Remark: Family 023. Patterson’s first moment for cubic Gauss sums

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most $X$, including both conjugates, have an explicit positive main term of order $X^{5/6}/\log X$. Every fixed nonzero prime-angle Fourier mode has smaller order.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.