For a while I dismissed 3 cycles per comparison to be not even worth testing -- surely it would be too expensive and complicated.

Then I realized that by placing Unification and Dispersion right, the atoms would seamlessly recycle back into the checker without any fiddling.  All that was left was to set it up.

Of course, this is probably still too expensive and complicated.  I'm guessing there's a 5-period machine with a similar recycling idea that gets around 800 wsum by destroying this one on cost and area.  I just can't be bothered to build it at this point.

I found this puzzle to be way more about delivery and disposal of atoms than the computation algorithm.  Quintessence logic is temperamental.  For this reason, all my solutions heavily featured hex arms.

Here's what this does:
First, it converts the match pattern into 3 anti-match shells.  These get held by arms 5, 6, and 7.  The key part of the conversion is shuffling the atoms on the glyph of dispersion so that no matter what element each atom is, you always end up with the right three atoms in the same places at the end.  There was some work done to pick the order of cardinals that are checked in this process -- I think this is near-optimal.

Second, it loads up the main wheels (hexarms 1, 2, and 3) with salt.  Now setup is complete, and we've only wasted 157 cycles!

Third, BRRRRR.  Every 3 cycles, it tests an atom.  If it doesn't match, it wands off the salt, and makes a 0-3 atom buffer.  Every 3 checks, the top salt (if it exists) wands gold out of the way.  I think checking for non-match is cleaner, because the machine doesn't have to count.  Also, checking for non-match makes sure that the machine has a salt available when it's time to output.  The other way, a sample that matches NONE of the input atoms will have to find another way to get salt.

Only two other things to draw attention to:
1. The goofy input delivery.  Amazingly, this worked out to the same cost and area as my stick-based input I had before.  However, it allowed me to get rid of pauses in between inputs, and run at max rate the whole time.  Hooray 5 metric points!
2. length-3 arm dodges.  Probably known to the wizards out there, but I found it in this puzzle.  Check out the berlo for what I mean.