, 0], [0, 1]], [-1, 1]), ([[-1, 1, 0], [1, -1]], [0, -1]), ([[-1, 1, 0], [1, 0]], [0]), ([[-1, 1, 0], [1, 1]], [0, 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[-1, -1, -1][-1, -1, -1]Passed
explicit oracle 1[0][0]Passed
explicit oracle 2[-1][-1]Passed
explicit oracle 3[-1, 1, -1][-1, 1, -1]Passed
explicit oracle 4[-1, 1, -1][-1, 1, -1]Passed
explicit oracle 5[-1][-1]Passed
explicit oracle 6[-1, -1, -1][-1, -1, -1]Passed
explicit oracle 7[-3, 3, -1][-3, 3, -1]Passed

SHA-256 / 4f629b4165161e47d9e9f5a48c4419d49e9e9fe7ff4ab985fe2b4454829cf015

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 2026-09-29T14:39:18.659814+00:00.

Case digest / e864bb7203733b70858f871b55fd4b5c3131f56e71aa366c2d39f6b9008e28c4