GENERAL IDEA:

This is a heavy modification of binary counting to convert between bases, interpreting each metal as a digit in base 6; the metals are however many quicksilver you need to promote lead into that metal (lead is 0, gold is 5). Imagine splitting the 6-by-6 square of possibilities into four quadrants--this consumes two bits, leaving three bits to be packed into each of the 3-by-3 squares. The base conversion problem becomes smaller--3 bits into 2 trits. You can add to a digit with the Glyph of Projection, and do carry math with quicksilver sticks. Everything is fine.

Now imagine drawing the borders of the quadrants differently in order to be cracked and save one (1) quicksilver--that's my encoding. The quadrants are 1) the even/odd parity of the left metal digit and 2) whether the right digit is closer to gold or to lead. So if you rearrange those to put them all together, you could see the base-3 math, but just enjoy the pretty chart Haxton made instead because this stream is getting long.

Instead of "carrying the 1" when a base-3 digit rolls over, I "carry the 2"; this lets me stay in the correct even/odd parity of the left metal. To do carry math, I build a stick of up to 4 quicksilver and conditionally disassemble it to project the correct digits. This is WAY more feasible with small sticks than it is with long base 6 ones; I'll spare you the details.

The reason I don't simply do the binary counting in base 6, where I get to use more convenient presence/absence conditionals on gold, is that this way requires less quicksilver and cycles are kinda important. The binary counting in base 6 can take as little as 13 quicksilver with good strategy; this weird slicing and double carrying only takes 10 quicksilver. I built several designs; pure base 6 came out to 399 wSum, and an 11-quicksilver idea with more sensible quadrants comes out to 377.

RATE NOTES:

This design's rate is limited to 3 per quicksilver, which is 30. My actual rate is 32 if we pretend this loops; the fire reader chokes slightly on the two bits that only inject one quicksilver each.