Online attention is commonly summarized using cumulative volume, peak activity, or arithmetic averages, but such measures can obscure differences between activity that is sustained over time, concentrated near the present, or renewed after dormancy. This paper introduces the Time-Horizon Engagement Effectiveness (TH-EE) framework, which constructs interpretable temporal profiles of online attention by distinguishing three related but non-equivalent properties: persistence, recency, and re-emergence. We evaluate the framework through controlled engagement traces, a proof-of-concept application to 1,850 YouTube videos across 37 topics (18 cohorts selected a priori as exemplars of persistent, acute, cyclical, and recently originating attention, and 19 cohorts corresponding to authoritatively debunked claims), and an event-level validation on eleven years of daily Wikipedia pageview series for the same topics. The controlled analyses show that the framework distinguishes distributed activity from concentrated bursts, introduces temporal-order sensitivity through recency weighting, and identifies renewed activity after a defined dormant interval. On the event-level series, the framework's reactivations co-locate with Kleinberg burst onsets, and PELT change points far more often than chance. The YouTube application shows that debunked-claim cohorts do not occupy a unique region of temporal-profile space; they exhibit heterogeneous patterns that overlap substantially with benign topics. These results support treating persistence, recency, and re-emergence as separate dimensions of online attention. TH-EE is a descriptive and comparative measurement framework, not a classifier of misinformation, coordination, intent, or content veracity.

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