Here are some raw numbers for gaps following twin primes in the integer ranges from 5 to n, where n is the first column in the table. The merit of each gap, given by g(p')/ln(p') where p' is the larger member of a twin prime and g(p') is the gap that follows p' to the next prime is calculated for each gap.

The second column provides the merit of the median gap in an array of all twin primes in the 5 to n range, ordered by the merit of the gaps

the third column provides the sum of the merits for all gaps following all twin pairs in the 5 to n range divided by the number of gaps in the range 5 to n - i.e. the average merit

The median result looks odd to me, especially the median at 1e7 compared to 1e8. Maybe its my program!

Code:

n median average
1e4 1.130409 1.174763
1e5 0.960076 1.140654
1e6 0.879785 1.12384
1e7 0.800977 1.10489
1e8 0.875686 1.091765
1e9 0.861240 1.084227

This is the same set of calculations for the primes up to 1e8:

Code:

n median average
1e4 0.790794 1.015308
1e5 0.778257 1.004082
1e6 0.780593 1.001669
1e7 0.765885 1.000514
1e8 0.744425 1.000131

And the percentage uptick for each set:

Code:

n median average
1e4 42.95% 15.71%
1e5 23.36% 13.60%
1e6 12.71% 12.20%
1e7 4.58% 10.43%
1e8 17.63% 9.16%

These are huge upticks! I was thinking 1%.

For the gaps preceding the twins, the results closely mirror the results for the gaps following the twins even down to the 1e7 median anomaly - this looks exceedingly odd

Code:

n median average
1e4 1.095597 1.16574
1e5 0.965776 1.161442
1e6 0.878152 1.113329
1e7 0.802137 1.10796
1e8 0.876182 1.092035