Perhaps the most surprising result is that if the primes have level of distribution in arithmetic progressions greater than 1/2 then one can prove there are infinitely often bounded gaps between primes. Since the Bombieri-Vinogradov theorem implies the primes have level of distribution 1/2, this is just beyond what can be proved unconditionally. If the primes have level of distribution 1 (the Elliott-Halberstam conjecture) then there are infinitely many pairs of primes with difference 16 or less. Unconditionally we can prove that there are pairs of primes much closer together than the average distance between consecutive primes.
There are many important questions concerning our method which we have not yet been able answer. Can one obtain bounded gaps between primes unconditionally by this method? The method is very good at detecting two primes close together, but it fails to detect three or more primes close together.