Accoring to this question and a linked duplicate, it's been verified empirically up to some number that all twin prime averages greater than six, are the sum of two smaller twin prime averages.
I was curious whether or not these formed a tree (with twin prime pairs at the vertices and the sum relation the edges). In order for these to form a tree, for every twin prime average, we would only be able to find one pair of smaller twin prime averages which summed to it.
Can you find a counterexample, namely any twin prime average which is a sum of two smaller twin prime averages, more than one way?
Number experiment finds a lot of counterexamples.
The smallest one is $71 + 73 = (11 + 13) + (59 + 61) = (29 + 31) + (41 + 43)$.
If we allow larger numbers, there can be a lot of ways: $$1877 + 1879 =\\ (5 + 7) + (1871 + 1873) =\\ (179 + 181) + (1697 + 1699) =\\ (269 + 271) + (1607 + 1609) =\\ (599 + 601) + (1277 + 1279) =\\ (827 + 829) + (1049 + 1051) =\\ (857 + 859) + (1019 + 1021)$$