A Sharp Obstruction to Uniform Low-Degree Tails over Finite Fields

For a prime power \(q\) and integers \(m\ge n\ge2\), let \(D_q(n,m)\) be the least degree bound on a polynomial \(g\in\mathbb F_q[X]\) for which \(X^m+g(X)\) has an irreducible factor of degree \(n\). We give the elementary sharp identity \(\max_{m\ge n}D_q(n,m)=n-1\): equality holds whenever \(m\equiv-1\pmod{q^n-1}\). More generally, the residue \(m\equiv-j\) forces \(D_q(n,m)\ge n-j\) for \(1\le j<n\). In particular, over \(\mathbb F_2\), no tail of degree at most \(6\) makes \(X^{254}+g(X)\) have an irreducible factor of degree \(8\). This disproves the unrestricted-exponent formulation of Question 7 in the AIM algorithmic number theory workshop list. It does not settle Gao's original conjecture, where the exponent is the least power of \(q\) at least \(n\), or the separate sparse-polynomial variants.