This is my first submission to the Opus Magnum tourney ever! Yay! I have followed the tournament on/off for the past couple years but just never participated till now.

My solution is very big. I focused only on rate, leading to 2000+ instructions. In fact, I had to copy+paste the entire solution to achieve max rate, which is not ideal.

=== Analysis ===

To find the max rate, I used CONSERVATION OF MASS. Every element can be given its own mass:
Quicksilver 8, Lead 1, Tin 2, Iron 4, Copper 8, Silver 16, Gold 24.

Specifically, the masses have to work out so that the glyphs can't make more mass out of nothing. Purification and projection either keep the mass the same, or make it decrease (for example, projecting silver into gold preserves mass, but purifying silver into gold wastes mass).

Since the output weighs 136 and the input weighs 15, the maximum possible rate is 2*136/15 = 18.1333 or worse -- you can't make something from nothing, and need at least 136 "stuff" to make an output. So the question is whether it's achievable.

=== Design ===

To get the max rate, we cannot waste any mass. So all of our lead, tin, and iron must be purified into copper.
Also, all of our gold will have to come from projecting silver (since 16+16 > 24 but 16+8 = 24).

But copper is ambiguous: both projection and purification are mass-preserving. So we have to do both!

136 inputs turn into 136 quicksilver and 119 copper. From this, we want to make 15 outputs, which are built from 90 silver and 75 quicksilver.

This means that we have 61 quicksilver left over to do projection ... which turns out to be exactly the right amount, leaving 58 copper to purify into 29 silver.

So this is how my solution works: make two sticks, one with 136 quicksilver and one with 119 copper. Cut 61 atoms off of each and zip them together in a projection. Purify the rest of the copper. Now we've got two chains: 75 quicksilver and 90 silver.

Split the chains into sticks of 6-silver and 5-quicksilver. Then grab one of each stick, and combine them into the output.

Oh, and do all of this twice... My design to build the 100+ stick cannot accept more than 1 atom every 4 cycles, but we need 2 atoms every 4 cycles, so we just have two of everything. C'est la vie.