# Inhomogeneous approximation assumption reference

Family 022: The weak inhomogeneous Duffin–Schaeffer conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Researchers can organize almost-everywhere approximation under the stated divergent totient-weighted sum.

## Applicability and commercial boundary

The claim is almost everywhere, has unrestricted numerators and no finite convergence rate; it does not guarantee a chosen real input or denominator budget.

## 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

Identify an effective deterministic subclass before software approximation 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).

[The Weak Inhomogeneous Duffin–Schaeffer Conjecture](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf).