# Prime-minus-one factor statistics reference

Family 011: Prime-factor statistics of $`p-1`$. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Number-theory researchers can compare normalized ordered prime-factor logarithms with the Poisson–Dirichlet law.

## Applicability and commercial boundary

An asymptotic distribution over primes does not guarantee a particular prime has desired factor structure or accelerate factoring. Totient multiplicity is a different consequence.

## 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

Inspect finite error rates and test a seeded arithmetic sample without security claims.

## 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).

[Weighted dilation graphs, smooth shifted primes and totient fibers](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf).