*ABOUT THE SOLUTION*
n-size of the string
m-size of the pattern
p-number of inputs
There are better algorithms like KMP that run in O(m+np), while the trivial one runs is O(mnp). Since m = 3 this is not worth it unless you find a way to implement it without a big constant.

Since the pattern is constant,it makes sense to preprocess it so we can compare with it faster, I do it by calculating the complement of the characters that are in it and putting them on the unification. Since each comparison consumes them,we need a fast and cheap way to reset them. I found this way to do it with period 8, area 40(considering waste) and cost 280G(if we swap the pistons to arms) and it seems like that going any faster will need double the gold or generate a lot of waste since it takes a while to do contitionals on the result of the unification. Also, 3 comparison modules seems to be the sweet spot.

I reused these comparison modules to do the preprocessing to save G and A.

The rest of the solve is kind of unoptimized, I just trewh arms until it was solved. My wishlist for this solve is:
*a way to give the hexarms the trash generated by the modules so we can merge it into the disposal
*a smaller and cheaper way to do the preprocessing
*a way to put the input of the modules closer together

I think that the best I could do with this solve is about 1500G/250C/150A = 700 wsum

*ABOUT THE TITLE*

I proved that deciding if a machine has a rate is not computable even with an HALT oracle, and I plan on writing a paper about it. So even if GOD himself gave panic a HALT oracle he wouldn't be able to stop my shenanigans.