Encoding explained: bit 2 and 4 are simple parity driving one quicksilver each and bit 1, 3, 5 influences 2 quicksilver each. In a sense this reduces the encoding space into 8 cases for 3x3 metals. In this corner of encoding space, the first two checks on bit 1 and 3 are straightforward: they flip by one each. The magic happens on the last check where the third pair of quicksilver may fall on 3 different places:
bit 1 | bit 3 | output so far | if bit 5 is salt | if bit 5 is fire
------+-------+---------------+------------------+-----------------
salt  | salt  | 10            | move 0, 10       | move 1, 20
fire  | salt  | 11            | move 1, 21       | move 2, 12
salt  | fire  | 00            | move 0, 00       | move 2, 01
fire  | fire  | 01            | move 1, 11       | move 2, 02
(0 = lead, 1 = iron, 2 = silver)
This as boolean functions are not too bad to implement. Arm 3's salt dupes from bit 1 and bit 3, and triplex bonds onto bit 5. This triplex bonded salt is carried to main checking station and arm 12 picks it up and moves the stick by 2. Arm 2's salt dupes from bit 1 and dupes onto bit 5, so when bit 1 is fire and bit 5 is salt, arm 8's known fire moves the stick by 1.

I'm too tired at this moment to explain why turing machine is a good idea. The fun thing here is that this solve is twice inverted turing machine. On where state and output meets, previously seen patterns have state stick "write" to output but here the stationary quicksilver projects metals on the stick, like a typewriter. On where input and state meets, previously seen patterns have input bits wand a stick of mostly known fires (or salts when a certain state doesn't want to be moved) but here known fires wand the input bits on state stick. With these two inversions the machine looks more like some assembly line.

Machine runs at 22R and the sole bottleneck is arm 7. It seems that taking its work away requires a new arm which gets some +20G/-5C/+1A.
 
===== 'TURING ASSEMBLY LINE' =====
 0.	🜔🜔🜔🜔🜔	 => 	♀ ♄
 1.	🜔🜔🜔🜔🜂	 => 	☉ ♂
 2.	🜔🜔🜔🜂🜔	 => 	♀ ♃
 3.	🜔🜔🜔🜂🜂	 => 	☉ ♀
 4.	🜔🜔🜂🜔🜔	 => 	♃ ♄
 5.	🜔🜔🜂🜔🜂	 => 	♀ ♂
 6.	🜔🜔🜂🜂🜔	 => 	♃ ♃
 7.	🜔🜔🜂🜂🜂	 => 	♀ ♀
 8.	🜔🜂🜔🜔🜔	 => 	♂ ♄
 9.	🜔🜂🜔🜔🜂	 => 	☽ ♂
10.	🜔🜂🜔🜂🜔	 => 	♂ ♃
11.	🜔🜂🜔🜂🜂	 => 	☽ ♀
12.	🜔🜂🜂🜔🜔	 => 	♄ ♄
13.	🜔🜂🜂🜔🜂	 => 	♂ ♂
14.	🜔🜂🜂🜂🜔	 => 	♄ ♃
15.	🜔🜂🜂🜂🜂	 => 	♂ ♀
16.	🜂🜔🜔🜔🜔	 => 	☉ ♄
17.	🜂🜔🜔🜔🜂	 => 	♀ ☽
18.	🜂🜔🜔🜂🜔	 => 	☉ ♃
19.	🜂🜔🜔🜂🜂	 => 	♀ ☉
20.	🜂🜔🜂🜔🜔	 => 	♃ ♂
21.	🜂🜔🜂🜔🜂	 => 	♃ ☽
22.	🜂🜔🜂🜂🜔	 => 	♃ ♀
23.	🜂🜔🜂🜂🜂	 => 	♃ ☉
24.	🜂🜂🜔🜔🜔	 => 	☽ ♄
25.	🜂🜂🜔🜔🜂	 => 	♂ ☽
26.	🜂🜂🜔🜂🜔	 => 	☽ ♃
27.	🜂🜂🜔🜂🜂	 => 	♂ ☉
28.	🜂🜂🜂🜔🜔	 => 	♄ ♂
29.	🜂🜂🜂🜔🜂	 => 	♄ ☽
30.	🜂🜂🜂🜂🜔	 => 	♄ ♀
31.	🜂🜂🜂🜂🜂	 => 	♄ ☉