Remark: Family 022. The weak inhomogeneous Duffin–Schaeffer conjecture

Proves that for every real shift $\gamma$ and finite-valued $\psi :\mathbb{N}\to [0,\infty )$, divergence of ${\sum }_q\varphi (q)\psi (q)/q$ implies $\Vert qx-\gamma \Vert <\psi (q)$ for infinitely many $q$, for almost every $x$. Here $\varphi$ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on $\gamma$ is needed.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.