A simple heuristic of the first million primes shows that no prime number can be bigger than the sum of adding the previous twin primes.

$7 < 5 + 3$

$11 < 7 + 5$

$17 < 11 + 13$

$23 < 17 + 19$

At larger numbers:

$4886639 < 4886489 + 4886491$

$5389451 < 5388869 + 5388871$

$3155597 < 3154757 + 3154759$

I assume that if it could be proved, it would prove the twin prime conjecture of whether twin primes exist forever.

So I am not exactly seeking for a proof, but rather for possible explanations or references for why it is assumed true (or not)?

Also as the list grows, there seems to be a range for how small or big can a prime be in comparison to the sum of adding the previous twin primes.

As the list grows, a prime is usually never bigger or smaller than slightly above $50\%$ of the sum of the previous twin primes. Any references for such a range will be appreciated too.

*Update: When mentioning "the previous twin primes", I am implying to: $(107, 109), 113, 127, 131, (137, 139)$.

$131 < 107 +109$