I seems to be very few even numbers that can't be written as a sum of two primes with twins or as a sum of two primes without twins.
That is, suppose that $\mathbb P'$ is the set of the primes not belonging to a twin and that $\mathbb P''$ is the set of the primes belonging to a twin, then all but a finite number of even natural numbers can be written as $2n=p'+q',\quad p',q'\in\mathbb P'$ or as $2n=p''+q'',\quad p'',q''\in\mathbb P''$, due to a conjecture.
So far the only exceptions are $0,2,96,98$ of even numbers $<100,000,000$.
I'm interested in counterexamples and references.
Below the number of ways to write $1000$ as sum of two twin primes is 3, and the number of ways to write $1000$ as sum of two non twin primes is $10$.
100 terms cardinality . cardinality . 1 2 ok
1000 terms cardinality . cardinality . 10 3 ok
10000 terms cardinality . cardinality . 63 10 ok
100000 terms cardinality . cardinality . 415 64 ok
1000000 terms cardinality . cardinality . 3455 167 ok
10000000 terms cardinality . cardinality . 27081 1259 ok
100000000 terms cardinality . cardinality . 210124 5854 ok
I see now that it seems to be a conjecture for the case $2n=p'+q',\quad p',q'\in\mathbb P'$, but then with a larger number of small exceptions: {0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,62,64,66,68,72,78,80,82,86,88,92,96,98,108,110,118,122,124,128,138,140,148,152,170,182,188,208,212,218,222,232,238,266,268,272,282,292,296,328,332,358,362,392,482,512,628}.
The corresponding set of small exceptions in the case $2n=p''+q'',\quad p'',q''\in\mathbb P''$ is: {0,2,94,96,98,400,402,404,514,516,518,784,786,788,904,906,908,1114,1116,1118,1144,1146,1148,1264,1266,1268,1354,1356,1358,3244,3246,3248,4204,4206,4208}.
And the intersection is {0,2,96,98}.