Earlier this year, Yitang Zhang published a proof of a weakened form of the Twin Prime Conjecture. I'm wondering if any of the new mathematical machinery he developed has uses in cryptography or could be adapted for future use in the field.
There are no known applications or implications to cryptography of that mathematical machinery. I've not seen any suggestion that the machinery would plausibly have implications for cryptography. The twin primes problem doesn't have any obvious connection to any hardness assumption or problem in cryptography.
(Yes, both Zhang's proof and cryptography have something to do with prime numbers, but that is extremely thin. Much of number theory can say it is somehow related to prime numbers, but only a small fraction of number theory has had applications to cryptography. It'd be like assuming that recent improvements in building fast sailboats for the America's Cup has implications for the possible future availability of an immortality potion, since they both involve fluids.)
In other words, this is about as close to a "No" answer as one is likely to be able to give, given the nature of the question (which makes it very unlikely that you are going to get a definite, resounding no).
Here are some surveys of the mathematics of this result and related results, if you want to take a look for yourself: