An idea occurred to me. OP is looking for a die that is approximately 13/64-24 which as he has already said is very close to #12-24. An ideal 13/64-24 calculated using modern thread tolerances is only ~.013 inches smaller in diameter than standard #12-24.
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Inexpensive thread dies you get from Home Depot and similar are solid and most often hex in shape. Thay are primary for chasing damaged thread not necessarily for the best cutting new threads in raw material (all though they will do it).
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A proper thread cutting die usually used by a machinist is round in shape and they are often (not always) are split on one side so that machinist using the internal screw in the die and the screws on the die holder can make fine adjustments to the resulting threads diameter to make it slightly larger or smaller thread than nominal. It might be possible to take a #12-24 dies (preferable one made on a small die blank like 13/16" or 1" OD so that is it more flexible and adjust it down to the point were it cuts something approximating a 13/64-24 thread.
It might be worth a try.
That sir is a very good idea.
But first I need to determine the angle of the thread. I understand that common imperial threads are 60 degrees while Witworth threads sre either 50 or 55 degrees. I'm not sure what metric threads are. It is likely that this thread is neither imperial or Whitworth as it predates Whitworth by 50 plus years or so.
I have not been able to determine the angle from the impression I received. Casting a positive from the impression has proved a bit more difficult than I thought. Once I have a positive the best I can hope for is to be able to compare the positive to a known 60 degree thread using very high magnification.
I'm trying real hard to not recut the threads on the bottom of the cock. I'm not sure how this will resolve but I'm in no rush.
Thanks for the 'outside the box ' thinking.