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Jack -- they are two different things.

Input lag measures the time between when a signal is sent to the projector and when that signal is actually displayed, compared to an external reference monitor. The signal that makes it to the projector is still in its native frame rate and can still have all kinds of problems with judder and motion.

Frame interpolation smooths out motion in film and video by analyzing the frames of the signal and creating new interstitial frames between them. This makes video appear smoother and drastically reduces judder, but it also increases processing time and thereby also makes input lag worse.

Here's an analogy: if you order a pizza from a national chain, they will have it to your door very quickly, but it probably won't be a very good pizza. On the other hand, you can get a really delicious pizza from an actual pizza parlor, but you won't have it in 30 minutes. Increasing quality (by creating interstitial frames) increases delivery time (input lag). The analogy has its limits, one of which is that I'm now hungry for pizza, but I hope that helps.