Problem 73. Minimal symmetric tuples from consecutive twin primes...
https://www.primepuzzles.net/problems/prob_073.htm

Definition
the tuples from k consecutive twin primes
p1, p1 + 2, p2, p2 + 2, ..., pk-1, pk-1 + 2, pk, pk + 2
is symmetric if the following condition is satisfied
p1 + pk + 2 = p1 + 2 + pk = p2 + pk-1 + 2 = p2 + 2 + pk-1 = ...

Diameter of tuple is the quantity d = pk + 2 - p1.

We will consider minimal symmetric tuples from k consecutive twin primes with minimal diameter.

k = 2
5, 7, 11, 13

We will write it down briefly
5: 0 2 6 8
d = 8

0 2 6 8
is the pattern of tuple.

k = 3
5: 0 2 6 8 12 14
d = 14
k = 4
663569: 0 2 12 14 18 20 30 32
d = 32
*k = 5
39713433671: 0 2 6 8 18 20 30 32 36 38
d = 38
**k = 6
5008751356547: 0 2 12 14 24 26 30 32 42 44 54 56
d = 56
*k = 7
2485390773085247: 0 2 24 26 30 32 42 44 54 56 60 62 84 86
d = 86

I found theoretical patterns for k = 8–13.
k = 8
0 2 30 32 42 44 54 56 60 62 72 74 84 86 114 116
d = 116
k = 9
0 2 18 20 30 32 42 44 60 62 78 80 90 92 102 104 120 122
d = 122
k = 10
0 2 12 14 30 32 42 44 54 56 90 92 102 104 114 116 132 134 144 146
0 2 12 14 42 44 54 56 60 62 84 86 90 92 102 104 132 134 144 146
d = 146
k = 11
0 2 12 14 30 32 42 44 54 56 72 74 90 92 102 104 114 116 132 134 144 146
d = 146
k = 12
0 2 30 32 42 44 60 62 72 74 84 86 120 122 132 134 144 146 162 164 174 176 204 206
d = 206
k = 13
0 2 6 8 36 38 48 50 66 68 78 80 108 110 138 140 150 152 168 170 180 182 210 212 216 218
0 2 6 8 36 38 48 50 66 68 90 92 108 110 126 128 150 152 168 170 180 182 210 212 216 218
0 2 6 8 36 38 66 68 78 80 90 92 108 110 126 128 138 140 150 152 180 182 210 212 216 218
0 2 36 38 48 50 66 68 78 80 90 92 108 110 126 128 138 140 150 152 168 170 180 182 216 218
d = 218

Question
find minimal symmetric tuples from k consecutive twin primes with minimal diameter for k > 7.

* The solution found in T. Brada Experimental Grid BOINC project
https://boinc.tbrada.eu/

** The solution contained in the OEIS sequence
https://oeis.org/A330278