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

For a prime power q and integers m >= n >= 2, let D_q(n,m) be the least degree bound on a polynomial g over F_q for which X^m + g(X) has an irreducible factor of degree n. We prove the elementary sharp identity max_{m >= n} D_q(n,m) = n-1: equality holds whenever m is congruent to -1 modulo q^n-1. More generally, the residue m congruent to -j forces D_q(n,m) >= n-j for 1 <= j < n. All successful tails at the sharp bound in the reciprocal class are characterized explicitly. In particular, over 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 main assertion of Question 7 in the AIM algorithmic number theory workshop list (AIM-COMPUTATION-0095 in the UnsolvedMath corpus). It does not settle Gao's original conjecture, whose exponent is the least power of q at least n, or the separate sparse-polynomial variants. Two exact implementations check every binary tail in the explicit example. The archive includes the English LaTeX source, proof, self-audit, exact checkers and their outputs. This AI-assisted, self-audited preprint is unrefereed. Novelty of the elementary correction is undetermined; no absolute-priority, independent human review or proof-assistant verification is claimed.