4 inches is deep for a standard dimensioned MERV-rated filter. It’s normal for a high quality but sadly usually proprietary fancy HVAC filter (Aprilaire, Lennox, etc).

That's true, but I want to know what they meant when they said the 4-inch pleating is designed to reduce air resistance. Since it's an unusually large amount of depth for a MERV filter, I would expect the opposite to happen?

The 4-inch pleating translates into a larger surface area of filter material, which in turn means the flow rate per area is reduced. So you have the same amount of air going through 4x the area of filter material compared to a 1" filter. Its analogous to using a thicker wire with the same current.

Air resistance goes down with thicker filters because filter area goes up.

Note, the actual filter material is not "thicker", but is just folded or "pleated".

Ok, yes. Do we know how the air resistance/air flow varies as a function of surface area? Is it almost linear? I'm wondering how important it is.

I've used both. I don't have hard numbers since they are different filters, but I replaced a 1" filter with a 4" filter and I can run my fan at a lower speed.

Feel the air coming out, vs without a filter. The slower the air is, the fewer opportunities you get to trap each dust grain. Overall miss rate improves exponentially with airflow speed.

That doesn’t make sense, the higher the flow rate the higher the energy each particle has so more chance of it making it through the filter.

You might imagine so, but that turns out not to be how it works. Each time through there is a chance the particle will be blocked. It doesn't matter how fast in went in, because the mv^2 of a dust mote is tiny at any plausible speed. What matters is that N trips through gives you (1-p)^N probability of not getting caught, which falls off very fast... in fact, exponentially.

A deeper-pleated filter is the way to reduce velocity in traversal.