In light of the previous post the following is obviously a long stretch, but I can’t resist mentioning it.
The way Jones-Sato-Wada-Wiens construct their polynomial is to start from the characterization of primes by Wilson’s theorem (namely that is prime iff divides ), to convert that into a system of exponential equations, and this is itself converted into a larger system of polynomial equations, which are finally assembled into one big polynomial.
Now twin primes can be characterized also by a Wilson-like statement, namely are twin primes iff divides .
So one could construct a polynomial in several integer variables along the same lines, and end up with the (difficult to exploit) reformulation of the twin prime conjecture : there are infinitely many twin primes iff takes on arbitrarily high positive values on (where is the number of variables). Said differently, a disproof of the conjecture would follow from an absolute bound.
By Polymath8b, a similar polynomial for the case of pairs is known to indeed take arbitrarily high positive values, while of course the same is true for the original Jones-Sato-Wada-Wiens polynomial since there are infinitely many primes. So those two “sandwich” somehow, and there just might be hope to say educated things about that (which I obviously can’t, but the learned reader is encouraged to do so, should that seem to be a good idea).
Update: for sake of completeness, I should have mentioned that the Wilson-like characterization of twin primes goes back to a paper of Clement in the Montly from 1949, and the analog for pairs of primes has been worked out by Lee and Park to be that divides .