2 atoms are tested at the purifier. 
If they are the same, arm 3 uses the resulting atom to send a quicksilver to the reserved lead, then sends it to arm 5, the "rejection station" that reduces its level by 1, back to what it was before it was purified. This atom then goes to the left side of the purifier to be tested against the next atom.
If they are different, arm 2 makes a stick with the left atom and the reserved lead, which then gets sent down to arm 16.
Because a purification glyph does not test if 2 gold are the same, arm 16 tests whether it receives multiple lead-gold sticks in a row. It does this by attempting to project the metal, and if the quicksilver is remaining, it uses it to send the stick downwards to the bonder.
The bonder holds a bit of information - If there is a quicksilver on the bonder, there is a gold stick to upgrade and the current one is waste, and if there is not, the current stick is the one to upgrade.
If the received stick was not gold, it needs to be rejected at arm 6, then it collects the current gold stick, clears the bonder bit, and goes to the output.