OpenAI · 023-01 · 2026-09-25 · Number theory
An unconditional first moment for cubic Gauss sums
preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026
Family 023. Patterson’s first moment for cubic Gauss sums.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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 , including both conjugates, have an explicit positive main term of order . Every fixed nonzero prime-angle Fourier mode has smaller order.
We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is , where . For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 023: Patterson's first moment for cubic Gauss sums. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.