# Bernoulli convolution parameter classification research

Family 153: Arithmetic classification and non-Pisot singularity for Bernoulli convolutions. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Probability researchers can study arithmetic overlap criteria and Salem-parameter singularity examples.

## Applicability and commercial boundary

The classification uses an infinite one-sided approximation condition, which may not yield a terminating efficient decision procedure. Finite histogram appearance is not absolute continuity.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Inspect effective parameter subclasses and an exact algebraic example.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf).