FAILURE MAP
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FA-15706 / Numerics / Open access

Binary field minimal polynomial: root constant coefficient product · case 01

The exact binary field minimal polynomial result violates the stated contract at root constant coefficient product.

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

ROOT CAUSE

The root constant coefficient product step uses d[j]^b instead of d[j]^mul(b,root).

VERIFIED REPAIR

Use d[j]^mul(b,root) at the root constant coefficient product step.

Unsuccessful approach: The partial repair d[j]+mul(b,root) still violates the root constant coefficient product invariant.

Case contract

Input [a,irreducible bitmask]; return ascending 0/1 coefficients of the minimal polynomial of a over F2, through its distinct Frobenius orbit.

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,poly=x;n=poly.bit_length()-1
    def mul(a,b):
     z=0
     while b:
      if b&1:z^=a
      b>>=1;a<<=1
      if a&(1<<n):a^=poly
     return z
    orbit=[];v=a
    for _ in range(n+1):
     if v in orbit:break
     orbit.append(v);v=mul(v,v)
    c=[1]
    for root in orbit:
     d=[0]*(len(c)+1)
     for j,b in enumerate(c):
      d[j]=d[j]^b
      d[j+1]=d[j+1]^b
     c=d
    return c
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 7], [0, 1]), ([2, 7], [1, 1, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([1, 7], [1, 1]), ([3, 7], [1, 1, 1]), ([0, 11], [0, 1]), ([1, 11], [1, 1]), ([2, 11], [1, 1, 0, 1])], [([2, 7], [1, 1, 1]), ([3, 11], [1, 0, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([5, 19], [1, 1, 0, 0, 1]), ([6, 19], [1, 1, 1]), ([7, 19], [1, 1, 1]), ([8, 19], [1, 1, 1, 1, 1])], [([3, 7], [1, 1, 1]), ([6, 11], [1, 1, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([6, 25], [1, 1, 0, 0, 1]), ([7, 25], [1, 1, 0, 0, 1]), ([8, 25], [1, 1, 1, 1, 1]), ([9, 25], [1, 0, 0, 1, 1])], [([0, 11], [0, 1]), ([3, 19], [1, 1, 0, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([7, 31], [1, 1, 0, 0, 1]), ([8, 31], [1, 1, 1, 1, 1]), ([9, 31], [1, 0, 0, 1, 1]), ([10, 31], [1, 1, 0, 0, 1])], [([2, 11], [1, 1, 0, 1]), ([6, 19], [1, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([8, 37], [1, 0, 1, 1, 1, 1]), ([9, 37], [1, 0, 0, 1, 0, 1]), ([10, 37], [1, 0, 1, 1, 1, 1]), ([11, 37], [1, 0, 0, 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][0, 1]Failed
explicit oracle 1[1, 0, 1][1, 1, 1]Failed
explicit oracle 2[1, 1, 0, 0, 1, 1][1, 0, 0, 1, 0, 1]Failed
explicit oracle 3[1, 1][1, 1]Passed
explicit oracle 4[1, 0, 1][1, 1, 1]Failed
explicit oracle 5[1, 1][0, 1]Failed
explicit oracle 6[1, 1][1, 1]Passed
explicit oracle 7[1, 1, 1, 1][1, 1, 0, 1]Failed

SHA-256 / f1a786802b0a838ec3e64402583a4f772766415e8ee402a89de773d1e88f0140

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,poly=x;n=poly.bit_length()-1
    def mul(a,b):
     z=0
     while b:
      if b&1:z^=a
      b>>=1;a<<=1
      if a&(1<<n):a^=poly
     return z
    orbit=[];v=a
    for _ in range(n+1):
     if v in orbit:break
     orbit.append(v);v=mul(v,v)
    c=[1]
    for root in orbit:
     d=[0]*(len(c)+1)
     for j,b in enumerate(c):
      d[j]=d[j]+mul(b,root)
      d[j+1]=d[j+1]^b
     c=d
    return c
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 7], [0, 1]), ([2, 7], [1, 1, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([1, 7], [1, 1]), ([3, 7], [1, 1, 1]), ([0, 11], [0, 1]), ([1, 11], [1, 1]), ([2, 11], [1, 1, 0, 1])], [([2, 7], [1, 1, 1]), ([3, 11], [1, 0, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([5, 19], [1, 1, 0, 0, 1]), ([6, 19], [1, 1, 1]), ([7, 19], [1, 1, 1]), ([8, 19], [1, 1, 1, 1, 1])], [([3, 7], [1, 1, 1]), ([6, 11], [1, 1, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([6, 25], [1, 1, 0, 0, 1]), ([7, 25], [1, 1, 0, 0, 1]), ([8, 25], [1, 1, 1, 1, 1]), ([9, 25], [1, 0, 0, 1, 1])], [([0, 11], [0, 1]), ([3, 19], [1, 1, 0, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([7, 31], [1, 1, 0, 0, 1]), ([8, 31], [1, 1, 1, 1, 1]), ([9, 31], [1, 0, 0, 1, 1]), ([10, 31], [1, 1, 0, 0, 1])], [([2, 11], [1, 1, 0, 1]), ([6, 19], [1, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([8, 37], [1, 0, 1, 1, 1, 1]), ([9, 37], [1, 0, 0, 1, 0, 1]), ([10, 37], [1, 0, 1, 1, 1, 1]), ([11, 37], [1, 0, 0, 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[0, 1][0, 1]Passed
explicit oracle 1[1, 5, 1][1, 1, 1]Failed
explicit oracle 2[1, 29538, 8785, 1669, 84, 1][1, 0, 0, 1, 0, 1]Failed
explicit oracle 3[1, 1][1, 1]Passed
explicit oracle 4[1, 5, 1][1, 1, 1]Failed
explicit oracle 5[0, 1][0, 1]Passed
explicit oracle 6[1, 1][1, 1]Passed
explicit oracle 7[1, 5, 12, 1][1, 1, 0, 1]Failed

SHA-256 / 0dd2c9a09edbbe813a9fa759c688b4d77019d550bcc79d53c1f3208d0fa05631

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,poly=x;n=poly.bit_length()-1
    def mul(a,b):
     z=0
     while b:
      if b&1:z^=a
      b>>=1;a<<=1
      if a&(1<<n):a^=poly
     return z
    orbit=[];v=a
    for _ in range(n+1):
     if v in orbit:break
     orbit.append(v);v=mul(v,v)
    c=[1]
    for root in orbit:
     d=[0]*(len(c)+1)
     for j,b in enumerate(c):
      d[j]=d[j]^mul(b,root)
      d[j+1]=d[j+1]^b
     c=d
    return c
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([0, 7], [0, 1]), ([2, 7], [1, 1, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([1, 7], [1, 1]), ([3, 7], [1, 1, 1]), ([0, 11], [0, 1]), ([1, 11], [1, 1]), ([2, 11], [1, 1, 0, 1])], [([2, 7], [1, 1, 1]), ([3, 11], [1, 0, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([5, 19], [1, 1, 0, 0, 1]), ([6, 19], [1, 1, 1]), ([7, 19], [1, 1, 1]), ([8, 19], [1, 1, 1, 1, 1])], [([3, 7], [1, 1, 1]), ([6, 11], [1, 1, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([6, 25], [1, 1, 0, 0, 1]), ([7, 25], [1, 1, 0, 0, 1]), ([8, 25], [1, 1, 1, 1, 1]), ([9, 25], [1, 0, 0, 1, 1])], [([0, 11], [0, 1]), ([3, 19], [1, 1, 0, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([7, 31], [1, 1, 0, 0, 1]), ([8, 31], [1, 1, 1, 1, 1]), ([9, 31], [1, 0, 0, 1, 1]), ([10, 31], [1, 1, 0, 0, 1])], [([2, 11], [1, 1, 0, 1]), ([6, 19], [1, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([8, 37], [1, 0, 1, 1, 1, 1]), ([9, 37], [1, 0, 0, 1, 0, 1]), ([10, 37], [1, 0, 1, 1, 1, 1]), ([11, 37], [1, 0, 0, 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[0, 1][0, 1]Passed
explicit oracle 1[1, 1, 1][1, 1, 1]Passed
explicit oracle 2[1, 0, 0, 1, 0, 1][1, 0, 0, 1, 0, 1]Passed
explicit oracle 3[1, 1][1, 1]Passed
explicit oracle 4[1, 1, 1][1, 1, 1]Passed
explicit oracle 5[0, 1][0, 1]Passed
explicit oracle 6[1, 1][1, 1]Passed
explicit oracle 7[1, 1, 0, 1][1, 1, 0, 1]Passed

SHA-256 / 99974dace44aac418dfce0bd7217a6502e3b5a6718e71f024b9906700667bf92

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 2026-09-29T14:39:29.474697+00:00.

Case digest / 83fd1167e4dd1ac0452bb3fee958dba174759f4ca1250034a0533a8f1dddfa25