I used this solve and my other showcases (gold duplication and partial gold duplication) to justify to myself which metal to duplicate, and these are the things I thought about when doing so, so feel free to just show one of them or to show them all together.
The number in the title refers to how may cycles it takes for 24 silvers to become utilizable, ie. not where they were first created but the first cycle where it could be bonded, so the theoretical minimum cycles count for this puzzle would be that+2, since after bonding you still need to move and drop. At first glance at the solution that number might appear to be 36 cycles, but you can grab and move the silver on the catalyst part of the glyph of projection, meaning you actually have an extra silver, just one you can only use for the last product, making the number 34 cycles. Silver duplication is much simpler, and since the glyph of purification is included in the puzzle, an obvious thing to try is to duplicate gold instead, rejecting a quicksilver from each one to create the next gold. This self-sustaining cycle adds a lot of complexity, but means that you only need three silver to jump-start the machine, and can pull silver from the input at full speed for use in products afterwards, and while I didn't do this in my gold duplication showcase, you would also reject the gold on the catalyst back down to a silver towards the end, meaning you don't have to reject the last few silver produced from projection and can also reuse the catalyst similar to the silver projection. Despite the seeming advantages, gold duplication makes 24 silver usable by cycle 35, losing out due to the delay to the first projection, and only wasting one less silver, since you have no glyph of projection so you need to use two silver to make the initial gold, and only get one back later, and you still need one other silver to bootstrap the gold loop. I also tried starting with silver to get the loop going, then quickly promoting the catalyst up to gold (partial gold duplication), but since it wastes a cycles of the gold loop and still requires an extra silver to promote, it (thankfully) still only hits 34 cycles. If the glyph of projection was available, then the silver loop could maybe hit 33 cycles to 24 silver usable since there exists enough waste metalicity among the two waste leads, remaining tin, and unused quicksilver to make one more silver, but I am once again thankful that isn't even an option. Overall, I'm 90% certain that I have minimum cycles on my cycles solve because of all this, but the glyph of purification not being used, along with it's application only making things more difficult plants a seed of doubt within my mind that I've missed something, though I guess I'll find out on Sunday.