My usual method for determining minimum Rate is to figure out the most efficient recipe for one product, see how much waste is left, and then see how many of those waste sets are needed to make an odd-out product.  That may not be the fastest way of finding the number, but I like it because it also reveals a construction recipe that I can understand and implement.

That didn't work for this puzzle.  10:1 is the minimal-input single-product recipe, but the waste is an even-more-awkward fraction of a product. Larger-scale attempts of the same strategy also did not resolve neatly; so, in the end (and for the first time ever) I had to set up a system of linear equations to figure out the target ratio.

And no wonder.  The numbers in this puzzle are *crazy*.  Here are the ones that I found to be the juiciest:

136/15  -- The ideal ratio of inputs to outputs.
119 -- The number of coppers created via purification per batch.  This is 7*17; a silly pair of numbers in their own right.
61 -- The number of coppers that must be projected per batch.  THIS IS A GOODAMN PRIME.

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Normally, there are two ways to resolve awkward ratios and strange numbers.  

The method I typically prefer is to program a long tape that enumerates all the necessary operations in the proper proportions. But the secondary metric here is instructions! By this method, simply pulling and dropping all of the inputs would add up to over 400i, with total instruction count likely to be well over 1000.  Not playing well to the metric at all.

Thus, I reluctantly reached for the other method: programming a short tape that relies on geometry-based conditionals. 

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You can see the realization of that choice in this solve.  The machinery surrounding the input builds sticks of copper and quicksilver to the desired lengths, stacks them into a brick, and pushes everything to the right for processing.  

Specifically: for copper, it builds a stick of 61 and attaches it to a stick of 58.  119 total, with the atoms due for projection and purification in separate rows.  For quicksilver, it builds two sticks of 45, for double-projection to gold, and attaches them to one stick of 46, for 16 projections to silver plus 30 output atoms.  The center-left machinery cluster handles all of this promotion, and rebricks the remaining atoms.

The center-right machinery rearranges the new brick into a form that can more easily be turned into products.  I was surprised and pleased to find that my newly-developed skills from building a P3 machine in Week 6 came in handy, as much of this rearrangement is accomplished via freespace-I style track rivers (with horrible lengths in multiples of 17!).

Finally, the top-rightmost machinery assembles and outputs the products.  Mercifully, this part of the machine works at a much-friendlier 1/2P cadence. 

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My initial build for this strategy took 4 days, and consisted of a half-rate machine at Period 4.  Duplicating, mirroring, and merging that yielded an instruction count of 422.  I was glad to have a stake in the ground, but did not like the high ratio of grabs & drops to useful operations.  Moreover, I was pretty sure that a single-pipe machine would score better.

With another 4 days effort, I was able to reimplement the same strategy as a single-pipeline Period 8, yielding the machine you see here at 262i.  Very much better on secondary!

Interestingly, the only portion of the machine that truly utilizes P8 is the mess of input mechanisms on the far left.  Most of the rest operates on pseudoperiod 4, but P8 allows it to halve the number of grab-drops.   

I feel comfortable betting that P8 will be the sweet spot for top solves, as the next highest option is....P17.  Ugh.