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FA-15701 / Numerics / Open access

Binary field minimal polynomial: monic polynomial product identity · case 01

The exact binary field minimal polynomial result violates the stated contract at monic polynomial product identity.

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

ROOT CAUSE

The monic polynomial product identity step uses [0] instead of [1].

THE FAILURE

The monic polynomial product identity step uses [0] instead of [1].

Unsuccessful approach: The partial repair [a] still violates the monic polynomial product identity 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=[0]
    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]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([1, 7], [1, 1]), ([2, 7], [1, 1, 1]), ([3, 7], [1, 1, 1]), ([0, 11], [0, 1]), ([1, 11], [1, 1]), ([2, 11], [1, 1, 0, 1])], [([1, 7], [1, 1]), ([0, 11], [0, 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])], [([2, 7], [1, 1, 1]), ([4, 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])], [([3, 7], [1, 1, 1]), ([7, 11], [1, 0, 1, 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])], [([0, 11], [0, 1]), ([3, 19], [1, 1, 0, 0, 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, 0][0, 1]Failed
explicit oracle 1[0, 0, 0, 0, 0, 0][1, 0, 0, 1, 0, 1]Failed
explicit oracle 2[0, 0][1, 1]Failed
explicit oracle 3[0, 0, 0][1, 1, 1]Failed
explicit oracle 4[0, 0, 0][1, 1, 1]Failed
explicit oracle 5[0, 0][0, 1]Failed
explicit oracle 6[0, 0][1, 1]Failed
explicit oracle 7[0, 0, 0, 0][1, 1, 0, 1]Failed

SHA-256 / 8b2e2d04bb2486b1cca16ebd536e3cb19d239296ad941595539b1e2f2785597a

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=[a]
    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]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([1, 7], [1, 1]), ([2, 7], [1, 1, 1]), ([3, 7], [1, 1, 1]), ([0, 11], [0, 1]), ([1, 11], [1, 1]), ([2, 11], [1, 1, 0, 1])], [([1, 7], [1, 1]), ([0, 11], [0, 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])], [([2, 7], [1, 1, 1]), ([4, 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])], [([3, 7], [1, 1, 1]), ([7, 11], [1, 0, 1, 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])], [([0, 11], [0, 1]), ([3, 19], [1, 1, 0, 0, 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, 0][0, 1]Failed
explicit oracle 1[31, 0, 0, 31, 0, 31][1, 0, 0, 1, 0, 1]Failed
explicit oracle 2[1, 1][1, 1]Passed
explicit oracle 3[2, 2, 2][1, 1, 1]Failed
explicit oracle 4[3, 3, 3][1, 1, 1]Failed
explicit oracle 5[0, 0][0, 1]Failed
explicit oracle 6[1, 1][1, 1]Passed
explicit oracle 7[2, 2, 0, 2][1, 1, 0, 1]Failed

SHA-256 / 82ccdbdff37887d4dd04d9e2b8742b038462554bea0c21f3b77df8e18729e7d0

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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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 2026-09-29T14:39:29.427902+00:00.

Case digest / 2394026961a57739615b7aba8d0c93bad876f4ecfbf3035de6ce83047627c180