GENERAL IDEA:

Convert a 5-bit base 2 number into a 2-digit base 6 number. 0 is salt, 1 is fire, and the metals are however many quicksilver you need to promote lead into that metal (lead is 0, gold is 5).

THING I DON'T WANT TO DO:

Division. Parsing a 31-atom-long stick of quicksilver sounds lame and hard.

THING I CAN DO INSTEAD:

Divide as I go along. Imagine a magical two-atom metal molecule where if you try to project the right side when it's already gold, it resets to lead and the left side projects instead. Instead of dumping tons of quicksilver into the right side, sometimes I'll know the bit is greater than 6 in advance. So, for the bit representing 8, I don't project the right side 8 times; we project the left side once and the right side twice. I use this shortcut for both the 16 and the 8 in the binary representation.

DIGIT TWIDDLING:

Also I interpret the 16 as an 18 instead. That still works! I end up with numbers from 0 to 33, but skip 16 and 17. That still fits in the 36 possibilities of metal representation.

CARRY MATH:

With that trick, the magical right atom only ever gets projected 9 times at most, so there will only ever be one carry operation for real molecules, which I can check for at the end. This is extra convenient by processing the bit representing 4 projections last--if we need to carry, we have an atom presence conditional on the final quicksilver that didn't project. This would break on any combination of bits that force a carry operation without the bit representing 4 projections, but there aren't any of those.

UN-MAGIC-ING THE MAGIC:

For the lower digit, I prepare chunky lead to soak up 9 quicksilver. We'll call the halves of the chunky lead the "lower digit" and the "rollover". Each quicksilver is tested against both halves, prioritizing the lower digit and spilling over into the rollover, all the way until the final quicksilver. That quicksilver is only tested against the lower digit, and then another arm regrabs where the quicksilver was, using that atom presence conditional. Then comes some wanding, debonding, more bonding, and drawing the rest of the owl. In parallel, I finish the upper digit; the bits are ordered in a weird way that complicates the instruction tape but saves wSum. The result is that if we didn't need to carry, we output the upper+lower. If we did need to carry, we promote the upper, then output the upper+rollover. Either way, we get a waste atom.

RATE NOTES:

Up to 13 quicksilver may be used in this design, and the conditionals can operate at pseudo-period 3 at best, limiting this design's rate to 39. My actual rate is 40 if we pretend this loops; the fire reader chokes slightly on the fastest bit.

PREDICTION:

This machine will be beaten by something using Ravari's Wheel, which could have a much better Rate. I investigated it as best I could and found algorithms with Rate possibly as low as 28, but I could never find a way to beat the area and cost overhead.