Read the Berke paper, thanks. I see a lot of work has gone into search functions and frequency functions for twin primes.
Is there a history of attempts based on symmetry?
Create two sets, a set A of composites dividing by primes from 2 to p_n, and a set B of composites _only_ dividing by primes from p_n+1 upwards.
A shows strict reflective symmetry between multiples of the LCM of the set. For example, if a 'gap' is two consecutive odd numbers neither of which is an A-set composite, then gaps between 2 times (22.214.171.124.126.96.36.199) and 3 times (188.8.131.52.184.108.40.206) will be mirrored in 3/2 times (220.127.116.11.18.104.22.168), such as 2 times (22.214.171.124.126.96.36.199) + 29, +31 being one such gap, and 3 times (188.8.131.52.184.108.40.206) - 31, -29 being another. Call this the 'range'.
B-set composites must be asymmetrical over this range, both in number and distribution.
Define 'filling a gap' as finding a B-set composite at either one or both of the members of the gap pair. Even though this means there is space for up to twice as many B-set composites in the upper half of the range as in the lower half of the range, their distribution as a set still cannot be symmetrical enough to suffice. Either all the gaps are filled in the upper half, in which case some must be unfilled in the lower half, or there are unfilled gaps in both halves.
Bearing in mind that the two sets cannot overlap, so that there is "nowhere else to go" for the B-set composites except spaces in the A-set mask, the B-set composites cannot overflow the gaps available and by lack of symmetry, some must remain unfilled. If we posit a symmetrical-enough distribution for set B, it should be simple to show that individual members of B stray outside the frame of gaps available.
Therefore, by adjusting the size of set A, there are infinitely many twin primes.
Obviously very crudely outlined, but surely someone must have tried a more rigorous version of this? Presumably there is a way of measuring 'degrees' of symmetry?