# Primitive-root prime generation assumption audit

Family 029: Primitive roots for every admissible integer base. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Finite-field library researchers can examine whether the new all-base infinitude statement removes assumptions from a relevant deterministic backend.

## Applicability and commercial boundary

The lower count has base-dependent constants and sufficiently large x; it does not provide a fast next-prime algorithm, Artin’s exact density or cryptographic parameter security.

## 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

Trace the exact primitive-root generation theorem and effective costs before modifying opportunity 016.

## Evidence scope

The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scope notes were also inspected. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Primitive roots for every admissible integer base](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf).