How it works: The goal was to try to solve this puzzle without gears. I started with the pyramid structure with 6 bits. This pyramid has 6 exit points. I mapped out exactly when which exit point gets taken, to see if some combination of exits had a timing pattern that matches “2”, “4”, “8” as needed in this puzzle. And indeed there is: a combination of the rightmost exit and one coming from the middle. One more bit to the blue side is added to skip every second one from the pyramid (so the timing inside it is really “1”, “2”, “4”), and on the red side two bits to have the 3th one go to the intercepter.
The bottom left bit of the pyramid can be replaced by a ramp, but it looks less aesthetic (I wouldn’t be surprised if a smaller structure than this 6-bit pyramid could do it and make a solution with less total parts, I didn’t try yet).