Encoding:
(1=Fire,0=Salt)
00000: Lead-Iron
00001: Lead-Tin
00010: Lead-Lead
00011: Lead-Gold
00100: Lead-Silver
00101: Lead-Copper
00110: Tin-Iron
00111: Tin-Tin
01000: Tin-Lead
01001: Tin-Gold
01010: Tin-Silver
01011: Tin-Copper
01100: Iron-Iron
01101: Iron-Tin
01110: Iron-Lead
01111: Iron-Gold
10000: Iron-Silver
10001: Iron-Copper
10010: Copper-Iron
10011: Copper-Tin
10100: Copper-Lead
10101: Copper-Gold
10110: Copper-Silver
10111: Copper-Copper
11000: Silver-Iron
11001: Silver-Tin
11010: Silver-Lead
11011: Silver-Gold
11100: Silver-Silver
11101: Silver-Copper
11110: Gold-Iron
11111: Gold-Tin

Notes:
This puzzle is unique in that any encoding is allowed as long as it's a one-to-one mapping, but I'm not very creative, so the only implementable mapping I could think of is to treat the input as a binary string and the output as its equivalent base-6 string. Initially made a solve using a 6-atom counting stick. However, it is apparent that the limiting factor in the weighted sum is cycles, so a method that does the counting faster will get a better score. This involves minimizing actions per count, and the fastest way I could think of is simply a held atom moving back and forth to move an object. This solve prepares a 36-long counting stick by unrolling Ravari wheels. The quicksilver is placed at intervals of 6 atoms and is used to promote a lead for the sixes digit. Since the ones digit only requires cycling through the 6 metals in some fixed order, this solve will produce a valid encoding for any orientation of the Ravari wheel input. In the submission, I've chosen iron to be value 0.

Overall, the counting part is pretty fast, but a similar number of cycles is needed to dispose of the uncounted atoms, so it's not ideal. I suspect the fastest solutions may not involve counting at all, but some kind of conditional algorithm that can quickly partition the output space. 

Thanks to Haxton for hosting this year's tournament, and see you guys in the weeklies!