--- DO NOT RED THIS PART OUT LOUD ---
Fell free to jump around the order of the explanation if some tech was alreadly  found before or if you are interested in an specific part of the machine.


THERE IS AN -1c SAVE VIA 6Ping IT, PLEASE COMENT ABOUT THIS IF I TIE OR WOLD TIVE WITH -1
-------------------------------------


---CONDITIONA PURIFICATION TECH---

I stole this tech from biggie's Alchemical Duplicator.

arm 8 is the center of this solve. It conditionally supresses the quicksilver with some ghost atoms. By loading it with 6 ghost atoms and contitionally purifying 10 times we get exactly the necessary quicksilver to get GOLD+GOLD, this makes 4 ghost atoms conditionally purify twice and the other 2 conditionally purify only once.

The arm can purify and be reloaded at period 3, unfortunally unloading takes period 4 so the total period is 3*10+6 = 3*4+4*6 = 36. This is an blessing in desguise because it allow us to process the input at period 4 with the possibility waiting for an ceirtain arm to catch up. 

theoretical rate optimizations(it may be worth to skip this): 
*if arm 8 hovered above an disposal, it would allow for an period 3 unloading via having an arm holding the atom to avoid the disposal, this would be -36-ish cycles(-6r) but it would put heavy restrictions on disposal that would likely make it hard to dispose of salts, costing up to +30a. 
*We could have 2 slower purification station at total period 20. But I wasan't able to imagine an cheap machine like this.
*we could get by with 9 quicksilver, but this would force an extra conditional beside the ghost atoms.
*Ravari's Wheel may be usefulk in order to implement an arithmetic mod 6 or something even weirder.

---CODIFICATION---

(Warning: depending on what you count as "0": (fire/salt, present/absent atom, gold/lead) the folowing logic may need to be negated to make sense, but the analysis will be exactly the same)

We have 4 ghost atoms who purify twice and 2 who purify only once. In order for each half of the string to be conditionally purified 5 times we need for half of each group to purify each half. Since the parities of the atoms of the output is defined by the ghost atoms who purify once, so we can just copy
2 bits from the input into those 2. The rest of the problem is to encode 3 bits into 2 trits(ternary digits) using an intermediary unary encoding. 

Extra explanation if needed: we are treating the seximal digits as an trit and an bit where the trit is more significant, and in the intermediary encoding(the ghostatoms in arm 8) they are both being encoded separatly in unary(witch is the same as binary for a single bit)

The intermediary binary encoding makes linear transformations natural by just copying bits into the encoding, in particular if we consider the transformation (a,b,c) ->(a+b,a+c) we get something that can be interpreted as the drawing of an cube in an oblique projection(drawing) where the front of the cube(0,1,1) is on top of the back(1,0,0).

2      *----*
      /|   /|
     / |  / |
    /  | /  |
   /   |/   |
1 *----*----*
  |   /|   /
  |  / |  /
  | /  | /
  |/   |/
0 *----*

 0    1    2

Now we just need to solve the problem of this projection not beeing injective, we can do this by trowing one of the points in (1,1) to (2,0). Is as follows:

Let A,B,C be the bits from the input(ignoring the easy 2 bits).
A generates 2 ghost atoms who will usually each afect an diferent trit.
B and C will each generate 1 ghost atom each afecting an diferent trit
One of the 2 ghost atoms from A will swap the atom it afects if neither B or C is present. This can easly be checked with an bonder+debonder conditional.

In other words, we compute A and not(B or C) and in this case swap the placement of te ghost atom.

---ARMS---

1+2: They deliver the input to arm 4.
obs: the extra unecessary track is for latency

4: It transforms the fire/salt bits into an present/absent ghost atom bit. It also pivot one of the atoms to generate 2 ghost atoms. Also Recycles fires on it's free time.

6:We can't instantly deliver the ghost atoms because we need to compute the positioning of the special case ghost atom. So we use arm 6 to delay the delivery.

5:In order:
*Grab the pair of ghost atoms
*Reuses the logic debonder to deliver one of teh ghost atoms to arm 6 while narrowly avoiding the next ghost atoms coming from arm 4, unfortunally it delivers the atom in the wrong position but arm 7 will fix it
*insert the special atom on logic bonder
*wait frot logic to happen from arm 6
*insert the special ghost atom in arm 6 with the position conditioned on B or C

7:Delays the first ghost atom and does the reloading of arm 8

8:Explained before
 
9:Choses witch of the atoms get's projecter for each ghost atoms.

10:Unloads arm 8

3:Trow away the salt.
obs:YOU MAY THINK BASED ON IT'S INSTRUCTIONS THAT IT IS CLOSELY RELATED TO ARM 2 BUT IT"S ACTUALLY A CLOSE RELATIVE TO ARM 10 WHO JUST RECENTLY LOST IT"S TRACK. THIS IS JUST CONVERGENT EVOLUTION.