An infinite family of counterexamples to Goss's conjecture on L-functions of cyclotomic function fields

Goss conjectured that the "congruent to one modulo p" part g(X, omega_P^i) of the Artin L-function of a cyclotomic function field K_P satisfies deg_X g <= 1 at every q-magic index i (an analogue of Vandiver's conjecture; recorded as open by Angles, arXiv:math/0502130). We disprove it. For p prime, a in F_p^*, P_a = T^p - T - a, and i = p^n - 1, we prove the exact congruence sumₘ₌₀ᵖ⁻¹ (sumdeg a'=m a'^i) X^m == (1 - X)^n in F_p[T]/(P_a), hence deg_X g(X, omegaP_ap^n-1) = n - 1. For p >= 5 and 3 <= n <= p-1 this gives deg_X g >= 2, contradicting the conjecture, with unbounded deficiency. This also yields the counterexample to the weak form A_P = {0} explicitly asked for by Angles. The proof is elementary: the Artin-Schreier identity theta^p = theta + a turns Frobenius into a translation, collapsing the power sums via a telescoping recursion. The smallest case (p=5, i=124) is confirmed by a direct p-adic computation of the reciprocal roots of L(X, omega_P^124).