# Verification and scope of the finite field counterexample

The complete argument is Theorem 1 and Proposition 2 of the accompanying
manuscript. For every prime power q and m >= n >= 2, let D_q(n,m) denote the
least tail-degree bound allowing X^m + g to have an irreducible degree-n
factor. If m is congruent to -j modulo q^n - 1, with 1 <= j < n, then
D_q(n,m) >= n-j. Polynomial remainders always give D_q(n,m) <= n-1.
Consequently max over m >= n of D_q(n,m) equals n-1. In the reciprocal
class m congruent to -1, all successful tails of degree at most n-1 are
exactly (F/F(0)-1)/X for the monic degree-n irreducibles F.

## Explicit counterexample

For q=2, n=8 and m=254, no polynomial g of degree at most 6 works. A
putative degree-eight factor would divide 1+Xg, a nonzero polynomial of
degree at most seven. This supplies a contradiction independently of all
computer checks. Degree seven is attainable.

## Exact arithmetic evidence

The bit-polynomial checker compares exhaustive trial division with the
Frobenius criterion: both identify precisely 30 monic degree-eight
irreducibles over F_2. All 128 possible tails of degree at most six are
tested against all 30 factors, giving 3,840 nondivisibility checks. Each
tail also passes a gcd test against the product of all such factors.
The 30 sharp degree-seven tails are enumerated.

A separate SymPy implementation repeats the same binary domain in dense
polynomial arithmetic. It is a different implementation, not independent
specialist review. The bit checker also performs 7,285 negative-window
checks over F_2 for degrees 2 through 12. SymPy performs 1,050 additional
checks over F_3 (degrees 2 through 5), F_5 (2 through 4) and F_7 (2 and 3).
Both retained runs pass. These bounded computations do not replace the
general proof and repeated domains are not double counted.

## Source scope and review status

The unrestricted main assertion of AIM Question 7, page 27, is disproved.
Gao's original Conjecture 1.3 restricts m to the least q-power at least n;
that conjecture is not refuted. The separate m=n, sparse-polynomial and
primitive-factor questions remain outside this result. Thus this is a
complete counterexample to the main assertion, not a solution of every
auxiliary question in the source record.

The author-directed originating research workflow includes a written
adversarial self-audit, source comparison and bounded literature search.
The elementary ingredients are standard and explicitly credited. Novelty
of this particular correction is undetermined; there is no absolute
priority claim. This is an AI-assisted, unrefereed preprint, with no
independent human review or proof-assistant verification claimed.