An unconditional first moment for cubic Gauss sums

We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is $(6/5)c_{\ast }{X}^{5/6}/\log X$, where $c_{\ast }=(2\pi {)}^{2/3}/(3\Gamma (2/3))$. For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.