# Cubic Gauss sum exact reference library

Family 023: Patterson's first moment for cubic Gauss sums. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Analytic-number-theory researchers can compare the explicit first moment and angular-mode behavior over primary Eisenstein primes.

## Applicability and commercial boundary

All-primary sampling, both conjugates and asymptotic error terms matter. This does not improve generic FFT or signal modulation without a separate computational bridge.

## Initial business decision

Research infrastructure or conditional engineering until a computable implementation, effective bounds and a recurring buyer need are demonstrated. No commercial novelty or profitability is established.

## Next verification action

Extract the main coefficient and exact finite sum implementation.

## Evidence scope

The catalog statement was individually reviewed. Manuscript proof and selected formal scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[An unconditional first moment for cubic Gauss sums](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf).