Lower bound for the 'de Polignac constant' Let's introduce the 'de Polignac constant'  $ K_{Pol} : =\sum_{i>0}2^{-k_{i}} $ , where  $ k_{i} $ is the  $ i $ -th positive integer such that  $ 2k_{i} $ is a Polignac number, i.e a number that is the difference of two consecutive primes in infinitely many ways. De Polignac's conjecture is equivalent to  $ K_{Pol}=1 $ .
Do we know a non trivial lower bound for  $ K_{Pol} $? 
Edit : a proof that  $ K_{Pol}>1/2 $ would entail the truth of the twin prime conjecture.

Thanks to the Polymath project we know that infinitely often $p_{n+1}-p_n \le 246$, which means that some gap {2, 4, 6, ..., 244, 246} must occur infinitely often. In the worst case, only 246 would occur infinitely often, contributing 2^-246 to the sum. From Theorem 1.4 we get more precisely:
$$
K_{Pol} \ge 2^{-246} + 2^{-398130} + 2^{-24797814} + 2^{-1431556072} + 2^{-80550202480} + \sum_{m=6}^\infty 2^{-Cm\exp((4-28/157)m)}
$$
for "an absolute (and effective) constant $C$".
For practical purposes the first term gives a sufficient approximation:
$$
K_{Pol} \ge 8.8434366\times10^{-75}.
$$
On the Elliott-Halberstam conjecture the bound improves to
$$
K_{Pol} \ge^? 2^{-12} + 2^{-270} + 2^{-52116} + 2^{-474266} + 2^{-4137854} + \sum_{m=6}^\infty 2^{-Cme^{2m}} \approx 0.000244140625
$$
and on a generalized EH it is further improved to
$$
K_{Pol} \ge^? 2^{-6} + 2^{-252} + 2^{-52116} + 2^{-474266} + 2^{-4137854} + \sum_{m=6}^\infty 2^{-Cme^{2m}} \approx 0.015625.
$$
[1] D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes
