We study how micro-level timing of push notifications affects user engagement. Industry practice assumes that delivering messages at screen-on moments—when users are actively using devices—maximizes effectiveness for mobile applications. Using large-scale data from a leading push notification platform in China, we exploit a natural experiment where a server-side ``listening window" mechanism generates quasi-random variation in whether a push is delivered instantly at screen-on. Our regression discontinuity design shows that screen-on push counterintuitively reduces same-day user logins in the sending app. To explain this puzzle, we estimate the causal, non-linear effect of delivery lag by instrumenting latency with whether the user's nearest screen-on signal occurs just after versus just before the listening window opens within a narrow time bandwidth, which mechanically shifts delay. Our results reveal an inverted-U relationship: modest delays (about 1-10 minutes) increase engagement relative to near-instant delivery (0-1 minute), whereas longer delays underperform. The findings suggest that hyper-precise timing can backfire in advertising: attention capture requires carefully calibrating for interruption costs, not just minimizing delay.