This cycles puzzle was fun! Usually cycles isn't my favorite because I can theorize all I want about what would need to be done, but then I can never make the geometry work. This one was approachable enough that I managed to do it.

My secondary doesn't seem great. It should be possible to combine arms 3 6 and 7 into two arms, and the top solves will surely have a lot more multi-purpose arms, but I'm still proud of what I got.

Here comes my analysis of the theory to get min cycles, feel free to skip it if it has already been discussed enough on stream:

Silver is the limiting factor, you can grab 1 every 2 cycles from the input and you can use Proliferation once every 3 cycles. There are two different possible approaches:

A. Proliferate Silver
In this case you need to regularly consume Silver to extract Quicksilver from it in order to keep the proliferation running. The earliest Silver from Proliferation can be grabbed on cycle 6.

B. Proliferate Gold
In this case you can reject the Gold to gain Quicksilver, so the Proliferation feeds itself. This is the correct approach for rate, but not necessarily better for cycles. The earliest possible Gold from Proliferation can be grabbed on cycle 10.

Some key insights I had along the way:
1. The atom that you use as proliferation template isn't lost, it can be used in the end when you're done with proliferation
2. With approach A, you can get additional Quicksilver from purifying your Lead
3. With approach B, you should still start by copying Silver, because you want to start Proliferation as early as possible

According to my calculations, there are 3 different ways to grab your 24th Silver on cycle 33, which means last output on 36:

A1: Use only 2 Silvers to extract Quicksilver, which means you can only proliferate 9 times even though you'd have time to proliferate 10 times.

A2: Use 3 Silvers to extract Quicksilver, which lets you proliferate 10 times until cycle 33 and you don't need to bother with purifying Lead.

B: Use 1 Silver to extract Quicksilver to start your Proliferation engine. Use 2 Silver to make a Gold template atom, but in the end you can get 1 Silver back from it by rejection. Start by proliferating Silver twice, then proliferate Gold. For the last proliferation, you can do Silver again to save the rejection step. In this case you could actually grab your 24th and 25th Silver both on cycle 33, but this doesn't help, the limit is still the same.

I tried A2 first, since it seemed conceptually the easiest, but it meant I grab my last silvers on cycles 31, 33 (from Reagent), 30, 33 (from Proliferation), 31 (template atom). This was too much at once for my assembly stations, which are designed to accept 1 Silver every 2 cycles. So I switched to A1, and I think this is the easiest concept to get min cycles. I'm curious to see if the top solves take the same route.