Hi dagr, to your questions

How do I know that the WG finished running. For 1H, I’m still on “Optimization started…”

The scan never concludes, simply because wheels optimization is an NP-hard type of problem. This means it is virtually impossible to definitely state that we have reached the absolute best solution (best covering), unless if it is somehow obvious but generally it is not. The only case of "obvious best" is if the covering has reached the theoritical minimum, when this can be computed of course. The rule of thumb is, if you follow the optimization guidelines and you get no further improvements, the optimization is considered complete. Check out this for optimization guidelines.

viewtopic.php?f=19&t=427
viewtopic.php?f=17&t=824
How do I know the minimum mathematically correct?

The Th.Min value indicated at the bottom of Wheel Generator (WG) is the theoretical minimum described here

viewtopic.php?f=11&t=35. Also check out this

viewtopic.php?f=17&t=349.

These results refer to non-filtered constructions, also known as unconditional. Typically filtered constructions require fewer blocks compared to the unconditional wheels, which is the primary reason you may want to use filtered wheels. An in-depth explanation of filtered vs unconditional coverings can be found here

viewtopic.php?f=17&t=654 which I highly suggest to read first.

For 20 numbers, what should be the minimum mathematically correct?

This question cannot be answered if you don't define at a minimum the t,k,m parameters.If you have read the above links however, it should be clear by now that even if we have established the theoretical minimum, also indicated by WG, it doesn't mean we can actually construct that wheel with that minimum of blocks.

I have also tested 5 if 6 with 15 numbers in GAT Help Documentation, Page 49. I choose a block of 6 numbers: 1, 3, 4, 9, 13, 15. I couldn't find 5 exact numbers in the 82 blocks

That is correct. Since the covering displayed there is a matrix one, it has some constraints to produce the guarantee we want. In this particular matrix, the additional constraints are "we have 3 groups 1-5, 6-10, 11-15 of non-overlapping numbers and we expect to have EXACTLY 2 correct numbers in each group". You picked as winning combination 1, 3, 4, 9, 13, 15 so right away you violate the constraint EXACTLY 2 correct numbers for group 1-5 and 6-10, since in group 1-5 you have 3 correct (1,3,4 numbers) and in group 6-10 you have only one correct number (9). If the constraints are not fulfilled, the matrix is not guaranteed to produce the desired 5if6.

Matrix explanation and construction in WG

viewtopic.php?f=17&t=81
viewtopic.php?f=17&t=706