# Arithmetic progression density reference

Family 159: Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Additive-combinatorics researchers can compare finite progression-free sets to the stated fixed-length density upper bounds.

## Applicability and commercial boundary

Reciprocal-sum divergence is an infinite condition; constants and exponents may be impractical, and no fast progression-search procedure is supplied.

## 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

Extract effective fixed-k constants before a finite planning tool.

## 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).

[Quasipolynomial Bounds for Arithmetic Progressions](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf).